16TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING AND 9TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING
COMPUTER-AIDED CHEMICAL ENGINEERING Advisory Editor: R. Gani Volume 1: Volume 2: Volume 3: Volume 4: Volume 5:
Volume 6: Volume 7: Volume 8: Volume 9: Volume 10: Volume 11: Volume 12: Volume 13: Volume 14: Volume 15: Volume 16: Volume 17: Volume 18: Volume 19: Volume 20: Volume 21:
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: Unite Operations (I. Pallai and Z. Fonyó, Editors) Part B: Systems (I. Pallai and G.E. Veress, Editors) Neural Networks for Chemical Engineers (A.B. Bulsari, Editor) Material and Energy Balancing in the Process Industries - From Microscopic Balances to Large Plants (V.V. Veverka and F. Madron) European Symposium on Computer Aided Process Engineering-10 (S. Pierucci, Editor) European Symposium on Computer Aided Process Engineering-11 (R. Gani and S.B. Jørgensen, Editors) European Symposium on Computer Aided Process Engineering-12 (J. Grievink and J. van Schijndel, Editors) Software Architectures and Tools for Computer Aided Process Engineering (B. Braunschweig and R. Gani, Editors) Computer Aided Molecular Design: Theory and Practice (L.E.K. Achenie, R. Gani and V. Venkatasubramanian, Editors) Integrated Design and Simulation of Chemical Processes (A.C. Dimian) European Symposium on Computer Aided Process Engineering-13 (A. Kraslawski and I. Turunen, Editors) Process Systems Engineering 2003 (Bingzhen Chen and A.W. Westerberg, Editors) Dynamic Model Development: Methods, Theory and Applications (S.P. Asprey and S. Macchietto, Editors) The Integration of Process Design and Control (P. Seferlis and M.C. Georgiadis, Editors) European Symposium on Computer-Aided Process Engineering-14 (A. Barbosa-Póvoa and H. Matos, Editors) Computer Aided Property Estimation for Process and Product Design (M. Kontogeorgis and R. Gani, Editors) European Symposium on Computer-Aided Process Engineering-15 (L. Puigjaner and A. Espuña, Editors) 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering (W. Marquardt and C. Pantelides)
COMPUTER-AIDED CHEMICAL ENGINEERING, 21A
16TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING AND 9TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING Edited by
W. Marquardt RWTH Aachen University, Lehrstuhl für Prozesstechnik, Aachen, Germany
C. Pantelides Process Systems Enterprise Ltd. & Imperial College London, London, UK
Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo
Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2006 Copyright © 2006 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email:
[email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 0-444-52257-3 978-0-444-52257-3(Part A) 0-444-52970-5 978-0-444-52970-1(Part B) 0-444-52969-1 978-0-444-52969-5(Set) ISSN: 1570-7946 For information on all Elsevier publications visit our website at books.elsevier.com
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Contents Part A Keynotes Innovation in the Chemical Industry: A Growth Engine S. Marcinowski ................................................................................................................. 1 Life Cycle Modelling in the Chemical Industries: Is there any Reuse of Models in Automation and Control J. Bausa and G. Dünnebier............................................................................................... 3 Hierarchical Multiscale Model-based Design of Experiments, Catalysts, and Reactors for Fuel Processing D.G. Vlachos, A.B. Mhadeshwar and N.S. Kaisare ............................................................9 Process Intensification and Process System Engineering: A Friendly Symbiosis J.A. Moulijn, A. Stankiewicz, J. Grievink and A. Gorak.................................................. 29 Recent Developments in the Risk Management of Offshore Production Systems D. Averbuch.................................................................................................................... 39 Challenges and Opportunities in Process Innovation L.R. Genskow.................................................................................................................. 45 Recent Developments and Industrial Applications of Data-based Process Monitoring and Process Control M. Kano and Y. Nakagawa ............................................................................................. 57 Model-Centric Technologies for Support of Manufacturing Operations J.A. Romagnoli and P.A. Rolandi ................................................................................... 63 The Systems Engineering of Cellular Processes V. Hatzimanikatis and L. Wang ...................................................................................... 71 Systems Biology and the Silicon Cell: Order out of Chaos H.V. Westerhoff .............................................................................................................. 81 Challenges for Process System Engineering in Infrastructure Operation and Control Z. Lukszo, M.P.C. Weijnen, R.R. Negenborn, B. De Schutter and M. Ilic ....................... 95 Supply Chain Design, Management and Optimization D. Kassmann and R. Allgor .......................................................................................... 101 Business Decision Making in the Chemical Industry: PSE Opportunities R. Srinivasan, I.A. Karimi and A.G. Vania ................................................................... 107
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Contributed Papers Topic 1 Oral Simulation of Mass Transfer in Reactive Absorption N. Asprion .....................................................................................................................119 Integration of Generalized Disjunctive Programming with Modular Process Simulators J.A. Caballero, A. Odjo and I.E. Grossmann ................................................................125 Large-scale Optimization Strategies for Zone Configuration of Simulated Moving Beds Y. Kawajiri and L.T. Biegler .........................................................................................131 Comparison of the Startup of Reactive Distillation in Packed and Tray Towers F. Forner, M. Meyer, M. Döker, J.-U. Repke, J. Gmehling and G. Wozny....................137 Parameter Estimation for Stochastic Differential Equations: Algorithm and Application to Polymer Melt Rheology B. Pereira Lo, A.J. Haslam and C.S. Adjiman...............................................................143 A “Targeted” QSPR for Prediction of Properties N. Brauner, R.P. Stateva, G. St. Cholakov and M. Shacham .........................................149 Global Bounds on Optimal Solutions in Chemical Process Design U.-U. Haus, J. Gangadwala, A. Kienle, D. Michaels, A. Seidel-Morgenstern and R. Weismantel.........................................................................................................155 Stochastic Grey Box Modeling of the Enzymatic Biochemical Reaction Network of E. Coli Mutants F.P. Davidescu, H. Madsen, M. Schümperli, M. Heinemann, S. Panke and S.B. Jørgensen........................................................................................................161 Validated Solution of ODEs with Parametric Uncertainties Y. Lin and M.A. Stadtherr .............................................................................................167 Optimal Experimental Design for Ill-posed Problems A. Bardow .....................................................................................................................173 Dynamic Oil and Gas Production Optimization via Explicit Reservoir Simulation D.I. Gerogiorgis, M. Georgiadis, G. Bowen, C.C. Pantelides and E.N. Pistikopoulos ..................................................................................................179 Multi-Scale Modelling and Optimization of Hydrogen Storage Systems Using Advanced Solid Materials E. Kikkinides, M.C. Georgiadis, M. Konstantakou and A. Stubos .................................185
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Risk Analysis and Robust Design Under Technological Uncertainty R.F. Blanco Gutiérrez, C.C. Pantelides and C.S. Adjiman ........................................... 191 Network of Three Catalytic Reactors with Periodical Feed Switching for Methanol Synthesis: Bifurcation Analysis M. Pota, L. Russo, E. Mancusi and S. Crescitelli ......................................................... 197 CFD Model of a Semi-batch Reactor for the Precipitation of Nanoparticles in the Droplets of a Microemulsion A.A. Öncül, B. Niemann, K. Sundmacher and D. Thévenin........................................... 203 Solution of the Population Balance Equation Using the Sectional Quadrature Method of Moments (SQMOM) M.M. Attarakih, H.-J. Bart and N.M. Faqir.................................................................. 209 A Global Parametric Programming Optimisation Strategy for Multilevel Problems N.P. Faísca, V. Dua, P.M. Saraiva, B. Rustem and E.N. Pistikopoulos ........................ 215 Modelling Deammonification in Biofilm Systems: Sensitivity and Identifiability Analysis as a Basis for the Design of Experiments for Parameter Estimation D. Brockmann, K.-H. Rosenwinkel and E. Morgenroth ............................................... 221 The Combined-Continuum-and-Discrete-Model (CCDM) for Simulation of Liquid-particle Flows K.F. Malone, B.H. Xu and M. Fairweather .................................................................. 227 Implementation of Efficient Logic-based Techniques in the MINLP Process Synthesizer MIPSYN M. Ropotar and Z. Kravanja......................................................................................... 233 Calculation of Three-phase Bubble Columns D. Wiemann and D. Mewes .......................................................................................... 239 A Framework for Model-based Design of Parallel Experiments in Dynamic Systems F. Galvanin, M. Barolo, F. Bezzo and S. Macchietto.................................................... 249 OPEN CHEMASIMTM: Breaking Paradigms in Process Simulation H. Hasse, B. Bessling and R. Böttcher.......................................................................... 255 Simulation of the Population Balance for Droplet Breakage in a Liquid-liquid Stirred Tank Reactor Using H-matrix Methods J. Koch, W. Hackbusch and K. Sundmacher ................................................................. 261 Simultaneous Dynamic Validation/Identification of Mechanistic Process Models and Reconciliation of Industrial Process Data P.A. Rolandi and J.A. Romagnoli ................................................................................. 267 A Model Discrimination Based Approach to the Determination of Operating Regimes for Chemical Reactors A. Yang, E. Martin, G. Montague and J. Morris........................................................... 273
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A Performance Comparison of some High Breakdown Robust Estimators for Nonlinear Parameter Estimation E.L.T. Conceição and A.A.T.G. Portugal ......................................................................279 Equivalent Dynamic Solution of an Industrial HDPE Slurry Reactor S. Nigam, K.M. Moudgalya and A.K. Pani....................................................................285 Dynamical and Stationary Analysis of an Electrolyte Diode and Comparison with Experiments Z. Slouka, M. Pribyl, J. Lindner, D. Snita and M. Marek ..............................................291 Stability Analysis of Differential-Algebraic Equations in AUTO_DAE B.C. von Clausbruch, E.C. Biscaia, Jr., and P.A. Melo.................................................297 Application of Particulate Models for Industrial Processes G. Skillas, C. Becker, M. Verduyn and J. Vorholz .........................................................303 Optimization of Operating Conditions for Ferrichrome Production in a Membrane Bioreactor Using Ustilago maydis A. Drews, H. Arellano-Garcia, M. Wendt, M. Kraume and G. Wozny ..........................309 Modelling and Simulation of MSF Desalination Process Using gPROMS and Neural Network Based Physical Property Correlation M.S. Tanvir and I.M. Mujtaba.......................................................................................315 A New Operation Mode for Reactive Batch Distillation in Middle Vessel Columns: Start-up and Operation I. Carmona, H. Arellano-Garcia and G. Wozny ............................................................321 Towards a Novel Optimisation Algorithm with Simultaneous Knowledge Acquisition for Distributed Computing Environments S. Yang, A. Kokossis and P. Linke .................................................................................327 Floating Index of Inequality Constrained DAE Systems D.F. de S. Souza, R.C. Vieira and E.C. Biscaia Jr. .......................................................333 Predictive Modeling of Ionic Permselectivity of Porous Media L. Seda and J. Kosek .....................................................................................................339 Development of a Multi-Compartment Dynamic Model for the Prediction of Particle Size Distribution and Particle Segregation in a Catalytic Olefin Polymerization FBR G. Dompazis, V. Kanellopoulos and C. Kiparissides ....................................................345 Mixing in a T-shaped Microreactor: Scales and Quality of Mixing D. Bothe, C. Stemich and H.-J. Warnecke .....................................................................351 Direct Modelling of Unit Operations on Molecular Level D. Babic and A. Pfennig................................................................................................359 Modelling and Simulation of Fe2O3/Aluminum Thermite Combustion: Experimental Validation L. Durães, P. Brito, J. Campos and A. Portugal ...........................................................365
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Topic 1 Poster Modelling of Self-ignition and Process Upsets in Industrial Gaseous Hydrocarbon Oxidation Processes H.J. Pasman and M. Fairweather................................................................................. 371 A Simplex Search Method for Experimental Optimization with Multiple Objectives E. Martinez ................................................................................................................... 377 Automatic Generation of Reduced Reaction Mechanisms for Hydrocarbon Oxidation with Application to Autoignition Boundary Prediction for Explosion Hazards Mitigation R. Porter, M. Fairweather, J.F. Griffiths, K.J. Hughes and A.S. Tomlin ...................... 383 Combining HAZOP with Dynamic Process Model Development for Safety Analysis S. Eizenberg, M. Shacham and N. Brauner................................................................... 389 Validation of a Digital Packing Algorithm for the Packing and Subsequent Fluid Flow Through Packed Columns R. Caulkin, M. Fairweather, X. Jia and R.A. Williams ................................................. 395 A Hybrid Global Optimization Scheme for Process Design and Dynamic Optimization C.-T. Chen, S.-T. Peng, Y.-J. Ciou and C.-L. Chen....................................................... 401 Parameter Identifiability Analysis and Model Fitting of a Biological Wastewater Model Q. Chai, S.H. Amrani and B. Lie .................................................................................. 409 Methodology for Decision Support Among Conflicting Objectives Using Process Simulators N. Ramzan and W. Witt................................................................................................. 415 Grey-box Stochastic Modelling of Industrial Fed-Batch Cultivation J.K. Rasmussen, H. Madsen and S.B. Jørgensen .......................................................... 421 Monitoring and Improving LP Optimization with Uncertain Parameters D. Zyngier and T.E. Marlin .......................................................................................... 427 Assessing the Performance of Batch Reactive Distillations Through Conceptual Models J. Espinosa ................................................................................................................... 433 An Integrated Stochastic Method for Global Optimization of Continuous Functions M. Srinivas and G.P. Rangaiah .................................................................................... 439 The ProMoT/Diana Simulation Environment M. Krasnyk, K. Bondareva, O. Milokhov, K. Teplinskiy, M. Ginkel and A. Kienle ....... 445
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Strategy and Framework for Solving Signal-based MIDO Problems R.H. Nyström, I. Harjunkoski and R. Franke ................................................................451 “Smart Models” - A Framework for Adaptive Multi-Scale Modelling E.S. Fraga, G. Wills, M. Fairweather and T. Perris......................................................457 Process Design Using Ionic Liquids: Physical Property Modeling A.E. Ayala, L.D. Simoni, Y. Lin, J.F. Brennecke and M.A. Stadtherr ............................463 Study of Non-linear Dynamics in Reactive Distillation for TAME Synthesis Using Equilibrium and Non-equilibrium Models A.M. Katariya, R.S. Kamath, S.M. Mahajani and K.M. Moudgalya ..............................469 An Agent-Oriented Architecture for Modeling and Optimization of Naphtha Pyrolysis Process X. Gao, B. Chen and X. He............................................................................................475 On Model Portability H.A. Preisig, T. Haug-Warberg and B.T. Løvfall ..........................................................483 Utility Systems Operational Planning Optimization Based on Pipeline Network Simulation X.L. Luo, B. Hua, B.J. Zhang and M.L. Lu ....................................................................489 Particle Swarm for the Dynamic Optimization of Biochemical Processes J. Zhang, L. Xie and S. Wang ........................................................................................497 A-priori Identification of Critical Points for the Design and Synthesis of Flexible Process Schemes Z.N. Pintaric and Z. Kravanja.......................................................................................503 Using Water Cascade Analysis to Synthesize Water use Network in Batch Process S. Wang, S. Zheng, X. Yang and Y. Li............................................................................509 Multiobjective Optimization of Multipurpose Batch Plants Using Superequipment Class Concept A. Mosat, L. Cavin, U. Fischer and K. Hungerbühler ...................................................515 Integrated Design of Energy-Saving Chemical Process Systems: Strategy, Methods and Implementation G. Ostrovsky, Y. Volin, D. Dvoretsky and S. Dvoretsky.................................................521 Generic Hybrid Models of Solvent-based Reactive Systems Combined with Membrane Separation System P.T. Mitkowski, G. Jonsson and R. Gani .......................................................................527 On the Numerical Calibration of Discrete Element Models for the Simulation of Bulk Solids T. Gröger and A. Katterfeld ..........................................................................................533 A Heat Transfer Model of a Scraped Surface Heat Exchanger for Ice Cream P.M.M. Bongers ............................................................................................................539
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Computer-Aided Forecast of Catalytic Activity in an Hydrotreating Industrial Process Using Artificial Neural Network, Fuzzy Logic and Statistics Tools F. Jiménez, V. Kafarov and M. Nuñez .......................................................................... 545 A Framework for Modeling Particle Size Effects in Emulsion Polymerization Systems Using Computational Fluid Dynamics Linked to a Detailed Population Balance Model R.C. Elgebrandt, D.F. Fletcher, V.G. Gomes and J.A. Romagnoli ............................... 551 Pricing Utilities for Large-Scale Chemical Production Site K. Hirata, P. Chan, H. Sakamoto and C.-W. Hui.......................................................... 557 Optimal Experimental Design for the Precision of a Subset of Model Parameters in Process Development A. Yang, E. Martin, G. Montague and J. Morris........................................................... 563 The Complex Distillation Column Network Systematic Optimization by Mathematical Programming S. Choi, H. Kim, C. Han and E.S. Yoon ........................................................................ 569 Modelling and Simulation of Coal and Petcoke Gasification in a Co-current Flow Reactor E.M. López, V. Garza and J. Acevedo........................................................................... 577 Simulation of (Electro)Chromatography by means of CFD D.-U. Astrath, T. Schneider and W. Arlt ....................................................................... 583 Modeling of Heat Transfer Processes in Particulate Systems Z. Süle, C. Mihálykó and B.G. Lakatos......................................................................... 589 A Comprehensive Investigation on High-Pressure LDPE Manufacturing: Dynamic Modelling of Compressor, Reactor and Separation Units P. Pladis, A. Baltsas and C. Kiparissides ..................................................................... 595 Sensitivity Analysis in the Simulation of Complex Solids Processes D. Schwier, A. Püttmann, E.-U. Hartge, G. Gruhn and J. Werther............................... 601 Identification of Parametric and Structural Models Based on RTD Theory via GAMS Package S. Hocine, L. Pibouleau, C. Azzaro-Pantel and S. Domenech....................................... 607 Hybrid Modeling for Continuous Production of Bioethanol E. Ccopa Rivera, I. Mantovaneli, A.C. da Costa and R. Maciel Filho .......................... 613 Prediction and Estimation Techniques for Modeling Pervaporation Process M.E.T. Alvarez, E.B. Moraes and M.R.W. Maciel ........................................................ 619 Model Discrimination and Parameter Estimation Through Sensitivity Analysis M. Sales-Cruz and R. Gani ........................................................................................... 625
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Solving MINLP Containing Noisy Variables and Black-Box Functions Using Branch-and-Bound E. Davis and M. Ierapetritou.........................................................................................633 Modelling and Simulation of High Pressure Industrial Autoclave Polyethylene Reactor E. Caliani, M. Cavalcanti, F.A.N. Fernandes and L.M.F. Lona....................................639 Energy Saving in Distillation Columns: The Linde Column Revisited G. Soave, L. Pellegrini, D. Barbatti, N. Susani and S. Bonomi .....................................645 Computer-Aided Modeling for Hydrodesulfurization, Hydrodenitrogenation and Hydrodearomatization Simultaneous Reactions in a Hydrotreating Industrial Process F. Jiménez, V. Kafarov and M. Nuñez ...........................................................................651 Modelling and Dynamic Simulation of Thermal Stresses in Brazed Plate-Fin Heat Exchanger F. Picard, D. Averous, X. Joulia and D. Barreteau.......................................................659 ReDrop - An Efficient Simulation Tool for Describing Solvent and Reactive Extraction Columns M. Altunok, T. Grömping and A. Pfennig ......................................................................665 Numerical Simulation of Micro Roughness Effects on Convective Heat Transfer S. Scholl and W. Augustin .............................................................................................671 Classical Models of Secondary Settlers Revisited R. David, A. Vande Wouwer, P. Saucez and J.-L. Vasel................................................677 An Approach to Implicit Modelling for Complex Process Optimization X.G. Yuan, W.Z. An, Y.J. Liu, Y.Q. Luo and C.J. Liu.....................................................683
Topic 2 Oral Structural Design of Polymers for Membrane Based Separation Processes Using Reverse Simulation Approach V. Soni, J. Abildskov, G. Jonsson, R. Gani, N. Karayiannis and V. Mavrantzas ...........689 Innovative Flowschemes Using Dividing Wall Columns M.A. Schultz, D.E. O’Brien, R.K. Hoehn, C.P. Luebke and D.G. Stewart .....................695 On the Rapid Development of New Products Through Empirical Modeling with Diverse Data-bases J.F. MacGregor, K. Muteki and T. Ueda.......................................................................701 Conceptual Design of Reactive Distillation Flowsheets G. Daniel, P. Patil and M. Jobson ................................................................................707
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LCA of a Spent Lube Oil Re-refining Process T.N. Kalnes, D.R. Shonnard and A. Schuppel ............................................................... 713 Effective Process Design Instruction: From Simulation to Plant Design D.R. Lewin, E. Dassau and A. Goldis ........................................................................... 719 Developments in the Sequential Framework for Heat Exchanger Network Synthesis of Industrial Size Problems R. Anantharaman and T. Gundersen ............................................................................ 725 Linking Experiments to Modeling in Biodiesel Production A.A. Kiss, A.C. Dimian and G. Rothenberg .................................................................. 731 Optimization Studies in Sulfuric Acid Production A.A. Kiss, C.S. Bildea and P.J.T. Verheijen .................................................................. 737 Integrating Advanced Thermodynamics and Process and Solvent Design for Gas Separation E. Keskes, C.S. Adjiman, A. Galindo and G. Jackson ................................................... 743 Integrated Approach to Crystallization Process Design for Fine Chemicals and Pharmaceuticals C. Wibowo, K.D. Samant and L. O’Young.................................................................... 749 Improved Solutions for Zebra Mussel (Dreissena polymorpha) Control – A Chemical Product Engineering Approach R. Costa, P.M. Saraiva, P. Elliott, D.C. Aldridge and G.D. Moggridge ....................... 755 Success Factors for CAPE in the Engineering Practice of a Process Plant Contractor G. Engl and A. Kröner.................................................................................................. 763 Polyurethane Design Using Stochastic Optimization J. Eslick and K. Camarda ............................................................................................. 769 Real-Time Imaging and Product Quality Characterization for Control of Particulate Processes Y. Zhou, X.-T. Doan and R. Srinivasan......................................................................... 775 An Engineering Company’s Approach to Filling “CAPE Gaps” in Process Simulation A. Kröner...................................................................................................................... 781 A Computer-Aided Methodology with Robust Design Criteria for Selection of Solvents for Reactions M. Folic, C.S. Adjiman and E.N. Pistikopoulos ............................................................ 787 Separation of Azeotropes in Batch Extractive Stripper with Intermediate Entrainer V. Varga, E.R. Frits, V. Gerbaud, Z. Fonyo, X. Joulia, Z. Lelkes and E. Rév ............... 793 Conceptual Steady State Process Design in Times of Value Based Management A. Wiesel and A. Polt .................................................................................................... 799
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Computer Aided Methods & Tools for Separation & Purification of Fine Chemical & Pharmaceutical Products M.B.C. Afonso, V. Soni, P.T. Mitkowski, L. d’Anterroches, R. Gani and H. Matos .......805 Integration Along the Lifecycle of Calcium Fluoride in the Fluorine Industry A. Garea, R. Aldaco, I. Fernández and A. Irabien ........................................................811 Design of Sustainable Processes: Systematic Generation & Evaluation of Alternatives A. Carvalho, R. Gani and H. Matos ..............................................................................817 Systematic Procedure for Designing a Microreactor with Slit-Type Mixing Structure O. Tonomura, T. Takase, M. Kano and S. Hasebe ........................................................823 Model-based Optimal Design of Pharmaceutical Formulations F.P. Bernardo, P.M. Saraiva and S. Simões..................................................................829 Scope for Process Systems Engineering Studies in Proton Exchange Membrane Fuel Cells (PEMFC): A Review of Opportunities R. Madhusudana Rao, T. Oh and R. Rengaswamy ........................................................835 A Framework for Innovation in Process Development for Heterogeneously Catalysed Gas-phase Reaction Systems D. Montolio-Rodriguez, D. Linke and P. Linke .............................................................841 Multi-Objective Optimization of Fixed-Bed Ion Exchange Processes for Phytopharmaceutical Production C.M. Silva, A.G. Barreto Jr. and E.C. Biscaia Jr. .........................................................847 Computer Aided Methodology for Simultaneous Synthesis, Design & Analysis of Chemical Products-Processes L. d’Anterroches and R. Gani .......................................................................................853 Correlation and Prediction of Drug Molecule Solubility with the NRTL-SAC Model C.-C. Chen and P.A. Crafts ...........................................................................................859 Dynamic Modelling of Complex Batch Distillation Starting from Ambient Conditions S. Gruetzmann, T. Kapala and G. Fieg .........................................................................865
Topic 2 Poster Genetic Algorithms Approach for Retrofitting Heat Exchanger Network with Standard Heat Exchangers R. Bochenek and J.M. Jezowski.....................................................................................871 A Discrete Interactive Graphical Method for Heat Exchanger Network Synthesis E.S. Fraga and G.W.A. Rowe ........................................................................................877
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Optimization of Nanosized Silver Particles Synthesis via Sequential Pseudo-Uniform Design Method J.-S. Chang and Y.-P. Lee............................................................................................. 883 Multiclass Molecular Knowledge Framework for Product and Process Design M. Korichi, V. Gerbaud, P. Floquet, A.-H. Meniai, S. Nacef and X. Joulia .................. 889 Quantitative Structure – Odor Relationship: Using of Multidimensional Data Analysis and Neural Network Approaches M. Korichi, V. Gerbaud, P. Floquet, A.-H. Meniai, S. Nacef and X. Joulia .................. 895 Simulation and Optimization in 1,3-butadiene Process from C4-Cut Using Genetic Algorithm F. Jalali and R. Saffari ................................................................................................. 901 Property Clustering and Group Contribution for Process and Molecular Design F. Eljack, M. Eden, V. Kazantzi and M. El-Halwagi .................................................... 907 A Combinatorial Formulation for Optimal Sizing, Scheduling and Shift Policy in Designing the Milling Section of a Ceramic Tile Industrial Plant B.P.M. Duarte, L.O. Santos and J.S. Mariano.............................................................. 913 An Automated Method for Synthesizing a Multi-Stream Heat Exchanger Network Based on Stream Pseudo-Temperature D. Yuan, Y. Wang, W. Xiao, P. Yao, X. Luo and W. Roetzel ......................................... 919 Flexible Heat Exchanger Network Design for Chemical Processes with Operation Mode Changes M. Noda and H. Nishitani............................................................................................. 925 Molecular Design Based on Enhanced Topological Descriptors A.A. Kiss and M.V. Diudea ........................................................................................... 931 Effects of Catalyst Activity Profiles on the Scale-up of Polymerization Reactors S. Nemeth, J. Abonyi, B. Feil, P. Arva, J. Tolveth, A. Janecska and G. Nagy ............... 937 Optimization-based Root Cause Analysis E. Dassau and D. Lewin ............................................................................................... 943 A Synthesis Procedure for the Design of Semicontinuous Reactive Distillation for Specialty Chemicals T.A. Adams II and W.D. Seider..................................................................................... 949 Multi-Objective Optimisation of Batch Distillation Processes T.M. Barakat, E.S. Fraga and E. Sorensen ................................................................... 955 Sustainable Production of Industrial Chemical Products from Bioresources J. Seay, M. Eden, R. D’Alessandro and C. Weckbecker................................................ 961 Cooling Crystallization: A Process-Product Perspective C.B.B. Costa and R. Maciel Filho ................................................................................ 967
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A Systematic Approach for Automated Reaction Network Generation S.-H. Hsu, B. Krishnamurthy, P. Rao, C. Zhao, S. Jagannathan, J. Caruthers and V. Venkatasubramanian ........................................................................................973 A Hybrid Methodology for Detailed Heat Exchanger Design in the Optimal Synthesis of Heat Exchanger Networks J.M. García, J.M. Ponce and M. Serna .........................................................................979 Optimal Design of Shell-and-Tube Heat Exchangers Using Genetic Algorithms J.M. Ponce, M. Serna, V. Rico and A. Jiménez..............................................................985 Integration of Process Design and Operation for Chemical Product Development with Implementation of a Kilo-Plant Y. Qian, Z. Wu and Y. Jiang ..........................................................................................991 Importance of the Selection of Feed Tray Location on the Optimum Design of a Heterogeneous Azeotropic Distillation Column with p-xylene Feed Impurity I-L. Chien, H.-Y. Lee, T.-K. Gau and H.-P. Huang .......................................................997 Supporting Waste Minimization Studies by Integrating Expert System with Process Simulators I. Halim and R. Srinivasan ..........................................................................................1003 Process Intensification for Systematic Synthesis of New Distillation Systems with Less than N-1 Columns B.-G. Rong and I. Turunen ..........................................................................................1009 Mixed-Integer Optimization of Distillation Column Tray Positions in Industrial Practice I. Thomas and A. Kröner.............................................................................................1015 A Chemical Process Design Framework Including Different Stages of Environmental, Health and Safety (EHS) Assessment H. Sugiyama, U. Fischer, M. Hirao and K. Hungerbühler ..........................................1021 Multi-Objective Reactor Network Synthesis for Industrial Mass Transfer Limited Processes F. Neves, D. Silva, N. Oliveira and F. Mendes............................................................1027 Synthesis of Separation Systems for Azeotropic Mixtures: Preferred Distillation Region S.K. Wasylkiewicz .......................................................................................................1033 Modeling and Designing Powder Mixing Processes Utilizing Compartment Modeling P.M. Portillo, F.J. Muzzio and M.G. Ierapetritou .......................................................1039 Design and Control of Homogeneous and Heterogeneous Reactive Distillation for Ethyl Acetate Process H.-Y. Lee, H.-P. Huang and I-L. Chien .......................................................................1045
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Decomposition Based Algorithm for the Design and Scheduling of Multipurpose Batch Plants T. Pinto, A.P.F.D. Barbósa-Póvoa and A.Q. Novais .................................................. 1051 A Knowledge-based Approach for Accident Information Retrieval M. Suzuki, R. Batres, T. Fuchino, Y. Shimada and P.W. Chung.................................. 1057 Feasibility Study of Batch Reactive Distillation in Hybrid Columns C. Steger, E. Rev, Z. Fonyo, M. Meyer and Z. Lelkes ................................................. 1063 Heat Integration Between Processes: Integrated Structure Using Stage-Wise Model A. Kovac Kralj and P. Glavic ..................................................................................... 1069 Generic Model Framework for the Synthesis of Structured Reactive Separation Processes G. Sand, M. Tylko, S. Barkmann, G. Schembecker and S. Engell ............................... 1075 Knowledge Extraction During the Design of Activated Sludge Systems X. Flores, M. Poch, I. Rodríguez-Roda, L. Jiménez and R. Bañares-Alcántara.......... 1083 Pharmaceutical Process Development Applying Automated Laboratory Reactors T. Chován, I. Markovits, B. Farkas, K. Nagy, L. Nagy, K. Nyíri and F. Szeifert......... 1089 Addressing the Design of Chemical Supply Chains Under Demand Uncertainty G. Guillén, F.D. Mele, A. Espuña and L. Puigjaner ................................................... 1095 Development of a Mixture Design Methodology for Problems with Incomplete Information. Application to PVC Heat Stabiliser Design U.G. da Cruz and G.A.C. Le Roux.............................................................................. 1101 Principles for Chemical Products Design L.A. Cisternas and E.D. Gálvez .................................................................................. 1107 Mathematical Development for Scaling-up of Molecular Distillators: Strategy and Test with Recovering Carotenoids from Palm Oil C.B. Batistella, E.B. Moraes, R. Maciel Filho and M.R. Wolf-Maciel ........................ 1113 Case Study on Design of Regulatory Policies for Sustainable Emission Reduction A. Malcolm, L. Zhang and A.A. Linninger .................................................................. 1119 A Decomposition/Reconstruction Algorithmic Procedure for Computer Aided Case Based Reasoning – Implementation in Biodegradation F.A. Batzias ................................................................................................................ 1125 Environmentally Conscious Design of Ethanol Fed Fuel Cell System L. Hernández and V. Kafarov ..................................................................................... 1131 Morphogenesis of Polyolefin Particles in Polymerization Reactors B. Horackova and J. Kosek......................................................................................... 1137 Modeling and Design of a Biochemical Process for NOx Removal C.S. Bildea, M.L. Oudshoorn, C. Picioreanu and A.C. Dimian .................................. 1143
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Multicriteria Design of Separation Sequences by Including HSE Criteria and Uncertainty K. Cziner, M. Hassim and M. Hurme ..........................................................................1149 Optimal Scheduling of Tests for New Product Development H.-R. Son, S.-K. Heo and I.-B. Lee ..............................................................................1155
Part B Topic 3 Oral State Estimation of a Molten Carbonate Fuel Cell by an Extended Kalman Filter M. Groetsch, M. Mangold, M. Sheng and A. Kienle....................................................1161 Real-Time Failure Prediction for Chemical Processes: Plantwide Framework A. Meel and W.D. Seider .............................................................................................1167 Multiscale Analysis and Monitoring of Paper Surface M.S. Reis and P.M. Saraiva ........................................................................................1173 A Real Time Adaptive Dynamic Programming Approach for Planning and Scheduling N.E. Pratikakis, J.H. Lee and M.J. Realff....................................................................1179 Theoretical Analysis and Experimental Studies of Mixed Product Run-to-Run Control Y. Zheng, M.-F. Wu, S.-S. Jang and D.S.-H. Wang .....................................................1185 Methods of State Estimation for Particulate Processes M. Mangold, C. Steyer, B. Niemann, A. Voigt and K. Sundmacher .............................1191 Simultaneous Scheduling and Optimization of a Copper Plant I. Harjunkoski, H.W. Borchers and M. Fahl................................................................1197 Coordinator MPC with Focus on Maximizing Throughput E.M.B. Aske, S. Strand and S. Skogestad.....................................................................1203 Fault Diagnosis Based on Support Vector Machines and Systematic Comparison to Existing Approaches I. Yélamos, G. Escudero, M. Graells and L. Puigjaner ...............................................1209 Explicit Parametric Controller for a Batch Polymerization System M. Asteasuain, K. Kouramas, V. Sakizlis and E.N. Pistikopoulos ...............................1215 An Effective MIDO Approach for the Simultaneous Cyclic Scheduling and Control of Polymer Grade Transition Operations A. Flores-Tlacuahuac and I.E. Grossmann .................................................................1221
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On the Global Dynamic Optimization of Highly Nonlinear Systems A. Flores-Tlacuahuac and L.T. Biegler ...................................................................... 1227 Integrating Stiction Diagnosis and Stiction Compensation in Process Control Valves R. Srinivasan and R. Rengaswamy ............................................................................. 1233 Differential Recurrent Neural Network Based Predictive Control R. Al Seyab and Y. Cao............................................................................................... 1239 Chance Constrained Programming Approach to Process Optimization Under Uncertainty P. Li, H. Arellano-Garcia and G. Wozny .................................................................... 1245 Support for Design of User Interfaces in Plant Operations X. Liu, H. Kosaka, M. Noda and H. Nishitani............................................................. 1251 Online Prediction of End-of-Batch Product Quality Using Phase-Specific PLS Models X.-T. Doan and R. Srinivasan ..................................................................................... 1257 Optimal Current Distribution Control for Parallel Electrolytic Baths H. Kugemoto, K. Ozakia, Y. Kutsuwa and Y. Hashimoto ............................................ 1263 Hybrid Model Predictive Control of a Sugar End Section D. Sarabia, C. de Prada, S. Cristea and R. Mazaeda ................................................. 1269 Systematic Methodology for Reproducible Optimizing Batch Operation S.B. Jørgensen and D. Bonné ..................................................................................... 1275 Discriminant Analysis and Control Chart for the Fault Detection and Identification X. Pei, Y. Yamashita, M. Yoshida and S. Matsumoto .................................................. 1281 Stability Analysis of Nonlinear Model Predictive Control: An Optimization Based Approach V. Dua ........................................................................................................................ 1287 An Optimization Framework to Computer-Aided Design of Reliable Measurement Systems R. Angelini, C.A. Méndez, E. Musulin and L. Puigjaner ............................................. 1293 An Approach to Linear Control of Nonlinear Processes T. Schweickhardt and F. Allgöwer.............................................................................. 1299 Control of the Synthesis Section of a Urea Plant by means of an MPC Controller O.M. Agudelo Mañozca, J.J. Espinosa and J. Vandewalle ......................................... 1305 Control of Thermal Runaway via Optimal Bifurcation Tailoring Aided Gain-Scheduling Feedback P. Altimari, L. Russo, E. Mancusi, M. di Bernardo and S. Crescitelli ........................ 1311
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Systematic Design of Logic Controllers for Processing Plants Starting from Informal Specifications S. Lohmann, O. Stursberg and S. Engell .....................................................................1317 Static/Dynamic Analysis and Controllability Issues in Reactive Distillation Columns T. López-Arenas, E.S. Pérez-Cisneros and R. Gani.....................................................1323 Predictive Control of Polymerization Batch Reactors Using Hybrid Models J. Espinosa and W. van Brempt...................................................................................1329 Dynamic Optimization of Molecular Weight Distribution Using Orthogonal Collocation on Finite Elements and Fixed Pivot Methods: An Experimental and Theoretical Investigation A. Krallis, D. Meimaroglou, V. Saliakas, C. Chatzidoukas and C. Kiparissides .........1335 Multivariate Statistical Batch Process Monitoring Using Dynamic Independent Component Analysis H. Albazzaz and X.Z. Wang.........................................................................................1341 Model-based Optimization for Operational Policies in Seeded Cooling Crystallization A. Abbas, S.M. Nowee and J.A. Romagnoli .................................................................1347
Topic 3 Poster Controllability Analysis of Thermally Coupled Distillation Sequences for Five – Component Mixtures M. Carrera-Rodriguez, M. Ledezma-Martinez, J.G. Segovia-Hernández and S. Hernández ........................................................................................................1353 Multiscale SPC in the Presence of Multiresolution Data M.S. Reis and P.M. Saraiva ........................................................................................1359 Nonlinear Model Predictive Control of the Wastewater Treatment Plant M.V. Cristea and S.P. Agachi......................................................................................1365 Branch and Bound Methods for Control Structure Design V. Kariwala and S. Skogestad .....................................................................................1371 A Mathematical Programming Approach Including Flexible Recipes to Batch Operation Rescheduling S. Ferrer-Nadal, C.A. Méndez, M. Graells and L. Puigjaner ......................................1377 Using Multi Sensor Data Fusion for Level Estimation in a Separator N.-O. Skeie, S. Mylvaganam and B. Lie.......................................................................1383 Product Quality Estimation Using Multi-Rate Sampled Data B. Lin, B. Recke, T. Jensen, J. Knudsen and S.B. Jørgensen .......................................1389
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A Novel Solution Approach for Quality-based Re-trimming Optimization I. Harjunkoski and M. Fahl ........................................................................................ 1395 An Integrated Framework Based on Data Driven Techniques for Process Supervision B. Bhushan and J.A. Romagnoli ................................................................................. 1401 Reliable Multi-Objective Optimal Control of Batch Processes Based on Stacked Neural Network Models A. Mukherjee and J. Zhang......................................................................................... 1407 A Thermodynamic Based Plant-Wide Control Design Procedure of the Tennessee Eastman Process L.T. Antelo, I. Otero-Muras, J.R. Banga and A.A. Alonso .......................................... 1413 Stochastic Optimal Control in Batch Reactive Systems: Developments on Engineering Applications of Real Option Theory V. Rico-Ramirez, J.F. Cambero-Benitez, H. Cañada-Jaime and S. Hernandez-Castro............................................................................................ 1419 A Web Service Based Online Optimization and Monitoring System for Chemical Processing Systems Xiangyu Li, Xiuxi Li and Y. Qian ................................................................................ 1425 Using the Process Schematic in Plant-Wide Disturbance Analysis S.Y. Yim, H.G. Ananthakumar, L. Benabbas, A. Horch, R. Drath and N.F. Thornhill ...................................................................................................... 1431 Constrained Control for Chemical Processes Using Reference Governor K. Kogiso, M. Noda and H. Nishitani ......................................................................... 1437 Agent-based Diagnosis for Granulation Processes R. Lakner, E. Németh, K.M. Hangos and I.T. Cameron .............................................. 1443 An Application of Metamodels for Process Optimization M.V.C. Gomes, I. David, L. Bogle, D. Odloak and E.C. Biscaia Jr. ........................... 1449 Time Scale Separation and the Link Between Open-Loop and Closed-Loop Dynamics A. Araújo, M. Baldea, S. Skogestad and P. Daoutidis ................................................ 1455 Fault Detection and Diagnosis of Pulp Mill Process G. Lee, T. Tosukhowong and J.H. Lee ........................................................................ 1461 Improving Observability of Large-Scale Systems by Iterative Weighting Adjustment R. Faber, H. Arellano-Garcia, P. Li and G. Wozny .................................................... 1467 Simulation Based Engineering – From Process Engineering to Automation Engineering H. Fischer and J.C. Toebermann................................................................................ 1473
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Large-Scale Dynamic Optimization of an Integrated Cryogenic Process M. Rodriguez and M.S. Diaz .......................................................................................1477 Scheduling of Storage and Transfer Tasks in Oil Refineries by Using Fuzzy Optimization L.C. Felizari and R. Lüders.........................................................................................1483 Degrees of Freedom Analysis for Process Control M. Rodríguez and J.A. Gayoso....................................................................................1489 Autothermal Reactors for Hydrogen Production: Dynamics and Model Reduction M. Baldea and P. Daoutidis ........................................................................................1495 Performance Assessment and Controller Design for Unknown Systems Based on Gain and Phase Margins Using Modified Relay Feedback J.-C. Jeng and H.-P. Huang ........................................................................................1501 Graphical Modeling for the Safety Verification of Chemical Processes J. Kim, Y. Lee and I. Moon..........................................................................................1509 Application of a Hybrid Control Approach to Highly Nonlinear Chemical Processes Y. Sakakura, M. Noda, H. Nishitani, Y. Yamashita, M. Yoshida and S. Matsumoto ....1515 Dynamic Optimization of Dead-End Membrane Filtration B. Blankert, B.H.L. Betlem and B. Roffel ....................................................................1521 Combined Nonlinear Model Predictive Control and Moving Horizon Estimation for a Copolymerisation Process M. Diehl, P. Kühl, H.G. Bock, J.P. Schlöder, B. Mahn and J. Kallrath ......................1527 Automatic Adjustment of Data Compression in Process Information Management Systems F. Alsmeyer .................................................................................................................1533 Virtual Plant, New Paradigm for Future Production Management H.A. Gabbar, K. Nishiyama, I. Shingo, T. Ooto and K. Suzuki....................................1539 Adaptive Monitoring Statistics Based on State Space Updating Using Canonical Variate Analysis C. Lee, S.W. Choi and I.-B. Lee...................................................................................1545 Scheduling of Make and Pack Plants: A Case Study C.-U. Fündeling and N. Trautmann ............................................................................1551 Detection of Abnormal Alumina Feed Rate in Aluminium Electrolysis Cells Using State and Parameter Estimation K. Hestetun and M. Hovd ............................................................................................1557 Thermodynamic Diagram Based Estimation Structure Design for Ternary Distillation Column A. Pulis, C. Fernandez, R. Baratti and J. Alvarez .......................................................1563
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Optimal Operation of a Mixed Fluid Cascade LNG Plant J.B. Jensen and S. Skogestad ...................................................................................... 1569 Multiplicity of Steady States in an UOP FCC Unit with High Efficiency Regenerator J.L. Fernandes, C.I.C. Pinheiro, N. Oliveira and F. Ramôa Ribeiro .......................... 1575 On-Line Data Reconciliation and Parameter Estimation for an Industrial Polypropylene Reactor D.M. Prata, J.C. Pinto and E.L. Lima ........................................................................ 1581 Optimal Reactive Scheduling of Multipurpose, Make-to-Order Industries M.C. Gomes, A.P. Barbosa-Póvoa and A.Q. Novais................................................... 1587 Optimal Steady-State Transitions Under Constrained Predictive Control D.K. Lam and C.L.E. Swartz ...................................................................................... 1593 Optimal Configuration of Artificial Neural Networks V. Dua ........................................................................................................................ 1599 Diagnosis of Oscillations in Process Control Loops Y. Yamashita............................................................................................................... 1605 Advances and Future Directions in Morphology Monitoring and Control of Organic Crystals Grown from Solution X.Z. Wang, K.J. Roberts and J. Calderon De Anda .................................................... 1611 Molecular Weight Control in Acrylonitrile Polymerization with Neural Network Based Controllers I. Atasoy, M. Yuceer and R. Berber ............................................................................ 1617 A New Approach to Chance Constrained Process Optimization and Control Under Time-Dependent Uncertainties H. Arellano-Garcia, T. Barz, W. Martini and G. Wozny............................................. 1623
Topic 4 Oral A Lab-on-a-Chip Simulation Framework A.J. Pfeiffer, X. He, T. Mukherjee and S. Hauan ........................................................ 1631 Two Level Control of the Sequence Fed Batch – Continuous Hybridoma Bioreactor I.D. Ofiteru, A. Woinaroschy and V. Lavric................................................................ 1637 Optimal Delivery of Chemotherapeutic Agents in Cancer P. Dua, V. Dua and E.N. Pistikopoulos ...................................................................... 1643 Dissipative Particle Dynamics Simulation of Ibuprofen Molecules Distribution in the Matrix of Solid Lipid Microparticles (SLM) C. Long, L. Zhang and Y. Qian ................................................................................... 1649
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An Integrative Systems Biology Approach for Analyzing Liver Hypermetabolism E. Yang, F. Berthiaume, M.L. Yarmush and I.P. Androulakis .....................................1655 Solid Fuel Decomposition Modelling for the Design of Biomass Gasification Systems D. Brown, T. Fuchino and F. Maréchal ......................................................................1661 Hybrid Metabolic Flux Analysis/Data-Driven Modelling of Bioprocesses A. Teixeira, C.L.M. Alves, P.M. Alves, M.J.T. Carrondo and R. Oliveira ...................1667 Rotavirus-Like Particle Production: Simulation of Protein Production and Particle Assembly A. Roldão, H.L.A. Vieira, M.J.T. Carrondo, P.M. Alves and R. Oliveira ....................1673 Prediction of Secondary Structures of Proteins Using a Two-Stage Method M. Turkay, O. Yilmaz and F. Uney Yuksektepe ...........................................................1679 Reconstruction of Transcriptional Regulatory Networks via Integer Linear Programming J.M.S. Natali and J.M. Pinto .......................................................................................1687 Systematic Design of Drug Delivery Therapies M. Xenos, L. Zhang, M.B.R. Somayaji, S. Kondapalli and A.A. Linninger ..................1693
Topic 4 Poster Mathematical Modelling of Three-Dimensional Cell Cultures in Perfusion Bioreactors. Part II F. Coletti, S. Macchietto and N. Elvassore..................................................................1699 Metabolic Regulatory Network Optimization Using an Information Guided Genetic Algorithm Approach Y. Zheng, C.-D. Yang, J.-W. Yeh and S.-S. Jang..........................................................1705 Minimal Reaction Sets and Metabolic Pathways for Cultured Hepatocytes H. Yang, M.L. Yarmush, C.M. Roth and M. Ierapetritou.............................................1711 Hybrid Modular Mechanistic/ANN Modelling of a Wastewater Phosphorous Removal Process J. Peres, F. Freitas, M.A.M. Reis, S. Feyo de Azevedo and R. Oliveira ......................1717 Modelling Morphological Change in Endothelial Cells Induced by Shear Stress R.J. Allen, D. Bogle and A.J. Ridley ............................................................................1723
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Topic 5 Oral Disturbance Propagation and Rejection Models for Water Allocation Network X. Feng and R. Shen ................................................................................................... 1729 Energy Planning in Buildings Under Uncertainty in Fuel Costs: The Case of a Hospital in Greece G. Mavrotas, K. Florios and P. Georgiou .................................................................. 1735 Modelling an Electricity Infrastructure as a Multi-Agent System — Lessons Learnt from Manufacturing Control K.H. van Dam, M. Houwing, Z. Lukszo and I. Bouwmans .......................................... 1741 Global Optimization of Multiscenario Mixed Integer Nonlinear Programming Models Arising in the Synthesis of Integrated Water Networks Under Uncertainty R. Karuppiah and I.E. Grossmann ............................................................................. 1747 Hierarchical Markov Reliability/Availability Models for Energy & Industrial Infrastructure Systems Conceptual Design A.N. Ajah, P.M. Herder, J. Grievink and M.P.C. Weijnen .......................................... 1753 Agent-Enabled Dynamic Management System for Process Plants A. Kokossis, Z. Shang and E. Gao .............................................................................. 1759 Methodology for the Design of Industrial Hydrogen Networks and the Optimal Placement of Purification Units Using Multi-Objective Optimisation Techniques L. Girardin, F. Marechal and P. Tromeur .................................................................. 1765
Topic 5 Poster Modelling and Simulation of a Tyre Gasification Plant for Synthesis Gas Production N.R. Mitta, S. Ferrer-Nadal, A.M. Lazovic, J.F. Perales, E. Velo and L. Puigjaner ............................................................................................ 1771 Library for Modeling and Simulating the Thermal Dynamics of Buildings J.I. Videla and B. Lie .................................................................................................. 1777 A New Method for Designing Water Network Based on Variable Removal Ratio of Treatment Process L. Song, J. Du, S. Cai and P. Yao ............................................................................... 1783 Environmental Life Cycle Impact and Cost Minimization in the Steam and Power Generation Plant P. Martínez and A.M. Eliceche ................................................................................... 1791
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Case Study of a Regional Network for the Recovery of Hazardous Materials J. Duque, A.P.F.D. Barbosa-Póvoa and A.Q. Novais .................................................1797 Optimisation of a Pertraction Process for Wastewater Treatment and Copper Recovery A.M. Eliceche, M.F. Orlandi, A.M. Urtiaga and I. Ortiz.............................................1803 Data-Centric Demand Forecasting for Utilities Z. Beran, K. Marík and P. Stluka ................................................................................1809 A Screening Tool for Exploring Production Chains L. Stougie, R.M. Stikkelman and M. Houwing .............................................................1815 Cost versus Network Length Criteria in Water Network Optimal Design P. Iancu, V. Plesu and V. Lavric .................................................................................1821 Synergy Analysis of Collaboration with Biofuel use for Environmentally Conscious Energy Systems M. Turkay and A. Soylu...............................................................................................1827 Process Optimization and Scheduling of Parallel Furnaces Shutdown in Large-Scale Plants E.P. Schulz, J.A. Bandoni and M.S. Diaz ....................................................................1833 Operational Optimization of the Thermoelectric System of an Oil Refinery S.R. Micheletto and J.M. Pinto....................................................................................1839 Water Reuse: A Successful Almost Zero Discharge Case R.M.B. Alves, R. Guardani, A.E. Bresciani, L. Nascimento and C.A.O. Nascimento ...............................................................................................1845 Model Development for the Optimal Water Systems Planning E. Kondili and J.K. Kaldellis.......................................................................................1851 Simulation of Electricity Production Systems in Autonomous Networks in Order to Maximize the Wind Energy Penetration J.K. Kaldellis and E. Kondili.......................................................................................1857 Heat Integration in Micro-Fluidic Devices T. Zhelev and O. Strelow .............................................................................................1863 Network Synthesis for a District Energy System: A Step Towards Sustainability C. Weber, I. Heckl, F. Friedler, F. Maréchal and D. Favrat.......................................1869
Topic 6 Oral Close Loop Supply Chains: Managing Product Recovery Portfolio A.C.S. Amaro and A.P.F.D. Barbosa-Póvoa ..............................................................1875
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Simulation Based Optimization for Risk Management in Multi-Stage Capacity Expansion X. Wan, J.F. Pekny and G.V. Reklaitis........................................................................ 1881 A Decision Support Tool for Process Optimization of Sulphur Free Diesel Production Z. Lukszo, M. Salverda and P. Bosman ...................................................................... 1887 An Attainable Region Approach for Effective Production Planning C. Sung and C.T. Maravelias...................................................................................... 1893 A Planning Support System for Biomass-based Power Generation N. Ayoub, K. Wang, T. Kagiyama, H. Seki and Y. Naka ............................................. 1899 Semantic Analysis for Identification of Portfolio of R&D Projects. Example of Microencapsulation A. Kraslawski ............................................................................................................. 1905 Dynamic Rule-based Genetic Algorithm for Large-Size Single-Stage Batch Scheduling Y. He and C.-W. Hui ................................................................................................... 1911 Application of Multi-Stage Scheduling P.M.M. Bongers and B.H. Bakker ............................................................................. 1917 Slot-based vs. Global Event-based vs. Unit-Specific Event-based Models in Scheduling of Batch Plants M.A. Shaik, S.L. Janak and C.A. Floudas ................................................................... 1923 A Unified Approach for Knowledge Modeling in Pharmaceutical Product Development C. Zhao, A. Jain, L. Hailemariam, G. Joglekar, V. Venkatasubramanian, K. Morris and G. Reklaitis.......................................................................................... 1929 A Multistage Stochastic Programming Approach with Strategies for Uncertainty Reduction in the Synthesis of Process Networks with Uncertain Yields B. Tarhan and I.E. Grossmann ................................................................................... 1937 A Framework for Capturing the Impact of Resource Allocation Policies in the Selection of a New Product Portfolio J.C. Zapata, V.A. Varma and G.V. Reklaitis .............................................................. 1943 Multi-Period Capacitated Lot Sizing with Variable Batch Sizes Y.C. See-Toh, S.P.K. Walsh and N. Shah .................................................................... 1949 Integration of Discrete-Event Simulation and Optimization for the Design of Value Networks M. Schlegel, G. Brosig, A. Eckert, K. Engelke, M. Jung, A. Polt, M. Sonnenschein and C. Vogt .................................................................................... 1955
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A CP Method for the Scheduling of Multiproduct Continuous Plants with Resource Constraints L.J. Zeballos and G.P. Henning ..................................................................................1961 Stochastic Integer Programming in Chemical Batch Scheduling: Evolution Strategies vs. Exact Algorithms J. Till, G. Sand and S. Engell ......................................................................................1967 Scheduling and Planning with Timed Automata S. Panek, S. Engell and O. Stursberg .........................................................................1973 Novel Continuous-Time Formulations for Scheduling Multi-Stage Multi-Product Batch Plants with Identical Parallel Units Y. Liu and I.A. Karimi .................................................................................................1979 Routing and Cargo Allocation Planning of a Parcel Tanker K.-H. Neo, H.-C. Oh and I.A. Karimi ..........................................................................1985 An Approximate Framework for Large Multistage Batch Scheduling Problems Focusing on Bottleneck Resources P.A. Marchetti and J. Cerdá .......................................................................................1991 On the Dynamic Management of Chemical Engineering Knowledge Using an Ontology-based Approach A. Kokossis, E. Gao and A. Kourakis ..........................................................................1997 Lagrangean-based Techniques for the Supply Chain Management of Flexible Process Networks P. Chen and J.M. Pinto ...............................................................................................2003 Restructuring Methodology in Process Engineering for Sustainable Development I. Koshijima, A. Shindo, Y. Hashimoto and T. Umeda.................................................2009 Development of a Multiobjective Scheduler for Semiconductor Manufacturing O. Baez Senties, C. Azzaro-Pantel, L. Pibouleau and S. Domenech............................2015 Ontology-based Information Management in Design Processes S.C. Brandt, J. Morbach, M. Miatidis, M. Theißen, M. Jarke and W. Marquardt ....... 2021 Workflow Support for Inter-Organizational Design Processes R. Hai, M. Heller, W. Marquardt, M. Nagl and R. Wörzberger ..................................2027 Rigorous Scheduling Resolution of Complex Multipurpose Batch Plants: S-Graph vs. MILP S. Ferrer-Nadal, T. Holczinger, C.A. Méndez, F. Friedler and L. Puigjaner ..............2033
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Topic 6 Poster Integrative Optimization of Refining and Petrochemical Plants C. Li, X. He, B. Chen, B. Chen, Z. Gong and L. Quan ................................................ 2039 Scheduling of Identical and Parallel on/off Production Units Under Uncertainty in Cost and Demand Prediction P. Pulkkinen and R. Ritala.......................................................................................... 2045 A Flexible Design of Logistic Network Against Uncertain Demands Through Hybrid Meta-Heuristic Method Y. Shimizu, S. Matsuda and T. Wada .......................................................................... 2051 A Flexible Framework for Optimal Biorefinery Product Allocation N. Sammons, M. Eden, H. Cullinan, L. Perine and E. Connor ................................... 2057 Systems for Decisions Support in Industrial use K. Coböken, G. Mogk, T. Mrziglod and U. Telle ........................................................ 2063 Medium Term Planning of Biopharmaceutical Manufacture Under Uncertainty K. Lakhdar, S.S. Farid, J. Savery, N.J. Titchener-Hooker and L.G. Papageorgiou .... 2069 A Multi-Criteria Optimization Model for Planning of a Supply Chain Network Under Demand Uncertainty C.L. Chen, T.Y. Yuan, C.Y. Chang, W.C. Lee and Y.C. Ciou ...................................... 2075 A View-based Information Model for Enterprise Integration in Process Industries P. Li, M.L. Lu, Y.S. Peng and B. Hua ......................................................................... 2081 Strategic Planning and Design Using MILP: An Industrial Application from the Tissue Manufacturing Industry J. Westerlund, P. Castro and S. Forssell .................................................................... 2087 Multiple Time Grid Continuous-Time Formulation for the Short Term Scheduling of Multiproduct Batch Plants P. Castro and I. Grossmann ....................................................................................... 2093 An Inventory Control Scheme for Simultaneous Production Planning and Scheduling Under Demand Uncertainty T. Nishi, H. Tominaga and M. Konishi ....................................................................... 2099 Implementation of an Integrated Platform of Process System Operations for Education and Research X. Li, Y. Qian and Y. Jiang ......................................................................................... 2105 Integration of Multi-Scale Planning and Scheduling Problems H. Stefansson, P. Jensson and N. Shah ....................................................................... 2111
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Plant-Wide Planning and Marginal Value Analysis for a Refinery Complex W. Li, X. Liang and C.-W. Hui ....................................................................................2117 Refinery Planning Under Correlated and Truncated Price and Demand Uncertainties W. Li, I.A. Karimi and R. Srinivasan ...........................................................................2123 An Integrated Model for the Design and Planning of Supply Chains with Product Return M.I.G. Salema, A.P. Barbosa-Povoa and A.Q. Novais ................................................2129 Pipeline Scheduling and Distribution Centre Management – A Real-World Scenario at CLC S. Relvas, A.P.F.D. Barbosa-Póvoa, H.A. Matos, J. Fialho and A.S. Pinheiro ...........2135 Scheduling Under Demand Uncertainty Using a New Multiparametric Programming Approach Z. Jia and M.G. Ierapetritou .......................................................................................2141 Information Modeling for Pharmaceutical Product Development C. Zhao, L. Hailemariam, A. Jain, G. Joglekar, V. Venkatasubramanian, K. Morris and G. Reklaitis ..........................................................................................2147 Decentralized Supply Chain Dynamics and the Quantity Flexibility Contract V. Subramanian, J.F. Pekny and G.V. Reklaitis ..........................................................2153 Planning and Scheduling of Multipurpose Continuous Plants C. Schwindt, S. Herrmann and N. Trautmann .............................................................2159 Priority-Rule Based Scheduling of Chemical Batch Processes N. Trautmann and C. Schwindt ...................................................................................2165 A Rigorous Approach to Coordinate Production and Transport Scheduling in a Multi-Site System C. Méndez, A. Bonfill, A. Espuna and L. Puigjaner.....................................................2171 Multi-Criteria Evaluation for the Chemical Industrial Parks J. Xiaoping, T. Zhang and L. Shi.................................................................................2177 Scheduling with High Accuracy at Low Maintenance Costs: An Approach Using Discrete Event Simulation M. Jung and C. Vogt ...................................................................................................2183 An Integrated Model for Planning in Global Chemical Supply Chains A. Sundaramoorthy, S. Xianming, I.A. Karimi and R. Srinivasan ...............................2189 Information Sharing in a Distributed Enterprise: Impact on Supply Chain Performance and Decision-Making I.B. Owusu and S. Hauan ............................................................................................2195
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Integration of Supply Chain Management and Logistics: Development of an Electronic Data Interchange for SAP Servers L. Jiménez and R. Muñoz............................................................................................ 2201 A Novel Combined Approach for Supply Chain Modeling and Analysis F.D. Mele, C.A. Méndez, A. Espuña and L. Puigjaner................................................ 2207 A Hybrid Approach Using CLP and MILP Applied to Tank Farm Operation Scheduling S.L. Stebel, F. Neves Jr. and L.V.R. Arruda ................................................................ 2213 PRoduct ONTOlogy. Defining Product-Related Concepts for Production Planning Activities D. Giménez, M. Vegetti, G. Henning and H. Leone .................................................... 2219 Recipe Informatics to Shorten the Lead Time from Product Development to Production in Batch Processes T. Fuchino, T. Kitajima, Y. Shimada, K. Takeda, S. Hashizume, T. Hamaguchi, R. Batres, A. Yamada, K. Kawano and Y. Hashimoto ................................................. 2225 Efficient MILP-based Solution Strategies for Large-Scale Industrial Batch Scheduling Problems P. Castro, C. Méndez, I. Grossmann, I. Harjunkoski and M. Fahl ............................. 2231 Innovation and Knowledge Management: Using the Combined Approach TRIZ-CBR in Process System Engineering G.C. Robles, S. Negny and J.M. Le Lann.................................................................... 2237 Decision-Making Tool for Scheduling of Batch Processes: The Dynamic Hybrid Simulation Kernel N. Olivier, R. Thery, G. Hétreux and J.-M. Le Lann ................................................... 2243 Multiobjective Multiproduct Batch Plant Design Under Uncertainty A. Dietz, A. Aguilar-Lasserre, C. Azzaro-Pantel, L. Pibouleau and S. Domenech ........................................................................................................ 2249
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Innovation in the chemical industry: a growth engine! Dr. Stefan Marcinowski Member of the Board of Executive Directors and Research Executive Director of BASF AG
Abstract This presentation addresses the opportunities and challenges of the chemical industry as an innovation motor in the global market place with examples from our daily business. About 80% of all chemical products are passed on as semi-finished products to other branches of industry, such as the automotive, construction, or microelectronics industry. Success in innovation is therefore determined by our ability to provide products that give our customers a competitive advantage in their respective market place. The objective of chemical R&D is to enable product and system innovations by putting market trends and ideas from science into practice as rapidly as possible. And to increase our competitiveness by continuously improving production processes. In order to provide leading-edge products and solutions, capturing technological excellence by cooperation is crucial. Cooperation with universities, institutes, and startup companies provide a ´window on technology´, such as biotechnology, or nanotechnology in the earliest stages of development. Collaboration with value-adding customers in form of strategic partnerships is important to identify relevant product solutions and market trends. Mutual trust, understanding of market needs and technological capabilities, as well as speed of innovation are key to success. The ever-shortening product-cycles in the end-consumer market keep challenging the pace of the innovation process in the chemical industry. Ultimately, it is essential for the chemical industry to continuously improve its cost structure by new processes and operational excellence.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Life Cycle Modelling in the chemical industries: Is there any reuse of models in automation and control? Jens Bausaa und Guido Dünnebierb a
BASF AG, Ludwigshafen, Germany Bayer Technology Services GmbH, Leverkusen, Germany
b
Abstract In the last two decades, simulation technology had a large influence on process industries. Today, modern numerical methods, powerful personal computers and convenient software packages facilitate the solution of complex engineering problems at every office workplace. The concept of model re-usage in the chemical industries and the supporting methods and tools are subject to current research and can be found in numerous publications. In practice, the integrated use of models for different applications, in particular in automation and control, can be found only rarely so far. This contribution concentrates on these applications in process optimisation and advanced control. By considering the different viewpoints of (academic) researchers, software providers and industrial users, the authors discuss potential reasons for the gap between the positions of these three groups. This contribution demonstrates the current state of industrial applications, the problems and limitations occurring therein, and the fact that these problems are no insurmountable obstacles for the application of model based methods in automation and control. Keywords: model based control, advanced process control, dynamic simulation, life cycle modelling.
1. Introduction Today, modern numerical methods, powerful personal computers and convenient software packages facilitate the solution of complex engineering problems at every office workplace. Typical tasks in the chemical industries are steady-state process design, dynamic process simulation for the development of control strategies and the design of model based control concepts [1,5,13]. In principle, existing models could be applied comprehensively to make use of the already available process knowledge. For instance, this aspect comprises the usage of steady state design process models for controller design based on dynamic models. The general concept of model re-usage and the supporting methods and tools are subject to current research and can be found in numerous publications. In practice, the integrated use of models in automation and control, as illustrated in figure 1, can be found only rarely so far. Since the main driver for application is not availability of a technology but the profitability, it might be concluded that the profitability of life-cycle-modelling and model re-use is at least not widely recognized. This contribution concentrates on integrated use of models in process optimisation and advanced control. By considering the different viewpoints of (academic) researchers,
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MPC
Operator Training Simulation
Observer / Dynamic Soft Sensor
Dynamic Optimisation
Data Reconciliation Dynamic Simulation / Controller Design
Flowsheeting Tools
Soft Sensor
Steady State Design Models
Planning
Online Optimisation
Operation
Fig. 0: Model reuse in automation and control
software providers and industrial users in sections 2 to 4, the authors discuss potential reasons for the gap between the positions of these three groups. Specifically, it is remarkable that after a wave of projects using dynamic simulation in the 1990s, the integrated use of models for automation and control has not become widely accepted in chemical industries yet. The mayor part of this contribution is a collection of industrial applications in chemical industry, both “historical” in section 5 and current in section 6, to illustrate the capabilities of the technology. The following discussion is, besides the references cited, based on the professional experience of the members of the NAMUR working group “Advanced Process Control”, representing a wide range of the German process industries.
2. Academic Perspective From the academic perspective, modelling and simulation in the chemical engineering field is widely accepted as technological mature and research is currently conducted mainly in very specialised areas. Recent research focuses on the integration of different simulation tools through standard interfaces (CAPE OPEN, which is already an established technology for physical properties,[7]), and software independent model repositories (ROME, [6]). Some even more specialised aspects are model reduction, efficient initialisation and efficient algorithms for dynamic optimisation. One possible summary for the academic perspective could be: A numerous amount of methods has been developed over the last decades, which theoretically allow the solution of most problems occurring in industrial reality for “classical” chemical processes. Methods for model re-use are developed and even the applicability has been demonstrated. The current research is addressing several application driven open issues, like solid and biological processes, and modelling of product properties (instead of processes), to mention a few aspects only. Some lessons might also be learned from the automotive industry where issues of model reusability and platform indepence for control applications has been addressed for quite a while (see eg. [22] for an overview).
Life Cycle Modelling in the Chemical Industries
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3. Software Provider Perspective The integration of steady state and dynamic simulation in a single product or a product family became a standard in many cases, and several common simulation tools offer the possibility to export process models in a general mathematical form to be used in other applications. Supplier independent model libraries should gain larger impact in the future, but economical interests of in particular the large suppliers in the market and the lack of power to effectively define standards retard this process. Current developments concentrate on application areas with high profitability, e.g. large continuous (e.g. petrochemical) processes. The extension to small scale applications (“keep it smart and simple”) is not visible. However, it has to be addressed that the process industry does not provide a clear wish list or roadmap to the software providers. Customer needs are hard to guess, if the discussion with different people from one company (e.g. plant manager, service provider automation, conceptual process deigner) does not lead to a consistent picture.
4. Industrial (End-) User Perspective Model centred technologies are most consequently applied in the large scale (petro-) chemical processes. Model predictive control [2,3], online optimisation and training simulation projects are executed by external engineering companies with dedicated software products, and supplier decisions are driven mainly by (short-term) economic consideration and not by model reusability. Due to the dynamic development of simulation technology, version and even product cycles are short. The disability of many software tools to easily import old model files often leads to the reimplementation.. A common, standardized simulation language would definitely help the software end-user to deal with tools of different vendors and to overcome the fast release cycles. However, current commercial simulation software became very efficient for standard unit operations, such that the reimplementation of models in different tools does not cause significant costs. To establish a company wide model library is a challenging task. Most often the units responsible for conceptual process design, equipment design and process automation are widely spread over the organization. To bring these units to complete agreement about the benefit of a common model library and to find a funding for these long-term activities often fails. Thus, company-wide model libraries have only been introduced in a few exceptional cases. To promote the implementation of model based methods in the future, the consciousness for the long-term value of process model needs to be raised significantly. The main challenge is to overcome the gap between the long-term benefit of concepts for the re-use of models on the one hand side and the short-term difficulties like the lack of continuity of simulation tools on the other hand side.
5. Historical Examples In the sequel, we refer to a set of previously published “historical” application examples mainly collected within Bayer and BASF to document the industrial research focus and implementations some ten or more years ago: Dating from already more than 40 years back, an online optimisation example for an oxo plant at BASF is documented. This application used an early digital computer, a process model derived by regression using 7 inputs and 5 outputs and a gradient based optimisation algorithm [15]. Considering the very limited capabilities of digital computers at this time, and the non-existence of comfortable programming helps, not to
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speak of modelling tools, this implementation was extremely ambitious and far ahead of times. Dating from approximately 10 years later, a data reconciliation application for the selectivity calculation of an ethylene oxide reactor is reported [16]. Model and optimisation algorithm are of similar complexity and the digital computer was certainly slightly more advanced using FORTRAN as programming language than in the previous example, but basically the methodological approach remained unchanged for more than 10 years and probably was still ahead of its time. The latter application has been extended recently and is still running today [17]. The peak of dynamic simulation research and development in the 1990’s is illustrated be the cooperation between Bayer and Cray to implement a dynamic simulation of a whole distillation train using supercomputers [18,19]. Even though models of comparable complexity now run on a PC and are much easier to engineer, applications with such ambitious scope today are only frequently reported in the operator training simulation area. The model usage along the lifecycle has never been a real topic until then, first attempts from industry to reuse dynamic simulation model for controller design are documented in academic collaboration in the mid 1990’s [20].
6. Recent Applications This section briefly lists some recent examples for applications and pilot projects collected in the NAMUR working group “Advanced Process Control”, mainly focusing on the reusability of process models along the lifecycle and towards automation and control applications. Operator Training Simulation to Advanced Control: The integration of operator training simulation and advanced process control support the start-up of a Bayer monomer plant to full load in automatic mode in two weeks only [14]. Steady State Design Model to Advanced Control: To derive a control concept of two coupled distillation columns, BASF compared the efforts needed for two different approaches, first to start from a steady state design model, and second, only reuse the physical property configuration and reimplement the remaining process model with a detail level tailored for the purpose [9]. Due to the high level of comfort and efficiency of commercial software products for this type of standard equipment, the usage of the existing model has proven to be the more expensive case here, which is mainly due to the fact of the different assumptions and details in the design model. Online Optimisation: Steam crackers are the most common application for online optimisation. In the BASF example, there was no existing process model for the old steam crackers which lead to a new implementation in a dedicated tool [10,11]. But even for units with existing design models, the direct transfer of existing models was nearly impossible due to new developments in the optimisation software. Dynamic Optimisation to Model Predictive Control: The integration of dynamic optimisation and model predictive control for a polymerisation process using an existing design model was one application within the INCOOP research project [4,12]. The feasibility and economic potential could be shown in simulations and plant experiments, but an online implementation could not be achieved during the project duration. This is certainly due to the fact that neither the developed technology nor the mindset of the responsible operating personnel is yet ready for a sustainable application. Dynamic Simulation to Model Predictive Control: A similar application was used in the POLYPROMS research project [8], whilst the design model available here had to be
Life Cycle Modelling in the Chemical Industries
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transferred manually since it was implemented in a software package dating back from the 1990’s. In this case, the model is being used for model predictive control, and the transfer to a modern software environment should only be considered as a little break in the lifecycle usage of the model towards automation and control. Some of the conclusions which can be drawn from this set of representative applications are: a) Model based methods today are indispensable in process automation. Many of these applications have proven to be profitable and reliable. However, process models typically are developed uniquely without making use of already available process models. b) Many achievements of academic research do not yet reach the industrial end-user, sometimes due to necessary extensions or refinements of the methods, but mostly due to the lack of a commercially available and affordable software implementations that are in line with the already company wide applied tools c) The economic break-even for model based application using the currently available methods and tools is still relatively high, tailor-made approaches for smaller processes and those extending the “classical” gas-liquid systems (e.g. batch/multipurpose units, solid processes, biotechnology, smaller life science processes) are urgently needed to promote the technology more widely.
7. Summary and Conclusions The discussion of the historical development and the perspectives of the three different interest groups shows (not surprisingly) partly different perspectives, which are, amongst others, driven by economic interests on the supplier side and organisational constraints on behalf of the end users. These conflicts in some case limit the technical developments. Research issues driven by industrial needs are towards small scale processes and those involving not only gas and liquid systems. Nevertheless, the list of examples shows the feasibility and the economic potential of model centred application in automation and control. Even in the far future, not every technically feasible solution will lead to economical applications, but the applicability of the methods needs to be extended by close cooperation between academic researchers, software providers and industrial end users. This contribution is based on an extended publication, which is recommended for further reading on this topic [21]. The authors gratefully thank the members of the NAMUR working group “Advanced process control” (A. Bamberg, Merck KgaA, J. Lamers, Henkel KgaA, U. Piechottka, Degussa AG, R. Piontek, Krupp Uhde GmbH, C. Vermum, Oxeno GmbH and O.Lorenz, Siemens AG) for numerous discussions and valuable input
References [1] W. MARQUARDT (1996). Trends in Computer-Aided Process Modeling COMPUTERS AND CHEMICAL ENGINEERING 20(6/7), S. 591-609. [2] QIN, S.J. AND T.A. BADGWELL (1996) An Overview of Industrial Model Predictive Control Technology PROCEEDINGS CPC-V,LAKE TAHOE,CA. [3] SEKI,H. ,M. OGAWA, S. OOYAMA, K. AKAMATSU, M. OHSHIMA, W.YANG (2001) Industrial application of a nonlinear model predictive control to polymerization reactors. CONTROL ENGINEERING PRACTICE 9, S. 819-828 [4] KADAM, J.V., W. MARQUARDT, M. SCHLEGEL, O.H. BOSGRA T. BACKX, P.-J. BROUWER, G. DÜNNEBIER, D. VAN HESSEM, A. TIAGOUNOV AND S. DE WOLF
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(2003). Towards integrated dynamic real-time optimization and control of industrial processes. In: Proc. FOCAPO 2003 (I.~E. Grossmann and C.~M. McDonald, Eds.). S.593-596. [5] W. MARQUARDT. (1990) Rechnergestützte Erstellung verfahrenstechnischer Prozessmodelle CHEMIE INGENIEUR TECHNIK 64(1), S. 25-40. [6] L. V. WEDEL, W. MARQUARDT (2000) ROME: A Repository to Support the Integration of Models over the Lifecycle of Model-based Engineering. In: Pierucci, S. (Hrsg.): Europ. Symp. on Computer Aided Process Engineering - 10, 535-540, Elsevier [7] THE CAPE OPEN LABORATORY NETWORK: Delivering the power of component software and open standard interfaces in computer-aided process engineering HTTP://WWW.COLAN.ORG/. [8] Targeted research action on polymer materials (TRA-PM) of the European Community: Development of advanced polymerisation process modelling, simulation, design and optimisation tools (polyPROMS), HTTP://WWW.TRA-PM.ORG/PROJECTS/GRD-25555.HTm. [9] J. BAUSA, J. BIRK: Development of dynamic process simulations using existent steady-state simulations - A user's perspective, ACHEMA 2003, Frankfurt. [10] E. STEIN, H. VEREECKEN: ROMeo-based closed-loop optimization of BASF ethylene plants, Simsci User Group Meeting, Heidelberg, Mai 2004. [11] O. ABEL, J. BIRK (2002) Echtzeitoptimierung verfahrenstechnischer Anlagen am Beispiel der Olefinproduktion, AT – AUTOMATISIERUNGSTECHNIK 50(12), S. 586-596. [12] G. DÜNNEBIER, D. VAN HESSEM, J.V. KADAM, K.-U. KLATT UND M. SCHLEGEL (2004) Prozessführung und Optimierung von Polymerisationsprozessen CHEMIE INGENIEUR TECHNIK 76(6), S. 703-708. [13] W. MARQUARDT (1992) Rechnergestützte Erstellung verfahrenstechnischer Prozeßmodelle CHEMIE INGENIEUR TECHNIK 64, S. 25-40. [14] R. GUHL (2005) Start-Up: Sicher und effizient mit Hilfe von Trainingssimulatoren und Advanced Process Control ATP-AUTOMATISIERUNGSTECHNISCHE PRAXIS 47(5), S. 128-140. [15] G. HELLER (1963): Optimierung einer Oxo-Anlage mit einem Prozessrechner. Vortrag auf der NAMUR Hauptsitzung, Bad Dürkheim. [16] H. E. MÜLLER (1976): Datenerfassung und online Berechnung in einer EthylenoxidAnlage. Anwendung von Prozessrechnern in der Verfahrensindustrie, Tagungsband, Florenz [17] H.-J. BISTER, A. WEISS, G. DÜNNEBIER (2002) Prozessüberwachung mit Datenvalidierung PATENT DE 102 51 192. [18] L. BRÜLL, L. LANG, R. ZELLER AND S. ZITNEY (1994) Bayer AG and Cray Research collaborate on plantwide dynamic process simulations CRAY CHANNELS 16(1), S. 2-7. [19] S. ZITNEY, L. BRÜLL, L. LANG AND R. ZELLER (1995) Plantwide dynamic simulation on supercomputers: Modeling a Bayer distillation process," in Proc. Fourth International Conference on Foundations of Computer Aided Process Design (FOCAPD '94), L. T. Biegler and M. F. Doherty, eds., AIChE Symp. Ser. 91 (304), pp. 356-359. [20] F. GROSS, E. BAUMANN, A. GESER, D.W.T. RIPPIN AND L. LANG (1998) Modeling, simulation and controllability analyis of a heat integrated industrial distillation system, COMPUTERS AND CHEMICAL ENGINEERING, 22(1), pp. 223-237 [21] J. BAUSA UND G. DÜNNEBIER (2005) Durchgängiger Einsatz von Modellen in der Prozessführung, CHEMIE INGENIEUR TECHNIK, 77(12), pp. 1873-1884 [22] P. STRUSS AND C. PRICE (2003) Model-Based Systems in the Automotive Industry, AI MAGAZINE, 24(4), pp. 17-34
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Hierarchical Multiscale Model-based Design of Experiments, Catalysts, and Reactors for Fuel Processing D. G. Vlachos*, A. B. Mhadeshwar, and N. S. Kaisare Department of Chemical Engineering and Center for Catalytic Science and Technology (CCST) University of Delaware, Newark, DE 19716
Abstract In this paper a hierarchical multiscale simulation framework is outlined and experimental data injection into this framework is discussed. Specifically, we discuss multiscale model-based design of experiments to optimize the chemical information content of a detailed reaction mechanism in order to improve the fidelity and accuracy of reaction models. Extension of this framework to product (catalyst) design is briefly touched upon. Furthermore, we illustrate the use of such detailed and reduced kinetic models in reactor optimization as an example toward more conventional process design. The ammonia decomposition on Ruthenium to produce hydrogen and the water-gas shift reactions on Platinum for converting syngas to hydrogen serve as illustrative fuel processing examples of various topics. Finally, opportunities for process design and control in portable microchemical devices (lab-on-a chip) are discussed. Keywords: Multiscale, Process and Product Engineering, Model-Based Design of Experiments, Reactor Optimization, Microreactors.
1. Introduction There is an ever increasing number of portable electronic devices, such as cellular phones, laptops, personal data assistants, personal transportation, night vision goggles, GPS, unmanned aerial vehicles, etc. that necessitate portable power generation. Traditional battery technology often results in power supply systems that either are too heavy, do not last long enough, or both. For military applications, the power requirements for special missions can often exceed the capacity of the dismounted soldier’s batteries [1]. Single-use batteries are often disposed of, resulting in heavy metals and other toxic substances being released. Hence, hydrocarbon-fuelled systems are envisioned to be replacements of current battery technology for civilian and military applications [2,3]. Table 1 shows different power sources and their mass-based energy densities. In general hydrocarbons possess two orders of magnitude higher energy densities than lithium ion batteries. Conversion of chemical energy of hydrocarbons into electricity *
To whom all correspondence should be addressed
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can then result in lighter or longer lasting portable devices. If greater than 1% of chemical energy could be converted into electricity, an improvement over batteries could be achieved. Additionally, hydrocarbons, if used properly, only release water and carbon dioxide. Often times it takes hours to recharge batteries, whereas hydrocarbonbased devices can be refueled quickly by simply adding more fuel. Successful commercialization of portable power systems depends on the development of robust fuel processing schemes that enable safe, efficient, economic, and convenient operation. Table 1: Energy densities of different sources. The energy density of combustion-based sources is based on complete combustion to carbon dioxide and liquid water at 25 ºC and 1 atm. Source
Energy Density [MJ/kg]
Lead acid Batteries
0.0792
Nickel cadmium batteries
0.158
Lithium ion batteries
0.468
Methanol combustion
22.7
Heating oil combustion
42.5
Gasoline combustion
45.8
Propane combustion
50.3
Methane combustion
55.5
Hydrogen combustion
142
In this paper, we first present an overview on multiscale simulation focusing on the idea of hierarchical multiscale modeling of chemical reactors that has recently been proposed for model development and/or parameter estimation [4,5]. Then we present examples of using these models for model-based design of experiments with the objectives of (1) maximizing the information content of a reaction model, (2) reduction of model complexity, (3) carry out catalyst design, and (4) optimal reactor design. These are some of the first demonstrations toward the direction of multiscale modelbased product and process engineering in the area of fuel processing for H2 production, which could, in conjunction with fuel cells, be used for portable power generation. Alternative routes of harvesting energy from fuels, such as thermoelectrics [6], thermophotovoltaics [7], or micro-engines [8,9] are not discussed here.
2. Multiscale Modeling: Process vs. Product Engineering Multiscale modeling is the enabling science that seamlessly and dynamically links models and phenomena across multiple length and time scales, spanning from quantum scales to macroscopic scales, in a two-way information traffic manner (see Fig. 1) [1014]. Macroscopic scales may include a process or an entire plant. The typical objective of multiscale modeling is to predict macroscopic behavior, such as selectivity, conversion, pollutant levels, hot spots, etc. from first principles. Multiscale modeling involves computing information at smaller scales and moving towards the top of the “simulation ladder” by coarsening degrees of freedom as one goes from finer to coarser scales. Prediction of large-scale process performance based on small-scale information
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is termed bottom-up approach or upscaling. Since it can be easily assimilated with Mesoscopic: Coarse-grained models Quantum: DFT
Macroscopic: Macroscopic CFD
Atomistic MD, Atomistic: KMC, TST
Figure 1: Schematic of multiscale simulation ladder with main scales and typical tools. Information flows up (bottom-up) and down (top-down) the ladder. The step narrowing indicates the loss or coarse graining of information as one moves from lower to upper scales. For more discussion, see [14]. DFT=Density function theory; CFD=Computational fluid dynamics; MD=Molecular dynamics; KMC=Kinetic Monte Carlo; TST=Transition state theory.
process alternatives, it is congruent with the traditional objective of process engineering. Recent reviews on multiscale modeling of chemical reactors, systems biology, and materials highlighting this view are given in [14-16] and references therein. A probably more important but relatively unexplored role of multiscale modeling is in product engineering. Coupling of models between scales provides a ‘descriptor’ or a ‘ladder’ linking atomistic scale information of materials with macroscopic scale processing. Such a descriptor provides a unique opportunity for product engineering. In the context of multiscale simulation, product engineering can be viewed as the possibility to define desirable performance (objective functions) at the macroscopic scale and then come up with better materials of suitable atomistic structure and possible synthesis protocols via the use of multiscale modeling. Examples can entail the identification of better (cheaper, more stable, more active and selective, etc.) catalysts, of optimal pore size distribution, of templates that produce a desirable zeolite, etc. Combined process-product engineering is obviously also very important. In particular one is often interested in manipulating variables at the macroscopic scale, e.g., change flow rates and composition, but achieve control at the nanoscopic length scale either by optimum design or model-based on-line control [17-19]. An example is the ability to control the particle size distribution, the particle shape, and the atomistic packing of materials in crystallization of proteins. Atomistic details of intermolecular forces and templating effects along with more traditional variables, such as local pH and supersaturation, significantly impact polymorphism and thus whether one gets the right material. Yet, macroscopically manipulated variables control the local (i.e., at the nanoparticle scale) supersaturation, concentration of templates, and pH, and therefore the local gradient in chemical potential that in turn affects growth rate and packing. Multiscale model-based control is currently plagued by the tremendous computational cost of multiscale simulation and the difficulty of having numerous nanoscopic sensors and actuators distributed in a system. The former can be handled using suitable reduced models. Model reduction of complex multiscale models is an important research
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direction [14] that will only be discussed briefly later in this paper. The prospect of using a small number of mobile sensors and actuators that can collect information from ‘optimal’ spatial and temporal locations is a promising avenue to overcome the latter and enable product-process system engineering. 2.1. Hierarchical Multiscale Simulation: Building on Ideas from Conceptual Process Design for Model Development The above multiscale science vision, while stimulating, is currently too ambitious to be of practical value for the design and control of complex systems, such as those encountered in microchemical systems for portable fuel processors. There are numerous reasons rationalizing this fact. Consider the example of quantum mechanics at the smallest scale. Density functional theory (DFT) is breaking new grounds in the parameter estimation front. Recent work sets a paradigm for DFT-based parameter estimation on single crystals [20-26]. While DFT is the only truly founded theoretical technique of practical interest for catalysis that has great potential, it is practically limited to small molecules, to single crystals, and to specific coverages and is semiquantitative (at best) in nature. First, even most of the best DFT calculations have an accuracy of ± 5 kcal/mol in predicting activation energies. As a result, reaction rates are not as accurate and this uncertainty is important in predicting activity and selectivity especially at low temperatures. Second, DFT simulations are carried out on idealized single crystals that are of interest in surface science studies but can be irrelevant for practical catalysts that are polycrystalline or defected nanoparticles spread on a support. Third, DFT calculations are carried out at certain coverages. The multicomponent nature of complex fuel processing reactions and the drastic variation of dominant coverages of surface species with varying operating conditions make parameterization of surface kinetics (as a function of coverages) a combinatorial problem of large dimension that is currently beyond the reach of computational capabilities. Forth, the number of reactions needed to describe the chemistry of complex reactions is large. For example for the water-gas shift (WGS) reaction discussed below, 46 elementary-like reactions may be considered [4,27], whereas for the partial oxidation of methane more than 100 reactions are employed [28]. These large reaction networks hint to the inability of expensive DFT calculations to deliver these many parameters. Fifth, it has been recognized that the active sites in many reactions involve steps, kinks, and other defects whose size and/or density is such that it is impossible to even fit them in the unit cell of a DFT calculation. Sixth, DFT is inaccurate for weak, e.g., van der Waals, interactions and cannot treat well small activation barriers. Some of these limitations are known as materials gap (inability of DFT to deal with multiple scales shown in Fig. 1); the rest are associated with the CPU intensive nature of DFT. At the mesoscopic scale, kinetic Monte Carlo (KMC) simulation with large kinetic mechanisms is still in embryonic stages [21,29]. KMC is seriously plagued by fast diffusion and more generally stiffness and the inability of reaching large length scales [30]. Coarse-grained KMC is a new tool that could overcome these problems [31]. At the reactor scale, computational fluid dynamics (CFD) simulations are employed when the continuum approximation is valid. Yet, CFD simulations are very intensive especially when flows are turbulent, when reaction networks are large, and when geometries are complicated. Process engineers use computationally efficient software,
Hierarchical Multiscale Model-based Design
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such as ASPEN and HYSYS, to carry out optimization and process control studies. This task is obviously impossible to achieve using CFD.
Figure 2. Hierarchy of chemical kinetic and reactor models at various scales. UBI-QEP: Unity Bond Index Quadratic Exponential Potential. See Fig. 1 for other abbreviations.
Instead of trying to simulate all phenomena at all scales with the highest accuracy, one realizes that only certain reactions, species, phenomena, and some of the scales are in reality crucial for accurate prediction of macroscopic properties. The idea of hierarchical multiscale modeling and simulation is then to start with the simplest possible “sound” model at each scale and identify the important scales and (‘active’) model parameters at each scale. Once this is accomplished, one assesses the model accuracy by comparison with data and potentially improves the model of the important scale(s) and the associated active parameters using a higher-level model or theory. For example, the simplest identification tool employed extensively and successfully in chemical kinetics is local sensitivity analysis [32]. Upon improvement of models and parameters, another iteration is taken until convergence is achieved, i.e., the important scales and parameters do not change between successive iterations. This approach is reminiscent of conceptual process design used for chemical flow sheets, where detailed design is done only after several iterations of calculations of increasing complexity are done [33]. Specific tools employed in hierarchical multiscale chemical reactor model development are depicted in Fig. 2. The model predictions at each scale become more accurate as one goes from the left to the right of the figure, at the expense of increasing computational intensity.
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2.2. Data Injection into Multiscale Models for Parameter Refinement or Scale-Model Replacement Irrespective of the power of multiscale modeling, model parameters, such as diffusivities and activation energies, and measured quantities, such as catalyst surface area, have always an uncertainty. As a result, models are almost never in perfect agreement with experimental data. In other instances the computational requirements are so large that one may have to completely bypass the modeling of a scale, typically of the quantum one. It is therefore desirable to estimate or refine the active parameters or fill in a missing model of a particular scale using experimental data instead of higherlevel theory/model discussed above. This injection of data into a multiscale model is needed to increase its predictive capabilities and can be done using data at one or more scales of the ladder (see Fig. 1). Parameter estimation or refinement and model replacement become then an integral part of multiscale model development. A complication is that multiscale models are typically complex and computationally intensive and involve discrete, often stochastic, models at some scales. Therefore parameter estimation can be very time consuming and with noisy models in comparison to traditional parameter estimation of deterministic models. Response surface methods (RSM) could be invaluable in achieving this objective at minimal computational cost [34]. Development of more accurate and efficient RSMs should be an important objective of the systems community. Hierarchical multiscale modeling can be extremely valuable also when parameters are completely unknown. For example, one uses a mean-field, continuum model (such a model assumes spatial homogeneity at the microscopic scale) to estimate parameters and then uses these parameters as a good initial guess in a KMC model (this model can naturally account for microscopic heterogeneity, surface diffusion, defects, etc.) [5,35]. As another example, one uses a deterministic continuum model to estimate parameters and these parameters are then refined using the corresponding stochastic simulation that considers fluctuations and correlations in species populations. The hierarchical multiscale modeling should be exercised with caution. Its success relies in the various models of a scale being ‘structurally’ the same. For example, a linear lower level model may not capture the behavior, such as bifurcations, of a nonlinear higher-level model. In these instances one may hope to be successful only locally or needs to develop better lower level models. 2.3. An example of NH3 decomposition on Ru for H2 production The specific hierarchical multiscale framework for chemical reactors is depicted in Fig. 2. At the lowest theoretical level (left column), detailed microkinetic models are developed for the surface chemistry consisting of elementary-like reaction steps. Preexponentials are set based on Transition State Theory (TST) and activation energies are computed using the semi-empirical Unity Bond Index-Quadratic Exponential Potential (UBI-QEP) theory [36], using heats of chemisorption as inputs. These inputs can be obtained from experiments (preferred), DFT, or estimated using the UBI-QEP method. The output of the UBI-QEP method is activation energies of all surface reactions as a function of surface coverages. Reaction rates are determined using the mean-field approximation and are passed into a suitable, simple reactor scale model that accounts for transport via standard mass and heat transfer correlations. The entire framework is an automatic ‘wrapper’ of Surface
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100 PFR model Model 80 with interactions PFR model Model without interactions
60 40
1.0 20 0
650
0.5 0.0 850
Expts. [Ganley et al.] T [K]
1050
1250
Figure 3. Comparison of predictions for NH3 decomposition on Ru with (solid line) and without (dashed line) adsorbate-adsorbate interactions in a CFD simulation, shown as inset [39], against data (symbols) of [40].
Chemkin [37] and allows users to simulate pseudo-homogeneous reactors, such as a fixed bed reactor, and compare different catalysts. At this stage one can inject data to refine parameters or use more advanced theoretical tools, such as DFT, KMC, or CFD depicted in the right column of Fig. 2, to improve the model and parameters at the scale(s) that appears most critical. In our work we have used data injection to refine pre-exponentials only and DFT to refine energetics. The latter has mainly been used to account for surface coverage effects that are nearly impossible to obtain experimentally but can be crucial in affecting reactivity and selectivity [38]. Instead of solving the combinatorial problem of computing all interactions between all species in a brute-force manner, we identify the most abundant surface species (typically 1 or 2) by running simulations and carry out only a small number of DFT calculations for those relevant interactions. Advantages of this theoretical framework include: (a) its high speed (sub-seconds), (b) reasonable predictive capabilities in most cases, (c) easy exploration of alternative reaction paths (this is important to ensure that most relevant chemistry is included), and (d) creation of insights into the important chemistry. An example of performance of a detailed kinetic model of NH3 decomposition on Ru, consisting of 6 reversible reactions, against data from a post microreactor is shown in Fig. 3.
3. Model Reduction The models obtained using the hierarchical multiscale framework are often very complex and computationally demanding. The aim of these models is the accurate prediction of macroscale properties, such as conversion. Ideal reactors (lower hierarchy at the reactor scale in Fig. 2) seldom represent the actual system accurately, and hence, more realistic CFD models need to be used. Using complex kinetic models (higher
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hierarchy at the kinetics scale) with complex CFD models (higher hierarchy at the reactor scale) represent a large computational burden. As a result, model reduction is required to obtain computationally tractable, physically meaningful models. Mathematical tools such as principal component analysis (PCA), approximate inertial manifold (AIM), etc. have been used for model reduction at various scales (for example, see [39]). Additionally, scaling analysis has been used to simplify the complexity of reactor models, whereas identification of the rate determining step (RDS) or the use of small scale asymptotics is useful at the kinetics scale [40]. For example, [41] simplified a transient CFD model using scaling laws, and solved a pseudo-steady 1D model in the gas phase and a transient 3-D model in the solid phase. [42] used boundary layer approximation and scaling analysis to reduce a 2D elliptic model into a more computationally tractable parabolic model, whereas, [43] reduced the kinetic model consisting of 6 reversible reactions (discussed in the previous section) for ammonia decomposition and used the resulting 1-step chemistry in CFD reactor modeling for design of integrated microdevices for hydrogen production [44]. These are just some examples of model reduction but model reduction is unquestionably an essential step in multiscale model development (Fig. 1) and in linking complex models to process and product optimization and control.
4. Model-Based Design of Experiments: Maximizing Chemical Information Content Experiments are typically carried out at certain conditions and it is often found that only a small number of kinetic parameters are active under those conditions. A natural question is whether one could design experiments based on a model, rather than statistical design, in order to increase the number of active model parameters and the accuracy of parameter estimation from data. The benefit of increasing the number of active parameters is that one could either validate or extract additional and possibly more accurate kinetic parameters. A parameter pj is most active when the response Ri of the model with respect to this parameter is highest, i.e., when the absolute value of a sensitivity coefficient | ∂ ln R i / ∂ ln p j | is largest. During the estimation of kinetic parameters, identifiability analysis [45] could determine the extractable ones. Once optimum operating conditions for maximizing the sensitivity coefficients of the responses with respect to the identifiable parameters in the mechanism have been determined, experiments need to be conducted to test the model. Correct prediction of the best operating conditions depends on how good the initial values of parameters of a model are. Therefore, an iterative approach may be needed. Given that lower level models are used to estimate parameters, model predictions are reasonable even in the first iteration and the search leading to better models and parameters is physically constrained, i.e., convergence is usually attained in 1-2 iterations. Next, we outline the elements of the proposed approach. Then we illustrate the procedure using our microkinetic mechanism for NH3 decomposition on Ru [38] as an example. 4.1. Identifiability Analysis One performs a sensitivity analysis with respect to the mechanism parameters to obtain a sensitivity matrix g
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Figure 4. (a) Schematic of global Monte Carlo search in experimental parameter space (represented as a 3D cube for graphical purposes; each (yellow) sphere represents a point randomly picked in space) to identify conditions that sensitize kinetics parameters. (b) and (c) Examples of an actual sensitivity analysis carried out under different conditions. The sensitive (active) parameters can vary considerably in parameter space.
g= ⎡⎣∂R i / ∂p j ⎤⎦ ,
(1)
where Ri is the vector of measured model response (e.g., NH3 conversion), p is the vector of parameters (e.g., pre-exponentials), n is the number of model responses, and m is the number of parameters. Then the Fisher Information Matrix (FIM) is calculated FIM=g T *g .
(2)
If the determinant of FIM is zero, some parameters are interdependent and not identifiable. These parameters have to be removed and the computation of the FIM repeated. Subsequently, one calculates a correlation coefficient (cc) to judge whether any two identifiable parameters can be estimated within the measurement error in the experiments. cc is given as cci,j =
FIM -1 ( i,j) FIM -1 ( i,i ) × FIM -1 ( j,j)
(3)
and can vary from –1 to +1. Larger absolute values (away from 0) indicate higher correlation between parameters. Every parameter is self-correlated (cci,i = 1). Even though some parameters are identifiable, based on the determinant criterion, they could be highly correlated, so it may be difficult to estimate them separately given measurement error. Such parameters should be removed and the analysis repeated, so that only the identifiable, less correlated parameters are estimated from the experimental data. 4.2. Global Stochastic Search We perform model-based design of experiments to maximize the number of active parameters and the values of sensitivity coefficients. In particular, a global search in experimentally feasible parameter space is conducted on the computer, using a Monte Carlo (MC) global search algorithm (see Fig. 4a). At each point in parameter space, a reactor simulation is run using the current detailed kinetic model along with a local sensitivity analysis of experimentally measured responses with respect to kinetic
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parameters. Our objective is to identify suitable combinations of experimental variables that sensitize the maximum number of kinetic steps, i.e., identify experimental conditions where the most abundant reactive intermediate (MARI) and the rate determining step (RDS) change, providing additional kinetic information. Herein the FIM is employed, following the methods of [46], to systematically screen and organize the results of the global MC search. 4.3. Illustration Using the NH3 Decomposition Reaction on Ru The microkinetic model of [38] for NH3 decomposition on Ru has 12 pre-exponentials. Using a continuous stirred tank reactor (CSTR) model, we carry out sensitivity analysis of the NH3 exit mass fraction with respect to the pre-exponentials at 700 randomly selected operating conditions within the ranges shown in Table 2. It is found that the determinant of FIM is non-zero. Therefore, all pre-exponentials are identifiable over the operating ranges. However, calculation of the correlation matrix shows that the backward pre-exponentials are highly correlated with the forward ones (an expected result since the forward and backward ones are related to each other via thermodynamic constraints). Therefore, the backward pre-exponentials are eliminated and the analysis is repeated. Table 2. Range and scaling type of operating variables used to convert them into the [0,1] interval. Operating variable
Min
Max
Scaling
Temperature, T [K]
500
1000
Linear
Pressure, P [atm]
0.1
10
Log
Residence time, τ [s]
0.05
5
Log
Catalyst area per unit reactor volume, A/V [cm-1]
150
15000
Log
Inlet H2 mole fraction
0.0
1.0
Linear
Inlet NH3 mole fraction
0.0
1.0
Linear
Inlet N2 mole fraction
0.0
1.0
Linear
With only the forward pre-exponentials, the determinant of FIM is non-zero and the correlations are not very high either; therefore, all six pre-exponentials are identifiable. Fig. 5 shows the correlation coefficients for all reactions based on 700 operating conditions. As expected, each parameter is completely correlated with itself (ccii=1). H2 adsorption and NH3 adsorption (cc16 and cc61) have ~80% correlation, indicating that independent extraction of pre-exponentials could be difficult and higher experimental accuracy might be required. The sensitivity coefficients change drastically within the parameter space, as shown in Figs. 4b, 4c, and 6 and so does the RDS (see Fig. 6). This implies that sufficient sampling of parameter space can indeed provide new chemical insights. Within parameter space, conditions with the largest normalized sensitivity coefficient for each identifiable parameter are found, simply by sorting the global search sensitivity data. To avoid non-interesting conditions of low NH3 conversion and to minimize
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Correlation coefficient
1
0.5
0
-0.5
(1,6) -1
(6,1)
1
2
3 4 Reaction number
5
6
Figure 5. Correlation coefficients for all identifiable pre-exponentials in the microkinetic mechanism for NH3 decomposition on Ru. Some reaction pairs are labeled for ease of visualization. experimental uncertainties, a threshold of 5% conversion is applied while selecting best operating conditions. Values of optimal operating conditions are depicted in Fig. 6. Subsequently, experiments must be conducted at the identified conditions to test predictions and further refine model parameters (if needed one can take another iteration to refine parameters). At this stage refinement of heats of chemisorption (another model input) and most sensitive pre-exponentials could simultaneously be carried out. 2
*
1.5
1
*
0.4
0 0
8 7
20 40 60 80 100 Ammonia conversion [%]
2
|NSC|
1.5
954 0.9 0.1 3427 0.0 0.6 0.4
1
|NSC|
722 5.9 0.1 5535 0.08 0.82 0.1
0.5
3 2
0 0
20 40 60 80 100 Ammonia conversion [%]
-0.5
0
20 40 60 80 100 Ammonia conversion [%]
*
984 0.1 0.2 1253 0.2 0.6 0.2
0
8 7
N2+2*=2N *
*
4 2 0 -2
20 40 60 80 100 Ammonia conversion [%]
*
NH3+*=NH3
6 5 4
1 0
-0.2 0
*
NH +*=N +H
10 8 6
0.2
0
|NSC|
|NSC|
0.5
-0.5
822 2.0 0.1 560 0.74 0.25 0.01
0.6
1
14 12
NH2*+*=NH*+H*
0.8
731 0.3 2.0 1541 0.02 0.50 0.48
|NSC|
|NSC|
*
NH3 +*=NH2 +H
20 40 60 80 100 Ammonia conversion [%] 817 4.6 2.4 6847 0.68 0.32 0.0 *
H2+2*=2H
6 5 4 3 2 1 0
0
20 40 60 80 100 Ammonia conversion [%]
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Figure 6. Absolute values of normalized sensitivity coefficients (NSC) from global MC search in parameter space vs. ammonia conversion. The values of optimum parameters of T [K], P [atm], τ [s], A/V [cm-1], and inlet mole fractions of H2, NH3, and N2 are displayed in this order at each maximum NSC.
With the growing success of high-throughput experimentation, the above framework could be applied for faster and more reliable development of microkinetic mechanism parameters that contain valuable chemical information about the adsorbates and the catalysts.
5. Toward Model-Based Catalyst Design By carrying out the above procedure for many catalysts, a library of kinetics models can be developed. We propose that this library can assist in catalyst design. This would then be an example of product design mentioned above. At the simplest level, the catalyst composition becomes a manipulated variable and optimization can lead to better catalysts formulations that can guide high throughput experiments by narrowing down the huge parameter space. This idea awaits experimental validation.
6. Use of Microkinetic Models for Reactor Optimization The design of any chemical system involves tradeoffs, and hence optimizing a process flow sheet is a frequently studied problem [47]. For microreactors, the objective function is cast as maximization of performance, such as yield or selectivity, or as a complex economic function. One of the more conceptually straightforward goals is to use the hierarchical multiscale reactor models to determine the optimal reactor network and operating conditions that optimize the objective function subject to new constraints arising at the microscale (see next section). The methods for reaction network synthesis can broadly be classified into two main types: attainable region (AR) methods and superstructure optimization methods. [48] defined the AR as a set of all physically realizable reactor outcomes for a given feed, and presented a geometric method to determine the AR in the concentration space. The reactor network that yields the maximum achievable performance can then be chosen in this AR. [49] presented an excellent overview of this method, while [50] have extended its applicability by proposing an optimization-based targeting method. On the other hand, superstructure methods consider a set of process design alternatives, which includes reactors, such as stirred tank reactors (CSTRs), plug flow reactors (PFRs), cross flow reactors (CFRs), with additional units, such as mixers, splitters, separators, etc. Given a reaction mechanism, kinetic data and physical properties, a mathematical model of the system is formulated and optimization is carried out in order to obtain the sizing and interconnections between the various units, inlet feed rates, stream compositions, and reactor temperatures. The resulting formulation is usually nonconvex, due to bilinearities arising from the mass balances and nonlinearities of the reaction kinetics, and hence, a method guaranteeing global optimum currently does not exist. Application of simulated annealing [51], genetic algorithms [52,53], or global optimization techniques, such as the αBB algorithm [54] can increase the chance of reaching a global optimum. Another issue in reactor network optimization using microkinetic models is the computational burden, as the model consists of tens to hundreds of reactions involving
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several species. While a solution for an idealized reactor (CSTR, PFR or CFR) requires a computational time less than 1 second, the overall optimization is computationally very demanding. Therefore, model reduction techniques described in the previous section can be vital in optimization. Additionally, the optimal reactor network and operating conditions should be physically realizable in the microreactor. Herein lies another opportunity for systems engineering researchers in areas of optimal sensor and actuator placement, and integration of system-wise design and control. 6.1. Example: Water Gas Shift (WGS) reaction WGS is an important reaction because it reduces the amount of CO – a fuel cell catalyst poison – as well as increases the amount of hydrogen in the reformed gas stream. The overall WGS reaction is: ZZX CO 2 + H 2 CO + H 2 O YZZ
(4)
WGS is a reversible, exothermic reaction; as a result, the CO conversion is equilibriumlimited at high temperatures and kinetically limited at low temperatures. The aim is to determine the optimal temperature profile and feed conditions to minimize the CO content in the effluent. In industrial practice, this is achieved through a two-stage WGS process: a high temperature WGS reactor converts most of the CO to CO2 (and H2O to H2), whereas a low temperature WGS reactor further reduces the CO content and increases the H2 content of the exit gases. While the two-stage design of WGS system is a standard practice, not much work has focused on actual optimization of this system, especially in the context of determining an optimum temperature profile. Recently, [55] used the AR method to geometrically determine the optimal reactor design. [56] extended this work to numerically generate the AR, specifically for the WGS reactor. [57], on the other hand, applied the superstructure-based approach to formulate the design problem and used a quasiNewton technique for optimizing the temperature. Here, we consider optimization of the temperature and the feed profile for a reaction network shown in Fig. 7. The reactor network consists of n-PFRs in series. The COrich stream is the feed, steam is fed as the side stream, and an optional recycle is possible. The microkinetic model developed by [27] for WGS on Pt catalyst is used. [58] performed a similar superstructure-based reactor network synthesis for methane acetylization using gas-phase chemistry consisting of 36 reversible reactions and 19 species; however, we are not aware of any reactor optimization work involving catalytic
Y k,s Y k,f
m s1 m f1
m s2 l 1, T 1
m f2
m d1
m sn l2 , T2
m d2
m fn
ln , Tn
m dn
Recycle Figure 7: A schematic of the reactor network superstructure consisting of n-PFRs in series.
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microkinetic models. One of the critical aspects in optimization of WGS using microkinetic model is to accurately capture the reaction equilibrium. In the absence of thermodynamic data for the surface-adsorbed species, the scheme proposed in [59] is used to ensure that all our models are thermodynamically consistent. The full model consists of 46 elementary-like reactions. [4] used PCA (principal component analysis) to reduce the model to 18 key reactions. The 18-reaction system was simulated for a wide range of operating conditions; the most abundant reaction intermediate (MARI) and the RDS were identified. Then, small parameter asymptotics was used to derive a 1-step global rate expression. In comparison to commonly postulated Langmuir-Hinshelwood rate expressions, an advantage of this a posteriori model reduction strategy is that the rate parameters are physically meaningful, no a priori assumptions were made in obtaining the model, the “loss of information” is well characterized and the model, being developed from a microkinetic model, is applicable over a wide range of operating conditions. As the simulation time for the reduced order expression is significantly lower than that for the 46-step mechanism, the reduced mechanism was used for optimization results presented here. Comparison of the results of the full and reduce chemistry models will be presented elsewhere. Note that the reduced expression still accounts for all the important surface phenomena, such as temperature and coveragedependent activation energies. Using the reduced-order model, we undertook reactor network optimization in two steps. First, we assumed an isothermal system and performed optimization using a gradient-based (quasi-Newton) optimizer. Based on these results, we were able to simplify the reactor network, as follows: recycle stream was not required since the recycle ratio was equal to or close to 0; the CO-rich stream is fed only at the inlet or PFR-1 (i.e., m fi = 0 for i > 1 ); steam may be split over the n-PFRs; no intermediate side-draw. The reactor network optimization problem was thus simplified to the one of optimizing the total reactor length, the temperature, and the feed rate of the side streams for each of the n PFRs. The resulting optimal temperature (a local minimum) profile with n = 10 reactors, shown in Fig. 8, indicates the expected trend: the temperature is high in the initial reactors, where the CO concentration is higher, and drops significantly as the CO
750 COout = 218 ppm
Temperature, T (K)
700
COout = 356 ppm
650 600 550
COout = 1174 ppm
500 450
2
4
6 # of PFR
8
10
Figure 8: Optimum temperature profile for a reactor network consisting of 10-PFRs in series obtained using a quasi-Newton scheme. The dotted line represents one isothermal reactor and the dashed line represents two-stage WGS reactors. In all cases, total reactor length was 2.0 cm
Hierarchical Multiscale Model-based Design
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conversion increases due to the system getting equilibrium limited. It is interesting to compare the optimization results with that used in industrial practice, and the ones obtained using AR by [56]. There are two different stages in WGS: high temperature stage with T ~ 700 K and the low temperature stage with T ~ 450 K. The higher limit represents a “break-off” point: an increase in temperature does not result in any significant increase in the reaction rate, but adversely affects the equilibrium conversion. The lower temperature limit is a trade-off between higher equilibrium conversion and a lower rate of reaction. Fig. 8 also provides a comparison between the non-isothermal system, a two-stage system described above, and an isothermal system. In all cases, the total reactor length was 2 cm and 40 sccm feed (dry basis) and 40 sccm steam. The optimized non-isothermal system results in significant improvement over two-stage and isothermal systems.
7. Integrated Engineering
Microchemical
Devices:
Opportunities
for
Systems
In the preceding section, discussion focused on reaction kinetics and on design and optimization of single microreactor(s). Production of power requires integration of reactors, heat exchangers, and separation units, much like in a chemical plant. The characteristic length scales of typical microscale devices are on the order of several hundred microns, and as a result, at high pressures the continuum approximation still holds for the reactor itself. Therefore, the conventional equations of motion and transport are still applicable for the device. Yet, at the catalyst scale, reaction and transport within pores require smaller scale, often non-continuum models, as shown in Figs. 2 and 1 and discussed in the previous sections. So one may ask the question of whether there are any differences between microscale devices and their large-scale counterparts even at the reactor scale. The answer to this is affirmative. First, due to their small scale the flows in microdevices are laminar and so mixing is slow. Yet one needs to achieve high performance in shorter residence times. This leads to the potential of break through and/or incomplete conversion. Furthermore, small particulates needed to fill a microdevice in order to give high surface area catalyst, cause huge pressure drops, and as a result the fixed bed paradigm for separation or reaction cannot be employed. Moveable, small parts break and can cause bypassing due to settling. These aspects point to the realization that different structures, possibly monolithic-like, need to be explored to overcome issues of mixing, high catalyst area, and pressure drop [60]. Operation is often transient, e.g., turning on and off a laptop, and thus, the catalyst must be active not only at steady state (common industrial situation). In addition, heat transfer must be sufficiently fast (orders of seconds or smaller) to achieve reasonable operation. Hybrid systems, where a small battery is used for start up, followed by a device converting chemical energy to electricity is a process alternative with most promise. Second, the increase in surface area per unit volume resulting from miniaturization results in an increase of transport rates, and thus, a microreactor has the potential to operate under kinetically controlled conditions. This is a major advantage in terms of process intensification (high throughput with small devices) and the ability to extract intrinsic kinetics from experimental data. However, hot spots could form due to higher rates. Furthermore, surface reactions are favored over gas-phase reactions. This fact has
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interesting implications for radical quenching of gas-phase combustion chemistry leading to inherent device safety, regarding flame propagation, but also to the inability of making workable gaseous microburners [42]. Heat losses become large, and thus designs that ‘trap’ energy inside the system [61] are highly desirable. The proximity of gas-phase chemistry to walls makes surfaces not only important for carrying out chemistry but the main conduits of heat transfer. As a result, the material makeup of the walls is crucial [62]. Miniaturization, in conjunction with heat losses requires compact, well-integrated designs with a very different layout (flow-sheet) than their large-scale counterparts. The different chemical and heat transfer characteristics found at microscales may render conventional wisdom originating from large scales inapplicable to the design of microdevices [63]. For example, co-currently and counter-currently coupled microreactors (multifunctional devices of carrying endothermic and exothermic reactions on opposite sides of a wall) hardly have any difference in their stability and maximum hydrogen produced when materials are highly conductive [64]. Thus, process design and control of microdevices (lab-on-a chip) need substantial rethinking [65] keeping in mind the aforementioned pros and cons of microchemical devices. Due to the strong coupling of various components, design and control of individual units is unlikely to work; interactions between various units need to be accounted for. This issue is further acerbated because these systems often run in transient operation. This is currently a relatively uncharged territory. Modeling of these systems needs PDEs, leading to infinite dimensional systems that are not easily amenable for control. Hence, model reduction methods are required to obtain control-relevant models. With the development of novel MEMS sensors and actuators, their optimal placement for estimation and fault diagnostics, and for improving flow and/or temperature control will receive more attention [66,67]. Finally, the shorter time scales, of the order of minutes to hours, make them suitable for "plant-wide" optimization and control schemes.
8. Summary and Outlook With rapid advances in nano- and micro-systems, multiscale simulation and analysis is emerging as a new paradigm in computational science that could facilitate a better understanding of the underlying physics, and enable improved design, optimization and control of these complex systems. The aim of this article was to highlight the progress achieved in this field in the last decade. This emerging field presents new challenges as well as new opportunities, and will benefit from an increased synergism between reaction engineering and process systems engineering communities. Specifically, this paper discussed the hierarchical multiscale modeling work done in our research group. We demonstrated how the various tools at different scales of the “multiscale simulation ladder” have been used to develop more accurate and physically meaningful microkinetic models that can be applied over a large range of operating and design conditions. Quantum mechanics, molecular dynamics, semi-empirical methods, Kinetic Monte Carlo (KMC), and coarse-grained KMC methods have been put to use to obtain those parameters that are unknown and where experimental data is lacking. System tools, such as parameter estimation, response surface method, identifiability analysis have been applied to improve the quality of models. Model reduction was used
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to obtain reduced-order models that are useful for tasks, such as CFD simulation/design of reactors, reactor network synthesis, etc. Process design and control of micro- and nano-scale systems needs careful rethinking since on the one hand system integration, thermal management, and water management are key, challenging issues that await solutions, and on the other hand measurements, sensing, actuation, and control are plagued by the large disparity of scales. Aside from modern applications, the systems community has also to offer lots to the development of multiscale simulation itself in terms of passing optimum information between models at various scales with minimal error, integrating data with models across scales, and developing reduced models. Some of these issues have briefly been touched upon above with examples from the fuel-processing arena and are also addressed in [12-14,68-70]. The low cost of Beowulf clusters renders multiscale simulation a reality. However, multiscale modeling requires substantial intellectual infrastructure, mainly in techniques that span a wide range of scales and is particularly demanding on students. In most cases, such research can be accomplished at a reasonable pace only via collaboration(s). In the long term, the creation of suitable training modules, courses, textbooks, and summer schools is needed for broad dissemination of multiscale modeling.
Acknowledgments This work was supported by the donors of the Petroleum Research Fund, administered by the American Chemical Society and by the Army Research Office under contract DAAD19-01-1-0582. Any opinions, findings, and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views of the Army Research Office.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Process Intensification and Process System Engineering: a friendly Symbiosis Jacob A. Moulijn1, Andrzej Stankiewicz2, Johan Grievink1 and Andrzej Gorak3 1
DelftChemTech, Delft Technical University, Delft, The Netherlands Laboratory for Process Equipment, Delft Technical University, Delft, The Netherlands, DSM Reserch, Geleen, The Netherlands 3 Department of Biochemical and Chemical Engineering, University of Dortmund, Germany 2
Process Intensification, is it a research area or a set of objectives [1]? In our view it is both. Process Intensification (PI) is an area in the discipline chemical engineering; taking the conventional, existing technologies as a frame of reference, it tries to achieve drastic improvements in the efficiency of chemical and biochemical processes by developing innovative, often radically new types of equipment, processes and their operation. One could argue that such objective and objects of study are the hallmark of chemical engineering for many decades. Figure 1 shows a striking similarity of plants in the past and in modern times, in spite of a gap of many centuries. It underlines the feeling that there might be room for breakthroughs in plant design. Conceptually, PI belongs to the discipline of chemical engineering but compelling examples suggest that there is something as a “PI approach” that gives it the character of a research area.
Figure 1. The modern plant is not that modern… Miniaturization of the plant or integration of reaction and separation within one zone of the apparatus, have become a hallmark of Process Intensification. But PI has also other sustainability-related dimensions, such as significantly increased material efficiency, reduced energy usage, reduced waste generation and increased process safety. Producing much more with much less is the clue to Process Intensification. It provides a new avenue to a better economy and ecology of industrial production clusters.
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Essential for Chemical Engineering is that it is a multi-scale (in space and in time) discipline. Traditionally, three spatial scales are considered, the process, the process unit and a compartment within a unit. The more refined scales are treated as part of the compartment in a lumped manner for reason of conciseness. The question then arises at what scale does PI takes place. In a top-down approach one might limit PI to the meso (process unit) and macrolevel (process). So, given the chemistry and physics, the chemical engineer designs the optimal intensified process. However, it is more rewarding to consider more scales. At the upper level of aggregation, the supply chain should be the reference level for setting life span oriented performance targets for an intensified plant; at the lower level, the molecules and catalytic sites are obviously instrumental in enabling the goals of PI. The particle and the intraparticle space are considered to belong to the mesolevel. What is a good strategy for PI? Miniaturisation and increased resource efficiency can be achieved by enhancing the target rate processes by an order of magnitude, while suppressing the rates of competing phenomena. Since there are many different, up to now unexplored ways to do so, it will be clear that the philosophy of PI (PI, what it is and how it can be done, what are the drivers?) is not yet mature and, as a consequence, examples are crucial. The lecture will focus on examples from chemical and biochemical processes and from these examples contributions to theory will be formulated. Contributions can be in the field of hardware, e.g., structured catalysts and reactors, and methods, e.g., (bio)reactive or hybrid separations. In a sense this division is analogous to that of IT in hardware and software. In the world of hardware high performance reactors and column internals have received most attention. A classical example of the former is the structured reactor. Structured reactors have fascinating characteristics. They enable high rates and selectivity. Figure 2 shows that at the same power input the mass transfer (G-L) in monolithic reactors under conditions of so-called Taylor flow is one to two orders faster than in turbulent contactors. In coated reactors gas transport from the gas phase to the (catalytic) wall is essential and it appears that the dominant resistance is in the film. From simple physics it is clear that the film is thicker, the higher the velocity. So, G-S mass transfer will be highest at lowest flow rates! So, in multiphase applications in the Taylor-flow regime structured reactors enable high rates of mass transfer at laminar conditions, defying the Chilton-Colburn analogy! In conclusion, in PI structured reactors and contactors are of great value.
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Figure 2. Taylor flow in monolithic channels results in an order larger mass transfer rates compared to stirred tank reactors Microreactors in general are examples of structured reactors. Microreaction technology promises breakthrough technology in many areas. Here, we can learn from life sciences where microarrays play a crucial role not only in analysis but also in synthesis. Due to the high surface volume ratio microreactors have the promise of extremely high process intensification coupled with the option of high heat transfer allowing isothermal conditions, even for highly exothermal reactions. Integrated heat exchanger reactors, where the heart source and sink are in direct contact, open up new ways for PI. Another example of intensified equipment are the structured catalytic packings, allowing the simultaneous chemical reaction and separation of the reaction products from the reaction environment (Fig. 3). It leads to the conversion enhancement, avoiding of by-products and energy saving. Later, under methods their functions will be discussed in more detail.
Fig 3: Catalytic distillation column (left); Examples of catalytic internals (right) [5]
Closely connected with the equipment are the materials. The operating conditions of the unit can be moved towards more extreme and favourable conditions by introducing more resistant materials for the walls and contact surfaces. At the methods side a wealth of opportunities suggest themselves. Several types of functional materials are available that can have a large impact on the design of a process for a desired (bio)chemical and physical transformations. An important representative of a (functional) material is a catalyst. Catalysts perform essential functions in most chemical conversion processes, in both classical and novel applications. With respect to PI it can be worthwhile to replace random packed bed reactors by structured reactors, containing catalytic coatings. Catalytic coatings are very attractive from the point of view of maximizing selectivity. For serial kinetics when the intermediate is the desired product, the well-defined thin coatings
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enable unprecedented high selectivity in a convenient fixed bed reactor. It is fair to state that for a good performance of any fixed bed reactor a stable catalyst is required. In practice, for structured reactors this usually is the critical point, in particular when catalytic coatings are applied. Alternative forms of energy, such as microwaves may accelerate chemical processes hundreds if not thousands times. Some of these alternative energy forms, such as electromagnetic or acoustic fields, allow for essentially 100% selective product formation, without any byproducts, unachievable with conventional technologies, or allow for synthesis of products that could not be synthesized at all with conventional methods. The application of photons in chemical engineering provides an additional degree of freedom with potential for PI. Not surprising, catalysis is instrumental in novel processes and photocatalysis is a new fast developing field, allowing for instance artificial photosynthesis, that might even (partially) solve the Greenhouse effect. Another option is the exploitation of non-linear dynamics by means of advanced control over a dynamic mode of operation (periodic, flow reversal). In multiphase reactors in the Taylor flow regime mass transfer is strongly enhanced by the local hydrodynamics. Many other options emerge for enhancing the key rate processes associated with the function of the unit. A classical example of utilizing a force field is the socalled Spinning Disk Reactor, which applied to an industrial, phase transfer-catalyzed Darzen reaction, resulted in 1000-fold reduction of the processing time, 100-fold reduction of equipment inventory and 12-fold reduction of the by-products level [1]. Conceptually, the Spinning Disk Reactor belongs to the category of multifunctional structured reactors. Structuring can be done not only at the scale of the reactor, but also on the scale of the catalyst particle. This gives fascinating degrees of freedom. Good examples are membranes covering catalyst particles allowing high selectivity or pores consisting of a hydrophobic wall in an aqueous environment, enabling chemical environments that are related to the remarkable world of enzymes. This can lead to high precision, enabling in a sense PI at the source. On the lowest scale the chemistry is dominant. Modification of the chemistry and the reaction path has the most profound effect of all, since it affects the nature and amounts of the chemical species in the units. New catalytic materials can lead to breakthroughs. Examples are multifunctional catalysts and enzymes. Many enzymes exhibit simultaneously high selectivity and high rates, providing a basis for intensified processes. Also in this case the rule holds: a superior catalyst usually deserves a structured reactor! The integration of reaction and separation into one unit (i.e. in a single piece of equipment) or the integration of several separations leads to reactive separations or hybrid separations, respectively. The reactive distillation application in Eastman-Kodak process is one of the most striking examples for the integration of reaction and separation [1]. But such integration may also lead to some disadvantages. One of them is the necessity to operate the reaction and separation at the same pressure and temperature what reduces the degree of freedom. Also equipment design influences the operating window of an integrated process. The degree of integration of both functionalities, reaction and separation, is another parameter for process optimisation. Therefore, it has to be checked in each individual case whether integration is advantageous or not. The well established PSE tools like heuristic rules (using e.g. PROSYN), reactive distillation residue curve maps, or MINLP methods can help in finding of optimal design of reactive separation processes [6]). These tools can also be applied to find the sequencing of hybrid separations (like combination of chromatography and extraction, distillation and crystallisation, distillation and pervaporation etc.) [7, 17]. Since hybrid
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Process Intensification and Process System Engineering separations replace energy intensive separation methods for isomer separation or bioethanol production, they lead to the real PI [8]. Figure 4 gives an overview illustrating the wealth of options in combining different functions.
(Bio)Reactive Separation
Separation
Hybrid Separation
reactive distillation
distillation
distillation + membrane
(bio)reactive absorption
absorption
distillation + crystallisation
(bio)reactive extraction
extraction
extraction + crystallisation
(bio)membra ne reactor
membrane separation
Figure 4. Separation and reaction options can be combined to multifunctional reactors and hybrid separations. PI is important for all sectors where chemical engineering is important: from pharma to the oil refinery. A special sector is biotechnology where the systems in general are very diluted and, as a consequence, PI can contribute a lot. In-situ removal of products e.g. extraction of metabolites or adsorption of enzymes has the potential of making a revolutionary contribution. An example may be the efficient oxygenase-based whole-cell catalysis of various commercially interesting reactions such as the biosynthesis of chiral compounds [9]. Critical issues such as reaching high enzyme activity and specificity, product degradation, cofactor recycling, reactant toxicity, and substrate and oxygen mass transfer can be overcome by biochemical process engineering and biocatalyst engineering. Both strategies provide a growing toolset to facilitate process implementation, optimization, and scale-up. A division in hardware and methods is in a sense artificial, the more so, when higher aggregation levels are considered. This may become clear from the following. At the level of the supply chain one might think of e.g. the consequences of transport of dangerous chemicals from one plant to the other. An example is the elimination of transport of phosgene. By microreactor technology small-scale on-site production can lead to PI. On the one hand, the microreactor is a piece of equipment, on the other hand it represents a novel processing method. Another example concerns functional materials. Photo-and electrocatalytic materials
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might be considered to belong to the category hardware but they are the basis of photo- and electro-catalysis, being processing methods. Process Intensification significantly increases safety of chemical processes. It is obvious that smaller is safer and making inventories smaller is the first fundamental rule of the Inherently Safer Process Design. As Trevor Kletz said: “what you do not have, cannot leak” [10]. The U.S. studies showed for instance that methyl isocyanate (MIC), the poisonous intermediate that had been released at Bhopal, could have been generated and immediately converted to final products in continuous reactors containing a total inventory of less than 10 kg of MIC [11]. In reality ca. 41 tons of MIC had been released in Bhopal causing almost 4,000 deaths. Process Intensification offers not only smaller equipment; it also offers much better possibilities for keeping processes under control. This can be done for example via extremely efficient heat removal using micro devices (heat transfer coefficients exceeding 20,000 W/m2K) or via a fully controlled gas-liquid flow in structured catalysts, preventing liquid maldistribution and hot-spot formation. The Bhopal disaster convincingly shows the potential benefit of minimising inventories by the choice of continuous instead of batch processing. Of course, other actions could be advisable. Also high heat transfer equipment could have reduced the damage. Let us now consider the relation between PSE and PI and the options for synergy. In PSE usually a top-down functional approach is taken. It is acknowledged that the intensification options at the upper scales have already been subject of thorough study within the PSE discipline. At the process plant scale the optimised use of common resources contributes to PI. The functional requirements (production capacity and quality, responsiveness to market dynamics, SHE requirements, ..) provide the reference conditions for the design of an effective network to distribute the various common physical resources in the plant (energy, exergy, solvents, water and other utilities) over the process units. Process Integration methods provide an established framework for tackling this resource issue [12]. Other concerns about critical resources at the scale of the plant involve the reliability and availability of the plant [13] as well as its capability to deliver on-spec product(s) [14]. Yet, at the scale of the molecules, structure of the catalyst, sites and local fluid dynamics, PSE has had less impact, traditionally, although it is recognized that the available PSE methods and tools can potentially have a very significant impact. In contrast, PI is very much focused on (bio)chemical engineering science aspects of the process units and the compartments within the units. In Figure 5 it is attempted to define PI in relation with PSE. The focus and action of Process Systems Engineering takes place along the product creation chain [15], marked by the pink arrow, while the focus and action of Process Intensification is on the separate boxes: it has a more analytical than integrating character and primarily aims at higher efficiency of individual steps in that chain. Also the scales considered are different; PSE focuses less on the scale of molecules, sites and (nano)structure, whereas PI explicitly includes this level but often gives less attention to the highest level. It is clear that PI has consequences for the “longitudinal” action of PSE; for instance, development and application of a reactive separation can influence the PSE over the whole chain, from molecule to site, if not to enterprise.
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Figure 5. The relation between PSE and PI As seen from the PSE philosophy the following points of attention for PI come forward. Process intensification, which aims at better utilisation of physical resources and an associated reduction in sizes of process equipment, is not risk free. While reduced storage of dangerous materials will greatly improve safety, the fast dynamics of the process (unit) can endanger the resiliency or stability of the process against disturbances [16]. Also, the operability and availability of the intensified process need to be investigated in order to secure the potential investment benefits by an anticipated flawless plant operation. Here a fruitful symbiosis between PSE and PI is essential. Another area for cross-fertilisation is in the application of synthesis methods (conceptual and computational) to the creation of novel processing structures at the micro-scale and below. While an intensified plant is economically a better plant, the issue whether it is also a better plant from sustainability point of view in every respect is not entirely settled. Intensification of rate processes by coupling and strengthening of driving forces will give rise to more entropy production and exergy losses. Although it may happen that at an integrated and intensified unit the exergy losses increase relatively to a conventional base case, the exergy losses at the overall plant can decrease, due to a drastic reduction in number of transfer and separation operations, so enhancing economics and sustainability in parallel. There might well be important open issues regarding process control: at certain conditions highly compact, intensified units may be poorly controllable or responsive to changing external conditions, like feed composition, desired product mix. What is the impact of modern smart control (e.g., new micro-scale sensors and actuators and advanced First Principles model-based control algorithms) on the optimal design of intensified plants? Are dynamic modes of operation better achievable in intensified plants? What is the impact from the option of applying more extreme conditions? Other questions to be addressed are in the multi-scale modeling area: What is the proper process modeling depth - from short-cuts to CFD applications – for each of the considered
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scales? What is the necessary accuracy of measured model parameters in connection with the chosen modeling depth? How predictive are the simulation methods of intensified processes? The answer to these questions can not be given for all PI operations but some general recommendations can be formulated for reactive separations [17]. Reactive absorption, distillation and extraction have much in common. First of all, they involve at least one liquid phase, and therefore the properties of the liquid state become significant. Second, they occur in moving systems, thus the process hydrodynamics plays an important part. Third, these processes are based on the contact of at least two phases, and therefore, the interfacial transport phenomena have to be considered. Further common features are multicomponent interactions of mixture components, a tricky interplay of mass transport and chemical reactions, complex process chemistry and thermodynamics. The most important model parameters are: VLE-equlibrium, reaction kinetics and mass transfer coefficients. The modelling approaches of reactive separations are given in Fig.6 Rate-based approach must be used for the modelling of reactive absorption. The use of the equilibrium stage model is usually accurate enough to predict the steady state and dynamic behaviour of reactive distillation columns. Recently CFD may become a powerful theoretical tool to predict the flow behaviour under different column unit and internals geometries for engineering applications. In particular, it can play an outstanding role in the development of the column internals for reactive separations. The optimal complexity of the model for reactive separations depends on one hand on the model accuracy, but on the other hand on the availability of the model parameters and efficiency of the simulation methods (Fig 7).
Reactive separation modeling approaches
Mass transfer
Reaction kinetics homogeneous system
Rate-based approach (Maxwell-Steffan)
Film & bulk reaction
heterogeneous system
External & internal resistance
Nonideal flow behavior
Plug flow (ideal mixed)
Rate-based approach (effective diffusion)
Equilibrium stage approach
Hydrodynamics (mixing)
Chemical equilibrium
Figure 6. Modelling approaches for reactive separations [5]
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Total costs Costs of parameters
Design costs Costs due to errors
Optimum Short-cutEquilibrium stage model Rate-based
Model complexity Computational Fluid Dynamics
Fig 7: Design costs as a function of model complexity for reactive separations It will be concluded that the approaches in PI and PSE are complimentary as indicated in Figure 5, indicating opportunities to intensify the interaction process between PI and PSE. The widening span of scales and the increasing diversity of processing methods call for a joint effort. A friendly symbiosis will be beneficial for innovative designs of future plants to save energy and resources, be it for the production of simple bulk chemicals, complex products, medicines or other consumer products. References 1.
Stankiewicz,. A.; Moulijn, J. A.: Re-engineering the chemical processing plant; Dekker; New York, 2004 2. Structured Catalysts and Reactors, 2nd ed., eds. A. Cybulski and J. A. Moulijn, CRC Press, 2006-03-17 3. M. T. Kreutzer, F. Kapteijn, J. A. Moulijn, S. Ebrahimi, R. Kleerebezem, M. C. M. van Loosdrecht, Ind. Eng. Chem. Res., 44 (2005) 9646-9652 4. M. T. Kreutzer, F. Kapteijn, J. A. Moulijn, J. J. Heiszwolf, Chem. Eng. Sci. 60 (2005) 5895-5916 5. C. Noeres, E. Y. Kenig, A. Górak, Chem Eng Process 42 (2003), 157-78 6. H. Schmidt-Traub, A. Gorak, “Integrated Reaction and Separation Operations”, Springer, Berlin, 2006 7. J. Bausa, W. Marquardt, Ind. & Engin. Chem. Res. 39 (2000) 1658-1672 8. A. Gorak, P. Kreis, “Reactive and hybrid separations”, PI Conference, Delft, 2005 9. B. Buhler, A. Schmid, Journal of Biotechnology, 113 (2004) 183-210 10. T. Kletz, Chem. Ind, 1978, May 6, 287 11. D. C. Hendershot, Chem. Eng. Prog., 96 (2000) 35 12. H. D. Goel, J. Grievink, P. M. Herder et al., Reliability Engineering & System Safety 78 (2002) 247258 13. R. Smith, “Chemical Process Design and Integration”, John Wiley, January 2005, pp 687 14. M.P. Harold, and B.A. Ogunnaike, AIChE Journal, 46 (2000) 2123-2127 15. Grossmann, I.E.; Westerberg, A.W., AIChE Journal 46 (2000) 1700-1703 16. W. L. Luyben, D. C. Hendershot, Ind. Eng. Chem. Res., 43 (2004) 384-396 17. 17. M. R. Eden, S. B. Jørgensen, R. Gani and M. M. El-Halwagi, Chemical Engineering and Processing, 3 (2004) 595-608
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Recent Developments in the Risk Management of Offshore Production Systems Dr. Daniel Averbuch Institut Français du Pétrole (IFP), 1 et 4 allée de Bois Préau Rueil-Malmaison, 92500 FRANCE
Abstract The development of offshore oil and gas fields involves important investments and operational expenditures to design, build and operate production facilities. In the context of deep offshore, several risks on the level of production have to be taken into account at the design phase. Various phenomena, such as gas hydrate plugs or wax deposit (named "flow failures"), which are related to the physical nature of the fluid and to the flowing conditions, can indeed lead to important reduction of production. Furthermore, the design of the system is mainly decided at a moment where information on the nature of fluids or the reservoir itself is incomplete, and when the prediction of those phenomena is hard to realize. A rational design of the production system should then take into account uncertainties present at the moment of the decisions, through an appropriate risk management of phenomena potentially leading to loss of production. This paper gives an outline of a methodology developed by IFP to manage risk related to the production performance of offshore oil and gas production systems. This original methodology allows to take into account risk caused by equipment failures as well as "flow failures" due to undesired physical phenomena resulting from the conditions of production. The approach is based on an explicit integration of production uncertainties relating to a lack of knowledge on the reservoir, into the mathematical models describing the undesired physical phenomena. Dedicated tools lead to an evaluation of the consequences of the occurrence of such "flow failures" on the availability and production availability of the production system. The approach may finally provide a global performance evaluation and a technical and economical optimization of the production system. Keywords: Risk management, production availability, offshore.
1. Introduction A current trend in the development of the oil and gas industry is the search for new reserves in deepwater. In a few decades, the offshore industry has moved from depths of a hundred meters to more than two thousand meters water depth. This evolution has been allowed by the improvement of the technology, associated to the discovery of important oilfields. Nowadays, the industry is aiming at drilling and production by three thousand meters with sub-sea pipelines of more than a hundred kilometers long. Following this trend, the investment in drilling in ultra-deep-water has been multiplied by thirty in fifteen years. This evolution involves however specific technical challenges related to ensuring the production. The low temperature of the sea can cause physical phenomena such as paraffinic solid deposits in the lines or obstruction of the lines by creation of hydrate plugs. Furthermore, the development of offshore oil and gas fields involve important investments and operational expenditures to design, build and operate production facilities. The design of the system is mainly decided at a moment where
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information on the nature of fluids or the reservoir itself is incomplete. A rational design of the production system should then take into account uncertainties present at the moment of the decisions, through an appropriate risk management of phenomena potentially leading to loss of production.
2. Risks involved Various phenomena, related to equipment failures or to the physical nature of the fluid itself and to the flowing conditions, such as gas hydrate plugs or wax deposit (named "flow failures"), can indeed lead to important reduction of production. As previously explained, they have to be explicitly taken into account. 2.1. Hydrate plugging In hydrocarbon production, solid particles can form under specific thermodynamic (high pressures, low temperatures) conditions that are encountered in deep offshore (Sloan, 1998). These particles are made of water and gas and can aggregate and plug the production lines. This can happen especially during the system shut-down phases (in case of maintenance or repair operation for instance) because the fluid temperature then quickly decreases. Hydrate plugging can be prevented by adequate insulation or by circulating inert oil (called dead-oil) with no dissolved gas, to remove the production fluid, or by chemical injection. These events are very important since they can fully stop the production. Moreover, removal of hydrate plugs (for instance by depressurization) is very complicated, time-consuming and may possibly be dangerous. Indeed field experience has shown that hydrate plugs can happen on important pipeline length and cause important production losses as well as important intervention costs. Hydrate plugging is then one of the major risks to take into account in the analysis of deep offshore production systems. 2.2. Wax deposits Paraffinic crude oils exhibit a tendency to create solid deposits along the walls of the flow lines (Burger et al., 1981). Such deposits are made of paraffinic components called wax. They appear when the temperature of the pipeline wall is lower than the temperature of the fluid. The wax deposit build-up is a slow continuous process that can progressively reduce the effective hydraulic diameter and eventually plug the lines. In order to deal with this problem, specific devices called "pigs" are sent into the pipe to clean the inner wall. The wax deposit is then retrieved when the pig is extracted at the end of the lines. This phenomenon can lead to two kinds of risks, namely a reduction of the production or a risk of blocking the pig itself in the line. 2.3. Equipment failures Many equipments are placed both on the sea-bottom and on the platform. Such equipments can be hydraulic or electric control equipment, sensors, or actuators. Their failures have of course to be taken into account in the analysis, especially if they are located under the water, where interventions are complicated and cost and time consuming. According to the nature of the equipment failures, repair or replacement can need the mobilization of remotely operated vehicles (ROV) which are small robots, or of dedicated vessels, which may not be immediately available onsite.
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3. Methodology When studying this problem, one quickly reaches the conclusion that it is necessary to evaluate all aspects of the situation in a coupled way. Indeed, equipment failure can cause shut-down situations, which themselves will modify the flowing conditions and therefore cause flow failures. IFP has then developed a methodology of simulation of the life of the production system (Dejean et al., 2005). This methodology covers several steps : 1. identification of the risks 2. representation of the dynamics of the system 3. simulation of the system dynamics 4. integration of the flow failures in the approach 5. uncertainty modeling 6. analysis of the results 3.1. Identification of the risks Of course, the identification of the risks to take into account is the first step of the approach. As examples, hydrate plugging and wax deposits are important phenomena that need to be taken into account into an analysis. In different cases, other physical phenomena (corrosion, slugging for example) could also be analyzed. A relevant modeling should include failures of the main equipments, which are represented on the Fig. 1.
Manifold : Valves Subsea Control System
Production platform Master control, Methanol Injection System
Well Well Head
Flowlines
Flying Leads Subsea Distribution System
Risers
Hydraulic and Electrical Umbilicals
Figure 1. Representative offshore production system
3.2. Representation of the dynamics of the system In order to simulate correctly the life of the system, it is necessary to represent its dynamics. More precisely, one needs to examine and describe the situation that the production system will encounter, when several events (such as failures) occur. Such an
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analysis may often be represented using so-called event-trees. An example of event-tree is given on Fig. 2., where a maintenance operation (by an ROV) of a sub-sea equipment is described. In order to derive quantitative values, one needs also to define numerical data such as delays of mobilization, of repair, probability of repair.
Figure 2. Event-tree for ROV intervention
3.3. Simulation of the system dynamics In order to simulate the system dynamics and evaluate the system performance in terms of reliability, availability, maintainability and safety, one has to describe the rules that define the change of states of the system. These rules may be derived from the events trees where delays and probability of success of the actions are detailed. Afterwards, one needs to use a stochastic simulator to evaluate the system performance, in terms of deferred production, production availability (the ratio of the total effective production to the maximum theoretical production). This may be done within the general mathematical framework of dependability. Dependability provides tools to model complex production systems and compute statistics such as availability or production availability. The most important problem for these tools is to take into account the dependencies that exist between some physical parameters (such as pressure, temperature, flow rate, etc.) of the production process and the nominal or dysfunctional behavior of some components of the production system. In our case, we have decided to use hybrid stochastic Petri nets, which provide powerful modeling tools. Petri nets (Petri, 1962) are widely known and were firstly introduced in the field of dependability for oil and gas applications in the nineties (Signoret, 1995).
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Stochastic Petri nets provide advantages over other classical techniques such as Markov chains, thanks to their ability to model precisely complex systems, and especially systems with dependencies (which cannot be modeled by Markov chains). Their modeling power also resides in their graphic representation, which enables to easily model complex systems by describing the system dynamics with simple elements. These are the main reasons for the use of stochastic Petri nets in this work. Calculations were made with the commercial software MOCA-RP V12.10 that allows all necessary facilities (Dassault Data Services, 2005). 3.4. Integration of the flow failures in the approach Integrating the events related to the "flow failures" in the modeling of the system presents several difficulties, because the physical behavior of the system may only be described by complex thermodynamics and multiphase hydrodynamics models. However, since Petri nets models require numerous Monte-Carlo simulations, one possibility that is being explored is to replace the complex models by Response Surface Models (RSM). The RSM mathematical framework helps deriving simplified models based on interpolation of complex models. In order to obtain an effective approach, experimental design techniques shall be used to limit the simulation time (Manceau et al., 2001). RSM modeling has been proven to solve this problem (Averbuch et al., 2005), and is being integrated at the moment in dependability modeling in a current international project. As an example, a given RSM will define for a given shutdown duration if a hydrate plug will form at a defined location, as a function of pressure, temperature and fluid composition. A key factor for the use of RSM is the fact that the flow failures are in limited numbers (since for instance, the locations of hydrate plugs are generally known in advance). During the simulation of the system dynamics by the stochastic Petri nets, for any shutdown situation, the stochastic simulator will then call the RSM to determine if an hydrate plug will form. Such an event would then be treated by the Petri nets exactly as an equipment failure. 3.5. Uncertainty modeling The principle of stochastic Petri nets is to use extensive Monte-Carlo simulations by running numerous histories (up to millions), each of these histories representing a random evaluation of all stochastic variables. This approach is widely used for dependability evaluations, where a lot of variables are assumed to be stochastic (time to failure and time to repair of equipments, mobilization durations for instance). This facility may also be used to model uncertainties on other variables. In our case, uncertainties on flow parameters and on reservoir characteristics may be treated in the same way. This involves to fire at random for each history the values of these additional parameters, which will be used afterwards in all Petri nets calculations of this history. 3.6. Analysis of the results The modeling of the dynamics of the system can provide very useful information with regard to the performance of the system. Classical performance indicators can be based on economical performance such as net present value or internal rate of return. Other information can be provided by the analysis of the history of the system. For instance, Fig. 3. represents the contributors to the production stops of the system. Such contributors can be either due to the equipment or to flow failures. The analysis of these results can then provide useful information in order to improve the system performance by adding redundancy or improving the system design.
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Figure 3. Contributors to the production system stops
4. Conclusion This paper gives an outline of a methodology developed by IFP to evaluate the risk related to the production of offshore systems. It consists in several steps including a risk identification, a description of the dynamics of the system and a simulation by stochastic hybrid Petri nets. Such a modeling can allow to evaluate the economical performance of the system, as well as identifying the main contributors to the system shutdowns. Current studies are being performed to integrate the flow failures in the approach. This implies a coupling of dependability models and of physical models.
References Averbuch, D., Pierchon, A., Gainville, M., Dejean, J.-P., 2005. Response Surface Models for Hydrate Prediction. Deterministic and Probabilistic applications. Multiphase Conference, Barcelone, Spain. Dejean., J.-P., Averbuch, D., Gainville, M., Doux, F., 2005. Integrating Flow Assurance into Risk Management of Deep Offshore Fields Developments. OTC, Houston, USA. Burger, E.D., Perkins, T.K.,. Striegler, J.H, 1981. Studies of wax deposition in the Trans Alaska Pipeline. Journal of Petroleum Technology, June, 1075-1086. Sloan, E.D., 1998. Clathrate hydrates of natural gases. Marcel Dekker inc., 2nd edition. Manceau, E., Mezghani, M., Zabalza, I., Roggero, F., 2001. Combination of Experimental Design and Joint Modelling Methods for Quantifying the Risk associated with Deterministic and Stochastic Uncertainties - An Integrated Test Study. SPE 71620, ATCE, New Orleans, USA. Petri, C.A., 1962. Kommunikation mit automaten. Technical report, Doctoral Thesis, University of Bonn, Germany. Signoret, J.P., Leroy, A.,1995. Probabilistic calculations of the production of a subsea production cluster. Proceedings of the Safety &Reliability Society Annual Symposium, Southport, UK Dassault Data Services, 2005. MOCA-RP V12.10 User manual.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Challenges and Opportunities in Process Innovation Larry R. Genskow Associate Director – Process and Emerging Technologies The Procter and Gamble Company 8256 Union Centre Blvd. West Chester, Ohio 45069 USA
Abstract This addresses key challenges and opportunities in process innovation. Important capability trends are identified and discussed. Key challenges and opportunities to be addressed include: 1) The challenge of learning at the smallest scale to increase innovation speed and decrease cost – and lessons from biotech. 2) The importance of identifying emerging technologies and disruptive innovations – innovations that can ultimately obsolete incumbent businesses with new to the world technologies. 3) The need for diversity to fuel diversity of thought – to nourish and enable creativity and invention. 4) The challenge and the promise of micro-technologies. 5) The role of modeling and simulation in process innovation. Keywords: process, innovation, disruptive, modeling, emerging technologies
1. Introduction 1.1 Overview The pace of innovation continues to increase not only in the developed world, but in developing markets. This is being driven by the consumer and enabled by increased technical capabilities. Consumers are demanding increased levels of personalization in their products. One of the most obvious examples of this is in the automotive industry. The number of automobile models has increased from 250 in 1999 to a projected 330 by 2008, according to Global Insight Inc., a Waltham, Mass., researcher. And the speed of innovation has increased. A decade ago the time from design concept to “on the showroom floor” was about 5 years. Today, best in class is well under 2 years. This reduction of innovation cycle times is largely a result of moving from a world of physical testing and prototype development to a virtual world enabled by modeling and simulation capabilities. We have not seen the same scale of progress in the chemical industries. The time from discovery to commercialization of a “new to the world” product is still measured in years. At the risk of over simplification, this is in large part due to the added complexity of chemical systems compared to mechanical systems. Certainly at P&G, we are still measuring innovation cycle times in years, even for some “new and improved” products. We have also seen added complexity as the variety of products (SKU’s) that we offer the consumer has increased at about 10% annually over recent years.
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46 As a result, better, faster, cheaper innovation is a top priority for us and for almost any company that hopes to remain in business. This paper addresses some of the key issues today, especially from the perspective of a consumer products company – but I believe these same issues are prevalent across the chemical industries. 1.2 What is Innovation? Let’s be clear on what innovation is before addressing challenges and opportunities. Innovation is: The practical translation of ideas and inventions into new and improved products, services, processes, or social interactions. As such it spans the earliest discovery work through commercialization. Innovation is creating something that others want and has commercial value. “Innovation is our lifeblood – new ideas and new products that make consumer lives better.” - AG Lafley, CEO, The Procter and Gamble Company “People implementing new ideas to create value.” – Joyce Wycoff, founding Director of Innovation Network It’s important to distinguish innovation from invention. Invention is also the creation of something new but some inventions are never commercialized to create value.
2. The Challenge of Learning at the Smallest Scale 2.1 Overview “Make your learnings at a small scale and your profits on a large scale.”1 Learning at a small scale used to mean learning at the pilot plant scale or at the bench scale. Today we talk about learning at the micro scale, the nano scale or the virtual scale. Generally, the smaller the scale of learning, the faster and cheaper it is. The reality is that we can’t afford to learn at a large scale today. Importantly, we can get more creative and take greater risk at this “smaller scale” because of the faster, cheaper learning. We can be more innovative. Another key driver to the challenges of learning, certainly in the consumer products sector, has been the increase in product complexity. As we add greater and greater product functionality from new ingredients, products become increasingly more complex. There are almost always, synergistic or harmful interactions between ingredients and these impact both product efficacy and stability. So much of our focus is new tools which enable faster and cheaper understanding of both efficacy and stability. It’s important to contrast this with the challenges of commodity chemicals where the focus is primarily cost. 2.2 Example from genomics/biotech The revolution in biotech and genomics should provide those of us in chemical related industries with some true inspiration in terms of what’s possible with small scale learning. It’s remarkable that the human genome was mapped just 6 years ago and even more remarkable that we’re using offshoots of this technology to understand and treat disease. The Human Genome Project itself is comparable to the invention of the first commercialized plastic, Bakelite, by Leo Hendrik Baekeland in 1906 or the discovery and invention of the transistor by John Bardeen, Walter Brattain, and William Shockley of Bell Labs in 1946. Bakelite led to the plastics revolution and the transitor led to the communications revolution. The Human Genome Project is enabling a genomics/biotech revolution.
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Technology projections are that we will all have our own personal genome chips in as little as 10 -15 years, i.e., they will be affordable in that timeframe. This will enable the assay of everyone for disease susceptibility and drug response; and enable selective prescription of pharmaceuticals to avoid ineffective treatment or adverse side effects, problematic with the prescription of drugs today.
DISPLAY OF GENES EXPRESSING
Figure 1 (a) Affymetrix GeneChip®
Figure 1 (b) Gene expression display
An example of this revolutionary GeneChip®2 is shown in Figure 1 (a). The GeneChip® contains all the genes in the human body – over 30,000. And they are all contained in a chip the size of a piece of Belgium chocolate. The possibilities enabled by the GeneChip® are profound. This chip enables us to screen new actives for treatment of disease – to determine which genes express, Figure 1 (b), when subjected to a specific chemistry. It enables companies like P&G and L’Oreal, to screen new actives to give us healthy and vibrant skin and hair. Recent patent applications3 by L’Oreal claim genes associated with hair graying. It enables us to screen new actives to improve our oral health and hygiene. And the cost of this chip is less than $500. The traditional alternatives for screening new actives have orders of magnitude higher cost. Clearly, this is a very powerful capability that has gone well beyond its initial applications in the pharmaceutical industry. It enables better, faster, cheaper innovation. You could say we have yet to develop the genome chip capabilities in chemical processing. While we could draw some analogies with the developing capabilities of microfluidic devices to facilitate learning, they clearly have not been developed or commercialized to the extent of the GeneChip.® This is in large part the result of the huge investment in genomics and biotech today. Genomics/biotech are the fourth broad scale technology revolution. (The first was the Industrial Revolution, the second the Chemical Revolution, and the third the Transistor/Microchip Revolution.) Certainly many parts of the chemical industry appear to have reached the plateau of the technology ‘S’ curve where margins are slim for the most part and many products have reached commodity pricing status. We have to question whether the Chemical Revolution itself has reached the plateau. Perhaps this is why many of the large, prestigious chemical companies are diversifying into biotech, agro-chemicals, and life science.
48 3. Identify Emerging and Potentially Disruptive Technologies 3.1 Overview Identifying emerging technologies, particularly those that can significantly disrupt current business models, is critical to long term successful innovation and the life of a company. And to do it well, an organization must dedicate resources to breakthrough innovation, i.e. create an organization whose objective is breakthrough innovation. Too frequently, rather than dedicating resources to breakthrough innovation, an organization will ask each employee to “allocate 10% of their time to innovation.” This very seldom works. The business problems of the day usually force this to 0%. 3.2 Capabilities to identify trends and key technologies It’s important to identify future technology and consumer trends as part of our innovation work. There are many approaches and certainly the capabilities that a diverse external focus provides are critical. Technology Futures nicely summarizes many of the important capabilities to view the future in “Five Views of the Future”4 as shown Figure 2 below. Most multi-national companies use some of the capabilities shown in this chart. And the capabilities may show up in various parts of the business. Some capabilities could be within a Strategic Planning organization and others within the technology organizations (Engineering, R&D, or even a Modeling and Simulation organization). And of course, they are also appropriate tools within Financial organizations.
Figure 2 Various approaches to “view” the future from Technology Futures, Inc. 3.3 TRIZ as a P&G example to identify trends We have found that an approach called Theory of Inventive Problem Solving (TIPS or TRIZ for the Russian acronym) to be very helpful. TIPS addresses the invention part of innovation in a very structured way. It was developed by Genrich Altshuller5 in Russia during the 1940’s and is taught rather extensively in Russia today – and now in Europe and North America. Altshuller was a patent examiner (like many other inventors) and noted some common “invention principles” and “technology trends” as
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he reviewed patent material. He eventually identified 40 inventive principles and 12 evolutionary trends. The evolutionary trend curves all exhibit the familiar ‘S’ shape for market adoption of technically superior new technologies – a pattern frequently characterized by mathematicians as a “logisitic curve” or a Fisher-Pry model. The model has been recently popularized by Malcolm Gladwell in a book titled Tipping Point6 – the “tipping point” being the first inflection of the ‘S’ curve when growth accelerates rapidly. While most of us as engineers feel a great technology can and should sell itself, this is in fact rarely the case. Gladwell explores the social and psychological aspects which can facilitate and even push a technology, concept or product past its tipping point. One of Altschuler’s “evolutionary trends”, the Structure of Fields and Forces7 is shown in Figure 3 below. It illustrates, as observed from Altshuller’s analysis of fields and forces patents that this trend moves from a constant field or force input ultimately to a standing wave. We might think of this input as a process variable, for example, the air flow in a dryer. This is in fact an area of research today and beneficial results have been found by a variety of researchers. These were some of the conclusions. A decrease of air consumption can be achieved in a fluid bed dryer with a pulsed fluidization technique without affecting drying time.8 An increase of range of velocities in a fluidized bed within the fluidized state can be achieved and there is a possibility of achieving the fluidized state at lower velocities.9 Optimized and more frequent airflow reversals in a kiln dryer can improve final moisture distribution, reducing in-process and residual lumber stresses.10
Figure 3 TRIZ Evolutionary Trend Curve for Structure of Fields/Forces/Actions
50 There is the opportunity to apply this evolutionary trend very broadly if we undertake a research program (academic or industrial) to explore this effect in key process unit operations. This is an example of how TIPS can provide a structured approach to innovation and how it can be used to leverage a well documented trend prediction. Generally, a company must have a program for sustaining “new and improved” innovation and a program that addresses breakthrough innovation and the possibility that a new disruptive technology could obsolete their products. Clayton Christiansen has popularized the concept of disruptive technologies in his best seller books11,12 on innovation. The basic premise is that the more successful your firm is in addressing the needs of your customers, the better your market research, the better your product is in meeting your best customers’ high end needs, the more likely your firm is to fail. You will have focused too much on incremental innovation and insufficiently on breakthrough innovation and the next technology ‘S’ curve. Someone, likely a new player in the industry, will jump frog your best products. So we need the appropriate balance between “new and improved” innovation and breakthrough innovation. The innovation focus on “new and improved” products or processes keeps the firm financially healthy over the short term. The innovation focus on breakthrough enables the firm to remain financially healthy over the long term.
4. Diversity Fuels Innovation 4.1 The power of a diverse organization to fuel innovation A diverse organization fuels diversity of thought. Diversity of thought nourishes and enables creativity and invention. One of America’s strengths has certainly been innovation, facilitated by the American “melting pot”. Diversity of race, religion, ethnic background, gender and education is a powerful enabler. If we all come out of the same mold (more or less), then we all think similarly. But if we come out of very different molds, then we have significant diversity of thought and are likely to be more innovative as an organization. 4.2 Xerox example Xerox’s Palo Alto Research Center (PARC) has long been noted for their creativity and invention. As the birthplace of technologies such as laser printing, Ethernet, the graphical user interface, and ubiquitous computing, PARC has an established track record for transforming industries and creating commercial value. It was not unusual for Xerox to staff a project with engineers, scientists, artists, and psychologists. While Xerox was noted for creativity and invention, they struggled with their ability to commercialize their inventions. And as pointed out earlier in this paper, innovation spans commercialization and this piece of the innovation cycle, in contrast to the discovery process, requires significant structure coupled with strong execution skills. So the challenge is to staff an organization, which is charged to create, invent and commercialize, with a diversity of thought and capability. Those charged with discovery need to be comfortable with uncertainty and change. Those charged with project execution need to be comfortable with tight schedules and cost control. Usually, these are very different people.
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4.3 P&G examples Personally, I have strived to staff my Engineering organization with diversity of disciplines not often found in a consumer products engineering organization. This organization includes expertise in ceramic engineering, radio engineering, plasma physics, ultrasound, biochemistry, polymer chemistry, rheology, and various disciples within chemical engineering (mixing, heat transfer, powders processing, drying, etc.). It also includes diversity from numerous regions of the world (North America, Latin America, Western Europe, Eastern Europe, and Asia). And of course it includes gender diversity. It is a melting pot. All that said, the other side of internal diversity is external diversity. AG Lafley, Procter and Gamble CEO, has noted “we will acquire 50% of our technologies and products from outside P&G”13. The approach we use is one of “connect and develop”, or C&D, to identify these “acquisitions” at any stage in the innovation cycle – discovery through commercialized and in the market (e.g. Gillette, which P&G recently acquired). The C&D concept has evolved as a best practice at P&G14 and has also become popularized as “Open Innovation” by Henry Chesbrough.15 The challenge is really to bring an external focus to innovation. At a minimum we need to know what is available externally that can solve our problem or meet a consumer need. We can agree there is a wealth of external ideas, inventions, and problem solutions and the challenge is to first decide to seek, and then to find these solutions. The good news is there are many companies today that specialize in finding and linking the needs of one client with the solutions of another client. Companies like NineSigma (which P&G helped create), and Innocentive (which was founded by Eli Lilly), specialize in connecting clients like Procter and Gamble, who are seeking problem solutions, with companies and individuals (inventors, academics, etc.) who may already have solutions to these problems. And NineSigma promotes open innovation as a way of making internal R&D more important, not simply an outsourcing of R&D. And there are other ways that P&G leverages diverse external capabilities. P&G joined several other Fortune 100 companies about six years ago to invest in Yet2.com, an online marketplace for intellectual property. Yet2.com connects globally across industries, universities, and national labs. A client such as P&G works closely with Yet2.com to write a “brief” describing the need or problem to be solved. This brief is then distributed to this global network. If there is a connection, then the client negotiates directly with the provider. Realizing the richness of retired technologist, P&G partnered with Eli Lilly, Boeing and other companies in 2003 to form YourEncore, a business that connects retiree mastery with short term company needs. It’s a simple, but powerful way to bring the diversity of thinking from another industry to bear on a particular problem or project. Lastly, P&G has developed a specific role to facilitate our C&D work. We have roughly 70 Technology Entrepreneurs around the world whose job responsibility is to identify external connections with real business value. A Technology Entrepreneur, exploring the Japanese market, identified a technology which enabled a new product, P&G’s Mr. Clean Eraser®.
5. The Challenge and Opportunities with Micro-Technologies
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52 5.1 Overview The promise of micro-technologies or more specifically, microfluidics for process engineers, continues to be an area of significant research effort. Much of the early work focused on parallel processing to enable smaller more efficient mixing, heat transfer and reaction processes. This also enables a simpler “number scale-up” instead of size scaleup. There have been some successes in this area, but we appear to still be in the low slope part of the ‘S’ curve. We haven’t reached the tipping point. The other area which seems to be getting more attention now is the capability of micro-technologies to enable us to “make our learnings on the smallest scale” – the concept of the lab-on-a-chip. An excellent review of this work was done by Stone16, et al. These devices enable us to develop a mechanistic understanding of various process phenomena (e.g., emulsification, crystallization, precipitation). And these fundamental understandings enable more effective discovery work, analysis and scale-up for these systems. They are innovation enablers. 5.2 Micro-technology example For example, Figure 4 shows a 200 micron jet in a microfluidic system. A surfactant dissolved in a solvent flows in the center tube and an aqueous polymer solution flows in the outer channel. Upon contact, lamellar liquid crystalline vesicles form and the interfacial viscosity rises significantly. As a result, chaotic flow occurs as the system forms recirculation zones. This type of system then, gives us some valuable insight as to the transformations occurring at this scale.
Figure 4 200 micron microfluidic jet
Micro-rheology is another interesting micro-scale capability. It enables us to study micro-scale phenomena that bulk rheological measurements may miss. This is done by following the Brownian motion (or externally induced motion) of tracer particles. It is also a fast growing area which can give some remarkable insights into product transformations during processing, stability, and product performance. Much of this work has been enabled by developments in genomics, biotech, and microchips. The fabrication techniques and the applications are quite diverse and our challenge in the process field is to understand what’s possible in terms of application of these capabilities to address process issues. Again, how can these capabilities enable better, faster, cheaper innovation.
6. The Role of Modeling and Simulation in Process Innovation As noted earlier, we could say that modeling and simulation can enable learning at the smallest possible scale – the virtual scale. And as we think about the entire innovation process, we need to apply modeling tools during each of the phases. Models can facilitate a broader range of options studied since they significantly reduce the risk of
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failure. We like to say “explore digitally and verify physically” with regard to reducing innovation cycle times. It’s also important to start with simple “pilot” models that provide quick learnings with an immediate payback. As more is learned, the models can get more detailed and complex. We can draw an analogy with the innovation cycle for physical prototyping. As noted earlier, much of the modeling and simulation capability today is with mechanical systems. This work was fueled in large part by the application of Computer Aided Engineering (CAE) in the automobile and aerospace industries. 6.2 Modeling challenges The technical challenges are even greater in the chemical processing industries – and if we break this down further we can say that solids processing challenges are greater than fluids processing challenges. For example, the modeling and prediction of solids (granular) flows is certainly in its infancy and well behind those capabilities for fluids. Modeling of unit operations such as agglomeration and spray drying is very difficult. The challenge of modeling a unit operation such as spray drying is modeling of the various transformations (e.g., droplet formation and agglomeration) in addition to powder and air flows and heat and mass transfer. Certainly computational modeling is a growth area in P&G as noted by our Chief Technology Officer, Gil Cloyd, in a recent issue of Industry Week – "Computational modeling once existed primarily in a computer-aided-engineering context to facilitate our manufacturing processes. Now we're using it to help with actual product design.”….It will enable us to be more creative and design better products”17 And he predicts that as much as 90% of P&G’s R&D will be done in a virtual world with the remainder being physical validation. The reality is we are using modeling and simulation very broadly today. For example, we’re realizing significant cost and time savings in consumer testing. Internet Product Panels enable 24 hour engagement with consumers. We’re adopting a multiscale modeling approach that spans discovery through commercialization. The challenges are many – consumer modeling, connecting models across scale, modeling of product performance and stability, modeling of the transformations that occur during processing, and effective management of the data that goes into the models - to note just a few. In many situations it is difficult to measure key variables necessary to validate a model. For example, it is a real challenge to measure powder and air velocity profiles in a spray dryer operating at high temperature and high humidity. And although we’ve been using CFD to model our spray dryers for years, the model is still incomplete in terms of modeling the agglomeration transformation that occurs in the dryer and which is a key morphological attribute of the dried powder. There is also frequently a misunderstanding of what can and can not be modeled with today’s capabilities. It is all too frequently assumed that we can model anything. The fact is that in many cases we do not have the physical systems and prototypes to enable us to get the data needed to begin even elementary empirical models – much less the
54 first principles models that we prefer to enable getting outside the box of current product and process designs, i.e., getting to the truly innovative designs. And frequently the challenge is our ability to get good data – not the modeling associated with that data. This is again a situation where small scale capabilities are very valuable. For example, while we can get data on a plant scale system, it is usually cost prohibitive to go outside the boundaries of normal operation – so the model has limited capability. Having the capability to get this data at pilot scale certainly lowers the cost and increases the range of operation that can be modeled. This can still be very expensive though. However, if the data can be obtained at a bench or micro-scale, then this cost may not be a critical issue in the overall data generation/model development picture. We can better fuel innovation.
7. Summary and Conclusions Innovation is a broad topic that is in the spotlight of most corporations today. There are many thoughts on innovation. There are many books on innovation. There is not one approach which works for all organizations or all situations in an organization. In the end we need to approach innovation as we do other technical issues or problems. Survey trends and what is available. Gather data and information. Test a specific method that looks appropriate for your situation on a pilot basis. Learn and move forward.
References 1
Genskow, L.R., “Challenges in Transferring Research into Industrial Innovation”, Nordic Drying Conference’05, Karlstad, Sweden 2 http://www.affymetrix.com/index.affx 3 4
WO04007742A2, WO04007764A2
http://www.tfi.com/rescon/five_views.html Altshuller, G., “Creativity As An Exact Science”', Translated by Anthony Williams, (New York, Gordon And Breach, 1988.) 6 Gladwell, M. (2000) “the Tipping Point”, Back bay Books/Little, Brown and Company 7 http://www.gen3partners.com/about.htm 8 Nitz, Marcello, et al, “Drying of Beans in a Pulsed-Fluid Bed Dryer – Fluid Dynamics and the Influence of Temperature an, Air Flow Rate Frequency of Pulsation on the Drying Rate”, Proceedings of IDS’2004. Sao Paulo, Brazil 9 Jinescu, Gheorghita, “The Romanian School Contributions on the Oscillations Influence in the Intensification of Process Transfer in a Gas Fluidized Bed” , Proceeding of IDS’2004, Sao Paulo, Brazil. 10 Pang, Shusheng, “Airflow Reversals for Kiln Drying of Softwood Lumber: Application of a Kiln Drying Model and a Stress Model, Proceedings of IDS’2004, Sao Paulo, Brazil. 11 Christensen, Clayton, (1998) The Innovator’s Dilemma – When New Technologies Cause Great Companies to Fail, Harvard Business School Press 12 Christensen, Clayton, Raynor, Michael, (2003) The Innovator’s Solution – Creating and Sustaining Successful Growth, Harvard Business School Press 13 Berner, R., P&G: “New and Improved”, Business Week, July 7, 2003, 52-63 14 Huston, L., and Sakkab, N., “Connect and Develop: Inside Procter & Gamble’s New Model for Innovation”, Harvard Business Review, March 2006, 58-66 5
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Chesbrough, Henry, Open Innovation, Harvard Business School Press, 2003 Stone, HA, Stroock, AD, Ajdari, A., “Engineering Flows in Small Devices: Microfluidics Toward a Lab-on-a-Chip”, Annu. Rev. Fluid Mech. (20004) 36:381-411 17 Teresko, John, (Dec. 2004) “P&G’s Secret: Innovating Innovation”, Industry Week, Vol. 253, Number 12, 26-32 16
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Recent Developments and Industrial Applications of Data-Based Process Monitoring and Process Control Manabu Kano,a Yoshiaki Nakagawab a
Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan Sumitomo Metals (Kokura), Ltd., Kokurakita-ku,Kitakyushu 802-8686, Japan
b
Abstract Statistical process monitoring and control are now widely accepted in various industries. In recent years, statistical techniques are expected to solve quality-related problems. The issue of how to improve product quality and yield in a brief period of time becomes more critical in many industries where the product life cycle becomes shorter. Examples include steel processes and semiconductor processes. These processes are totally different in appearance, but the problems to solve are highly similar: how to build a reliable model from a limited data, how to analyze the model and optimize operating condition, and how to realize an on-line monitoring and control system and maintain it. In this paper, the problems and solutions are described with our application results in steel facilities. Keywords: Statistical Quality Control, Statistical Process Control, Multivariate Analysis, Iron and Steel Process.
1. Introduction How can we improve product quality and yield? More than ever, the answer to this question is vital as product life cycles are getting shorter and international competition is getting keener. Since this question arises repeatedly when a new product is developed, quality improvement should be achieved faster and in a more systematic way. Statistical quality control (SQC) has been widely used to address this issue and to search for an operating condition that can achieve the desired quality through designed experiments. However, designed experiments are impractical in more than a few industrial processes, because they require considerable time and cost. Jaeckle and MacGregor (1998) proposed a data-based method for determining the operating condition that can achieve the desired product quality. Kano et al. (2004) extended the method to cope with qualitative quality as well as quantitative quality and applied it to steel making and finishing processes. The proposed method is referred to as DataDriven Quality Improvement (DDQI). On the other hand, in various industries, run-torun (R2R) control has been widely used to control the product quality by manipulating operating conditions between batches (Castillo and Huriwitz, 1997), and multivariate statistical process control (MSPC) has been widely used to detect and diagnose faults (Kourti and MacGregor, 1995). In this paper, a newly developed process control and monitoring system for product quality and yield improvement, referred to as hierarchical quality improvement system (HiQIS), is presented. HiQIS consists of DDQI, R2R control, local control, and MSPC. Among these elements, DDQI, which is based on a statistical model, plays the most important role. It can cope with qualitative as well as quantitative variables, build
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a partially nonlinear model, determine the operating conditions that can achieve the desired product quality, optimize operating condition under various constraints, select manipulated variables suitable for R2R control, and thus can provide useful information to improve product quality. This paper aims to give an outline of DDQI and HiQIS and to show their usefulness via industrial case studies.
2. Hierarchical Quality Improvement System (HiQIS) In the process industry, a hierarchical control system has been widely accepted. The most famous one would be a model predictive control system which is integrated with a steady-state optimizer and local controllers. Qin et al. (2004) proposed a hierarchical fab-wide control framework in the semiconductor industry. The fab-wide control system is analogous to the model predictive control system. The hierarchical quality improvement system (HiQIS) is also an analogue to them. A schematic diagram of HiQIS is shown in Fig. 1. DDQI is a process analysis system located at the top of the hierarchy. It constructs a statistical model from operation data, analyzes the cause of inferior quality and low yield, selects manipulated variables, and optimizes the operating conditions that can achieve the desired quality. R2R control updates operating conditions or operation profiles for the next batch and gives set-points to local controllers on the basis of information provided by DDQI. In addition, MSPC detects and diagnoses faults on the basis of the statistical model built in DDQI. In this section, R2R control and MSPC are briefly reviewed.
Figure 1. A schematic diagram of hierarchical quality improvement system (HiQIS).
2.1. Run-to-Run Control Run-to-Run (R2R) control is a form of discrete control in which the product recipe is modified ex situ, i.e., between runs, so as to minimize process drift, shift, and variability. There are several R2R control algorithms. One widely used R2R controller is based on the exponentially weighted moving average (EWMA) statistic to estimate process disturbances. Although EWMA has been used for a long time for quality monitoring, its use for R2R control is relatively recent. Since the early 1990's, R2R control techniques have been developed and used to control various semiconductor manufacturing processes (Castillo and Huriwitz, 1997).
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2.2. Multivariate Statistical Process Control For the successful operation of any process, it is important to detect process upsets, equipment malfunctions, or other special events as early as possible and then to find and remove the factors causing those events. In industrial processes, data-based process monitoring methods, referred to as statistical process control (SPC), have been widely used. To improve the monitoring performance, multivariate statistical process control (MSPC) has been developed. The original Shewhart-type control chart for correlated variables is the Hotelling T2 control chart. Later, PCA was used as a tool of MSPC, and the control charts were introduced for the sum of squared errors (SPE) as well as T2 of principal components retained in a PCA model. In the last decade or so, various extensions of MSPC have been proposed (Kourti and MacGregor, 1995). When an outof-control signal is detected, it is necessary to identify the process variables that cause the out-of-control signal. This information helps operators to further diagnose the actual cause of a fault. For this purpose, contribution plots are widely used.
3. Data-Driven Quality Improvement (DDQI) In this section, Data-Driven Quality Improvement (DDQI) is focused. Jaeckle and MacGregor (1998) proposed a product design method based on linear/nonlinear multivariate analysis. Although their method can derive the operating conditions that can achieve the desired product quality, it does not account for qualitative variables. DDQI can handle qualitative as well as quantitative variables in a unified framework. In addition, DDQI has several additional important functions. 3.1. Modeling Quality and Operating Conditions DDQI is based on a statistical model that relates operating conditions with quality. To cope with a collinearity problem, principal component regression (PCR) or partial least squares (PLS) are usually used. The derived coefficient matrix shows the influence of operating conditions on product quality. Although PCR and PLS are useful for building a quality model, they cannot cope with process nonlinearity. On the other hand, nonlinear modeling methods such as artificial neural networks are not always desirable because limited samples make it difficult to build a reliable nonlinear model and also its interpretation is difficult. Therefore, in DDQI, analysis of variance (ANOVA) is integrated with statistical modeling method. First, a linear regression model is built by using PCR or PLS. Then, ANOVA is applied to operation data after operation data of each input variable are classified into two or more levels. ANOVA clarifies whether significant interaction exists between specific variables. If it exists, then an interaction term is introduced into the quality model. In addition, nonlinearity between prediction error and each input variable is analyzed, and a quadratic term is introduced if necessary. This approach can generate a simple quality model with minimum nonlinear terms. As a result, the derived model is easy to analyze and interpret. 3.2. Optimizing Operating Condition To determine the operating conditions that can achieve desired product quality, an inverse problem of the statistical model is solved. In general, the number of quality variables is less than that of principal components, and thus, the operating condition cannot be determined uniquely. However, it can be optimized when an objective function is provided. The objective function is optimized under the following four constraints: 1) the desired product quality is achieved, 2) the operating condition exists in the space spanned by principal components, 3) all operating condition variables exist within their upper and lower bounds, and 4) T2 statistic of scores is below its upper control limit or approximately 4') all scores exist within their upper and lower bounds.
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The last constraint is necessary for finding a new optimal operating condition within the region where the statistical model is valid. In other words, extrapolation should be avoided by using the last constraint. If there is no solution that satisfies all constraints, i.e., the imposed specifications on quality are too severe, the operating condition that achieves as desired quality as possible should be determined. 3.3. Handling Qualitative Variables In the previous subsection, the method to optimize the operating condition that can achieve the desired product quality is explained. However, it is applicable only to cases where all quality variables are quantitative. When the quality variables are qualitative, e.g., good and bad, the desired product quality cannot be specified quantitatively. To cope with this problem, a novel quantification method was proposed (Kano et al., 2004). To build a quality model by using PCR, qualitative variables should be quantified. As is well-known, for example, qualitative information such as good and bad can be quantified and denoted by 1 and 0, respectively. This conventional quantification method is useful for building a quality model, but not for solving its inverse problem. A serious problem is that the physical meaning of the quantified variable is not clear at all. For example, what does 0.6 mean? Is it acceptable or not? Nobody can answer this question. For a qualitative quality variable, the yield, i.e., the percentage of good products to all products, can be specified instead of the quality itself on the basis of the histogram for each category. Each histogram can be obtained from operation data, and it can be drawn as the frequency distribution of good or bad samples against the discriminant axis defined by PCA-LDA, which is the integration between principal component analysis (PCA) and linear discriminant analysis (LDA). Then, the yield against the discriminant axis can be derived. Once the desired yield is specified, operating conditions that can achieve the desired yield can be found by following the above-mentioned approach.
4. Applications to Industrial Iron and Steel Process In the iron and steel industry, process control systems have been designed by using mathematical models that describe the relationship between controlled product quality variables and manipulated variables. However, the relationship of operating condition to product quality such as surface flaws and internal defects is not clear. In general, these qualities have been maintained by skilled operators on the basis of their experience and intuition. It is empirically known that the occurrence of surface flaws and internal defects is affected by operating conditions of a furnace in a rolling process and a continuous casting equipment in a steel making process. However, it is not clear which operating condition has an influence on qualities to what extent. In addition, since internal defects are checked by using ultrasonic testing after a rolling process, it may take a few days to get control results in a steel making process, and thus real-time control cannot be applied to this type of quality control problem. To improve product yield, it is important to predict final product qualities. Due to these characteristics, surface flaws and internal defects have not been the target of process control for many years. However, business situation is changing. To meet customers’ requirements for higher product quality, to realize higher product yield, and to cope with decrease in skilled operators (Year 2007 problem in Japan), most iron and steel companies have started to cope with qualities such as surface flaws and internal defects within a framework of process control. Recently, the authors have investigated a statistical approach to address this problem and succeeded in improving product yield in iron and steel processes shown in Fig. 2 (Kano et al, 2005).
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Figure 3. Results of PCA-LDA. (left) Discrimination between good products and bad products. (right) Top six regression coefficients.
4.1. Reduction of Surface Flaws Reducing surface flaws is crucial for improving product yield. Surface flaws are checked after cold rolling as one of the key product qualities, and their shape and size are varied depending on their factors and steel grades produced. In this application, surface flaws frequently occurring in a specific alloy steel product are investigated. It is empirically known that the occurrence of surface flaws is affected by operating conditions of a rolling process and a steel making process. A large amount of defective steel is produced if steel making causes surface flaws, because surface flaws cannot be detected before the final inspection process. Therefore, it is important to clarify the cause of surface flaws, to find the operating condition that can minimize them, and to realize setup control. Here, application of PCA-LDA to this problem is described. First of all, a model relating operating conditions in steel making and hot rolling as input variables to surface flaws inspection results as an output variable was developed. Input variables include contents of various additive elements in a steel making process, temperature and residence time in each heating zone in a hot rolling process, and temperature at the exit of each stand in a hot rolling process. A total number of input variables selected is 55. The sample number is 138 consisting of 122 samples with surface flaws, classified into bad, and only 16 samples without surface flaws, classified into good. The dimensionality was reduced from 55 to 6 via PCA after all variables were normalized. Then, LDA was used for discriminating between two classes, i.e., good and bad. The developed PCA-LDA model was able to successfully discriminate between good products and bad products along the discriminant axis as shown in Fig. 3. This PCA-LDA model can relate operating conditions with the product yield through the proposed quantification method. Six variables having the largest influence on the product yield are listed in Fig. 3 with their regression coefficients. On the basis of this
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result, process engineers selected to manipulate alloy element e1 considering both operation cost and operability. Figure 3 suggests that surface flaws can be reduced by increasing the content of alloy element e1. It is confirmed from the verification experiments that surface flaws can be significantly reduced by increasing the content of alloy element e1. 4.2. Reduction of Internal Defects The objective in this application is to minimize internal defects. The target process consists of a steel making process, a blooming process, and a bar rolling process. Internal defects are checked by using ultrasonic testing (UST) after the bar rolling process. In this application, 40 operating condition variables are selected as input variables. The sample number is 740 consisting of 208 samples with internal defects and 532 samples without internal defects. The number of principal components retained is five. The developed PCA-LDA model was able to discriminate between good products and bad products along the discriminant axis. On the basis of the PCA-LDA model, process engineers selected to manipulate two factors in the steel making process and one factor in the blooming process considering both operation cost and operability. The next step is to optimize operating conditions via DDQI. The optimal operating condition that can improve the product yield by 20% is searched. To verify the results, test experiments were performed at the operating condition close to the optimal point, and it was confirmed that the percentage of defective product was reduced by half.
5. Conclusions To date, HiQIS and DDQI have been tested in the steel, the semiconductor, and the liquid crystal display industries, and have succeeded in finding new operating conditions to achieve higher product quality. As the product life cycle becomes shorter, the issue of how to improve product quality and yield in a brief period of time becomes more critical in many industries. How can we improve product quality and yield? From the authors' experience of applying HiQIS and DDQI to several industrial processes, the author hopes to develop a unified framework that can answer to this question on the basis of data-based methodologies. Of course, process knowledge is the key to success. Although different knowledge and models are required for coping with different processes, a data-based quality improvement framework could be applied to any process in various industries.
References C.M. Jaeckle and J.F. MacGregor, 1998, Product design through multivariate statistical analysis of Process Data, AIChE J., 44, 1105-1118. M. Kano et al., 2004, Data-driven quality improvement: handling qualitative variables, IFAC DYCOPS, CD-ROM, Cambridge, July 5-7. E.D. Castillo and A.M. Huriwitz, 1997, Run-to-run process control: literature review and extensions, J. Qual. Technol., 29, 184-196. T. Kourti and J.F. MacGregor, 1995, Process analysis, monitoring and diagnosis, using multivariate projection methods, Chemometrics and Intelligent Laboratory Systems, 28, 3-21. S.J. Qin et al., 2004, Control and monitoring of semiconductor manufacturing processes: challenges and opportunities, IFAC DYCOPS, CD-ROM, Cambridge, July 5-7. M. Kano et al., 2005, Product quality improvement using multivariate data analysis, IFAC World Congress, CD-ROM, Tu-M03-TP/22, Prague, Czech Republic, Jul. 3-8.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Model-centric technologies for support of manufacturing operations J. A. Romagnoli1, P. A. Rolandi2 Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA, 70803 USA 2 Process Systems Enterprise Ltd., 107a Hammersmith Bridge Road, London W6, UK
1
Abstract In this work we discuss the impact of a series of technologies for analysis and improvement of industrial manufacturing operations. These technologies are fused in a model-centric framework for integrated simulation, estimation/reconciliation and optimization of large-scale/plant-wide industrial process systems. A continuing industrial case-study is used to illustrate the viability of these technologies and their impact on the industrial workplace.
1. Introduction Throughout the 1990s, the computer-aided process engineering (CAPE) community made considerable progress in two strategic areas: the technical development and commercialisation of general-purpose modelling, simulation and optimisation environments; and the standardisation of open interface specifications for componentbased process simulation. Contemporary commercial modelling technologies and academic research have largely engaged in the developing frameworks and methodologies for tackling the model development process; however, rigorous mechanistic process models are just one of the many components of any sophisticated software tool targeting industrial applications. In order to succeed in their insertion in the industrial environment, model-based software tools must overcome a series of challenges limiting their ability to meet the needs of the Process Industries. First, a series of novel mechanisms and advanced software tools must be devised so that the definition of complex model-based problems is simplified. Concurrently, complementary model-based technologies must be integrated seamlessly into a single framework so that points of synergy between modelbased activities for process analysis and improvements are explored systematically. In light of these facts, and considering the increasing need for comprehensive process modeling and growing scope for model-based applications (Braunschweig et al., 2000), it is clear that further research in methodologies and technologies enabling a greater sophistication of manufacturing operations would certainly welcomed by governments, the academy and industry. In this work we discuss the impact of a series of technologies for analysis and improvement of industrial manufacturing operations. These technologies are fused in a model-centric framework for integrated simulation, estimation/reconciliation and optimisation of large-scale/plant-wide industrial process systems. A continuing industrial case-study focussing on the pulping section of a pulp and paper plant is used
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to illustrate the relevance of these technologies and their impact on the industrial workplace.
2. Evolution of general-purpose modelling languages By the late 1990s, high-level equation-oriented declarative modelling languages abandoned their conventional scope as modelling and simulation software and embraced a more powerful and promising paradigm as unified modelling, simulation and optimisation environments (MSOEs). gPROMS is early and prominent example of such evolution. Ever since, these MSOEs have gained an increased acceptance as the most appropriate tools to tackle the modelling process when full control over the scope and detail of the process model is required (Foss et al., 1998). State-of-the-art MSOEs provide the modeller with a series of sophisticated mechanisms that contribute enormously to increase the efficiency of the modelling process. Moreover, high-level equation-oriented modelling languages benefit from the intrinsic independence between mathematical models and solution methods; thus, by segregating the mathematical definition of any given model from structural, symbolic or numerical solution algorithms, a single model description can be used to accommodate for a large number of complementary activities. As the CAPE community continues developing and validating process models, the incentive behind developing and implementing modelbased applications grows. Today, the widespread adoption of these powerful multipurpose process-engineering software tools has both motivated a genuine interest in the novel area of model-centric technologies (MCTs) and created novel and opportunities (and challenges) for advanced support of manufacturing operations. By the mid 1990s, developers and end-users of CAPE software were confronted with the reality that commercial and proprietary process-engineering tools severely restricted the accessibility and usability of model descriptions embedded within these generalpurpose modelling software. To address this problem, the CAPE-OPEN (CO) and Global CAPE-OPEN (GCO) projects were initiated. CO focussed on providing standard mechanisms to support a long-term vision according to which: process modelling components (PMCs) built or wrapped upon the standard could be incorporated into process modelling environments (PMEs) straightforwardly; and model descriptions declared within PMEs supporting the standard would be accessible to external modelling tools. This way, developers would be able to assemble software components from heterogeneous sources to solve complex model-based problems. Today, this emerging paradigm for open system architectures facilitates the development of complex mathematical process models needed to deliver even more complex modelbased applications solving real-life process-engineering problems.
3. A model-centric framework as a layer for support of process operations 3.1. Characteristics Figure 1 provides a schematic representation on how the different components of the model-centric framework described in this work are expected to support the operation of an industrial process system.
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Decision Makers
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Figure 1: Schematic representation of the framework for model-centric support of process operations. The environment for reconstruction of process trajectories precedes all modules that make use of raw plant data since it is imperative to obtain a consistent set of data for the robust execution of any subsequent tasks. The estimation/reconciliation environment incorporates dynamic parameter estimation and dynamic data reconciliation activities, which make use of consistent data sets for the estimation of process operating parameters and evaluation of process measurement biases. The information gained from these activities is presented to the decision-makers, who then have a chance to make informed decisions on issues such as process instrumentation and equipment maintenance and inventory analysis. Consistent data sets are also provided to the simulation environment, which extracts meaningful information from past operating conditions. An increased mechanistic understanding of fundamental transformations within the process is used for assessment of existing operative practices and training of plant personnel. The insight gained at this stage triggers the exploration of future operating conditions, which are materialised through extensive parametric sensitivity studies and assessment of novel operating policies via the customised graphical user interface of the simulation environment. The optimisation environment incorporates nominal process optimisation and dynamic transition planning activities. The former facilitates the investigation of operating conditions for process improvement at nominal production levels by simultaneously considering realistic operative constraints and allowing for diverse conflicting performance objectives. The latter provides a means to find operating conditions/policies to ensure the optimal transition between nominal operating levels. The knowledge gained from these activities is used by the decisionmakers to update operating guidelines and manuals. Finally, the advanced process control environment incorporates the fundamental process model into a model-based control algorithm for on-line real-time assurance of optimal process performance under scheduled production rate changes and unexpected process disturbances and constraints.
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3.2. Architecture In order to meet the expectations of the manufacturing industries, MCTs are required to deliver high-performance model-based solutions while hindering unnecessary complexities and providing additional support to the end-users. Although MSOEs have gained increased acceptance in the academy and industry as multi-purpose modelling and solution engines (MSE), they have not been designed to facilitate the definition of realistic industrial model-based problems on-site. Therefore, MCTs targeting industrial applications should incorporate mechanisms to ease the conceptualisation of modelbased problems and their implementation in the corresponding modelling language. 3.2.1. Problem Definition Component In this work, we suggest materialising the framework for model-centric support of process operations proposed above by means of a Problem Definition Component (PDC). As sketched in Figure 2, the PDC manages the definition of advanced modelbased problems by interacting with both the PMEs and the user, while the PME executes the corresponding model-based activity by coordinating the calls to several PMCs. These PMCs contain the mathematical description of the process model, and they also provide other services such as physical property calculations and numerical solution algorithms (Braunschweig et al., 2000). Note that the standardisation of open interfaces for the PME and PMCs has been the focus of the CO/GCO projects. On the other hand, the communication between the PDC and other actors of the proposed architecture is regulated by a series of mechanisms discussed in this work and in Rolandi & Romagnoli (2005). These mechanisms entail the manipulation of the socalled Data Model Templates (DMTs) and Data Model Definitions (DMDs). 3.2.2. Data Model Templates and Definitions Rolandi & Romagnoli (2005) argue that, in order to ease the high-level definition of realistic model-based problems in the industrial workplace, it is necessary to provide additional meaning to end-users by including complementary information that transforms pure mathematical variables of process models into physical variables of process systems. DMTs are data models that contain this additional qualitative and quantitative information. DMTs have been conceived as extensible data structures which define the subset of process and model variables available for subsequent manipulation, their possible structural function, their nominal numerical values, and a series of advanced relational properties. In other words, DMTs determine how the process model and process-instrumentation data information can be used at higher levels of the hierarchy to define complex and realistic model-based problems driven by genuine process data. According to the characteristics of the framework described in previous sections, DMTs corresponding to simulation, optimisation and estimation/reconciliation activities as well as plant data were derived. At this point we should stress that DMTs do not necessarily represent valid model-based activities; instead, they are conceived as a kind of macro-structures from which valid simulation, optimisation, reconciliation/estimation problems can be defined. DMDs are also data models representing entities of interest to process-engineers (i.e. plant data sets and simulation, optimisation and estimation activities); however, on the contrary to DMTs, these data structures correspond to valid (although not necessarily feasible) model-based activities. In brief, DMDs are a consistent mapping of the
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problem definition which is derived by refining and overriding the original macromapping obtained from the corresponding DMT. Overall, the DMT/DMD mechanism creates an innovative means to embed process knowledge and expertise on the definition of model-based problems, as well as increased opportunities for documentation and re-use of case-studies. structure of the mathematical model
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structure of the mathematical definition of the model-based problem
Figure 3: Sequence of definition tasks
4. An industrial case-study We shall illustrate the benefits of the framework for integrated model-centric support of process operations by means of a continuing case-study; a state-of-the-art continuous pulping system of an industrial pulp and paper mill is selected for this purpose. Modern pulp and paper mills pose operability and controllability issues due to tightly coupled and closed processes. The mill used in this particular study is an excellent example of this situation, since reduced fixed capital costs were achieved at the expense of eliminating redundant (back-up) equipment and reducing inventories (hold-ups). The continuous pulping system is the heart of a pulp and paper mill; it is a network of interconnected units comprising: a feed line, a continuous cooking digester, and a heat exchange and recovery network. The daily operation of the continuous pulping process is affected by upstream and downstream disturbances that negatively influence its performance. Seasonal variations in the composition of raw materials, changes in the concentration of cooking chemicals and moisture of wood chips, and swings in the pressure of service steam are some of the conditions most often encountered, and they are likely to occur in a time scale that goes from months to hours. Concurrently, production rate changes in order to meet operational and inventory constraints and market demand fluctuations are common daily practices that negatively impact on quality indicators. The unavailability of on-line measurements of some key process variables such as selectivity and yield of the pulping reactions greatly affect the controllability of the process. Some important process disturbances such as white liquor composition and wood moisture are also unmeasured. Due to the number of units within the network of interconnected process equipment comprising the continuous pulping systems and the complexity of the physical and chemical phenomena occurring simultaneously within these units, the mechanistic process model of this industrial manufacturing system has resulted in a large-scale mathematical formulation. Using a large-scale process model to study an industrial process system under genuine
J.A. Romagnoli and P.A. Rolandi
68
conditions of operations is a non trivial task, which has been tacked successfully with the proposed framework for integrated model-centric support.
5. Results Due to space limitations only a small number of scenarios studies will be presented here. 5.1. Process simulation: performance assessment of past operating scenarios The objective of the continuous cooking digester is to deplete lignin from the wood matrix, which is a multi-component substance that also contains cellulose and hemicelluloses. In industrial continuous cooking processes, the selectivity and yield of the pulping reactions are the key process variables indicating the production of pulp and degree of delignification, respectively. Let us examine the yield and selectivity profiles throughout the length of the continuous cooking digester as given by historic operating conditions. These results are shown in Figures 4 and 5. A feature that deserves our attention is the qualitative shape of these profiles during the early stages of the cooking process, at the entrance to the digester (known as the impregnation zone). Figure 4 shows that the wood-chip yield decreases, indicating that degradation and solubilisation of wood components is taking place. Figure 5 shows that kappa number increases in the impregnation zone, a phenomenon that has also been reported by Wisnewski et al. (1997). A simultaneous decrease of pulp yield and increase of the kappa number (a measure of selectivity) in the impregnation zone indicates that undesired solubilisation of cellulose is taking place. This is usually associated to high impregnation temperatures and high alkali concentrations. Since chemical pulping target degradation of lignin-like components but not the solubilisation of cellulose and hemicelluloses, this snapshot of the operational status of the continuous pulping system leads us to the conclusion that the performance of this industrial system could possibly be improved. This important result leads us to continuing with the following case-studies. 10 0
16 7
90
14 7
80
12 7
70
10 7 87
60 0
10
20
30
40
50
Figure 4: yield vs height
0
10
20
30
40
50
Figure 5: kappa number vs height
5.2. Process optimization: improvement of nominal operating performance Here, two alternative scenarios are investigated aiming at improving the nominal operating conditions of the continuous pulping system: (CS1) maximum production and (CS2) maximum overall net profit. The operating conditions suggested by CS1 increase production by 1.2% (an improvement of the pulp yield at constant selectivity) which, in turn, boosts the overall
Model-Centric Technologies for Support of Manufacturing Operations
69
profit of the continuous pulping system by approximately 1.04US$/min. An analysis of the economic performance shows that the 1.68US$/min revenue increase from a higher pulp throughput is counterbalanced by a higher flow of black liquor for evaporation (0.64US$/min) while other sources of revenue and costs are less relevant to this study. Surprisingly, the operating conditions found in CS2 also boost the economic performance of the continuous pulping system, although this time the overall profit increases by approximately 3.00US$/min. Interestingly, these conditions lead to a reduction in the pulp yield of approximately 0.15% with respect to CS1; however, the lower pulp throughput (0.17US$/min) also results in a considerably lower flow of black liquor for evaporation, which translates into a 3.18US$/min expense decrease. Compared with the original operating policy of the mill, CS2 may potentially result in 2.0 million US$/yr additional revenue. 5.3. Transition management: assessment of the control structure and procedure Again, two possible scenarios are presented investigating the effect of different transition policies during production rate changes: (CS1) manipulation of both the lower circulation heater and the wash circulation heater controllers and (CS2) manipulation of the circulation heater controller solely. Figures 6 to 8 illustrate the trajectories of some key process variables under these two possible transition management scenarios. 90.04
64.91
CS1
90.02
CS2
64.89
90.00
64.87
89.98
CS1
64.85
89.96 89.94
64.83
CS2
64.81
89.92 -1
1
3
5
7
Figure 6: kappa number vs time
9
11
-1
1
3
5
7
9
11
Figure 7: yield vs time
From Figure 6 we can appreciate that there are no sensible differences in the trajectory of the blow-line kappa number for CS1 and CS2. Hence, the sole manipulation of the temperature set-point of the lower circulation heater during a production rate change is sufficient to reduce the variability of this key process variable significantly (i.e. CS2). In spite of this, Figure 7 shows that these transition management procedures are not equivalent, since CS2 gives rise to a more efficient use of the raw materials (note the pulp yield increase). Figure 8 demonstrates that the temperature profiles along the continuous cooking digester are not the same during the transition. In CS1, part of the increased cooking load has been shifted to the lower cooking zone and upper section of the wash zone (the bottom of the vessel, where temperature differences of more than 0.5ºC are observed). On the contrary, in CS2 the lower cooking zone (towards the center of the vessel) has accommodated the increase in temperature counteracting the
J.A. Romagnoli and P.A. Rolandi
70
reduction in residence time, which is know to have a positive effect on the selectivity and yield of the pulping reactions as confirmed in Figure 7. In summary, not only have
we achieved similar quality control performance, but we have also found an alternative transition management procedure which uses the raw materials (both cooking liquor and wood chips) more efficiently. 3.5
170.0 160.0 150.0 140.0 130.0 120.0 110.0 100.0 90.0
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2
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6
8 10 12 14 16 18 20
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170.0 160.0 150.0 140.0 130.0 120.0 110.0 100.0 90.0
2.5 1.5 0.5 -0.5 0
2
4
6
8 10 12 14 16 18 20
Figure 8: Reaction temperature profiles: (a) CS1and (b) CS2; Primary y axis (lhs): absolute temperature [ºC] (0hr); secondary y axis (rhs): relative temperature (difference) [ºC] (4hr, 12hr);
6. Conclusions Supporting the manufacturing operations of industrial process systems requires the consistent and integrated solution of a series of process-engineering problems. Conventionally, these problems would comprise the use of a model of the process system to solve model-based activities such as process simulation and optimization, estimation/reconciliation and advanced process control. The execution of these activities requires the definition of the corresponding model-based problem. The framework discussed in this work proposes a software architecture and methodology that eases the definition of advanced model-based process-engineering problems for support of process operations and promotes the transfer of knowledge between complementary model-based applications, especially in the context of large-scale/plant-wide industrial process systems. This research brings model-based applications for support of manufacturing operations (and model-centric technologies in general) to an unparalleled level of integration with the industrial workplace.
7. References Braunschweig, B.L., Pantelides, C.C., Britt, H.I. and Sama, S. (2000), “Process modeling: The promise of open software architectures”, Chemical Engineering Progress, 96, 65-76. Foss, B.A., Lohmann, B. Marquardt, W. (1998), “A field study of the industrial modeling process”, Journal of Process Control, 8, 325-338 Wisnewski, P.A., Doyle, F.J. and Kayihan, F. (1997), “Fundamental continuous-pulpdigester model for simulation and control”, AIChE Journal, 43, 3175-3192. Rolandi, P. A., Romagnoli, J. A. (2005), “Integrated model-centric framework for support of manufacturing operations”, submitted to Computers and Chemical Engineering.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
71
THE SYSTEMS ENGINEERING OF CELLULAR PROCESSES Vassily Hatzimanikatis* and Liqing Wang Northwestern University Evanston, IL 60208
Abstract Mathematical description of metabolic systems allows the calculation of the expected responses of metabolism to genetic modifications and the identification of the most promising targets for the engineering of cellular processes. Metabolic control analysis (MCA) provides such a description in the form of sensitivity coefficients, called control coefficients. These indices are determined by perturbation experiments, or through (log)linear analysis of nonlinear mathematical models, around a reference steady state, and, therefore, the predictive power of MCA is limited to small changes in the metabolic parameters. We present here the Nonlinear Metabolic Control Analysis (NMCA), a framework that allows accurate description of the metabolic responses over wide range of changes in the metabolic parameters. The performance and capabilities of NMCA are illustrated using a model of the yeast glycolysis.
Keywords Metabolic engineering, (log)linear model, metabolic control analysis, nonlinear metabolic control analysis, glycolysis. Introduction Complex systems are composed of interacting elements that give rise to properties which cannot be predicted by knowing the properties of their constituting elements alone (Ottino 2003). Metabolic pathways are such complex systems, where many different enzymes and proteins interact through common reactants, substrates, cofactors, and regulators to yield properties which cannot be described simply by the knowledge of the individual enzymes in the pathways. However, understanding the emerging properties of metabolic pathways is central to problems and questions in several life sciences disciplines, such as medicine, evolutionary biology, systems biology, and metabolic engineering. On the other hand, there is an exponential growth in the development of new technologies that provide a very good qualitative and quantitative description of metabolic systems. Genome sequencing and metabolic reconstruction allow the identification of the stoichiometry and pathway topology, and metabolic flux analysis (MFA) methodologies, developed within the field of metabolic engineering, provide with estimates of the fluxes (reaction rates) in metabolic networks (Forster et al. 2003; Varma and Palsson 1993a; Varma and Palsson 1993b). In this article we employ a mathematical and computational framework that allows the integration of the available information from these fields and the prediction of the responses of metabolic networks to changes in metabolic parameters. From a systems perspective, metabolite concentrations and reaction rates are the metabolic variables, stoichiometry, enzyme kinetic parameters, and environmental conditions are the metabolic parameters. In studying the systemic origins of disease, or in identifying targets for genetic and metabolic engineering, we are interested in understanding how changes in metabolic parameters impact the variables of a metabolic system. Such an understanding will require a quantitative characterization of these effects, and the identification of the most important parameters, e.g., the parameters that have the highest impact on certain metabolic variables. The latter is what is in many cases called the rate-limiting steps in a metabolic pathway. Quantitative characterization of this kind can be achieved either through sophisticated experiments that involve perturbation of the metabolic parameters and precise measurements of possibly every metabolic variable, or through simulations of detailed kinetic models of the metabolic pathways. However, both of these approaches have certain drawbacks. Namely, the experimental approach requires not only a large number of experiments, but also many advanced analytical techniques. Yet still, they are subject to a great degree of uncertainty. And for the computational approach, construction of such models entails a large amount of information, including but not limited to kinetics of the enzymes of
* To whom all correspondence should be addressed
72
V. Hatzimanikatis and L. Wang
the pathway and metabolite concentrations. Therefore a quantification framework that requires a minimum amount of information about enzyme kinetics and takes into account of the inherent uncertainty will significantly improve our understanding of the functions of metabolic pathways. Among the developed mathematical frameworks, Metabolic Control Analysis (MCA) quantifies the link between genetic modifications or environmental changes and cellular process responses. MCA introduces the control coefficients, similar to the concept of transfer function in the system control theory, to quantify the fractional change of cellular output (metabolite concentrations and metabolic fluxes) in response to a fractional change of system parameters (e.g. enzymatic activities and growth conditions) (Fell and Sauro 1985; Hatzimanikatis and Bailey 1996; Hatzimanikatis and Bailey 1997; Heinrich and Rapoport 1974; Kacser and Burns 1973; Kholodenko and Westerhoff 1993; Reder 1988). An immediate application of MCA on rational metabolic engineering design is the ranking of potential targets based on the values of the control coefficients of the flux leading to the desired cellular product (Bowden 1999; Cascante et al. 2002; Schuster 1999; Westerhoff and Kell 1996). However, MCA, being a “local sensitivity” analysis framework, does not guarantee an accurate estimation of the system responses to large changes in enzyme activities and growth environment. In order to address this issue we have previously developed a method called Nonlinear Metabolic Control Analysis (NMCA) that allows prediction of metabolic responses to large changes in metabolic parameters based on information about the properties of the system around a reference steady state (Hatzimanikatis 1999). We apply here this method on yeast glycolysis and illustrate how this method is applicable to the cases where partial information about the kinetics of the system is available. Generalized (Log)linear MCA Formalism For cells growing in a batch culture, the mass balances of intracellular metabolites can be described as
dx (1) = Nv ( x , p e , p s ) , dt where x is the metabolite concentration vector, N is the stoichiometric matrix, v is the metabolic flux vector, p e is the enzyme activity parameter vector, which includes both kinetic parameters and enzyme concentrations, and p s is the vector of other system parameters such as temperature and pH. Due to the presence of conserved moieties in the cellular metabolism, i.e. groups of compounds such as ATP, ADP, and AMP, whose total amount is assumed to remain invariant over the characteristic response time of the metabolic network, we can divide the original set of metabolite concentrations x into two categories: an independent metabolite concentration vector,
x i , and a dependent metabolite concentration vector, x d (Reder 1988). A third parameter set,
pm , is also introduced into the system to represent the total concentration of the metabolites in each moiety group (Wang et al. 2004). The reduced form of mass balances with respect to independent metabolites can be represented as
dx i = N R v ( x i , x d ( x i , p m ), pe , p s ) , dt where N R consists of the rows in N corresponding to the independent metabolites.
(2)
In the previous work, we have demonstrated the calculation of control coefficients in a intracellular metabolic system based on the (log)linear model formalism (Hatzimanikatis and Bailey 1996; Hatzimanikatis and Bailey 1997; Hatzimanikatis et al. 1996; Wang et al. 2004). Assuming a stable steady state for the system in Equation (2), and after linearization and scaling we obtain the following equations for the control coefficients:
[N R VΠm # N R VΠe # N R VΠs ] , + [Π m # Π e # Π s ] .
Cpxi = −(N R VEi + N R VEdQi )
−1
C = ( E i + E d Q i )C v p
xi p
Concentration control coefficients,
(3) (4)
C px , and flux control coefficients, C vp , are defined as the fractional change of
metabolite concentrations and metabolic fluxes, respectively, in response to fractional changes of system parameters. In this formalism, V is the diagonal matrix whose elements are the steady state fluxes; E i and E d are the matrices of the elasticities with respect to metabolites, defined as the local sensitivities of metabolic fluxes to independent and dependent metabolite concentrations, respectively; Π m , Π e , and Π s are the matrices of the elasticities with respect to parameters, defined as the local sensitivities of metabolic fluxes to system parameters, correspondingly; and
pm , pe , and p s ,
Qi is a weight matrix that represents the relative abundance of dependent metabolites with respect
73
The Systems Engineering of Cellular Processes to the abundance of the independent ones. A second weight matrix,
Qm , is also defined, for the relative abundance of
dependent metabolites with respect to the levels of their corresponding total moieties, which leads to the following expression for the matrices of elasticities with respect to parameters, Π m :
Π m = E d Qm
(5)
It is worth mentioning that the estimation of control coefficients using this framework does not require explicit knowledge of the values of the concentration of metabolites. Nonlinear Metabolic Control Analysis (NMCA) In the previous work (Hatzimanikatis 1999), we have demonstrated how we can use the “local” information from MCA (Equations (3)-(5) above) to calculate responses to large changes in metabolic parameters. The method is based on the observation that the dependency of metabolite concentrations and metabolic fluxes can be formulated as the following ordinary differential equation problem:
dzi = C pxik = f ( x, v ,E ( x ) ) dqk dw j v = C pkj = g ( x , v , E ( x ) ) dqk
(6)
(7)
where:
zi = ln
xi xi ,o
⇒ xi = xi ,o ⋅ e zi
(8)
and
w j = ln
vj v j ,o
⇒ v j = v j ,o ⋅ e
wj
(9)
with xi,o and vj,o the initial, reference steady-state values of metabolite xi and metabolic flux vj , respectively. Equations (6)-(9) can be now solved simultaneously using any ODE solver and could provide the dependency of steady-state values of metabolites and metabolic fluxes on large changes in the metabolic parameter pk, with
pk = pk ,o ⋅ e qk . For every integration point, we also check the local stability characteristic of the system in order to
guarantee that the system transitions to new steady states through a sequence of stable steady states. In case the system crosses into an unstable steady state, we terminate the integration. Details of the method will be provided elsewhere (Wang and Hatzimanikatis, in preparation). Analysis of the Rate Limiting Steps in Yeast Batch Fermentation We applied the NMCA framework on the central carbon metabolism of yeast S. cerevisiae growing in a batch reactor based on the model by Teusink et al. (Teusink et al. 2000) (Figure 1). Teusink et al. provided information about the kinetic parameters of most of the enzymes in the system and the concentration of most of the metabolites. A total of 14 parameter and metabolite values were missing. In order to overcome this limitation we consider the following three cases: I. The missing parameters and metabolites were chosen such that the corresponding enzymes were at high saturation at the reference, initial steady state. II. The missing parameters and metabolites were chosen such that the corresponding enzymes were at low saturation at the reference, initial steady state. 14 III. The missing parameters and metabolites were chosen with all possible 2 combinations of the corresponding enzymes between high and low saturation at the reference, initial steady state. In all cases we studied the responses of the metabolites and metabolic fluxes to large changes in the activity of the glucose transport enzyme, HXT. In case I we have been able to calculate the responses of metabolites and metabolic fluxes for a 100% change in HXT activity (Figures 2 and 3). In case II though, beyond 40% changes in the activity of HXT the system became unstable going through a saddle node bifurcation (Figures 4 and 5). This transition is also demonstrated through the sharp increase in the relative concentrations of some of the metabolites, such as PEP and PYR (Figure 4). In both cases, we compared the results from NMCA analysis with the changes in the values of metabolites and metabolic fluxes if we had used the control coefficients from the reference steady states using the following equations:
V. Hatzimanikatis and L. Wang
74
trans
Gin ATP HK
ADP
ATP
ADP
ATP
trehalose
PGI
ATP
G6P ADP
glycogen
F6P PFK
ADP FdP
ATP
ALD
TPI
NAD
DHAP
GAPDH
NADH
glycerol NAD
GAP
ADP
NADH
ATPase
BPG ADP PGK
ATP 3PG PGM
2 ADP AK
AMP
2PG ATP
ENO
PEP ADP PYK
ATP PYR PDC
CO2 AcAld
succinate
ETOH
3 NADH 4 ADP
ADH
NAD
3 NAD 4 ATP
NADH
Figure 1. Anaerobic glycolytic pathway model of nongrowing yeast, Saccharomyces cerevisiae, with glucose as the sole carbon source. Chemical species: Gin, intracellular glucose; G6P, glucose-6-phosphate; F6P, fructose-6phosphate; FdP, fructose 1,6-diphosphate; GAP, glyceraldehydes-3-phosphate; DHAP, dihydroxy acetone phosphate; BPG, bisphosphoglycerate; 3PG, 3-phosphoglycerate; 2PG, 2-phosphoglycerate; PEP, phosphoenolpyruvate; PYR, pyruvate; AcAld, acetaldehyde; ETOH, ethanol; ATP, adenosine triphosphate; ADP, adenosine diphosphate; AMP, adenosine monophosphate; NADH, nicotinamide adenine dinucleotide. Pathway steps and enzymes (in bold): trans, glucose cross-membrane transport; HK, hexokinase; PGI, phosphoglucose isomerase; PFK, phosphofructokinase; ALD, fructose 1,6-diphosphate aldolase; TPI, triose phosphate isomerase; GAPDH, glyceraldehydes-3-phosphate dehydrogenase; PGK, phosphoglycerate kinase; PGM, phosphoglycerate mutase; ENO, enolase; PYK, pyruvate kinase; PDC, pyruvate decarboxylase; ADH, alcohol dehydrogenase; ATPase, net ATP consumption; AK, adenylate kinase.
75
The Systems Engineering of Cellular Processes X * = X *,o ⋅ (PHXT PHXT ,o )
X CHXT
(10)
and
V* = V*,o ⋅ ( PHXT PHXT ,o )
V CH XT
(11)
In both cases the MCA prediction is quite accurate for up to 25% increase in the HXT activity. The MCA fails as the change in the activity increases, and as expected, it cannot capture the metabolite accumulation associated with the bifurcation observed in case II. It also appears that MCA can predict the fluxes much better in case II for up to almost 40%, while it fails to do so for the same range in case I. This observation suggests that the accuracy of MCA depends on the values of the metabolites at the reference, initial steady state. This effect is also illustrated in the studies of case III (Figure 6). In this case we have studied all the possible combinations of the 14 unknown metabolite concentrations that could support stable steady states up to 100% changes in the activity of HXT. In general, it appears that the MCA overestimates the responses of the fluxes relative to NMCA. Probably the most interesting result is the fact that the propagation of the uncertainty due to the uncertainty in the reference, initial parameters is not significant and it appears that this uncertainty propagation is smaller in the NMCA predictions. Future studies could provide important insights on the conditions that reduce, or increase, the uncertainty propagation. 4
1.4 GLC
3 2 1
1
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2
1.4
X ∗ X ∗,o
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2
1
1.5
2
PHXT PHXT ,o Figure 2. Relative changes in metabolite concentrations for 100% increase in the activity of HXT, case I. Chemical species notation same as in Figure 1. Solid (blue) line: NMCA predictions; dashed (black) line: MCA predictions.
76
V. Hatzimanikatis and L. Wang
1.15
1.15
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HXT
HXK
1.1
1.03
1.06 Treh
PGI
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Glycogen
1.02
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ENO
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ADH
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ATPase
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Glycerol
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AK
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0.99 1.05
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2
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1 1
2
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1 1
2
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GAPDH
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2
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ALD
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1 1
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1
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1 1
2
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2
1
0.98 1
1.5
2
0.97
1
PHXT PHXT ,o Figure 3. Relative changes in metabolite fluxes for 100% increase in the activity of HXT, case I. Metabolic reaction notation same as in Figure 1. Solid (red) line: NMCA predictions; dashed (black) line: MCA predictions.
2
1.4 GLC
1.3
1.5
1
1
1.2
1.4
3
X ∗ X ∗,o
1.2
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1 1
1.2
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10 PEP
PYR
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ACALD
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6
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4 F6P
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1 1
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1.4
0.7
1
1.2
1.4
1
1
1.2
1.4
0.9
1
1.2
1.4
PHXT PHXT ,o Figure 4. Relative changes in metabolite concentrations for 40% in the activity of HXT, case II. Chemical species notation same as in Figure 1. Solid (blue) line: NMCA predictions; dashed (black) line: MCA predictions.
77
The Systems Engineering of Cellular Processes
1.4
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PHXT PHXT ,o Figure 6. Relative changes in metabolite fluxes for 100% increase in the activity of HXT, caseIII. Metabolic reaction notation same as in Figure 1. Red line: NMCA predictions; Grey line: MCA predictions. Error bars quantify the standard deviation over all the samples. It also expected that increase in the activity of an enzyme will lead to redistribution in the value of the control coefficients. In cases I and II, the control coefficient of ethanol production, i.e., flux through enzyme ADH, with respect to HXT is very small in the initial state and therefore there is no significant redistribution in the values of the control coefficients (Figure 7). This also explains the relative small changes in metabolic fluxes to changes in the HXT activity (Figures 1, 2, and 6).
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C∗ADH Figure 7. Control coefficient of ethanol production with respect to the enzymes in the network. The subscript “*” correspond to the enzymes labeled on the Y-axis with notation same as in Figure 1. Panels (a) control coefficients at the reference, initial state, case I; (b) control coefficients after 100% increase in HXT activity, case I;. (c) control coefficients at the reference, initial state, case III; (d) control coefficients after 100% increase in HXT activity, case III. However, in case II, the initial control coefficient of ethanol production with respect to HXT activity is relative large (Figure 8a) and therefore the system experiences significant changes even with 40% increase in HXT activity (Figures 4 and 5). However, after 40% increase in HXT activity, the control coefficient of ethanol production with respect to HXT activity has vanished, and PDC became the rate limiting step as indicated by the significant increase in control coefficient of ethanol production with respect to PDC activity (Figure 8b). It also appears that in the reference state ATPase, a process associated with energy balancing in the cell, is the second most rate limiting process. However, its control also vanishes after 40% increase in HXT activity (Figure 8). Therefore NMCA analysis suggests that in a strategy to further improve the ethanol production using the overexpression of two enzymes, HXT and PDC are the most important targets. However, MCA analysis would have led us to choose HXT and ATPase, a choice that would have led to lower improvements. Conclusions We presented an analysis of yeast glycolysis using a metabolic control analysis framework, Nonlinear Metabolic Control Analysis (NMCA) that takes into account large changes in enzyme activities. This analysis demonstrated the power of the framework in identifying targets for genetic and metabolic engineering towards improvement of cellular processes. We also demonstrated the ability of the method to perform metabolic control analysis even in the absence of complete information about the system. Future efforts will combine NMCA with our previously developed method for metabolic control analysis under uncertainty (Wang et al. 2004a). Finally, the method appears to be ideally suitable identifying multiple targets for genetic and metabolic engineering. Acknowledgments The authors are grateful for the financial supported provided by the Department of Energy (DE-AC36-99GO103), the National Aeronautics and Space Administration (NAG 2-1527), and DuPont through a DuPont Young Professor Award to VH. LW received partial support by the Chinese Government through the State Excellence Scholarship program for students studying overseas.
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References Bowden AC. 1999. Metabolic control analysis in biotechnology and medicine. Nat Biotechnol 17(7):641-3. Cascante M, Boros LG, Comin-Anduix B, de Atauri P, Centelles JJ, Lee PWN. 2002. Metabolic control analysis in drug discovery and disease. Nature Biotechnology 20(3):243-249. Fell DA, Sauro HM. 1985. Metabolic control and its analysis. Additional relationships between elasticities and control coefficients. Eur J Biochem 148(3):555-61. Forster J, Famili I, Fu P, Palsson BO, Nielsen J. 2003. Genome-scale reconstruction of the Saccharomyces cerevisiae metabolic network. Genome Res 13(2):244-53. Hatzimanikatis V. 1999. Nonlinear metabolic control analysis. Metab Eng 1(1):75-87. Hatzimanikatis V, Bailey JE. 1996. MCA has more to say. J Theor Biol 182(3):233-42. Hatzimanikatis V, Bailey JE. 1997. Effects of spatiotemporal variations on metabolic control: Approximate analysis using (log)linear kinetic models. Biotechnology and Bioengineering 54(2):91-104. Hatzimanikatis V, Floudas CA, Bailey JE. 1996. Analysis and design of metabolic reaction networks via mixed-integer linear optimization. Aiche Journal 42(5):1277-1292. Heinrich R, Rapoport TA. 1974. A linear steady-state treatment of enzymatic chains. General properties, control and effector strength. Eur J Biochem 42(1):89-95. Kacser H, Burns JA. 1973. The control of flux. Symp Soc Exp Biol 27:65-104. Kholodenko BN, Westerhoff HV. 1993. Metabolic channelling and control of the flux. FEBS Lett 320(1):71-4. Ottino JM. 2003. Complex systems. Aiche Journal 49(2):292-299. Reder C. 1988. Metabolic control theory: a structural approach. J Theor Biol 135(2):175-201. Schuster S. 1999. Use and limitations of modular metabolic control analysis in medicine and biotechnology. Metab Eng 1(3):232-42. Teusink B, Passarge J, Reijenga CA, Esgalhado E, van der Weijden CC, Schepper M, Walsh MC, Bakker BM, van Dam K, Westerhoff HV and others. 2000. Can yeast glycolysis be understood in terms of in vitro kinetics of the constituent enzymes? Testing biochemistry. Eur J Biochem 267(17):5313-29. Varma A, Palsson BO. 1993a. Metabolic Capabilities of Escherichia-Coli .1. Synthesis of Biosynthetic Precursors and Cofactors. Journal of Theoretical Biology 165(4):477-502. Varma A, Palsson BO. 1993b. Metabolic Capabilities of Escherichia-Coli .2. Optimal-Growth Patterns. Journal of Theoretical Biology 165(4):503-522. Wang L, Birol I, Hatzimanikatis V. 2004a. Metabolic Control Analysis under Uncertainty: Framework Development and Case Studies. Biophys. J. 87(6):3750-3763. Westerhoff HV, Kell DB. 1996. What bio technologists knew all along...? J Theor Biol 182(3):411-20.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Systems Biology and the Silicon Cell: Order out of chaos Hans V. Westerhoff Manchester Center for Integrative Systems Biology, The University of Manchester, UK, EU and Molecular Cell Physiology, FALW, Free University, BioCenter Amsterdam, Amsterdam, The Netherlands, EU
Abstract This paper gives an overview of the thermodynamics and kinetics background to the silicon cell (SiC!) approach. SiC! makes precise mathematical models of components of systems inclusive of their interaction properties. It then puts these component models together into a computer program and integrates the behavior. For metabolic pathways, SiC! takes the ensembles of enzyme molecules as the components. It takes the ensemble averaged metabolite concentrations as the dependent variables these components work on. We show how this approach depends on principles of non equilibrium thermodynamics and kinetics. Metabolic control analysis is an early and characteristic approach to systems biology. Using silicon cells one can do this control analysis in silico. Also this analysis also has a number of theoretical foundations, which are again close to those of non equilibrium thermodynamics. We propose that Metabolic Control Analysis is in fact the extension from equilibrium thermodynamics to non equilibrium systems that so many searched for in the second half of the previous century.
1. Non-equilibrium thermodynamics In 1931 (1,2) Onsager published two seminal papers. They revealed that there should be a remarkable symmetry in cross-cause effects relationships in coupled processes. To obtain the symmetry property, coupled processes have to be described in a certain way, a way that has since been called non equilibrium thermodynamics (3). Describing each process in terms of a driving force equal to the free energy difference across that process, and a flow equal to the steady state rate of the process, the cross dependence of the two processes on the two forces had to be equal in the limit to equilibrium. The proof given was based on kinetics or a probabilistic version thereof, and therewith married mass-action kinetics with thermodynamics. Yet, it depended on the generic rather than the specific aspects of the kinetics and was therewith mechanism independent. Because this was also true for equilibrium thermodynamics, this mechanism independence was long thereafter considered an essential property, also of non equilibrium thermodynamics. This non equilibrium thermodynamics (NET) was often formulated as a systems of linear equations relating all steady state fluxes in the system to all thermodynamic forces through proportionality relations, for which the matrix of proportionality constants then had to be symmetrical for the Onsager reciprocity relations to be satisfied. Because Biology tends to look at functional processes that
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involve a number of coupled molecular processes, many biologists and biophysicists were attracted to this non equilibrium thermodynamics (4). Non equilibrium thermodynamics was also useful to Biology because it helped resolve the Schroedinger paradox (5). This paradox held the development of the order and structure of well developed biological organisms out of unordered food supplies, to be in conflict with the second law of thermodynamics. The usual formulation of this law in physics is that entropy can only increase, never decrease, where entropy s a measure of chaos. Non equilibrium thermodynamics then served to resolve this paradox, by reformulating that really the entropy production needed to be positive; by exporting lots of entropy, an organism could actually increase its order (3, 5). The entropy production function then became object of additional searches for general thermodynamic principles. Prigogine and coworkers showed that in the limit to equilibrium entropy production should be minimal at steady states (3). The minimum was with respect to variation of the independently variable thermodynamic forces. Entropy production was not minimal with respect to systems parameters (6), but again there was little interest in those systems parameters as they would carry mechanism specific information. These derivatives were thought not to lead to general results therefore. Understanding the coupling between processes in bioenergetics was an area where NET had some additional useful contributions. It enabled the definition of a coefficient that could quantify the degree of coupling between distinct biochemical processes (7). Defining this coefficient increased the awareness that coupling would not have to be complete, and that uncoupling or slippage should be a possibility. Up to that time and also subsequently, the unfounded notion that biological systems were necessarily ideal and therefore would not waste any free energy, made biologists only consider networks where coupling would be complete. Here the emergence of the chemiosmotic coupling mechanism was important. In this mechanism a membrane that was likely to have at least some passive permeability for protons was supposed to sustain the relevant free energy intermediate, i.e. the transmembrane electrochemical potential difference for protons (8). This mechanism was one of the early examples of systems biology, where only through the integration of at least two completely different types of processes (i.e. transport and chemistry) free energy could be transduced, between two chemical processes. Further consideration of the degree of coupling in terms of how its magnitude could contribute to the partly coupled process being optimal for certain functions, led to the conclusion that neither the degree of coupling nor the thermodynamic efficiency needed to be maximal for a number of relevant output functions to be optimal (9). Indeed it was calculated that many biological processes, including microbial growth (10) were highly inefficient, where some of the observed efficiencies could be understood in terms of the system being optimal with respect to both growth rate and power production in terms of biomass. Non equilibrium thermodynamics continued to be successful in non biological sciences where it helped explain cross-correlations between different types of phenomena, such as heat conductance and volume flow. Notwithstanding its apparent ability to function as an early systems biology approach being able to integrate multitudes of processes in its symmetrical linear equations, NET did not develop much further however. The reason was that much of what had been accomplished was valid only for processes that were less than a couple of kJoules per mole displaced from equilibrium. Biological reality is that the free energy of hydrolysis of ATP exceeds 40 kJ/mol, and the dissipation of free energy in many processes exceeds 10 kJ/mol (10).
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Therewith none of the proofs of the above principles derived by non equilibrium thermodynamics holds for many realistic biological systems and indeed there is some evidence that the relations themselves do not hold either (10). Rottenberg (11) and subsequently we (12, 10) then retraced some steps of NET and realized that one could translate well-accepted kinetic relationships into non equilibrium flow-force relationships. This led to the discovery that there was a basis for the linear flow-force relations often postulated for non equilibrium thermodynamics. That linearity was likely to be at a range away from equilibrium that was most relevant for the regulation of processes. However, in that range there needed be no Onsager reciprocity (10), continuing to take away the basis of the validity of the minimum entropy production principle (6). Importantly, here the paradigm was left that by definition non equilibrium thermodynamics should be devoid of mechanisms; the coefficients relating flows and forces were expressed into enzyme kinetic properties. And, using this new, ‘Mosaic Non Equilibrium Thermodynamics (MNET)’, the systemic implications for failing mechanisms of coupling could be predicted (10). A systems Biology approach, relating important systems function to molecular action and properties, had been born, avant la lettre. Paradoxically, another, in fact older, branch of non equilibrium thermodynamics thrived on the non-linearities in and amongst the processes in biology, and certainly on the substantial distance of many biological systems from equilibrium. The self organization addressed by this type of non equilibrium thermodynamics cannot occur in the Onsager domain where flow-force relations are symmetrical (3, 13). The resolution of the Schrödinger paradox described above merely stated that export of entropy could resolve that paradox, but it had not yet been clarified how that entropy export would be coupled to the entropy decrease held characteristic of developmental biology. Mechanisms were sought for pattern formation from initially symmetrical conditions, and found, e.g. by Turing (14, 3, 15). Symmetry breaking in time was also found to occur in chemical reaction schemes and held as model for the cell cycle in living organisms. Further developments included the discovery and analysis of sets of equations that could generate even more complex phenomena such as aperiodic selfexcitation and deterministic chaos (16). These analyses brought home the message that for some of these phenomena to occur quite special parameter values were needed. This reinforced the question whether indeed in biological reality those parameter values would reign, or if alternatively completely different mechanisms might be responsible for the observed complex phenomena to occur. In the mechanisms proposed by the fields of non equilibrium thermodynamics and nonlinear dynamics, there was frequently another limitation, i.e. lack of robustness. Symmetry breaking could occur but the precise version of the asymmetry (e.g. left-right versus right-left) depended on fluctuations and would therefore be random. Yet the observation that our right foot is usually on our right-hand side is quite convincing in showing that actual developmental biology is more robust than this. The argument then became that instead of a fluctuation, a well-controlled external condition would set the symmetry breaking in motion, now reproducibly. The requirement of such an external ordering factor was in line with the more general observation that the structures of living cells do not arise completely anew in every generation: the replication of DNA is semi-conservative, the plasma membrane of newborns cells are pinched off parts of the plasma membrane of their mother cells, and most of their proteins have been and are being made by ribosomes inherited from the
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mother cell. A view, in which biological structure was nothing but a perpetration of a complex dynamic structure that once got into existence, became an option. Meanwhile molecular biology found more and more factors that co-determine molecular biology, and important predictions of the simplest versions of the self-organization theory of segmental organization in Drosophila turned out to be wrong: proteins alternating their expression between segments were not directed by precisely the same promoter elements in the stripe in which they were expressed (17). Self-organization may still play a partial role in developmental biology, but it will be a partial role only. These developments have taught us that the attractiveness of a concept such as self organization should not lead to the non-critical implicit assumption that a process is self-organized. Even though self-organization may be the simplest mechanism for pattern formation in early development, that by itself has no value; there is no place for Occam’s razor in Biology. Critical experimental testing is required, probably through detailed modeling and checking whether the predictions made by the model for experimentally determined actual parameters values, are in actual agreement with the behavior of the system. Likewise, hypotheses that developmental processes are due to pre-specification will need to be so concrete as to be testable, or falsifiable in Popper’s sense (18).
2. Silicon cells The suggestion that hypotheses in Biology should be testable and indeed be tested would seem to be superfluous. Would any biologists accept that her/his science should not adhere to the criteria devised for the natural sciences? On the other hand Biology is a complex science and this has had the effect that at the truly biological level, few theories have actually been testable. Because of the complexity and nonlinearity of the networks in biology, the behavior of their components is a strong function of the molecules around them. Accordingly, failure of a set of molecules to act precisely as predicted by a theory, could always be attributed to the presence of an as yet unidentified additional factor, somewhat altering the mechanisms that would otherwise work as proposed. Accordingly many biologists working at the physiological level, are satisfied with theories that allow for exceptions even when if these are not made explicit. Other biologists took the opposite stance. They decided that if at the physiological level theories could not be falsified, they should refrain from working at that level and turn to model systems that were completely controlled, notably in vitro systems with purified molecules. There the hard scientific criteria could be met in principle. Genomics has altered the situation. Now, living systems such as some unicellular organisms, are completely characterizable in terms of the sequence of all their genes, and the concentrations of all mRNAs, proteins and (soon) metabolites. These concentrations can also be manipulated, enabling a large number of independent experimental tests. The physiologist can no longer propose that failure of the system to behave according to his hypothesis is due to an unidentified molecules; if there is such a failure, he should either reject the hypothesis or identify the perturbing molecule and extend his model to incorporate that molecule. The molecular biologist need no longer refrain from studying the actual functioning of his molecules, in the intact system or suitable models thereof. This new interface between molecular biology and physiology is called Systems Biology (19).
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Systems Biology focuses on the functional properties that arise in the interactions between the components of biological systems (20). The cell cycling and the self-organization discussed above are examples: none of their molecules cycles or forms spatial patterns in the absence of interaction with the other molecules. Systems biology also realizes that it should reach beyond mathematical biology in that it should not devise models that might explain biological function: it should devise models that do explain those phenomena, for the parameter values that are real. The silicon cell program (cf. www.siliconcell.net) is an epitome of this systems biology (21). It puts together the actual kinetic and interaction properties of the components of the biological system into a computer replica and then uses a computer program to calculate the system’s behavior of that replica. If that behavior corresponds with the experimentally observed functional behavior of the biological system, then the mechanisms present in the replica should be the explanations of the emergent functional behavior. Other than what their name may suggest, these silicon cells do not yet correspond to replica of entire cells. They correspond to replica of hopefully sufficiently autonomous parts of (pathways in) living cells to be testable. The strictness with which they adhere to the principle of the silicon cell that all component properties should have been determined experimentally, is also variable, but this is to improve in the future. The models in the silicon-cell program have gone through the quality control of international journals, some of which collaborate explicitly with the program.
3. At what level should one pitch the silicon cell? When pronouncing to make precise models of functioning systems of the living cell in terms of their components, it is not immediately obvious what the components should be. The silicon cell focuses on the whole cell as the ultimate system but begins with the limited focus of pathways in those cells as the systems. The components are the catalysts in the pathway, mostly the proteins, and the ‘micro-molecules’ (‘metabolites’) through which they communicate. This does not completely specify yet the level of modelling however. To pitch the right level, both siliconcell and systems biology learn from non equilibrium thermodynamics and kinetics. One could take the point of view that a precise replica model of what happens in a metabolic pathway in the living cell should consider each individual molecule explicitly in terms of its position, state, appearance and disappearance, and these as functions of time. However, the complexity accompanying such a point of view is unmanageable. Let us consider just 20 types of molecule such as ATP in the living cell. At concentrations of approximately 10 mM these would each number 6 million molecules per E. coli cell. Supposing that each of these molecules could be in either of two states and each at any of 500 locations, then the entire systems would have some 100020000000 possible states. Modeling how such a system proceeds its biased random walk through these states is not only impossibly time consuming, but it is also useless in terms of the detailed information it would give. We are simply not interested in the particular behavior of such a system; we would not even know whether it corresponds to a particular experimental system we are studying, because we could not know in what precise state that system is. Inevitably we are interested in trends of behavior; in behavior that is reproducible between incarnations of a system of interest, which may all be different in terms of their precise microscopic states but are expected to behave similarly macroscopically. Lack of direct interest is however insufficient
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reason not to engage in detailed modeling. The average trend of behavior of a system might depend crucially on the detailed processes and in this case one would need to model in complete detail to obtain the average behavior of interest (22). What then is the level of detail at which we need to enter the components into the silicon cell? Here statistical mechanics and kinetics have shown the way. Consider the simplest possible reaction i.e. the degradation of a molecule A. The probability that the degradation occurs at some location in the cell should be proportional to the probability to find molecule A at that location and the rate should again be proportional to that probability:
v = k '⋅P(1 | N = n) = k ⋅ n Here P(1|N=n) represents the probability to find a molecule A within a given small time interval at the location of interest, if the total number N of molecules A equals n. For the average rate of the process this implies:
v = k ⋅n = k ⋅n
which corresponds to the deterministic rate equation for this situation. With this the average behavior of the system is described in terms of the ensemble average concentration (if one also divides by the volume) of molecules of the same type. If the mixing in the system is much faster than the reactions, then that ensemble averaged concentration is the same for the entire cell (or compartment thereof) and this leads to an enormous simplification. Now the state of the system can be described by only 40 state variables, i.e. the ensemble averaged concentrations of the 20 molecules in their two internal states. The situation becomes more complicated in essence whenever the kinetics is nonlinear. We here take quadratic kinetics as the example:
⎛ ⎛ σ 2 − n ⎞⎞ ⎟⎟ v = k '⋅P(1 | N = n) ⋅ P(1 | N = n − 1) = k ⋅ n ⋅ (n − 1) = k ⋅ (n) 2 ⋅ ⎜1 + ⎜ ⎜ ⎜ n 2 ⎟⎟ ⎠⎠ ⎝ ⎝
()
where σ2 is the variance in the particle number. This equation shows that only under certain conditions the deterministic rate equation is followed. One is the case where the variance equals the mean, which occurs when the particle number follows a Poisson distribution. Poisson distributions occur in cases with unit stoichiometries (10) and should not be expected to be standard in biological systems. In most systems the variance may not be equal to the average number of particles, but is nevertheless of the same order of magnitude (10). Then:
⎛ ⎛ 1 ⎞⎞ ⎟⎟ v = k ⋅ ( n ) 2 ⋅ ⎜1 ± O ⎜ ⎜ ⎟⎟ ⎜ ⎝ n ⎠⎠ ⎝ This leads to the second condition, which should apply more frequently: deterministic kinetics applies whenever the number of particles exceeds 100. The above argumentation is classical. Yet we repeat it here for two reasons. First, one now often encounters research programs where modeling is done stochastically rather than by using the deterministic equations, but without rationalization of why the former approach is chosen. At least one should ensure that the particle number is low or the distribution is vastly different from Poisson. Second, a
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number of cases have been noted meanwhile where variances have indeed been much larger than the average particle number and some of these cases carry a truly biological signature. An exemplary case is that of the expression of a gene through mRNA to the protein level. Because a single mRNA readily leads to the synthesis of hundreds of protein molecules, a variance in mRNA close to the mean number of mRNAs may translate into variance and average also being similar at the protein level. Then kinetics at the protein level (or for similar reasons at the metabolic level) may differ highly from that predicted by the deterministic rate equation. Ehrenberg and colleagues have been working of highly relevant such cases of extremely large variance (23). A highly relevant case is of course dictated by the digital encoding of genetic information, allowing one or two copies of a gene per cell only. When mutations occur in the haplotype, the variance is of the order of magnitude of the mean. Then nondeterministic kinetics should be used, not only at the DNA level but also at the mRNA, protein and metabolic levels, when the mutating gene is an enzyme. In case a population of cells is genetically homogeneous, and the number of mRNA molecules encoding the enzymes is large or quasi-Poisson distributed, cellular processes will follow deterministic kinetics. It is these cases that the silicon cell approach has been limiting itself to until now (21). Hence the silicon cell approach describes the processes in the cell as processes that are carried out by enzymes the activity of which can be described by their ensemble-averaged activity, and their ensemble averaged kinetic properties which depend on the ensemble averaged concentrations of micromolecules (‘metabolites’). This limiting case is the same as the one proposed by non equilibrium thermodynamics (15) for the description in terms of average concentrations, or in fact chemical potentials:
⎛n⎞ ⎝V ⎠
μ = μ 0 ' + R ⋅ T ⋅ ln ⎜ ⎟ Here the rate equations becomes 0' v = k ⋅ e 2⋅(μ − μ )/ RT
where the number 2 refers to the case of quadratic kinetics, and should be replaced by 1 in the case of linear kinetics. Deterministic kinetics and non-equilibrium thermodynamics that is not restricted to the near equilibrium domain are really two expressions of the same thing. The advantage of the deterministic kinetics/non-equilibrium thermodynamic approach is the tremendous simplification. For the 20 types of molecules that can each occur in two states and at 500 locations in the cell, the number of state variables is now 20 000, which although large is no longer unmanageably large. In practice a further simplification is possible provided the situation is that of a reasonable homogeneous space, diffusion being much more rapid than reactions, or the enzymes being distributed homogeneously over space. Then only 40 state variables suffice. The ensembleaveraged concentrations or the corresponding chemical potentials, correspond to the functions of state of thermodynamics, adding to energy content and volume for isothermal, isobaric systems (10). Indeed, at this moment all silicon cells are spatially homogeneous within welldefined compartments and the following simple description is used (www.siliconcell.net ). For each process that occurs in a cellular compartment, one formulates what it actually does. This is the transformation of molecules of one chemical nature to
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molecules of a different chemical nature, as indicated by the reaction equation. In the case of transport, the molecules can be of the same chemical nature but in different explicit compartments. The reaction equation can also be encoded as a set (vector) of stoichiometries (positive for products and negative for substrates). This vector becomes a column of the stoichiometry matrix N, of which each column corresponds to a process in the cell. Often the stoichiometry matrix is formulated from the alternative point of view of the time dependence of the metabolite concentrations as a balance between of all the process rates. The end result is the same, but the former method is more in keeping with the silicon cell philosophy that the process should be independent of any knowledge about the system. Processes are not only characterized by what they do, but also by the rate at which they do it, and by the dependence of that rate on the state of the system, i.e. the concentrations of the molecules. For each process therefore an enzyme kinetic rate equation is formulated, which is typically of the form:
v = g (e) ⋅ f ( S , X , Y , K eq , kcat , K S , K P , K X ,...) Where g(e) is often a mere proportionality, indicating that the rate is proportional to the concentration of the catalyst. Often g(e) and kcat are combined into the single parameter Vmax. Usually, X and Y are variable metabolite concentrations, corresponding to functions of state of the system, as discussed above. When the list of all processes in the system has been compiled with heir stoichiometric and rate equations, the lists of the arguments of the functions in the rate equations contain two types of properties. The one type is that of the variables. These also occur in the lists of molecules produced or consumed by the processes. For these variable properties balance equations are then written using the expression:
dX = N ⋅v dt Where N is again the stoichiometry matrix, v is a vector of the process rates, and X is a vector of all the concentration variables. The other type of properties in the lists of the arguments of the rate equations is called parameters. The parameters are not altered by actions of the processes in the system studied, but set by external conditions or by properties that cannot be changed by the systems (e.g. the Michaelis constants of the enzymes, and sometimes the pathway substrate and product, S). This is almost (see below) all the biologist/biochemist formulating a siliconcell does: characterize the components of the system. The computer program does the rest, which importantly includes the computation of the system behavior. It integrates the set of differential equations:
dX = N ⋅ diag ( g (e)) ⋅ f ( S , X , Y , K eq , kcat , K S , K P , K X ,...) dt Where the biologists still has to specify the initial conditions. The specification of these is actually something that requires some knowledge about the system, but not knowledge on how and why it behaves. Alternatively, one is interested in the steady state and asks the computer to solve equations for time independence of the metabolite concentrations. These two options are available for just a click on the siliconcell model base of live models: http://www.jjj.bio.vu.nl A third option of this silicon cell live ‘modelbase’ calculates control coefficients, i.e. the dependence of steady state properties on all the process activities. Through the world-wide web anyone can now engage in in silico experimentation with refereed silicon-cell models of pathways of living organisms.
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With this the silicon cell comes to predictions and descriptions of systems behavior at the functional state, i.e. away from equilibrium, in terms of its thermodynamic properties, i.e. chemical potentials or ensemble-averaged concentrations, and ensemble averaged fluxes. This was one of the aims of non equilibrium thermodynamics.
4. New non equilibrium thermodynamics? Yet, what the siliconcell delivers may not be quite recognized as non-equilibrium thermodynamics, as it is always rather specific information which depends on the precise magnitudes of the kinetic parameters. No principles, no laws of general validity are produced by the siliconcell. Indeed, precisely because it aims at generating a computer replica of the real pathways, no reduction of complexity, no generalizations are produced without further activities. In addition the output is formulated in terms of fluxes and concentrations rather than in terms of chemical potentials. For a while it seemed that perhaps biological systems lack general principles other than the ones valid close to equilibrium and discussed above. Few if any of the properties and principles valid near equilibrium could be extrapolated successfully to systems displaced from equilibrium to the extent that regular biological systems are. In the late sixties of the previous century in Edinburgh (24) and Berlin (25) a new way of looking at biological systems came about, partly inspired by genetics and partly by silicon cell type of modeling of metabolism. The new approach was called metabolic control analysis (MCA). Until 1987 (10), little reference to a link between MCA and non equilibrium thermodynamics was made, even though the latter discipline was still in development. Yet, even though this was not agreed on by its original progenitors, metabolic control analysis led to new laws and principles for biological systems, and especially for networks of biochemical reactions. We shall here discuss the summation laws of metabolic control analysis from this perspective. To do this we shall first retrace our steps and recall the derivation of the Gibbs equation and the Gibbs-Duhem equations of equilibrium thermodynamics. We first recall the balance equation for ordinary energy U, which reads as follows (15, 10): n
dU = d eU = d eQ + d eW + ∑ μ j ⋅ d e n j j =1
Subscript e refers to exchange of the system with its environment. The first law of thermodynamics has been used here so as to require that no energy U can be produced or consumed. Accordingly energy in the system can only increase by the addition of heat, work or chemicals from the outside, where the latter carry a partial molar energy equal to their chemical potential. The addition of heat is equal to the reversible addition of entropy (exclusive of the entropy carried by the molecules) and volume (exclusive of the volume increase due to the addition of the molecules): n
dU = T ⋅ d e S − P ⋅ d eV + ∑ μ j ⋅ d e n j j =1
Assuming that the system is at equilibrium no entropy is produced. Because then also chemical reactions inside the system are absent, or their total contribution equals zero, internal volume changes are absent, this equation becomes the Gibbs-Duhem equation:
H.V. Westerhoff
90 n
dU = T ⋅ dS − P ⋅ dV + ∑ μ j ⋅ dn j j =1
This equation shows that energy U can be calculated from its initial value and the changes in entropy, volume and molecule numbers, provided that temperature, pressure and chemical potentials are known. We shall here consider isothermal, isobaric systems, freely exchanging matter with their environment which has constant chemical potentials for all molecules. T, P and chemical potential are intensive properties and energy, entropy, volume and molecule numbers are extensive properties, meaning that the latter do not change, and the latter do change proportionally with the size of the system. In other words, when changing the size of the system by the factor λ:
U (λ0 ⋅ T , λ1 ⋅ S , λ0 ⋅ P, λ1 ⋅V , λ0 ⋅ μ , λ1 ⋅ n) = λ ⋅ U (T , S , P, V , μ , n) Or in other words energy U is a homogenous function of order 1 of entropy, volume and molecule number (and of order zero of Temperature, pressure and chemical potential). Euler’s theorem then rules that:
1=
∂ ln U ∂ ln U n ∂ ln U + +∑ ∂ ln S ∂ ln V j =1 ∂ ln n j
The partial derivatives are given by the Gibbs-Duhem equation and inserting these, one obtains the Gibbs equation: n
U = S ⋅ T − P ⋅V + ∑ n j ⋅ μ j j =1
This inspired Gibbs to define the Gibbs free energy, as:
G ≡ U + P ⋅V − T ⋅ S
Which then leads to: n
G = ∑ nj ⋅ μ j j =1
Establishing the chemical potential also as the partial molar free energy and a the energy function of interest for isothermal, isobaric systems exchanging matter with their environment. The functions of state entropy, volume and molecule number describe a system that is at equilibrium, and only partly systems that are away from equilibrium. For the latter systems the aspect of time or fluxes is missing. When searching for thermodynamic descriptions of systems away from equilibrium, it may be useful to consider the phenomena that keep the system away from equilibrium. These are the Gibbs energy dissipating processes, and more precisely the activities of these. In biochemical networks, virtually all these processes have material counterparts, i.e. the enzymes that catalyze them. These are in turn encoded by genes, constituting a further relationship with nucleic acids. The properties of biochemical networks at steady state can be considered functions of all process activities (here denoted by ei) and many other properties. For such a property Z we write:
Z = z (e1 ,e 2 ,...,e n , S , P, T , K M , K eq ,k cat ,....) where one recognizes all the parameters of the enzyme kinetic rate equations. Z refers to a function that delivers the steady state value of Z. S refers to the concentrations of
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external pathway substrates and products, which are parameters to the system of interest. Above we transformed the system in terms of physical size, essentially by copying it lambda times, we here consider a transformation in terms of all activities: we consider the situation that all processes are simultaneously accelerated by the factor λ. Because we are at steady state, all processes should be balancing each other, such that there are no changes in time anymore of any property. As all processes are activated by the same factor this balance will be maintained, and nothing will change, except that all processes will proceed λ times faster. If Z refers to a flux, e.g. J, this implies that Z is a homogenous function of order 1 of all the processes rates. Using Euler’s theorem one then obtains (10, 26):
1=
∂ ln J ∂ ln J ∂ ln J + + + .... = C1J + C2J + C3J + ... ∂ ln e1 ∂ ln e2 ∂ ln e3
Where the coefficients denoted by capital C correspond to the flux control coefficients of MCA (10). One may here recognize the well-known summation law for flux control coefficients (24, 25). Similarly, realizing that the steady state concentrations are not changed, one sees that these are zero order homogeneous functions leading to the concentration control summation law:
0=
∂ ln X ∂ ln X ∂ ln X + + + .... = C1X + C2X + C3X + ... ∂ ln e1 ∂ ln e2 ∂ ln e3
We here have two fundamental laws of non equilibrium biochemical networks that have been derived in much the same way as the Gibbs equation was derived in equilibrium thermodynamics. We therefore propose that the summation laws are aspects of the non equilibrium thermodynamic theory that was long sought after. The concentration-control coefficients are the derivatives of the ensembleaveraged concentrations of the substances in the system with respect to the process activities. Because of the definition of the chemical potential, the concentration summation law can also be written as: n
∂μ X j =1 ∂ ln e j n
0 = ∑ C [j X ] = ∑ j =1
which now also shows as a thermodynamic law for non equilibrium steady state. The logarithm of the enzyme activity could also be written as the chemical potential of the enzyme: n
∂μ X j =1 ∂μ e j n
0 = ∑ C [j X ] = ∑ j =1
5. Discussion We here discussed fundamental aspects surrounding the silicon cell approach. These are related to statistical thermodynamic properties of biological systems. The present silicon cell approach is suited for biochemical, signal-transduction, and gene-expression networks that fulfill a number of conditions. These entailing that the fluctuations in them are limited or follow the Poisson distribution. This assumption corresponds to the one required for the use of deterministic rate equations and is therefore quite acceptable for most biochemical networks.
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Interestingly, the realization that these assumptions need to be made, suggests that the silicon-cell approach may be one of the types of approaches that nonequilibrium thermodynamics was looking for: it describes systems away from equilibrium. It does this by inserting the details of every molecular phenomenon at the ensemble level of enzyme-catalyzed reactions. The results of silicon-cell calculations are thereby also highly specific, i.e. they give the concentrations of all molecules as a function of time and all precise parameter values. Although the silicon cell therewith describes non-equilibrium processes, it may not qualify as thermodynamics, because it lacks any aspect of generality. Parts of metabolic control analysis (MCA) on the other hand do describe non equilibrium systems in terms of generic properties. We have here shown that important principles of MCA are analogous to principles derived in equilibrium thermodynamics, and so are the derivations of these principles. The same may be true for the Hierarchical Control Analysis (27), which generalizes MCA to systems with signal transduction (28) and systems with variable gene expression. MCA also has other, famous laws/principles, i.e. the connectivity theorems. Also these have strong thermodynamic connotations including an origin in stability vis-à-vis fluctuations (10). We here therefore postulate that HCA and MCA correspond to the non equilibrium thermodynamics that is most suited for most biological systems. We expect that taking this perspective and that of the silicon cell, and combining these more with thermodynamic considerations, even more new systems biology principles will emerge.
6. Acknowledgements We thank the EU-FP6 program, NWO, Ecogenomics-EZ, IOP-genomics, for supporting our Systems Biology work, enabling us to obtain deeper insights also into the scientific position of systems biology.
References 1. Onsager L. (1931) Phys. Rev. 37, 405 - 426 2. Onsager L. (1931) Phys. Rev. 38, 2265 - 2279 3. Glansdorf P. and Prigogine I. (1971) Thermodynamic Theory of Structure, Stability and Fluctuations, John Wiley & Sons, New York, 4. Katchalsky A. & Curran P.F. (1967) Non-equilibrium thermodynamics in biophysics, Harvard University Press, Cambridge MA 5. Schroedinger, E. 1944. What is Life? The Physical Aspect of the Living Cell. Cambridge: Cambridge University Press. 6. Juretic D., and Westerhoff H.V. (1987) Biophys. Chem. 28, 21-34. 7. Kedem O. & Caplan S.R. (1965) Trans. Faraday Soc. 21, 1897 – 1911. 8. Mitchell P (1961) Nature 191, 144 – 148. 9. Stucki J.W. (1980) Eur. J. Biochem. 109, 257 – 283. 10. Westerhoff H.V. & Van Dam, K. (1987) Thermodynamics and Control of Biological Free-Energy Transduction, Elsevier, Amsterdam 11. Rottenberg H. (1973) Biophys. J. 13, 503 –511. 12. Van der Meer R., Westerhoff, H.V. & Van Dam, K. (1980) Biochim. Biophys. Acta 591, 488 – 493. 13. Cortassa S., Aon M.A., and Westerhoff H.V. (1991) Biophys. J. 60, 794 - 803 14. Turing A.M. (1952) Philos. Trans. R. Soc. Lond. B. 237, 37-72.
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15, Nicolis G & Prigogine I (1977) Self-Organization in nonequilibrium systems, Wiley & Sons, New York. 16. Bakker R, Schouten JC, Giles CL, Takens F, & van den Bleek CM. (2000) Neural Comput. 12, 2355-2383. 17. Lawrence P.A. (1972) The making of a fly: (the genetics of animal design), Blackwell Science. 18. Popper K. (1963) Conjectures and Refutations, Routlegde and Kegan Paul, London 19. Westerhoff H.V. and Palsson B.O. (2004) Nature Biotechnol. 22, 1249-1252. 20. Alberghina L. & Westerhoff H.V. (2005) Systems Biology, Definitions & Perspectives, Springer 21. Snoep JL, Bruggeman F, Olivier BG & Westerhoff HV. (2006) Biosystems. 83, 207-216. 22. Westerhoff H.V., and Chen Y. (1985) Proc. Natl. Acad. Sci. USA 82, 3222 - 3226. 23. Elf J & Ehrenberg M. (2003) Genome Res. 13, 2475-2484. 24. Kacser H. & Burns, J. (1973) In Rate control of biological processes (Davies, D.D., ed.) pp. 65 – 104, Cambridge University Press. 25. Heinrich R & Rapoport T.A. (1974) Eur. J. Biochem. 42, 89 – 105. 26. Giersch C. (1988) Eur J Biochem. 174, 509-513. 27. Westerhoff H.V., Koster J.G., Van Workum M., & Rudd K.E. (1990) In Control of Metabolic Processes (Cornish-Bowden A., ed.), pp. 399 - 412, Plenum, New York. 28. Kahn D., & Westerhoff H.V. (1991) J. Theor. Biol. 153, 255 - 285. .
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Challenges for process system engineering in infrastructure operation and control Z. Lukszoa, M.P.C. Weijnena, R.R. Negenbornb, B. De Schutterb, Marija Ilićc a
Faculty of Technology, Policy and Management, Delft University of Technology, NL Delft Center for Systems and Control,Delft University of Technology, NL c Electrical and Computer Engineering and Engineering Public Policy, Carnegie Mellon University, USA b
Abstract The need for improving the operation and control of infrastructure systems has created a demand on optimization methods applicable in the area of complex sociotechnical systems operated by a multitude of actors in a setting of decentralized decision making. This paper briefly presents main classes of optimization models applied in PSE system operation, explores their applicability in infrastructure system operation and stresses the importance of multi-level optimization and multi-agent model predictive control.
Keywords: infrastructures, optimization, multi-agent systems, model predictive control.
1. Introduction Our society and economy have come to rely on services that depend on networked infrastructure systems, like highway and railway systems, electricity, water and gas supply systems, telecommunication networks, etc. Recent events such as large-scale power blackouts have contributed to a renewed awareness of the critical role of infrastructures in our economies. Malfunctioning and service outages entail substantial social costs and hamper economic productivity. Instead of installing additional capacity, more intelligent control of the existing capacity seems a more affordable and promising strategy to ensure efficient and reliable operation of critical infrastructures which, moreover, stimulates the creation of innovative value-added services such as dynamic congestion pricing. However, the multitude and variety of nodes and links in these networks as well as the multitude and variety of owners, operators, suppliers and users involved have created enormously complex systems. This complexity hampers the optimization of the overall system performance, due to our limited understanding of infrastructure systems as well as to practical limitations in steering the actors’ operational decision making. The process systems engineering (PSE) area defined by Grossmann and Westerberg (2000) is concerned with the improvement of decision making for the creation and operation of the chemical supply chain. As chemical process systems are networked systems and the PSE field has enabled tremendous advances in their optimization, it is intersting to explore to what extent the methods from PSE may be applied to infrastructure system operations. The urgent need for improving the performance of infrastructures creates a great demand for innovative optimization and control methods. This is the focus of this paper.
2. Infrastructure definition The physical network of an infrastructure system and the social network of actors involved in its operation collectively form an interconnected complex network where
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the actors determine the development and operation of the physical network, and the physical network structure and behavior affect the behavior of the actors. An infrastructure can thus be seen as a complex socio-technical system, the complexity of which is defined by its multi-agent/multi-actor character, the multi-level structure of the system, the multi-objective optimization challenge, and the adaptivity of agents and actors to changes in their environment. Their non-linear response functions in combination with the complex system structure often lead to unpredictable dynamic behavior of the system. Similar to the hierarchical decomposition of, e.g., the operation of an industrial plant in planning, scheduling, and processing functions, infrastructure systems can be viewed as multi-level systems, whether hierarchically interconnected or decentralized, with a number of operational regimes at the various system levels. Usually, at each level of the decomposed system local performance objectives are defined which should, preferably, not be restricted to the optimization of local goals, but rather aim at optimally contributing to the overall goal. However, the relation between local and overall system performance objectives may be rather fuzzy, especially since the overall objective is often not defined in detail and concerned with a longer time horizon. The local objectives are generally more detailed, concerned with a shorter time horizon and often with the specific interests of an individual actor. To facilitate an overall optimization of the performance of the system as a whole, a kind of coordinator may be required to supervise local decision making in its relation to the overall goal. In the practical situation of many infrastructure industries in liberalised markets, however, such central co-ordination or supervision no longer exists. Especially in these situations it is a challenging task to develop a method for decentralized optimisation that can be implemented, e.g., by a regulatory authority, to influence local decision making by individual actors in respect of societal interests. As a conceptual model of infrastructures as socio-technical systems we will use the concept of multi-agent systems composed of multiple interacting elements (Weiss, 1999). The term agent can represent actors in the social network (e.g. travelers taking autonomous decisions on which route to follow to avoid road congestion or companies involved in the generation, transmission and distribution of electricity)as well as a component (e.g. a production plant, an end-use device, a transformer station) in the physical network. In all these cases we see that the overall system – considered as a multi-agent system – has its own overall objective, while the agents have their own individual objectives.
3. Decentralized Decision Systems In a decentralized decision system the objectives and constraints of any decision maker may be determined in part by variables controlled by other agents. In some situations, a single agent may control all variables that permit him to influence the behavior of other decision makers as in traditional hierarchical control. The extent of the interaction may depend on the particular environment and time dimension: in some cases agents might be tightly linked, while in others they have little effect on each other, if any at all. For decision making in such systems two important aspects can be distinguished: a set of individual goals and ways of how to reach them, and a set of linkages allowing agents to interact. The individual decision-making step usually takes the form of single-criterion optimization as often applied in PSE. Optimization is one of the most frequently used tools in PSE decision-making to determine, e.g., operational and maintenance schedules, the sizing of equipment, pricing mechanisms, allocation of capacity or
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resources among several units, etc. For a detailed review of optimization methods, see e.g. Edgar (2001). 3.1. (Multi-criteria) Optimization problem Each optimization problem contains two elements: at least one objective function, or criterion, to be optimized, and constraints. The type of the ultimate optimization function(s) together with the specified constraints determines the type of optimization problem. The individual goals of each agent often represent a variety of criteria that, more often than not, turn out to be conflicting: an improvement in any one of them may be accompanied by a worsening in others. For the sake of simplicity it is assumed here that there is only one decision maker (i.e., one agent), which is actually searching for a satisfactory compromise rather than for a hypothetical numerical optimum. In principle, a multi-objective optimization problem can be formulated as follows: min J ( x ) = min ( J 1 ( x ), J 2( x ),..., J k ( x )) T x∈ X
x∈ X
where: Ji: ℜn → ℜ is an individual objective, i=1,2,…,k, X={x∈ ℜn: gj(x) ≥ 0, j=1,…,m} is the feasible area determined by constraints. Four classes of solution methods for multi-objective optimisation problems can be distinguished, see Verwater-Lukszo (1996): • Methods based on some measure of optimality, • Interactive methods, • Methods searching for Pareto-optimal solutions, • Lexicographic methods. Methods based on a measure of optimality make an attempt to measure alternatives in one way or another, by weighting each objective and then optimizing their weighted sum, or by replacing multi-objective optimization by optimizing only one criterion with the greatest preference. Therefore, methods of this category translate a multi-criteria problem into a single criterion. The second group of methods uses the information obtained from the decision maker in an iterative process to assign appropriate priority levels, e.g., weights, to all individual objectives. Pareto methods of the third group use the notion of Pareto optimality to achieve a balance between objectives. Here the optimal solution appears to be the natural extension of optimizing a single criterion, in the sense that in multi-objective optimization any further improvement in any one objective requires a worsening of at least one other objective. Finally, the lexicographic methods assume that the individual objectives may be ranked by their importance, so that a sequential optimization of the ordered set of single criteria is possible. In this way a multi-objective problem is translated into a multi-level optimization problem. This brings us to another important optimization approach applicable for decision problems in the world of infrastructure system operation: multi-level optimization. 3.2. Multi-level optimization In a multi-level optimization problem several decision makers control their own degrees of freedom, each acting in a sequence to optimize own objective function. This problem can be represented as a kind of leader-follower game in which two players try to optimize their own utility function F(x,y) and f(x,y) taking into account a set of interdependent constraints. Solving multi-level problems may pose formidable mathematical and computational challenges. In recent years, however, remarkable progress was made in developing efficient algorithms for this class of decision problems (see Bard, 1998). Interesting applications from the world of energy infrastructure
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operation concern the supplier-household interaction resulting from an introduction of micro CHP, see Houwing (2006). Another example concerned with dynamic road pricing aimed at better use of road capacity is described by Lukszo (2006); the upper level describes the overall road performance and the lower level the user-specific objective function. The simplest problem representation of a hierarchical optimization problem is the bilevel programming problem concerning the linear version of hierarchical optimisation, alternatively known as the linear Stackelberg game. x = [x 1 ,..., x n ] T
min F( x , y ) = c 1 x + d 1 y x∈ X
y = [y 1 ,..., y m ] T subject to : A1 x + B 1 y ≤ b1 min f ( x, y ) = c 2 x + d 2 y y∈Y
subject to : A2 x + B 2 y ≤ b 2
It should be stressed, that even in the linear case the bi-level programming problem is a non-convex optimization problem which is NP-hard. Generally, infrastructure systems pose multi-level programming problems with an arbitrary number of levels, in which the criteria of the leader and the follower can be nonlinear and/or discrete, which are even more challenging to solve. 3.3. Optimal Control Optimal control is another important, though hard to apply, technique to be used in infrastructure system operation. When modeling a system by a set of differential equations, an interesting type of dynamic optimization problem can be formulated, also referred to e.g. by Leonard (1992) as an optimal control problem. An optimal control problem is formulated and solved by an agent to find those inputs to the system that minimize the objective function over the running time of the system. A general optimal control problem is formulated as: tF
min J = u(t)
∫ f (x(t) , u(t), t)dt + Φ( τ
0 ,τ F
)
t0
subject to : d x (t ) / dt = g ( x (t ), u (t ), t )
ϕ i ( u(t)) ≥ 0 κ j ( x(t)) ≥ 0
i = 1,2,..., p j = 1,2,..., q
ν k ( τ 0 , τ F ) ≥ 0 k = 1,2,..., r
where: x(t) = [x1 (t), x 2 (t),..., x n (t)] T u(t) = [u 1 (t),u 2 (t),...,u m (t)]
T
is the state vector is the control vector
τ 0 = [t 0 , x1 (t 0 ), x 2 (t 0 ),..., x n (t 0 )] T τ F = [t F , x1 (t F ), x 2 (t F ),..., x n (t F )] T Φ( τ 0 ,τ F ) are the initial cost / final value function. The following features can make an optimal control problem extra hard to solve:
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• Besides a final value function the criterion may contain an initial cost function. • Final time can be a free variable, which in many cases may have to be chosen optimally; • not only final states, but also initial states can be free variables, which must be chosen optimally. • The optimization problem usually involves constraints on state variables, which are notoriously difficult to handle. • Constraints may be imposed (lower/upper bounds, linear and non-linear constrains) on initial and final states variables. • Integral constraints may be imposed on control variables; these constraints may also involve initial and final states, and possible final time. Optimal control methods can be solved by variational methods or, alternatively, by discretization converting the original problem into a large-scale static LP or NLP optimization problem. Variational methods use the optimality conditions given by the Maximum Principle of Pontryagin resulting in a so-called two-point boundary value problem, which is often hard to solve. If discretization methods are applied to an optimal control problem, then standard static NLP solvers may be used, e.g., the conjugate gradient method, or the sequential quadratic programming algorithm SQP, see Edgar (2001). In the following section we consider a particular control scheme that approximates the dynamic control problem with static control problems. 3.4. Model Predictive Control A particular approach to solve optimal control problems as introduced in Section 3.3 is Model Predictive Control (MPC), see e.g. Maciejowski (2002), Morari (1999). This method from the PSE area has become an important technology for finding optimization policies for complex, dynamic systems. MPC has found wide application in the process industry, and recently has also started to be used in the domain of infrastructure operation, e.g., for the control of road traffic networks, power networks, and railway networks.MPC approximates the dynamic optimal control problem with a series of static control problems, removing the dependency on time. Advantages of MPC lie in the fact that the framework handles operational input and state constraints explicitly in a systematic way. Also, an agent employing MPC can operate without intervention for long periods, due to the prediction horizon that makes the agent look ahead and anticipate undesirable future situations. Furthermore, the moving horizon approach in MPC can in fact be considered to be a feedback control strategy, which makes it more robust against disturbances and model errors. 3.4.1. Multi-Agent Model Predictive Control The main challenge when applying MPC to infrastructure operation stems from the large-scale of the control problem. Typically infrastructures are hard to control by a single agent. This is due to technical issues like communication delays and computational requirements, but also to practical issues like unavailability of information from one subsystem to another and restricted control access. The associated dynamic control problem is therefore typically broken up into a number of smaller problems. However, since the sub-problems are interdependent, communication and collaboration between the agents is a necessity. A typical multi-agent MPC scheme therefore involves for each agent the following steps, see Camponogara (2002): 1. Obtain information from other agents and measure the current sub-system state. 2. Formulate and solve a static optimization problem of finding the actions over a prediction horizon N from the current decision step k until time step k+N. Since the sub-network is influenced by other sub-networks, predictions about the behavior of
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the sub-network over a horizon are more uncertain. Communication and cooperation between agents is required to deal with this. 3. Implement the actions found in the optimization procedure until the beginning of the next decision step. Typically this means that only one action is implemented. 4. Move on to the next decision step k+1, and repeat the procedure. In particular determining how agents have to communicate with one another to ensure that the overall system performs as desired is a huge challenge that still requires a substantial amount of research. Negenborn describes many possible approaches (2006).
4. Conclusions In this paper we have considered challenges for process system engineering in infrastructure system operation and control. The relevance of optimization models as decision-supporting tools is very high for many players in the world of infrastructure. In all systems that exhibit interactions and interdependencies between subsystems, where multiple functionality plays a role, where capacity allocation in a complex and dynamic environment is an issue, feasible concepts of decentralized optimization are called for. As a particular challenge we pointed out the application of multi-level optimization and model predictive control in a multi-agent setting of decentralized decision making on infrastructure system operation. Besides computational complexity, a formidable challenge here is posed by the design of communication and cooperation schemes that enable agents to come to decisions that are both acceptable locally and ensure an overall system performance in respect of social and economic public interests. The design of markets and an appropriate legislative and regulatory framework to steer individual actors’ decision making towards public goals and to enforce adequate communication and collaboration schemes may be beyond the world of PSE, but will certainly be inspired by applicable PSE optimization strategies. Acknowledgements This research was supported by the BSIK project “Next Generation Infrastructures (NGI)”, and the VIDI project “Multi-Agent Control of Large-Scale Hybrid Systems” (DWV.6188) of the Dutch Technology Foundation STW.
References Camponogara, E., D. Jia, B.H. Krogh and S. Talukdar, Distributed model predictive control, IEEE Control Systems Magazine, 1:44.52, February 2002. Edgar T.F, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes, McGraw Hill, Boston, 2001 Grossmann I..E. and A.W. Westerberg, Research Challenges in Process Systems Engineering, AICHE J., 46 (9), 2000. Houwing, M.. P. Heijnen and I. Bouwmans, Deciding on mico CHP, IEEE Int. Conf. Networks, Sensing, and Control, Ft. Lauderdale, Florida, April 2006. Kirk, D.E., Optimal control theory: An introduction. Prentice Hall, Englewood Cliffs, N.J., 1970. Leonard D. and N. van Long, Optimal Control Theory and Static Optimization in Economics, Cambridge Univ. Press, 1992. Lukszo, Z. and D. Joksimovic, Optimization of the operation of infrastructures, IEEE Int. Conf. Networks, Sensing, and Control, Ft. Lauderdale, Florida, April 2006. Maciejowski, J.M., Predictive Control with Constraints. Prentice Hall, Harlow, England, 2002. Morari, M. and J.H. Lee, Model predictive control: past, present and future, Computers and Chemical Engineering, 23, 1999. Negenborn, R.R., B. De Schutter and J. Hellendoorn, Multi-Agent Control of Transportation Networks, IEEE Int. Conf. Networks, Sensing, and Control, Ft. Lauderdale, Florida, April 2006. Verwater-Lukszo, Z. A practical approach to recipe improvement and optimization in the batch processing industry, PhD Thesis, Eindhoven University of Technology, 1996.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Supply Chain Design, Management and Optimization Dean Kassmann and Russell Allgor Amazon.com; P.O. Box 81226; Seattle, WA 98108-1226
Abstract Modeling and optimization are the traditional workhorses of supply chain management. The techniques have been used by many companies for planning, manufacturing, and logistics decision making. These techniques generally rely heavily on approaches grounded in operations research that excel at capturing stochastics and the discrete nature of these problems. Approaches fundamental to the process industries such as identification, dynamic simulation, model-based control, and more generally, operationalbased decision making are often not understood or fully embraced by supply chain practitioners. This paper discusses the challenges and opportunities in using modeling and optimization techniques in supply chain management at Amazon. We will also discuss the application of control and feedback to supply chain systems, and discuss theoretical and practical challenges as well as opportunities in applying these ideas to realworld supply chain decision systems. Keywords: optimization, modeling, control, supply chain management.
1. Introduction Supply chain management refers to the decision technologies and business processes used to manage the logistics and operations of complex supply-demand networks. Rich sets of research and development opportunities associated with solving this class of problems exist. In this paper we touch on a subset of these problems, providing concrete examples from the online retail industry, specifically Amazon.com. Amazon sells millions of unique products to millions of customers world wide for billions of dollars in annual sales. The examples, although simple, represent some of the challenges and opportunities common to large scale supply-demand networks. Demand management issues, although important, will not be addressed here. This paper is outlined as follows. Section 2 provides a high level overview of the Amazon supply-demand network with needed background and context required for the discussion. Section 3 covers several of the challenges in more detail including capacity planning, inventory planning and control, customer order assignment, and demand forecasting; we conclude in Section 4.
2. The Amazon Supply-Demand Network The Amazon supply chain differs from traditional supply chains in several respects. First, it is shallower than many supply chains. Inventory is procured from suppliers,
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received in the fulfillment centers (FCs), and packed and shipped directly to customers. No large scale and complex manufacturing component exists in the supply chain. Likewise, no brick and mortar retail outlets are present, and each FC can serve each customer in a given marketplace. Second, the number of products available for sale is huge compared with most supply chains. Amazon offers wide selection of products, spanning dozens of product lines. At any point in time approximately ten million unique items may be up for sale on the Amazon website. Some of those items are sold by Amazon; some are sold by third party merchants via the Marketplace and Merchants@ programs[1]. Amazon holds a large number of these items in its fulfillment network at any given time. This contrasts many supply-demand networks that deal with hundreds of products or up to tens of thousands of products. The basic order and inventory flows are relatively simple. Customer orders are placed on the website and enter the fulfillment network through an order assignment system that assigns the units in each order to FCs across the network in order to minimize the fulfillment costs. When the time arrives to fulfill the order, a request is sent to the appropriate fulfillment center. The items are picked by one or more associates who walk the fulfillment center floor, retrieving the physical items for incoming orders. The picked items are sorted according to customer order and placed in a box for packing. The box is then labeled and directed to an outbound dock for shipping to the customer. The inventory flow is similar to the order flow. Using demand forecasts, inventory planning systems determine how much inventory to hold in each location for each product along with the frequency and quantity in which to purchase it. Once inventory arrives and is put away, it can be picked for a customer order as explained above. The supply chain is a dynamic system consisting of many manipulated, feed forward, and control variables. The primary states in the supply chain are orders, inventory, and labor (e.g., staffing level) each of which evolve on different time scales. The orders typically remain in the system from hours to days, inventory from days to weeks, and labor from months to years. Applying the appropriate technology to design, manage, and optimize this dynamic system results in many research opportunities. The remainder of the paper discusses specifics about the above problems and relates the use of modeling, optimization, and control to each area.
3. Optimization and Modeling Opportunities The management of the supply chain could be represented by an infinite horizon stochastic optimal control problem containing discrete and continuous decisions. We do not attempt to solve that problem directly. In practice, we break down the problem into smaller subproblems that are handled by individual systems. We highlight the following areas, which are critical to the operation of our supply chain in this section: capacity planning, inventory management, customer order assignment, and demand forecasting. We present an overview of the areas and some of the optimization and modeling opportunities. We then compare the techniques which have commonly been applied to these areas with those that are more prevalent in the process industries. 3.1. Capacity Planning Capacity planning and network management determine appropriate capital investments and the best way in which to fulfill anticipated demand through the fulfillment network.
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The objective is to determine what physical infrastructure in which to invest and how much of each product to fulfill from each facility. The goal is to minimize expected fulfillment costs subject to throughput, labor, and storage capacity constraints. The challenge is to balance capital investment with the variable costs of the fulfillment network subject to a highly seasonal pattern of customer demand. The primary decisions are the capital investments (e.g., new facilities, new storage modules, conveyor, etc.), where to fulfill the customer orders for various products at every instant in time, and how to manage staffing levels in each facility into the future. These plans must include commissioning and decommissioning of facilities, recommendations for construction projects, and deal with the highly seasonal nature of the retail business. This problem is analogous to the process design problems faced in the process industries, yet the uncertainty and seasonality of the customer demand distinguishes it from problems more commonly addressed in that area. Thus, our design problem differs from the classical design of a continuous process. The problems can be classified as a mixed nonlinear stochastic combinatorial financial optimization problems. For most real world instances, a full stochastic formulation is not tractable. Instead, most practitioners, including us, cast the problem as multi-period LPs or MILPs; stochastic elements are ignored, and various scenarios are analyzed. For these multi-period (i.e., finite horizon) problems, the choice of horizon length and the manner in which final time constraints are imposed is important. The capacity planning problem that we face does not reach steady-state conditions because of the growth and seasonality of our business. We address this by considering a time horizon that looks far enough into the future to ensure that constraints are no longer active and the ‘cost to go’ at the end of the time horizon can be sufficiently discounted. For example, the labor plans must ensure that the horizon includes the staffing requirements through the labor ramp and decay of the peak season. This ensures that enough trained staff are available to handle peak production volumes and deal with the situation in which demand increases at a faster rate than staff can be hired and trained. In addition, the horizon should look far enough past the end of the peak season to ensure that the proper balance of temporary and permanent staffing is maintained. This ensures that too many permanent workers are not hired at peak. Similar considerations impact the horizon for capital investment. In contrast to many of the process design and control problems that have been addressed, most approaches to planning do not include explicit disturbance models or state estimation techniques. Perfect state estimation is assumed, and little, if any, attention is given to stability of the closed loop system. Most research is focused on investment decisions and the minimization of operating costs under each investment. Detailed operational models and control strategies are not typically included in these formulations; instead, aggregate models of the operating costs are incorporated. This is similar to separating the process design and control problems. 3.2. Inventory Management Inventory planning is a function of many things. It depends upon the statistics of future uncertain demand, capacity constraints, product lead times and other factors. The problem includes the optimization of both the purchasing and the reverse logistics. For each item, there are multiple vendors and sourcing channels for procurement into a given location. Each has different costs and dynamics. For example, bulk discounts may exist when purchasing a product from a vendor (e.g., the per unit cost is greater when ordering single units instead of case or pallet quantities), freight and payment
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terms will vary. Different channels to remove overstock product also exist. For example, disposition options will often include vendor returns, liquidation, and markdown strategies, and the available options may depend not only upon the product, but also on the terms under which it was purchased. The inventory management problem is the analog to the level control problem. The fundamental operational question in inventory management is how many units of a given product should be held at a given location based on current and forecasted demand subject to the dynamics of the supply-demand network. The academic literature on inventory management is large. See [5,8,9] for a collection of papers and reviews. One focus in the literature is on finding optimal inventory control policies under different assumptions. The control policy is generally a closed form analytic formula that relates the current demand and inventory level to a purchase quantity. Different assumptions on demand (deterministic vs. stochastic; stationary vs. nonstationary, constant vs. time-varying), the number of products (single vs. many), and number of locations and structure of the supply-demand network all yield different models. The classical economic order quantity[11], dynamic lot size model[11], (Q,r) and (S,s) models[4,6], and other model variants are the result. The (S,s) policy is the rough analog to LQR theory and the Kalman gain for unconstrained single product systems with stochastic time varying demand. The inventory literature assumes that the models are correct and that the only variation comes in through the demand and lead time variation. A common theme in these problems is to simultaneously determine the optimal inventory levels along with the dynamic control policy. This contrasts the process industries where the problems are normally separate. The problems tend to be fairly simple because closed form solutions are sought for these inventory policies. Most problems deal with only a single product. Constraints are rarely considered. Even though these problems don’t deal with much of the real world complexity required to solve practical problems, they provide useful insight into the general dynamics of supply chains. These insights lead to rules of thumb that can be used by practitioners and analysts for policy making. The main theoretical tool for most of the above analysis is dynamic programming (DP); the Hamilton-Jacobi-Bellman (HJB) equation provides the foundation for provably optimal policies[3]. At the other extreme is an attempt to fully formulate the detailed inventory control problem and solve the resulting stochastic optimization problem. Quite often much of this work is application centric; relying on solution techniques rather than an overarching theory to provide commonality between application instances. Here again the workhorse is DP or approximate dynamic programming (ADP); however, in this instance it is used as a computational tool rather than a theoretical tool. Closed loop stability of the control policies is never really considered as the DP policy will be nominally stable as long the HBJ equation admits a solution. Robustness is not generally discussed. Model predictive control, although suboptimal compared to DP, has not been actively investigated. Most of the difficulty with these problems stems from the combinatorial size of the problem, the stochastics and non-stationary nature of demand, variable lead times, and the complex, often discontinuous, cost structures. Practical problems contain economies of scale and constraints that involve multiple products, such as inbound freight costs and vendor PO minimums, that are not often addressed by the inventory management literature. Finally, one time purchase opportunities may arise. As a result of these difficulties, there are many insights and opportunities that can be gained from
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taking a broader perspective to the control and the inventory management space. To date, no research provides a comprehensive approach to address the complexity of the problems faced in practice. 3.3. Order Assignment Based on the customer expectations set during the checkout process[2], the units in a customer order must be assigned to fulfillment centers across the network for processing. Each unit in an order has the potential to be fulfilled from a number of different physical locations based on inventory levels and the ship and delivery estimates provided to the customer at checkout. In its simplest form this is an exact set cover problem in which the units in the order are partitioned into sets, and each set of units is assigned to a feasible location whose cost can be calculated. The goal is to minimize the expected fulfillment cost of a feasible assignment. These problems are NP-hard for which no provably good approximation algorithms are available. The problem can also be modeled as a fixed cost multi-commodity network flow problem, in which nonlinear concave costs are easily handled. Although the shipping cost structure faced by Amazon is typically non-concave, we have found that it is reasonably approximated by concave functions in most cases. In addition, we have found that a large fraction of orders equate to problem instances that are small enough to employ exact solution techniques, and that variants of some approximation algorithms[7] perform well in practice. In our business, though, order assignment is even more complex. Network level operational goals must be satisfied in addition to the minimum cost order assignment. Every assignment of a unit to a facility consumes resources at that facility, such as labor capacity to pick, pack, and ship the items and processing capacity to move items within the facility. Poor assignment can starve or swamp a facility with work, so active control of the workload assigned to a facility is important. Many of the operational objectives of our fulfillment network can be recast as set points, zone constraints, and limits in a control-oriented formulation. Control research provides a formalism in which to represent these objectives in an online system. Practically, this is still very difficult. The challenge is to cast the network level objectives into a form that can be implemented in the decision logic of a single- or multi-order optimization algorithm. Our approach has been to solve the online problem using optimization and control techniques; we revisit these decisions to take advantage of new information and opportunities. The multi-order problems can become very large (over one hundred million variables) which provides challenges for both the solution algorithms and the software architecture [10]. This leads to research opportunities to provide better models of the business problem and improved solution techniques that can provide real-time answers within acceptable levels of accuracy. 3.4. Demand Forecasting Most of the above methods rely on predictions or forecasts of demand. It is a key feed forward component in many of the decision algorithms. The forecasts are produced by forecasting systems for different time ranges in the future (daily, weekly, etc.) Forecasts are created for every single product and can be represented at national and local levels. Much of the forecasting infrastructure relies on standard time series modeling techniques. ARX, ARIMA, and exponentially weighted moving average models are often used to model demand. Product lifecycle affects the model choice. New release
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items, not yet published items, and well established items all have markedly different demand dynamics. A single model can have difficulty remaining accurate for the entire life of a product. Practical approaches like switching techniques and mixture modeling are used to overcome these problems. State space formulations are not standard practice. Thus, it is much less common to see the use of Kalman filters and other estimation techniques in demand forecasting. Model identification for a small set of products can be accomplished manually. However, automatic identification and subsequent prediction for over a million products poses algorithmic and software challenges. External causal events, such as promotions, availability changes, competitor pricing, etc., also impact the forecasts. Capturing the multitude of these data signals, then screening and cleaning the data for problems is a technical challenge on its own.
4. Conclusion Effective planning and operation of a complex supply-demand network is a difficult and rewarding problem. There are significant opportunities for the classic operations research techniques as well as the optimization, simulation and control techniques to add value. Each discipline touches on different aspects of the problem, and the existing body of research does not address the exact problems commonly faced by industrial practitioners. Many outstanding challenges lie in the modeling of these problems, the development of efficient algorithms to solve the models, and the architecting of software systems to implement these solutions and manage data and workflow. Addressing these problems and questions holds enormous potential for bringing tangible economic benefit and competitive advantage to a business.
5. References 1. Amazon.com, Inc., 2004, Annual Report 2004, http://www.amazon.com. 2. M. Ball, C.Y. Chen and Z.Y. Zhao, 2004, Available to Promise, in Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era, Simchi-Levi, D., D. Wu, and M. Shen, eds., Kluwer Academic Publishers, Boston, pp 447-484. 3. D. Bertsekas, 2000, Dynamic Programming and Optimal Control, 2nd ed. 4. A. Clark and H. Scarf, 1960, Optimal policies for a mulit-echelon inventory problem, Management Science, 6, pp 475-490. 5. S. Graves, Rinnooy Kan and P. Zipkin, 1993, Handbooks in Operations Research and Management Sciences: Logistics of Production and Inventory. 6. G. Hadley and T.M. Whitin, 1963, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs. 7. K. Jain and V. V. Vazirani, 2001, Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation, Journal of the ACM 48, pp. 274--296. 8. T. de Kok and S. Graves, 2003, Handbooks in Operations Research and Management Sciences: Supply Chain Management, Design, Coordination and Operations. 9. S. Tayur, R Ganeshan, and M. Magazine, eds. 1998, Quantitative Models for Supply Chain Management, Kluwer Acad. Pub., Boston. 10. P. J. Xu, R. Allgor, and S.C. Graves, 2006 (forthcoming), The Benefits of Re-Evaluating Real-Time Fulfillment Decisions, Manufacturing and Service Operations Management. 8(1). 11. P. Zipkin, 2000, Foundations of Inventory Management, McGraw-Hill, Boston.
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Business Decision Making in the Chemical Industry: PSE Opportunities Rajagopalan Srinivasan*†, IA Karimi* *
Department of Chemical & Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576
†
Institute of Chemical and Engineering Sciences 1 Pesek Road, Jurong Island, Singapore 627833
Aspi Gave Vania Singapore Refining Corporation, Jurong Point P. O. Box 225, Jurong, Singapore 916408
Abstract The chemical enterprise of today faces a complex, global, and increasingly competitive environment, one with numerous market prospects and fraught with endless uncertainties. All enterprise-level decisions related to project, product as well as process selection, supply chain design and management, manufacturing, and logistics must carefully consider the various opportunities as well as the uncertainties and risk. In this paper, we examine the role that the Process Systems Engineering community can play at this interface of business and engineering.
1. Introduction The realization that a butterfly flapping its wings in the Amazon could result in a thunderstorm in Australia is usually attributed to the meteorologist Edward Lorenz who observed in simulations of weather patterns that a small change in the initial conditions can lead to a massive turmoil further down the line. This anecdote is typically used to highlight the interrelatedness of the complex meteorological system and the resulting complexity. The same is becoming true of enterprises in the globalized “flat world” of today [1], where intercontinental connectivity is prevalent. As exemplified by recent events – the soaring crude oil prices; the declaration by Chevron Oronite of force majeure in 2005 and the consequent oil additives rationing by many suppliers; and the spread of the avian flu and the Tamiflu shortage – various types of “hurricanes” buffet most businesses regularly. “Business Decision Making” involves managing the enterprise in the face of such “hurricanes”. Any manufacturing business can be considered to be an amalgamation of at least four intertwined networks: • Manufacturing network (dealing with production) • Services network (dealing with support services such as logistics)
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• Innovation network (dealing with development of new products/services) • Finance network (dealing with capital investment & working capital) Historically, the four networks have been operated largely in isolation and have been the subject of study of different communities. The following classes of decisions are common to each as summarized in Table 1: - Structural (design) decisions - Resource allocation decisions - Operational decisions Table 1: Similarity of business decisions among the networks Decisions
Manufacturing / Services Network
Innovation Network
Finance Network
Structural
Plan / develop network Plan product discovery Plan structure & raise / development capacities, structure capital (equity vs. debt, dividends, M&A) (production, distribution centers, supplier / customer selection)
Resource allocation
Allocating production to plants; Allocating resources to Portfolio management; leads Capital budgeting Allocating manufacturing resources to products
Operational Measurement, control, planning, Measurement, control, scheduling, monitoring & planning, scheduling, disruption management; E.g.: monitoring & Demand forecasting management of innovation tasks E.g.: Market research, clinical trials
Measurement, control, monitoring & management of financial resources E.g.: treasury functions, currency, asset hedging
The rest of this paper focuses on decisions involved in managing the four aforementioned networks. Specific emphasis is on the manufacturing / services networks which have received the most attention to date in PSE.
2. NETWORKS IN THE CHEMICAL INDUSTRY At first sight, businesses in the chemical industry seem to have many features in common with other high-tech manufacturing industries such as electronics. As highlighted by Chandler [2] there are however major differences: (1) the infrastructure that made mass production and distribution possible in the chemical industry – transportation (steamships, steam-powered railroads) and communication (telegraph and the transatlantic cable) – came of age in the 1880s; the infrastructurefor the electronics industry began much later in the 1950s. (2) A small number of companies were initially engaged in commercializing new products in the electronics industry – vacuum tube, transistor, integrated circuits and the microprocessor. Inntrast, co a much larger number
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of technologies based on chemical and biological sciences became available for the chemical industry and these were commercialized by at least fifty chemical and thirty pharmaceutical companies. (3) The products from the chemical industry utilized the new technologies to create new materials and medicine that replaced natural ones in contrast to the electronics industry which developed “novel” products for the consumer markets and thus reshaped the nature of life and work. These historical quirks of the chemical industry endure thus far and lead to some unusual features in chemical enterprises. Because of these, the plethora of research in enterprise optimization and supply chain management for other discrete manufacturing industries, does not port very well to the process-based chemical industry. In the next section, we review some of these distinguishing factors, particularly in the context of process industry supply chains. A primary feature of chemical supply chains is the huge variety of non-discrete, immiscible, incompatible, non-substitutable, and huge-volume products, each of which has its own unique characteristics. The concepts of “discrete parts” and “assembly” simply do not exist in chemical manufacturing. The industry is highly capital-intensive with long and divergent supply chains with recycle loops. The indu stry is the biggest consumer of itself and many of its businesses are high-volume and low-margin. Maritime transport is the workhorse of chemical supply chains and the hazardous nature and huge volumes of chemicals necessitate the use of highly expensive and sophisticated transport equipment and storage facilities that require complex and expensive cleaning procedures and maintenance, and result in long lead times. The logistics costs in the chemical industry could be as high as 20-30% of the purchase cost [4]. Huge inventories that are critical to the continuity and profitability; need for safetyfirst; sensitivity to oil prices, sociopolitical uncertainties, environmental regulations; and extensive trading are the other key features of the chemical industry, which set them apart easily from the other supply chains. Needless to say, the general supply chain research that has mainly focused on the computer industry has been oblivious to most of these complexities and features. While the above broad features distinguish the chemical industry as a whole, there are further fine-grained differences even among its various segments such as refining, lubricant additives (as one example of specialty chemicals), and pharmaceuticals. These distinguishing features are summarized in Table 2. These essential differences reflect themselves in two of the most essential aspects of any business – logistics and economics – as summarized in Table 3.
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Table 2: Key differentiating factors among the principal sectors of the chemical industry
Nature of product Form of product
Refinery
Lube Additives
Pharmaceuticals
Undifferentiated
Partial differentiation
Differentiated
Fluids
Predominantly viscous Predominantly solids fluids
Specs based (undifferentiated)
Performance centric
Molecule centric
Flammable
Mixed
Safe
Nature of industry
Low margin, high throughput
High-value, low throughput
High-value, low throughput
Core competency
Process & technologies
Formulation
Product innovation
Uniqueness of process / technology
Mostly similar between competitors
Unique intellectual properties
Unique intellectual properties
New product
New grades through New products through blending blending
Product definition Product hazards
New molecule
Batch; low throughput Batch; low throughput
Nature of processing
Continuous; high throughput
Nature of operation
Separations centric
Blending centric
Reaction centric
Complexity of operation
High & automated
Low (no reactions)
Low-Medium
Wastewater treatment
Limited
Water; heavy metals incineration
High
Low
Low-Medium
Consumer / business
Business
Consumer / government / business
Waste treatment Asset costs Type of customer
3. DECISIONS AND SUPPORT TECHNIQUES Arising from the differing characteristics, the key business decision problems summarized in Table 1 are of varying importance in the three sectors. Table 4 summarizes the nature and type of some of the common business decisions. Many of these have received substantial attention in the literature. Special-purpose decision support techniques broadly based on control theory, optimization, and artificial intelligence have been developed as reported in several excellent reviews [5]-[7].
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Table 3: Logistics and economic factors in the principal sectors of the chemical industry Refinery
Lube Additives
Pharmaceuticals
Sea most prevalent
Mix of land & sea
Air is predominant
Packaging
Bulk
Drums & Isotankers
Mix
Supply chain operation mechanism
Push
Pull
Push
High volume; short term
Low volume; short term
Low volume; long term
Procurement cycle length
Months
Weeks
Weeks
Delivery lead times
Weeks
Month
Days-Weeks
Prevalent
Some
Uncommon
Product Variety (# of SKUs)
Small
Medium
High
Barriers to cross-border supply
Low
Medium
High (Regulatory factors)
Key supply chain player
Oil suppliers
Customer
Company
Inventory
Customer satisfaction
Mix
High
Medium
Low-medium
Crude price
Mixed
R&D (product innovation) & Marketing
Critical
Important
Important
Operation costs
High
Low
Medium
Pricing variations across countries
Low
Medium
High
Nature of product pricing
Cost + margin based
Performance based
Market & innovation based
Mode of logistics
Inventory
Product Trading across Competitors
Supply chain KPIs Business Growth Rate Predominant cost factor
Raw material costs
Supply chain problems were traditionally considered in isolation before the supply chain perspective came into vogue. Optimization-based approaches have been the workhorse for a variety of the constituent problems in supply chains including planning, scheduling, inventory management, transportation, capacity expansion, etc. However, classical optimization techniques have been less successful in dealing with large-scale,
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integrated, dynamic, uncertain problems in real supply chains. Therefore, at present, simulation remains the predominant methodology for dealing with such systems. Specifically, with the widespread availability of easy-to-use simulation tools, most companies resort to Monte Carlo type studies. It should however be noted that neither of the two is a panacea for all enterprise optimization problems; each has its own distinct yet complementary advantages, which motivate their synergistic union. The simulationoptimization framework [8] provides one way of achieving this. Agent-based approaches [9] provide an alternative scalable solution to seamlessly integrate heterogeneous methodologies. We have explored some of the above avenues to address various problems in chemical supply chains. For instance, mathematical programming approaches seem best suited for well-determined deterministic problems such as refinery scheduling [10], capacity expansion [11] & [12], logistics [13], etc. On the other hand, simulation methodologies [9] & [14] are ideal when a decision has impact an across the supply chain and must be considered in its entirety (for e.g. crude procurement). 3.1. Refinery Supply Chain Simulator IRIS: As a first step in modeling oil and gas supply chains, we have developed a dynamic simulator, called Integrated Refinery In Silico (IRIS), for refinery supply chain simulation and analysis. Figure 1 shows a detailed block diagram of IRIS with various blocks representing supply chain entities and the connections represent information or material flow. The different types of blocks in IRIS are: • Refinery external entities (Supplier, Customer, Port) • Refinery functional departments (Procurement, Operations, Sales, Storage) • Refinery processes/units (Pipeline, inventory, CDU, Reformer, Cracker) • Refinery SC Policies (Procurement policy, Planning, Scheduling, Storage policy) • Refinery Economics IRIS has been implemented in Simulink and is an effective tool for evaluating real what-if scenarios. It can serve as a holistic test-bed for the evaluation of supply chain methodologies and algorithms. The interested reader is referred to [14] for a detailed description of this dynamic supply chain model.
4. Role of PSE in Business Decision-Making Process systems engineering vis-à-vis business decision making is today at the same juncture that it was in the 1970s with computing and process control. The emphasis in those days [15] was in developing dynamic models of the process and general-purpose dynamic simulation algorithms. The availability of the dynamic models led to the mushrooming of research in advanced process control, monitoring, diagnosis, etc. with emphasis on the development of techniques and tools to handle processing disturbances and uncertainties. PSE has arrived at a similar doorstep today vis-à-vis business decision making. With the availability of dynamic models of business networks, various types of business decisions can be supported using PSE tools and techniques. The
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emphasis on handling uncertainties through systematic approaches would continue and extend to this new domain as well. We outline a few possibilities next. Table 4: Key business challenges in the principal sectors of the chemical industry Refinery
Lube Additives
Pharmaceuticals
Capacity planning
New facilities Debottlenecking & adding units in existing facilities
Production planning
Supply chain integration with process complexity
Integrating production New product with delivery introduction
Production scheduling
Crude / product blending; throughput scheduling
Order scheduling
Campaign scheduling
Control
Advanced / modelbased
Manual
Manual
Measurement
Real-time
Lab-based
Lab-based (ref to PAT)
Fault diagnosis & recovery
Complex
Easy
Not allowed (ref to PAT)
Low importance
Low importance
Equipment monitoring High importance
New facilities
Process optimization
High
Low
Medium
Demand Forecasting
Critical
Low
Medium
Risk management & uncertainties
Raw material pricing
Production uncertainties
R&D
Logistics
Ship routing; pipeline Multi-modal network optimization design
Solution Methodologies Process simulation Business simulation; (Steady & dynamic); SPC; Spreadsheet; Business simulation; Manual / experience Model-based control; based /Heuristic approaches SPC; Spreadsheet; Math Programming; AI approaches; Manual / experience based /Heuristic approaches
Integrated optimization of supplier selection and logistics Business simulation; Batch control; SPC; Spreadsheet; Manual / experience based /Heuristic approaches
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Disruptions can be defined as any event or situation that causes deviation from normal or planned operations. Among the possible causes of disruption are operational difficulties, emergency shutdown, natural disasters, terrorist incidents, industrial actions (e.g. strikes, protests), and accidents (in-plant, or during transportation). Root causes for disruptions are often human error, wrong information, and poor planning or forecasting. Disruptions bring about adverse effects such as blockage of material flow, loss of ability to deliver the right quantity of the right product at the right place and at the right time, inability to meet quality requirements, loss of cost efficiency, under- or over-supply, process shutdown. All of these translate into financial losses, directly or indirectly and motivate the development of simulation models and decision support systems for managing disruptions in the supply chain. Some common disruptions in a refinery supply chain include delays in crude oil arrivals, crude oil being out-of-spec, unexpected changes in product distribution, unavailable or constrained plant units, and demand fluctuations. Such disruptions are not infrequent. For example, every month there are four to five occasions on average, when crude transpor tation by sea to the refinery is delayed. Similarly, use of crude oil from storage is constrained 4-5 times each month due to entrained rainwater. The liberal outsourcing of logistics activities has broken supply chains into independent entities that are inherently different. In many instances, this can introduce delays and disruptions in material and information flows. Thus, disruptions are a fact of everyday life in all supply chains and preventing them or mitigating their impact has become an important issue in supply chains. Simulators serve as a ready tool for managing business network disruptions. They can provide decision support in the face of uncertainties [16]. Thus, uncertainties in events such as crude delivery or in information such as product demands can be dealt with by embedding the appropriate decision processes within the same framework. Also, the simulation approach can be naturally integrated with optimization especially through the agent-based framework. With the availability of dynamic models, process control methodologies which have historically sought to eliminate the effect of process disturbances can be extended to handle business disruptions. Feedback, feedforward, and advanced control systems can be designed for enterprises [17]. Process identification techniques would be necessary for business process identification. Sensor selection and network design techniques would find analogues in business metrics and key performance indicators. Process monitoring, fault diagnosis, and abnormal situation management technologies can be extended to disruption management. The above examples dealt with operational decisions. However, business network models can play a major role in structural decisions as well. One such example is systematic risk identification and management in business networks. Approaches such as Hazard & Operability (HAZOP) studies, fault tree analysis, and other process safety management techniques commonly used in the PSE community can be extended to business networks as well. Once all important risks have been identified, sensitivity studies can be performed and the supply chain structure (eg: identification of alternate suppliers) and policies optimized for robustness.
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Finally, a key strength of the PSE community has been its ability to continually imbibe new ideas and concepts from other domains (eg: bio-, nano-, etc). This ability would become essential once more in the context of financial networks. With a few exceptions [18], the PSE community has shied away from the fundamentals of financial management. These considerations in various forms – structural, resource allocation, and operational – would be essential if PSE seeks to serve a central role in the businesses of the 21st century. 4.1. Impact of PSE : The truly widespread use of process simula tors such as Aspen-Plus, Hysys, Pro II, gPROMS, etc. in continuous chemical plants is an irrefutable example of the impact of PSE techniques on plant operations. The impact on business decision making, on the other hand, is relatively less documented. This is certainly not because the potential impact is lower. Two excellent examples of the scale of the impact exist in the operations research literature. First, Camm et al. [19] reported an optimization study for restructuring the supply chain network of Proctor & Gamble. Based on this study, P&G reduced its North American plants by almost twenty percent, wrote off over a billion US$ in assets & people transition costs, and saved well over US$250 million (before tax) per year. Second, Lee & Chen [20] reported a web-based production planning tool at BASF, which cut down production scheduling time from several days to a few seconds and reduced inventory and improved BASF’s use of production and storage capacities. So far, we have not found any similar success story in the PSE literature. This paucity could be because the PSE research on business decision-making is still in its infancy. Alternatively, this may be due to the cloak of confidentiality that surrounds business procedures, decisions, and impacts. For instance, the refining industry has been the single largest user of PSE techniques such as linear programming for several decades now. But well-documented impact reports are hard to find. Of course, nobody can doubt the impact of optimization on business decision-making in that industry. Recently, Kelly & Mann [21; 22] estimated that the use of advanced optimization techniques can save as much as $2.85 million/year in crude oil scheduling alone. It has been estimated [23] that even this number is easily dwarfed by the potential impact on crude transportation, which can run in tens or even hundreds of millions. Recently, we performed a logistics study [24] as part of a consulting project for a major multinational company, which concluded that roughly $750,000 per year (24%) could be saved through a systematic optimization of the company's inbound logistics operations alone. The company used our study and analysis as the basis for a major business decision. The above examples, based merely on our own knowledge, probably represent only a small fraction of the impact stories in the literature. But surely, a huge number of impact stories go unreported and undocumented. It is in the interest of the PSE community and the chemical industry to widely report such success stories and case studies in the PSE literature in particular and thereby stimulate further research and increase awareness of PSE tools and techniques for business decision making.
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ACKNOWLEDGMENTS The authors would like to extend their sincere appreciations to Mr. Jean-Luc Herbeaux of Degussa South-East Asia and Mr. Kenneth Bradley, Mr. Ivan Low, and Ms. Chua Poh Suan of Pfizer Asia-Pacific for sharing their valuable. We also gained from numerous discussions with Dr. Manjeet Singh, Mr. Arul Sundaramoorthy and Mr. Arief Adhitya of the Institute of Chemical and Engineering Sciences.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
T. L. Friedman, The World is Flat: A Brief History of the Twenty-first Century, Farrar, Straus & Giroux (2005) A.D. Chandler, Shaping the Industrial Century: the Remarkable Story of the Evolution of the Modern Chemical and Pharmaceutical Industries, Harvard University Press (2005) F. Boyd,Making Business Decisions: Re al Cases from Real Companies, Addison Wesley (1994) I.A. Karimi, R. Srinivasan and LH. Por, Chem. Engng Prog , May (2002), 32-38 Grossmann, I.E., Comput. Chem. Engng, 29 (2005) 29-39 N. Shah, Comput. Chem. Engng. 29 (2005) 1225-1235 V. Venkatasubramanian, R. Rengaswamy, K. Yin, and S.N. Kavuri, Comput. Chem. Engng. 27, (2003) 293 – 311. JY June, G. Blau, JF Pekny, GV Reklaitis, D. Eversdyk, Comput. Chem. Engng. 28 (2004) 2087-2106 N. Julka, I.A. Karimi and R. Srinivasan, Comput. Chem. Engng. 26 (2002) 1771-1781 P.C.P Reddy , I. A. Karimi , and R. Srinivasan, Chem. Eng, Sci. 59 (2004) 1325-1341 H-C. Oh and I.A. Karimi, Ind. Eng. Chem. Res. 43 (2004) 3364-3380 P. K. Naraharisetti, I. A. Karimi, and R. Srinivasan, Paper presented at INFORMS Annual Meeting, 13-16 November 2005, San Francisco, United States. I.A. Karimi, M. Sharaf, and M. Hari, AIChE Journal, 51, 1 (2005) 178-197 S. S. Pitty, R. Srinivasan, and I. A. Karimi,. Paper presented at AIChE Annual Meeting, 30 October - 4 November 2005, Cincinnati, United States, # 104d. R.W.H. Sargent, Comput. Chem. Engng. 28 (2004) 437-139 M. Bansal, A. Adhitya, R. Srinivasan, and I.A. Karimi, Computer-aided Chemical Engineering, Vol 20, Ed: L Puigjaner and A Espuna, (2005) 985-990 E. Perea-Lopez, E. Ydstie, and I. E. Grossmann, Comput. Chem. Engng. 27 (2003) 1201-1218 G. Guillén, F.D. Mele, M.J. Bagajewicz, A. Espuña and L. Puigjaner, Chem. Eng, Sci. 60 (2005) 1535-1553 J. Camm, T. Chorman, F. Dill, J. Evans, D. Sweeney, and G. Wegryn. Interfaces 27, (1997) 128. Y.Lee and E Chen, Interfaces, 32, (2002) 15. J. D. Kelley and J. L. Mann. Hydrocarbon Processing, 82, 6, (2003) 47. J. D. Kelley and J. L. Mann. Hydrocarbon Processing, 82, 7, (2003) 72. M. Duran, Personal communication, 2003. I. A. Karimi,; R Srinivasan; and L Por. Chem Eng Prog, 104, 5 (2002) 32-38.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Simulation of Mass Transfer in Reactive Absorption Norbert Aspriona a
BASF Aktiengesellschaft, Carl-Bosch-Str.38, 67056 Ludwigshafen, Germany
Abstract The discretization of the column and the films plays a significant role in mass transfer calculations and changes results significantly. The use of special grid distribution for the discretization helps to reduce the computational effort and guarantees reasonable results. The performance of these grid distributions will be demonstrated for a known and solved problem. The differences in calculation result will be shown for a column simulation. Keywords: mass-transfer, discretization, Stefan-Maxwell, reactive absorption.
1. Introduction Heat and mass transfer calculations based on Stefan-Maxwell equations are nearly stateof-the-art for chemical absorption. Unfortunately, there are only a few investigations about the discretization depth needed to obtain reasonable results. Here, predictions of a rigorous model are compared with analytical solutions for a well-known problem. In particular, the impact of a different number of film segments, and of different grid distributions, is investigated. Furthermore, a method for choosing a meaningful grid distribution to obtain accurate results with a low number of film segments is proposed, which helps to reduce the computational effort of these calculations.
2. Modeling The model normally used for simulation of heat and mass transfer in a counter-current column is quite similar to the model of theoretical stages. The column is also divided vertically into segments as can be seen in Figure 1. In each of these non-equilibrium segments, gas and liquid are exchanging heat and mass.
Fig.1. Scheme of the simulation model
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The two-film model is used for the description of the heat and mass transfer. In this model, one assumption is that the bulk phases are ideally mixed with uniform concentrations and temperatures. At the interface, vapor and liquid are in equilibrium. The heat and mass transfer resistances are assumed to lie in the film layers separated by the gas-liquid interface. The two films are stagnating so no convective mass transfer, but only diffusive mass transfer, is considered. As with mass transfer, for the case of heat transfer, only the conductive heat and not the convective heat transfer has been considered. The films are further divided into film segments. With these film segments, it is possible to calculate more accurately the concentration profiles in the film. This is essential for use in reactive systems, where as a consequence of the reaction the mass transfer can be enhanced. The simulation model allows for chemical reactions in the holdup of the bulk phases, and also within the film. The diffusion of the different components in the film layers is described with the Stefan-Maxwell equations.
3. Influence of Film discretization For reactive systems it is known that the description of the concentration profiles in the film is essential (cf. Danckwerts (1970) and Astarita et al. (1983)). The description of mass transfer enhanced by chemical reaction is investigated for a solved problem to test the model. Similar investigations are reported from Frank et al. (1995a,b) and Versteeg et al. (1989,1990). Here the aim was to find out the required film discretization. Furthermore, the use of different grid distributions within the film was investigated, since it has been reported (Versteeg et al. (1989)) that they can reduce the computational effort. To check the results of the mass transfer simulation with chemical reactions a simple, well-known problem (cf. Danckwerts (1970) and Astarita et al. (1983)) was used. The example investigated consists of a gas A which is absorbed into a liquid. The liquid is consisting of a solvent S and a component B, which is able to react with A. An irreversible reaction of the following type is considered: A+2B→3C
(1)
The reaction rate r of this reaction is second order. Dependent on the order of the kinetic constant k and the mass transfer coefficient β ′ in the liquid the mass transfer of the component A may be enhanced compared to a physical absorption due to the reaction. A measure of the enhancement is the enhancement factor E which is the ratio of the mass transfer rate with chemical reaction to mass transfer rate for a physical absorption.
E=
nchem n phys
(2)
The influence of the kinetic and the mass transfer coefficient on the enhancement is shown in Figure 2. Here the Hatta number is defined as
Ha =
k ⋅ cB ⋅ DA β′
(3)
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and the enhancement for an infinitely fast reaction kinetic (instantaneous reaction, E infinity) for the film model is given by
E ∞ = 1+
cB DA 2 c A DB
(4)
Herein cA the molar concentration (moles per volume) of A at the interface, cB the molar concentration of B in the bulk, DA and DB the diffusivities of A and B. 100000
E infinity
Enhancement
10000
n=1 n=4
1000
n=40 theory
100
10
1 0,1
1
10
100
1000
10000
Hatta number
Fig.2. Enhancement for different numbers of film segments
In Fig. 2 the full line represents the results of the numerical solution of the exact differential equations. The lines with the symbols show calculation results with the model for different numbers (1, 4 and 40) of equidistant film segments. As can be seen not even 40 equidistant film segments are sufficient to describe the enhancement correctly for the complete Hatta-range given in Fig. 2. Only 2 points are described well, independent of the number of films: The physical absorption (E=1) and the instantaneous reaction (E=E∞). For the rest the enhancement predicted with the model is significantly overestimated. For the practical use of the model this has 2 severe consequences: 1. The use of the model to evaluate reaction kinetics will lead to reaction kinetics, which are too low. 2. The use of literature kinetics will in general (also dependent on other physical properties) result in an overestimation of the enhancement. In addition, the use of very large numbers of film segments is unfavourable for column simulations or simulations of large flow sheets in terms of convergence stability and computational effort. In Figure 3 the concentration profiles of a simulation with 40 film segments is shown for a Hatta number of 101. As can be seen although the number of film segment is quite high only a few segments lie within the sharp decrease of component A at the interface. Therefore, the description of the concentration gradient at the interface will not be very accurate. The best option would be to have an adaptive grid distribution with a high number of grid in the film region where the biggest changes occur. Unfortunately, there is as yet no information available on this method.
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1,5E-06
1,0E+00
xA / (mol/mol)
A 1,0E-06
1,0E-02
B
1,0E-03
C
1,0E-04
5,0E-07
1,0E-05
xB, xC / (mol/mol)
1,0E-01
1,0E-06 0,0E+00
1,0E-07 0
0,2
0,4
0,6
0,8
1
x/δ
Fig. 3. Concentration profile for Ha=101
Alternatively an unsymmetrical grid distribution with more grids near to the interface will help. One possible kind of grid distributions is 1/ m
⎛i⎞ xi = δ ⎜ ⎟ ⎝n⎠
(5)
with xi the position of the i-th grid in the film with the total thickness δ. n denotes the number of films, and m is a grid distribution parameter. For m equal to one, this results in an equidistant grid distribution. For higher numbers of m, the grid distribution will shift towards the interface. If m is chosen in the following way
m = Ha ⋅ ln 2
(6)
then a sufficiently high proportion of all grids should lie within the region with the biggest changes. In Figure 4 it is shown that with this method, the theoretical enhancement curve is described well, even though only 6 film segments are used.
4. Influence of discretization on column simulation In the following, a column simulation of the operating data of Pintola et al. (1993) will be discussed. Pintola et al. investigated the CO2 absorption into an aqueous MEA solution in a packed column (3 beds with 4,6 m 2” pall rings, diameter 1,9 m). Of this data set only the first one will be taken into account. For a symmetrical, non-equidistant grid distribution with 6 film segments, Kucka (2002) investigated the agreement between simulation and experimental data. In his investigations he found out that 12 segments for the column height should be sufficient. The agreement between his simulation (6,5 ppm CO2 in the treated gas) with the operational data (8 ppm CO2) is excellent.
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100000
theory n=1
Enhancement
10000
n=6, m=Ha*ln 2 1000
100
10
1 0,1
1
10
100
1000
10000
100000
Hatta number
Fig.4. Enhancement for 6 film segments with variable grid distribution.
Since the model used here is similar, the reproduction of this result with a similar (but slightly different) grid distribution was possible and resulted in 5,6 ppm CO2 at the top of the column. However, in contradiction to the results of Kucka’s investigations a higher number of height segments decreased the concentration at the top until the equilibrium value of 4 ppm CO2 at the top was reached (compare Fig. 5). 14
column height / m
12 10
64 height segments 48 height segments 32 height segments
8
16 height segments 12 height segments
6 4 2 0 1,0E-06
1,0E-05 1,0E-04 1,0E-03 1,0E-02 CO2 vapor concentration / (mol/mol)
1,0E-01
Fig.5. Column profiles for different numbers of height segments (6 film segments)
Of course there is also an influence of the film distribution. In this example the Hatta number is about 50. According to eq. 6 a grid distribution with m=35 should be used. With this grid distribution it was possible to get a good description of the gradient at interface of CO2. As can be seen in Figure 6 the profiles with this grid distribution show a completely different CO2 treated gas specification (54 ppm with 32, 41 ppm with 64 and 39 ppm with 80 height segments). This result shows again that the wrong grid distribution will lead to a overestimation of the enhancement. Unfortunately, the more accurate calculation result is not in agreement with the operational data.
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14
column height / m
12 10 8 6 4 2
32 height segments, m=35 64 height segments, m=35 80 height segments, m=35 32 height segments, Fig.5
0 1,0E-06
1,0E-05
1,0E-04
1,0E-03
1,0E-02
1,0E-01
CO2 vapor concentration / (mol/mol)
Fig.6. Column profiles for different discretizations (6 film segments)
5. Conclusions The importance of the discretization on the results of mass transfer calculations with a Stefan-Maxwell approach has been demonstrated. A grid distribution was presented which helps to reduce computational effort and guarantees reasonable results.
References Astarita, G., Savage, D.W., Bisio, A., Gas Treating with Chemical Solvents, John Wiley & Sons, Inc. 1983. Danckwerts, P.V., Gas-Liquid Reactions, McGraw-Hill, Inc. 1970. Frank, M.J.W, Kuipers, J.A.M., Versteeg, G.F., Van Swaaij, W.P.M.. Chem. Eng. Sci. 1995a, 50 (10), 1645-1659. Frank, M.J.W., Kuipers, J.A.M., Krishna, R., Van Swaaij, W.P.M. Chem. Eng. Sci. 1995b, 50 (10), 1661-1671. Kucka, L., Ph. D. Thesis, University of Dortmund, 2002. Pintola, T., Tontiwachwuthikul, P., Meisen, A., Gas Sep. & Purif. 1993, 7, 47-52. Versteeg, G.F, Kuipers, J.A.M., Van Beckum, F.P.H., Van Swaaij, W.P.M., Chem. Eng. Sci. 1989, 44(10), 2295-2310. Versteeg, G.F, Kuipers, J.A.M., Van Beckum, F.P.H., Van Swaaij, W.P.M., Chem. Eng. Sci. 1990, 45(1), 183-197.
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improves the numerical robustness of these simulators (maybe at the expense of slowing down the convergence of the system). Nowadays practically all modular process simulators include optimization capabilities; however, they are constrained to optimize only operational conditions for a fixed flowsheet topology with only smooth external constraints. Capabilities like structural optimization, or the possibility of using discontinuous external constrains like costs functions defined in terms of size variables or in general any conditional or discrete constrain are not included. In this paper we show different algorithms to integrate GDP and Process simulators, not only at the level of structural decisions, but with any conditional constraint as for example discontinuous costs functions. The use of process simulators in a modular environment for solving MINLP has been addressed by Diwekar et al. (1992) Reneaume et al.(1995) and Díaz and Bandoni (1996). All these works are based on the augmented penalty/equality relaxation outerapproximation algorithm. Kravanja and Grossmann (1996) followed a similar approach, adapting the modeling/decomposition (M/D) strategy developed by Kocis and Grossmann (1987) that can be considered a precursor of generalized disjunctive programming. 2. GDP formulation in a Modular Process Simulator Environment. When we defined an optimization model with conditional equations in a modular environment we can differentiate three kinds of variables: 1. the design or independent variables (x). These are equal to the degrees of freedom of the problem and form the set of variables over which the optimizer has full control. 2. Variables that are calculated by the simulator (u) and that in general can only be read. 3. Variables that must be fixed for a given topology in the flowsheet – for example number of trays in a distillation column, binary or integer variables, etc- but that can be manipulated by the solver in each iteration. In the same way, we can differentiate two classes of equations: 1. implicit equations that are all the equations solved by each of the modules in the process simulator (or any other external module added to the program). These equations are usually considered “black box input-output” relationships because we have not access either the equations or the way in which those equations are internally treated. However, there is an important danger hidden in the equations introduced in a gradient based optimization environment: They could include points in where some of these equations are non differentiable, therefore we cannot consider these systems like completely black boxes but we should have at least a general knowledge of the system of equations in order to anticipate this behavior and correctly model the process. 2. External or explicit equations over which we have a complete control. The disjunctive formulation of the problem can then be written as follows:
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⎡ ⎤ Yi , j ⎢ ⎥ ⎢hI i , j ( x, u , p ) = 0⎥ ⎢ ⎥ ⎢ hEi , j ( x, u , p) = 0 ⎥ ∀ j ∈ J ⎥ i ∈D ⎢ ⎢ g Ei , j ( x , u , p ) ≤ 0 ⎥ ⎢ ⎥ p = pi , j ⎢⎣ ⎥⎦
∨
Ω ( Y )= TRUE x∈ X ⊆ ℜ n ;
(1)
Y ∈{True, False}
m
Where the index I makes reference to implicit equations and the index E makes reference to the explicit ones. Starting with the problem formulated in equation (1) it is possible to follow different alternatives. 1. - If there are no implicit equations inside the disjunctions then the problem can be directly reformulated to an MINLP using a big-M or a convex hull reformulation, Grossmann, (2002). If the resulting MINLP (or even the original GDP) is solved using a branch and bound strategy then no major problems are expected because the problem is reduced to solve a set of NLPs with some subset of binary (Boolean) variables fixed at each branch of the tree during the search. However, this approach could be very time consuming due to the potential large number of NLPs that must be solved (remember that each NLP include the convergence of the flowsheet inside the process simulator and although it is ‘acyclic’ because all the recycles are specify as external constrains, it could take an important amount of time). If an algorithm like outer approximation is used then we must generate a Master MILP problem. In this case, it is necessary to perform equation linearizations in terms of independent variables. In the case of explicit equations it can be done analytically if the functional relationship between x and u variables is known, but usually that relationship is not known and then a numerical approach in where the value of the output variables (u) are calculated for an adequate perturbation of the input variables (x) must be used. 2.- If the implicit equations appear also in the disjunctions then the direct reformulation to an MI(N)LP problem has very bad performance. The reason is that zero flows in non-existing units could prevent the simulator to converge and then the entire optimization procedure fail as well. The best option in this case is developing an initialization strategy in order to get information for the first master problem. Two options are available: 1 solving a ‘set covering’ problem to get the minimum number of feasible flowsheets that include all the alternatives (Turkay and Grossmann, 1996) or develop a sub-Lagrangian optimization (Kravanja and Grossmann, (1996)). In both alternatives, the Master problem is solved in the projection space of independent variables, where the relation between implicit and independent variables can be obtained directly from the simulator if the jacobian matrix is available or by an adequate numerical perturbation in other case.
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3. Example This example is a modification of the problem proposed by Seider et al, (1999) and consists of the design of a natural gas plant. It is required to process a natural gas stream at 5000 kmol/h, 20ºC and 1000 kPa. The gaseous product is required at 1500 kPa with at least 4900 kmol/h of nC4 and lighter products and a combined mole porcentage of at least 99.5%. In this example the process simulator, HYSYS.Plant©, performs the basic calculations at the flowsheet level, including all mass and energy balances and properties estimation. However, size and cost calculations, that depend on the type of equipment are calculated as implicit external functions developed in Matlab©, but with all basic data extracted from HYSYS through its COM communication capability. All the process is controlled from Matlab that contain the disjunctive formulation and controls all the solution procedure. The optimizer is also external to the process simulator and controlled by Matlab as well. Note that although in the process simulator some equipments are represented by a general unit operation (i.e. heat exchanger) the cost and size of those equipments depend on actual equipment; an air cooler is different from a floating head tube and shell exchanger. Therefore there are two kinds of implicit equations over which we have different control. The implicit equations associated to the basic flowsheet and the size and cost equations over which we have full control. The reasons of using these equations as implicit are : 1. They decrease the dimensionality of the problem at the optimization level and 2. the numerical behavior is better when the model is solved with a decomposition algorithm because linearizations are constrained to the input-output variables and not to all the intermediate non-convex equations reducing the possible effects of cutting parts of the feasible region due to linearizations. A Disjunctive conceptual representation of the model showing the different alternatives is as follow: min : TAC = 0.2 (investment cost ) + Utilities _ cost ⎡Y _ electric _ driver ⎤ ⎡Y _ combustion _ driver ⎤ ⎢ ⎥ ∨⎢ ⎥ ⎣cos t _ driver = f ( power )⎦ ⎣cos t _ driver = f ( power )⎦
⎤ ⎡ ⎥ ⎢ Y _ floating _ head _ cooler1 Y _ air _ cooler1 ⎤ ⎢ ⎡ ⎥ ⎥ ⎢ ⎢ ⎥ = = t cooler f A t cooler f A cos _ 1 ( ) cos _ 1 ( ) 1 1 ⎥ ⎢ ⎢ ⎥ ⎢ size _ cooler1 = f (TS 0, TS1, Tinair ,⎥ ∨ ⎢ size _ cooler1 = f (TS 0, TS1, Tinair , ⎥ ⎥ ⎢ ⎢ ⎥ Toutair , Qc1 ) Toutair , Qc1 ) ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ CUc1 = 0 ⎤⎥ ⎦ ⎢⎢ ⎡ Z _ cool _ water ⎤ ⎡ Z _ water ⎣ ⎥⎥ ⎢ ⎥∨⎢ ⎢⎣ ⎣CUc1 = Qc1 · Ccw⎦ ⎣CUc1 = Qc1 · Cw⎦ ⎥⎦
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⎤ ⎤ ⎡Y _ cooler 2 _ R3 ⎤ ⎡Y _ cooler 2 _ R 2 ⎡Y _ cooler 2 _ R1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ CUc CR Qc CUc CR Qc CUc CR Qc = = = 2 · 2 · 1 · 2 2 2 2 2 2 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎢cost _ cooler 2 = f ( A2 ) ⎥ ∨ ⎢cost _ cooler 2 = f ( A2 ) ⎥ ∨ ⎢cost _ cooler 2 = f ( A2 ) ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎢ size _ cooler 2 = f (TS1, ⎥ ⎢ size _ cooler 2 = f (TS1, ⎥ ⎢ size _ cooler 2 = f (TS1, ⎥ ⎢TS 2, TinR1, ToutR1, Qc )⎥ ⎢TS 2, TinR 2, ToutR 2, Qc )⎥ ⎢TS 2, TinR3, ToutR3, Qc )⎥ 2 ⎦ ⎣ 2 ⎦ ⎣ 2 ⎦ ⎣ ⎤ ⎡Y _ heater1 _ HotWater ⎤ ⎢ ⎡Y _ heater1 _ LP ⎥ ⎥ ⎢CUh1 = CHW · Qh1 ⎢ ⎥ ⎥ ∨ ⎢cost _ heater1 = f ( A ) ⎢CUh1 = CLP· Qh1 ⎥ h1 ⎥ ⎢ ⎢cost _ heater1 = f ( Ah1 ) ⎥ ⎥ ⎢ size _ heater1 = f (TS 3, TS 8, TinHW ,⎥ ⎢ ⎢⎣ size _ heater1 = f (TS 3, TS 8, TLP , Qh1 )⎥⎦ ⎢ ⎥ ⎦ ⎣ToutHW , Qh1 ) ⎡Y _ condenserW ⎤ ⎡Y _ condenserR1 ⎤ ⎡Y _ condenserR2 ⎤ ⎡Y _ condenserR3 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ CUcond Cw Qcd = · ⎢ ⎥ ⎢CUcond = CR1·Qcd ⎥ ⎢CUcond = CR 2·Qcd ⎥ ⎢CUcond = CR3·Qcd ⎥ ⎢cost _ Cd = f ( Acond )⎥ ∨ ⎢cost _ Cd = f ( Acond ) ⎥ ∨ ⎢cost _ Co = f ( Acond ) ⎥ ∨ ⎢cost _ Cd = f ( Acond )⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ size = f (TV 1, TL1, ⎥ ⎢ size = f (TV 1, TL1, ⎥ ⎢ size = f (TV 1, TL1, ⎥ ⎢ size = f (TV 1, TL1, ⎥ ⎢Twin, Twout , Qcd ) ⎥ ⎢TR1in, TR1out , Qcd ) ⎥ ⎢TR 2in, TR 2out , Qcd ) ⎥ ⎢TR3in, TR3out , Qcd ) ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡Y _ reboilerMP ⎤ ⎡Y _ reboilerHP ⎤ ⎢ ⎥ ⎢ ⎥ · = = CUreb CMP Qreb CUreb CMP Qreb · ⎢ ⎥∨ ⎢ ⎥ ⎢cos t _ Re b = f ( Areb ) ⎥ ⎢cos t _ Re b = f ( Areb ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ size _ reb = f (TL 2.TV 2, TMP , Qreb)⎥⎦ ⎢⎣ size _ reb = f (TL 2.TV 2, T HP , Qreb)⎥⎦
cos t _ compressor = f ( power ) size _ flashi = f (Volume flow and density of output streams) i = 1,2 cost _ flash = f ( H i , Di ) i = 1,2 cost _ column = cos t vessel ( Hc, Dc) + cos t int ernals
Figure 1. Basic flowsheet of the Gas Natural Plant in the example
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Previous problem was solved using a mixed big-M convex hull (for the linear equations) reformulation to a MINLP problem. Cost correlations were taken from Turton el al (1998). Data for utiliy costs and heat transfer coefficients was taken from the Database of the program DISTILL©. The relaxed problem initial problem produce an objective value of 102.15·104 $/year, only a 12.7% lower than the optimal value 117.01·104 $/year. Convergence is obtained in only one major iteration. The optimal solution include driver of the compressor must be a combustion engine. Refrigeration in coolers must be done with cool water and R1 refrigerant respectively. Hot water is used as heating media in heater 1. The condenser in the distillation column should use refrigerant R3 and the reboiler must be heated with medium pressure vapor steam. References 1. Díaz, M. S.; Bandoni, J. A. A Mixed Integer Optimization Strategy for a Large Chemical Plant in Operation. Comput. Chem.Eng. 1996, 20 (5), 531-545. 2. Diwekar, U.M.; Grossmann, I.E.; Rubin, E.S. An MINLP Process Synthesizer for a Sequential Modular Simulator. Ind. Eng. Chem. Res. 1992, 31, 313-322. 3. Grossmann, I.E. Review of Nonlinear –Mixed Integer and Disjunctive Programming Techniques. Optimization and Engineering, 3, 227---252, 2002 4. Kocis, G. R.; Grossmann, I. E. Relaxation Strategy for the Structural Optimization of Process Flowsheets. Ind. Eng. Chem. Res. 1987, 26, 1869-1880. 5. Kravanja, Z.; Grossmann, I. E. A computational Approach for the Modeling Decomposition Strategy in the MINLP Optimization of Process Flowsheets with Implicit Models. Ind. Eng. Chem. Res. 1996, 35, 2065-2070. 6. M. T¨urkay and I. E. Grossmann, “Alogic based outer-approximation algorithm for MINLP optimization of process flowsheets,” Computers and Chemical Enginering vol. 20, pp. 959–978, 1996. 7. Reneaume, J.M.F.; Koehret, B.M.; Joulia, X.L. Optimal Process Synthesis in a Modular Simulation Environment: New Formulation of the Mixed Integer Nonlinear Programming Problem. Ind. Eng. Chem. Res. 1995, 34, 4378-4394. 8. Seider, W.D.; Seader, J.D.; Lewin, D.R.; Process Design Principles. Analysis Synthesis and Evaluation. Ed by Jhon Willey and Sons. 1999. 9. Turton, R.; Bailie, R. C.; Whiting, W. B.; Shaeiwitz, J. A. Analysis, Synthesis and Design of Chemical Processes. McGraw- Hill: New York, 1998.
Acknowledgements The authors gratefully acknowledge financial support from Spanish “Ministerio de Ciencia y tecnología” under project CTQ2005-05456.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Chapter 1
Large-scale optimization strategies for zone configuration of simulated moving beds Yoshiaki Kawajiri and Lorenz T. Biegler Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA
Abstract Simulated Moving Bed (SMB) processes are widely used in sugar, petrochemical, and pharmaceutical industries. However, systematic optimization of SMB, especially finding the optimal zone configuration, is still a challenging problem. This paper proposes a simultaneous, fully discretized approach with an SMB superstructure using an interior-point solver. In case studies of the linear and bi-Langmuir isotherms, optimal zone configurations have been successfully obtained without introducing discrete variables. Keywords: Simulated Moving Bed, zone configuration, dynamic optimization, interiorpoint method, superstructure
1. Introduction Simulated Moving Bed (SMB) chromatographic process, originally developed and commercialized by UOP, performs a continuous and pseudo-countercurrent operation. SMB has been gaining more attention in food, sugar, and petrochemical industries. In recent years, SMB has been widely used as an enantiomeric separation technique in the pharmaceutical industry. An SMB system consists of multiple columns connected to each other in a circle, as shown in Fig. 1. The feed and desorbent are supplied continuously, and simultaneously the raffinate and extract products are withdrawn also continuously. Here, instead of actual movement of the adsorbent, the countercurrent operation is “simulated” by intermittently switching the four streams, desorbent, extract, feed, and raffinate, in the direction of the liquid flow. The operation of an SMB system is uniquely determined by the switching interval (step time) and the four velocities of the four zones, I, II, III, and IV. Furthermore, in SMB processes with more than 4 columns, the relative positions of the four streams are not unique, as shown in Fig. 2. This creates a large number of different zone configurations. As a consequence, we need to deal with quite a large number of choices in designing SMB systems.
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Figure 1. Schematic diagram of SMB: 8 column type with (NI,NII,NIII,NIV)=(2,2,2,2)
Some optimization techniques have been found to be useful in finding successful designs of SMB systems. Newton-based nonlinear programming approaches as well as meta-heuristic optimization approaches have been applied to SMB systems. In our previous work [1], we reported advantages of optimization with a spatial and temporal discretization using interior-point methods for SMB and PowerFeed processes, but did not consider optimal configuration of zones. Zhang et al. [2] reported the multiobjective optimization of SMB and VARICOL processes of up to 6 columns with finding optimal zone configurations. They employed a genetic algorithm to explore every possible zone configuration. In addition, superstructure formulations have been considered in Karlsson [3], where system of up to three columns system were optimized, and Toumi [4], where a relaxed Nonlinear Programming (NLP) formulation was developed for SMB and VARICOL. Nevertheless, the general problem of optimal zone configuration, determined for multiple columns with a fast NLP algorithm, still remains. In this work, we develop such an optimization approach for zone configuration by using a superstructure of SMB alternative systems. We apply a full discretization formulation, where a central finite difference is used for the spatial discretization and Radau collocation on finite elements is used for the temporal discretization [1]. The discretized equations are incorporated within a large-scale NLP problem, which is solved using an interior-point solver, IPOPT [5]. The reliability and efficiency of our approach are demonstrated with several case studies in Section 4.
2. Mathematical model We employ the following model: Mass balance in the liquid: ∂C n,i ( x, t ) ∂q n,i ( x, t ) ∂C n,i ( x, t ) εb + (1 − ε b ) + um =0 ∂t ∂t ∂x Mass balance in the adsorbent:
(1)
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Figure 2. Examples of different zone configurations; with 8 columns, there are 35 configurations.
liquid phase based: (1 − ε b ) or solid phase based:
∂q n ,i ( x, t ) ∂t
∂q n ,i ( x, t ) ∂t
= K appl ,i (C n ,i ( x, t ) − C neq,i ( x, t )) (2)
= K apps ,i (q neq,i ( x, t ) − q n ,i ( x, t ))
(3)
Isotherm: Liquid phase based: f (C neq,i ( x, t ), q n ,i ( x, t )) = 0
(4)
or solid phase based: f (C n,i ( x, t ), q ( x, t )) = 0
(5)
eq n ,i
where
εb
is the void fraction, C n ,i ( x, t ) is the concentration in the liquid phase of
component i in column n, q n ,i ( x, t ) is the concentration in the solid phase,
u m is the
superficial liquid velocity in Zone m, C neq,i ( x, t ) is the equilibrium concentration in the liquid phase, q neq,i ( x, t ) is the equilibrium concentration in the solid phase, Kapps,i and
Kappl,i are the solid-phase based and liquid-phase based mass transfer coefficient, respectively. The subscripts i correspond to chemical components, n the index of columns, and m the zone number, I, II, III, and IV, as shown in Fig. 1. Ncolumn is the number of columns, Nm is the number of columns in Zone m with NI+NII+NIII+NIV=NColumn. The cyclic steady state is given by: C n ,i ( x,0) = C n +1,i ( x, t step ) n = 1,..., N Column -1 , C N Column ,i ( x,0) = C1,i ( x, t step )
(6)
q n ,i ( x,0) = q n +1,i ( x, t step ) n = 1,..., N Column -1 , q N Column ,i ( x,0) = q1,i ( x, t step )
(7)
Also continuity constraints of concentrations and velocities between columns are enforced. For further details of the modeling, refer [1].
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3. Problem formulation We construct a superstructure of SMB that covers all possible zone configurations with the assumption that there is at least one column in each zone (Fig. 3). Then the following optimization problem is considered:
minimize
F (u D , u Dj , u Ej , u Fj , u Rj , u m , C n ,i ( x, t step ), q n ,i ( x, t step ) ) N Column −3 t step
(Extract Product Purity) =
∑ ∫u j =1
Ej
(t )C Ej ,k (t )dt
0
N Column −3 N C t step
∑ ∑ ∫u j =1
i =1 0 − 3 t step
Ej
N Column
(Extract Product Recovery) =
∑ ∫u j =1
N Column
Ej 0 − 3 t step
∑ ∫u j =1
(8)
≥ Purmin
(9)
≥ Rec min
(10)
(t )C Ej ,i (t )dt (t )C Ej ,k (t )dt
Fj
(t )C F ,k dt
0
ul ≤ uj(t) ≤ uu , j = 1...N Column (11) where tstep is the valve switching interval, or step time, Purmin and Recmin are the purity and recovery requirements of the desired product which should be recovered in the extract stream respectively. The desired product is denoted by the index k. CF,k is feed concentration of component k, and CEj,k(t) is concentrations of component k in the j-th extract stream. uu and ul are the upper and lower bounds on the zone velocities, respectively. The variables are constrained by the model equations discussed in Section 1. We now extend the approach in [1] to the more extensive problem stated above and shown in Fig. 3.
4. Case studies As the first case study, we consider the separation of fructose and glucose, which is typically modeled by the linear isotherm [6]:
q n ,i ( x, t ) = K i C neq,i ( x, t )
(12)
In this example, the total length of columns N Column × L is set to 8 m, and we choose ul=0 m/h, uu=8 m/h, PurMin=0.9, and RecMin=0.8. Also, as a more challenging example, we consider the bi-Langumuir isotherm which is suitable for modeling of enantiomer separations [7]:
q neq,i ( x, t ) =
H 1,i C n ,i ( x, t ) 1 + K 11C n ,1 ( x, t ) + K 12 C n , 2 ( x, t )
+
H 2 , i C n ,i ( x, t ) 1 + K 21C n ,1 ( x, t ) + K 22 C n , 2 ( x, t )
(13)
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Figure 3. 8 column superstructure for optimal zone configuration problem.
Table1. Feed velocity maximization of 1,1'-bi-2 naphthol enantiomeric separation (bi-Langmuir isotherm) for 12 columns with different feed compositions. For uFj, only the dominant flow is shown in each case. The discretized NLP formulation has 27139 variables and 27110 equalities. Feed composition (fructose:glucose)
80:20 %
50:50 %
20:80 %
uFj [m/h]
uF6=1.9008
uF6=1.2366
uF6=0.8712
(NI, NII, NIII, NIV)
(3,4,4,1)
(2,5,4,1)
(2,5,4,1)
Number of iterations
61
74
115
CPU time [min]
3.28
3.80
6.92
with N Column × L = 0.84 m, ul=0 m/h, uu=6.424 m/h, PurMin=0.97, and RecMin=0.8. After discretization, we implement the above optimization problems within the AMPL modeling environment. The optimization problem is solved using IPOPT. All computations are carried out on a 3.2 GHz Xeon processor. As our first case study, we consider maximizing the sum of all feed streams, ∑ u Fj . j
The results of the nonlinear isotherm with different feed compositions are tabulated in Table 1. As can be seen in the table, the optimal zone configuration is dependent on the feed composition. It is interesting that only one stream of each kind (uFj, uEj, and uRj) has a dominant nonzero flow and the rest have negligible values, typically less than 0.01% of the dominant flow. This is observed throughout our case studies, as long as appropriate tolerance and scaling are chosen. Therefore we are able to obtain the optimal zone configuration with an NLP formulation and without the addition of logic constraints or integer variables. This follows because streams of different types should be placed as far away as possible from each other to prevent contamination; the optimizer successfully finds such configurations. We also investigate how the objective function influences the optimal solution. (Table 2). As the minimization of the desorbent velocity is introduced in the objective function, NIV increases. This is because reducing the desorbent velocity requires a corresponding increase of the recycle stream, which could lead to elimination of the faster component at the downstream end of Zone IV that would contaminate the extract. To compensate, the optimizer also increases NIV to prevent the elimination of the faster component. Again, only one stream of each kind is dominant and the rest are negligible.
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Table 2. Optimization of fructose and glucose separation (linear isotherm) for 8 columns with different objective functions. For uFj, only the dominant flow is shown in each case. The discretized NLP formulation has 34731 variables and 34714 equalities. Objective function F
− ∑ j u Fj
− ∑ j u Fj + 0.5u D
uD *
uFj [m/h]
uF3=1.1584
uF3=0.6233
uF3=0.5000
uD [m/h]
3.50 (fixed)
0.6878
0.4773
(NI, NII, NIII, NIV )
(1,3,3,1)
(2,2,2,2)
(2,2,2,2)
CPU time [min]
1.73
2.81
4.00
Number of iterations
57
88
113
(* constrained with
∑u j
Fj
= 0.5 )
5. Conclusions and future work A fully-discretized NLP formulation with an SMB superstructure and interior-point solver has been found to be efficient and reliable for the optimal zone configuration problem. Moreover, discrete variables seem not to be required. In our future work, multi-component separations and more complex operations will be investigated.
•
Bibliography
[1] Y. Kawajiri, L.T. Biegler, AIChE J., (2006) to appear. [2] Z. Zhang, K. Hidajat, and A.K. Ray, AIChE J., 48(12) (2002) 2800 [3] S. Karlsson, Optimization of a sequential-simulated moving-bed separation processes with mathematical programming Methods, Ph.D. thesis, Åbo Akademi University, Åbo, Finland (2001) [4] A. Toumi, Optimaler Betrieb und Regelung von Simulated-Moving-Bed-Prozessen. PhD thesis, Universität Dortmund, Shaker Verlag, Aachen, Germany (2005). [5] A. Wächter, L.T. Biegler, Math. Prog. A, 106 (1) (2005) 25. [6] K. Hashimoto, S. Adachi, H. Noujima, H. Maruyama, J. Chem. Eng. Jpn. 16(5) (1983) 400. [7] M. Minceva, L.S. Pais, A.E. Rodrigues, Chem. Eng. Process. 42 (2003) 93.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Comparison of the Startup of Reactive Distillation in Packed and Tray Towers Florian Forner,a Michel Meyer,b Michael Döker,c Jens-Uwe Repke,a Jürgen Gmehling,c Günter Woznya a
Institute of Process and Plant Technology, Technische Universität Berlin, Str. d. 17. Juni 135, 10623 Berlin, Germany b ENSIACET/ Laboratoire du Génie Chimique, 5 rue Paulin Talabot,31106 Toulouse, France c Technische Chemie, Carl von Ossietzky Universität Oldenburg, PO Box 2503, 26111 Oldenburg, Germany
Abstract The startup of distillation towers and in particular reactive distillation (RD) towers is a very complex, time and energy consuming process. To analyze and optimize this process, a dynamic simulation model is developed which takes into account the changes of thermodynamic and hydraulic variables during the startup starting from a cold and empty state. Different aspects in modeling as well as in managing of the startup process for packed and tray towers are discussed and special startup strategies are analyzed considering as an example the methyl acetate synthesis in towers with different internals. Experimental validation results are presented showing good agreement between the measured and simulated temperature profiles during the whole startup. Keywords: startup, reactive distillation, dynamic simulation, esterification
1. Introduction The combination of reaction and distillation in one reactive distillation (RD) unit can lead to significant reductions of investment and operational costs. Conversion can be increased for equilibrium limited reactions. Heterogeneous reactive distillation in packed towers is of special interest because the catalyst does not have to be removed from the product and different reactive and non-reactive sections can be realized. At the same time the interactions of reaction and separation increase the complexity of the process and thus require a better understanding of the process dynamics. In this contribution the whole startup of RD towers from the first feed entering the cold and empty tower to the final operating point is analyzed. For the case of tray towers Reepmeyer et al. (2004) have developed a dynamic startup model. Based on this model further developments for the simulation of heterogeneous RD in packed towers have been carried out. In this contribution the differences of the startup of tray and packed towers concerning both modeling aspects and startup strategies are discussed.
2. Modeling For the simulation of RD, both equilibrium (eq) and nonequilibrium (neq) stage models have been considered and implemented in gPROMS®. Since the focus of this study is on general startup dynamics, sufficient accuracy for the description of these phenomena is required. For the considered case study (methyl acetate synthesis) it could be initially shown that the eq model predicts very well the experimental dynamic data published by
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Noeres et al (2004). Thus in the following the eq model is used which consists of the MESH equations, reaction kinetics and hydraulic correlations. Both liquid and vapor phase are modeled. Non-idealities of the liquid phase are considered using activity coefficients γi calculated from the UNIQUAC model, vapor phase association is taken into account by considering fugacity coefficients ϕi from the chemical theory according to Marek (1955). The modeling of packed and tray towers differs in the hydraulic correlations for pressure drop and liquid holdup. The pressure drop in tray towers is modeled as the sum of dry and hydrostatic pressure drop. The liquid holdup on a tray is calculated with the Francis weir formula. For pressure drop and liquid holdup in packed towers the empirical correlations presented by Engel et al. (2001) are implemented. In addition different forms of catalysis (heterogeneous or homogeneous) require different models for reaction kinetics. Heterogeneous catalysis is described using both pseudohomogeneous and adsorption-based approaches (Pöpken (2001)). All further model equations are unaffected by the choice of the internals. 2.1. Modeling of the startup During the startup of a distillation tower the hydraulic variables (flow rates, holdups) and thermodynamic variables (temperatures) undergo large changes (Ruiz et al. 1988). Due to these transitions it is not possible to describe the whole startup from a cold and empty state to the operating point with the eq stage model. Different sets of equations are needed for the different distinguishable phases of the startup requiring a switching between these model equations at certain points: The above-mentioned holdup correlations are applied only if on the considered section j a certain level (on the tray) or static holdup (in the packing) is reached. Otherwise the liquid flow rate is set to zero. At low temperatures Tj the phase equilibrium equation can not be used because conditions are far from boiling and the pressure pj can therefore not be calculated as the sum of the partial pressures of the components in the mixture. In the startup model the pressure is therefore first set to a constant value pinitial until the temperature calculated from the balance equations is equal to the additionally calculated bubble point temperature TjLV(pj,xij) at this initial pressure and current composition, Eq. (1).
(
)
IF T j ≥ T jLV p j , xij THEN ⎛ ϕ LV ⎞ p j = ∑ ⎜ xij γ ij p0LVij 0ij ϕij ⎟⎠ i ⎝ Tj
(1)
ELSE p j = pinitial
In the following the equilibrium equation is used for the correlation of temperature and pressure. When consequently the pressure on this section is getting higher than the pressure on the section above then vapour starts to ascent and the vapour flow rate is correlated to the pressure drop. Otherwise the vapour flow rate is set to zero. This modeling procedure has been explained in detail by Reepmeyer et al. (2004). In the special case of startup simulations, eq and neq modeling approaches only differ in the last phase of the startup simulation, when all stages have reached boiling temperature.
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139
2.2. Model validation Before using the newly developed process model for simulation studies, validation with dynamic experimental data from RD columns with different internals is required. Since, especially for the startup from a cold and empty state, such data can hardly be found in the literature, experiments have been carried out with different laboratory scale columns. For the validation of the model for heterogeneously catalyzed reactive distillation in packed towers a glass column with an inner diameter of 50 mm and a packed height of 6m has been used. Further details on this RD column can be found in Steinigeweg and Gmehling (2004). The esterification of acetic acid with isopropanol forming isopropyl acetate and water has been studied as an example system. Data for the adsorption-based kinetics has been published by Pöpken (2001). The experimental setup is shown in Fig. 1 together with a comparison of the simulated and experimentally measured temperatures during startup at the reboiler and three points in the column.
Figure 1. Packed column setup for the esterification of acetic acid with isopropanol and comparison of simulated and experimentally measured temperatures during startup.
The operating point has been reached without manipulation of reflux ratio or reboiler duty. The simulation reproduces very well the heating of the liquid in the reboiler and the ascent of the vapor in the column. Due to a first condensing of the vapor when heating up the liquid film and the column material (both column wall and internals are included), the rising of the temperatures at higher points in the column is delayed. The simulation model has also been validated with experimental data from a homogeneously catalyzed transesterification process in a 100 mm tray column. Both temperature data (for the startup) and concentration data (steady state and dynamic) have been used. The validation results have been published in Reepmeyer et al. (2004).
3. Startup strategies The startup can be carried out following different strategies in order to reach the desired steady state as fast as possible in compliance with given constraints. Different startup strategies for conventional distillation have been proposed in the literature. Besides conventional startup (with manipulated variables fixed to their final values), alternatives such as startup with total reflux (Kister (1990)), total distillate removal (Flender (1999)) or with different manipulated variables (Löwe et al. (2000)) have been discussed. Optimal strategies have been presented for different processes by Wozny and Li (2004).
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Reepmeyer et al. (2004) have proposed new strategies for reactive distillation in tray towers. In simulation studies average savings in startup time of about 45% compared to conventional startup were possible by initially charging product with different compositions (depending on the process). By recycling the off-spec top or bottom product with the feed during the startup, a reduction of disposal or processing costs could be achieved for some processes without significantly prolonging the startup time. Due to the different hydrodynamics, the time-optimal strategy for tray towers (initial charging) can not be applied directly to packed towers. These differences in startup strategies have been studied for the methyl acetate synthesis.
4. Case study and results The application of the above-mentioned strategies to packed and tray towers has been analyzed for the well known methyl acetate synthesis as introduced by Agreda and Partin (1984), following the design proposed by Al-Arfaj and Luyben (2002). The process design and the specifications of the studied tray and packed towers are presented in Fig 2. Holdups in sump and distillate drum are similar for both designs, so that the influence of the different column holdups on the dynamics can be compared. For the homogeneous process reaction takes place on every tray below the sulfuric acid supply on stage 28, including the reboiler. In the case of the packed tower, reaction is limited to the column sections between the two feeds which is equipped with Sulzer Katapak SP. Kinetic parameters for the homogeneous reaction have been taken from Rouzineau et al. (2003) and for the heterogeneous reaction (pseudo-homogeneous and adsorption-based approach) from Pöpken (2001). UNIQUAC parameters have been published by Pöpken (2001). Tray Tower
Packed Tower
specifications diameter
2m
reactive sections
0-28
11-28
catalyst
H2SO4 (5ml/l)
Amberlyst 15
conversion
96.6 %
98.1 %
XD,MeAc
0.956 mol/mol
0.962 mol/mol
XB,H2O
0.958 mol/mol
0.977 mol/mol
simulation results
Figure 2. Setup for methyl acetate synthesis in tray and packed tower and simulation results.
The simulation results in Fig. 2 show that for the two chosen configurations the product purities and the conversion are relatively close. To evaluate the startup time, the MX function has been calculated which gives the summation of the deviations between the current concentrations and their steady state values at the top of the column, where methyl acetate is produced. A first comparison between the simulation results for the packed tower using both pseudo-homogeneous and adsorption-based kinetics showed only very little differences that were due to the slightly different steady state results
Comparison of the Startup of Reactive Distillation in Packed and Tray Towers
141
(Fig. 3 left, curves 2 and 3). Therefore the simpler pseudo-homogeneous kinetic model can be used for further studies of the startup of the packed tower. To analyze the process dynamics for the two different designs, first the startup according to the conventional strategy has been simulated. Due to the different holdups in the towers, the startup times are very different (Fig. 3 left). The startup of the tray tower with fixed reboiler duty requires a large reboiler holdup because it takes longer to fill up the upper part of the column with the reflux so that a lot of product from the bottom with a high water fraction is evaporated before reflux reaches the bottom. This leads to high water concentrations in the lower part of the column during startup before real separation by counter current distillation can take place, so that the settling of the concentrations to their steady state values takes very long (Fig. 3 right). This phenomena cannot be observed for packed towers, since in this case reflux reaches the bottom faster and the counter current flow is established earlier. In addition, the steady state concentration profiles are quite different for the two designs (although close at the bottom and top), for the packed tower the water fraction in the lower part of the column is considerably higher.
Figure 3. left: Comparison of startup times for packed and tray towers following different strategies. (1): tray tower, conventional; (2): packed tower with pseudo-hom. model, conventional; (3): packed tower with ads.-based model, conventional; (4): packed tower with pseudo-hom. model, methyl acetate feed; (5): tray tower, initial methyl acetate charging; right: Comparison of the water fraction on section 7
The described behavior of the tray tower can be changed by initially charging top product (methyl acetate) on the trays (curve 5). Even without changing the feed specifications, this leads to a significant reduction of startup time, since in this case very little water is produced during the startup due to the relatively high methyl acetate fractions throughout the column. Comparable effects can be achieved by supplying the catalyst later to the system. Initial charging of product is not possible for packed towers. Alternatively feeding with a different composition is simulated until the reflux is turned on (curve 4). It is found that because of the smaller holdup the influence of the feed concentrations during the first part of the startup is not so important. At the top the product specifications can be reached faster but it takes longer for the whole tower to reach steady state. For all the studied cases the bottom product meets the specifications later than the top product as can be seen from the different time scales of the two graphs in Fig. 3. This behavior is due to the large reboiler volume.
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5. Conclusions and future work A dynamic startup model for homogeneously and heterogeneously catalyzed reactive distillation in packed and tray towers has been developed and validated. The dynamic behavior of packed and tray towers during startup has been analyzed and differences have been pointed out. Especially the different size of the liquid holdup on the different internals has a large influence on the startup time so that for tray towers startup time can be minimized by initially charging product to the column. For packed towers different feed concentrations during startup affect the startup time only slightly. In a next step experimental investigations of the methyl acetate system in a laboratory scale column equipped with Katapak SP will be carried out to further validate the simulation results for the startup. To draw more general conclusions concerning the different dynamics of RD in packed and tray towers, additional systems will be studied for both configurations.
References Agreda, V.H., Partin, L.R. 1984. Reactive distillation process fort he production of methyl acetate. United States Patent 4,435,595 Al-Arfaj, M. A., Luyben, W. L. 2002. Comparitive control study of ideal and methyl acetate reactive distillation. Chemical Engineering Science, 57 (24), 5039-5050 Druart, F., Reneaume, J.-M., Meyer, M., Rouzineau, D. 2004. Catalytic distillation simulation by a new transfer model - application for production of methyl acetate. Canadian Journal of Chemical Engineering 82 (5), 1014-1028 Engel, V., Stichlmair, J., Geipel, W. 2001.Fluid Dynamics of Packings for Gas-Liquid Contactors. Chemical Engineering & Technology, 24 (5), 459-462 Flender, M. 1999. Zeitoptimale Strategien für Anfahr- und Produktwechselvorgänge an Rektifizieranlagen. VDI Verlag, Düsseldorf, ISBN 3-18-361003-5 Kister, H. Z. 1990. Distillation Operation. McGraw Hill, New York, ISBN 0-07-034910-X Löwe, K., Li, P., Wozny, G. 2000. Chemical Engineering & Technology, 23 (10), 841-845 Marek, J. 1955. Vapor-liquid equilibria in mixtures containing an associating substance. II. Binary mixtures of acetic acid at atmospheric pressure. Collection of Czechoslovak Chemical Communications, 20, 1490-1502 Noeres, C., Dadhe, K., Gesthuisen, R., Engell, S., Górak, A. 2004. Model-based design, control and optimisation of catalytic distillation processes. Chemical Engineering and Processing 43(3), 421-434 Pöpken, T. 2001. Reaktive Rektifikation unter besonderer Berücksichtigung der Reaktionskinetik am Beispiel von Veresterungsreaktionen. Aachen: Shaker Verlag, ISBN: 3-8265-8638-7 Reepmeyer, F., Repke, J.-U., Wozny, G. 2004. Time optimal start-up strategies for reactive distillation columns. Chemical Engineering Science, 59 (20), 4339-4347 Rouzineau, D., Meyer, M., Prevost, M. 2003. Non equilibrium model and experimental validation for reactive distillation, Escape13, Finland, Computer Aided Chemical Engineering Ruiz, C.A., Cameron, I.T., Gani, R. 1988. A generalized dynamic model for distillation columns – III. study of startup operations. Computers & Chemical Engineering, 12 (1), 1-14 Steinigeweg, S., Gmehling, J. 2004. Transesterification processes by combination of reactive distillation and pervaporation. Chemical Engineering and Processing, 43, 447-456 Wozny, G., Li, P. 2004. Optimisation and experimental verification of startup policies for distillation columns. Computer & Chemical Engineering, 28 (1-2), 253-265
Acknowledgements We gratefully acknowledge the financial grant from the BMWA through the AiF (Arbeitsgemeinschaft industrieller Forschungsvereinigungen), Grant No. 14183N/1, as well as the support from the DAAD through the PROCOPE program.
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Parameter estimation for stochastic differential equations: algorithm and application to polymer melt rheology Bernardino Pereira Lo, Andrew J. Haslam, and Claire S. Adjiman* Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
A parameter estimation algorithm for stochastic differential equation (SDE) systems is proposed. The problem is formulated using the maximum likelihood (MLE) objective function, and a modified Levenberg-Marquardt (LM) algorithm is developed for its solution. Stochastic sensitivity equations are derived and used in order to obtain reliable parameter estimates. Computational efficiency is addressed by varying the simulation size according to the proximity of the current iterate to the optimal solution, as indicated by the magnitude of the trust-region radius. To evaluate the confidence intervals of the parameters, a global uncertainty analysis is proposed, which is based on sampling and accounts for experimental uncertainty and model noise. The algorithm is applied to a stochastic model of polymer rheology. 1. Introduction SDEs have gained popularity in recent years, for their ability to model systems that are subjected to fluctuations. The general form of an SDE is:
dX t = μ (t , X t ;θ )dt + σ (t , X t ;θ )dWt
(1)
where t is time, Xt is the state variable of interest, μ and σ are the drift and diffusion term respectively, θ is a vector of model parameters and Wt is a Gaussian N(0,Δt1/2) noise term (a stochastic process). Applications of SDEs include material modelling (e.g. polymer rheology), environmental pollution *
Author to whom correspondence should be addressed; email:
[email protected]
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models, reactor systems, and finance. [1-4] Due to the existence of the noise term it is difficult to obtain closed-form solutions for most SDEs, and numerical methods such as the Euler and the Taylor schemes are required to calculate discrete-time trajectories of the state variables. [5] The accuracy and cost of integrating an SDE system depends on the number of stochastic processes generated (size of simulation). A key issue in using SDEs for practical applications is the estimation of model parameters. This is hindered by the stochastic nature of the model, which makes the computation of gradients unreliable, and by the high computational cost of integrating SDEs by numerical methods. The objective of this work is to develop a gradient-based parameter estimation algorithm for SDE models, that provides reliable values of the parameters and their confidence intervals at reasonable computational cost. In section 2, the parameter estimation algorithm is outlined. The application of the algorithm to a model of polymer rheology is demonstrated in section 3, and in section 4, a global uncertainty analysis method for evaluating confidence intervals is described. 2. Methodology and algorithm The algorithm is a modification of the LM algorithm [6], which takes into account the stochastic nature of the problem by careful consideration of the reliability of the gradients, and by using a variable simulation size. The parameter-estimation problem is formulated using the MLE objective function:
⎛ NE NM i ⎡ ( X ijexp (t ) − X ijmodel (t ;θ )) 2 ⎤ ⎞⎟ N 1 2 ⎜ Φ = ln(2π ) + min ∑ ∑ ⎢ln(σ ij ) + ⎥⎟ 2 ⎜ i =1 j =1 ⎢ σ 2 2 ij ⎣ ⎦⎥ ⎠ ⎝
(2)
where NE is the number of experiments performed and NMi is the number of measurements in the ith experiment; the LM algorithm is specifically designed to solve least-square problems. The algorithm requires reliable gradients to successfully identify optimal parameters. The issue of gradient calculation is addressed by deriving the sensitivity equations for the SDE model. The original SDEs are differentiated with respect to the model parameters, and the augmented SDE system is then integrated to obtain the sensitivities of the state variables. The importance of this is illustrated in the following example, using the stochastic model of polymer-melt rheology (described in the next section) as a representative stochastic process. A comparison of analytical gradients, derived using the sensitivity equations, with numerical gradients, derived using the central finite-difference method, for different step sizes h (Figure 1) reveals that the numerical gradients are noisier. The numerical gradient with h = 0.1 appears unstable as the small step size amplifies the noise of the model predictions, and
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results in a gradient with the wrong sign for part of the trajectory. For the larger step size h = 0.5, even though the trajectories follow the same shape as the analytical gradient, the same accuracy cannot be achieved. Moreover, the analytical gradient calculations, which involve one integration of the augmented SDE, require about 40% less computational time than the numerical gradient calculations which involve two integrations of the SDE model. As a result, the analytical gradients are more reliable, less noisy and faster to compute.
Figure 1: Numerical gradients (h = 0.1, 0.5) compared with analytical gradients from sensitivity equations
The stochastic nature of the problem reduces computational efficiency; a large simulation size is required to obtain reliable model predictions. This issue is addressed by varying the simulation size from iteration to iteration. In the LM algorithm, the size of the step to the next iterate is determined by the trust-region radius Δ. The magnitude of Δ is kept constant or increased after successful iterations, while it is reduced after unsuccessful iterations. In this work, upon each successful iteration the simulation size is kept constant or decreased; when the contrary happens, the simulation size is increased to improve the accuracy of the predictions of the state variables and the reliability of the gradients so as to increase the probability of identifying the optimal solution. As a result, at each iteration the simulation size is computed as an inverse function of Δ, increasing computational efficiency. The function used in this work is size = 5000 ||D θ ||Δ-1, where ||·|| denotes the Euclidean norm, and D is a diagonal scaling matrix (for details, see page 111 of Reference 6). 3. Application of algorithm to a stochastic model of polymer rheology The parameter estimation algorithm has been applied to a stochastic model of polymer melt rheology [1]. In this model, the role of the SDEs is to mimic the random orientation of polymer chains under flow conditions, known as
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reptation. The model is used to predict transient viscosity under different shear and extensional flow conditions. The key stochastic variable of the model is the random orientation vector u; its stochastic process takes the form:
⎡⎛ u u ⎞ ⎤ ⎛ uu dut = ⎢⎜1 − t 2t ⎟ ⋅ κ ⋅ ut − 2 Dut ⎥ dt + 2 D ⎜1 − t 2t ⎜ ⎢⎣⎜⎝ ⎥⎦ ut ut ⎟⎠ ⎝
⎞ ⎟ ⋅ dW t ⎟ ⎠
(3)
where κ is the transpose of the velocity gradient tensor and D is the orientation diffusion coefficient associated with the reptation motion. The stress tensor, τ, is then obtained as a function of the expectation of the dyadic product uu at time t. The transient shear viscosity η+ and extensional viscosity μ+ are then given by simple functions of the appropriate components of the stress tensor and the shear rate or strain rate (respectively). The maximum likelihood objective function is then computed as a function of the least-square of the difference between model-predicted viscosity and experimental data. There are three model parameters which are related to the dynamic properties as well as the architecture of polymer chains. They are: • the plateau modulus, G0N: this is the plateau value of the shear relaxation modulus, and it characterises the transition of the dynamics of polymer chain motion from vibration at short time scales to reptation at long time scales. • the reptation time, τd: this is a characteristic relaxation time for polymer chains to diffuse away from an imaginary confined tube imposed by surrounding polymer chains. • the maximum stretching ratio, λmax: this is the ratio of the contour length of a fully stretched polymer chain to the length when it is in its equilibrium state. The ability of the algorithm to identify known parameter values is tested by considering a “model” polymer. Pseudo-experimental data are generated from simulations using known parameter values at three different extensional rates, and noise, representing experimental error, is added to the data. Starting from parameter values some distance away from the true values, the algorithm estimates parameter values that are close to the true ones, both for fixed and variable-size simulations (see Table 1). The quality of fits to the data (Figure 2) is very high. However, the computational expense is 50% smaller when using a variable size, compared to the case of fixed size. Table 1: Parameters used to generate pseudo-experimental data, starting point of the algorithm, parameters estimated by the algorithm and the computational expense (runs were performed on a hyperthreaded Pentium 4 3.4GHz computer running on RHEL 3 system).. Parameter “True” parameter Starting point Fixed size (= 100,000) Variable size (min 10, max 100,000)
G0N (Pa) 9876 1000 9894 9774
τd (s)
λmax
54.3 100 53.44 55.00
2.1 10 2.130 2.090
CPU (s) --------49104 99754
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Figure 2: Model-predicted viscosity (lines) and pseudo-experimental data (symbols) for a model polymer at three different extensional rates. The fits for both fixed and variable size are shown, but they are so close that they are indistinguishable.
The algorithm is then applied to polystyrene data [7] at four different shear rates. Figure 3 shows the parameters estimated and the fits to the experimental shear viscosity. The model-predicted and experimental trajectories are in qualitative agreement while, quantitatively, the fits are better at shear rate = 1.0 s-1 than at higher rates, as is expected for this model.
Figure 3: Parameter estimates for a polystyrene sample and the fits to experimental data
4. Evaluating the confidence intervals of the parameters Uncertainty in the parameters estimated arises from experimental error and from the stochastic nature of the model. To obtain confidence intervals, a global uncertainty analysis is proposed in this work. A number of normally distributed experimental trajectories are first sampled, and the cumulative probability that a given set of parameters is optimal is then estimated, based on knowledge of the simulation noise (±0.5% for size = 100,000). This is repeated for a set of uniformly distributed parameters values, and a distribution of the probability
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that any given parameter value is optimal is obtained. This then gives an estimate of the expected parameters values and their confidence intervals. Table 3 shows the results of the global uncertainty analysis for a sample of polyacrylate at extensional rate = 1.0 s-1. The optimal parameters predicted by the algorithm are close to the expected value of the parameters, and the 95% confidence intervals are small, ranging between 2 to 7% of the parameter values. Table 3: Results of global uncertainty analysis for polyacrylate at extensional rate = 1.0 s-1 Parameter G0N (Pa) τd (s)
λmax
Optimal value 24413 69.71 15.70
Expected value 24677 69.48 15.62
95% confidence interval [23208, 26475] [68.27, 71.13] [15.06, 16.87]
5. Conclusions SDEs have found many applications in the modelling of complex systems subjected to randomness, but pose problems for parameter estimation due to their stochastic nature. In this work, a reliable parameter estimation algorithm for SDE models has been developed and implemented. This is based on a modified Levenberg-Marquardt algorithm, in which the simulation size is varied to improve computational performance. The gradients required for the successful identification of the parameters are derived from stochastic sensitivity equations. To quantify the uncertainty in the parameters due to experimental error and the stochastic nature of the model, a global uncertainty analysis is proposed. The application of this algorithm to a stochastic model of polymer rheology has been successfully demonstrated. Acknowledgement The financial support from the Engineering and Physical Sciences Research Council (UK) and the EU (Framework V PMILS: G5RD-CT2002-00720PE0586) is gratefully acknowledged. References 1. 2. 3. 4. 5.
J. Fang, M. Kröger and H. C. Öttinger, J. Rheol., 44(2000) 1293 R. Leduc, T. E. Unny and E. A. McBean, Appl. Math. Modelling, 12(1988) 565 A. Bhave and M. Kraft, Siam J. Sci. Comput., 25(2004) 1798 J. C. Hull, Options, Futures and Other Derivatives, Prentice Hall, New Jersey, 2005 P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, New York, 1992 6. J. J. Moré, Lecture Notes in Mathematics, 630(1977) 105 7. T. Schweizer, J. van Meerveld and H. C. Öttinger, J. Rheol., 48(2004) 1345
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A "Targeted" QSPR for Prediction of Properties Neima Braunera, Roumiana P. Statevab, G. St. Cholakovc and M. Shachamd a
School of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel Institute of Chem. Eng., Bulgarian Academy of Sciences, Sofia 1113, Bulgaria c Dept. of Organic Synthesis and Fuels, Univ. Chem. Technol., Sofia 1756, Bulgaria d Dept. of Chem. Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel b
Abstract In order to improve the reliability of the Quantitative Structure-Property Relationships (QSPR) for property prediction, a "targeted" QSPR (TQSPR) method is developed, from a training set, which contains only compounds structurally similar to the target compound. Structural similarity is measured by the partial correlation coefficients between the vectors of the molecular descriptors of the target compound and those of the predictive compounds. The available properties of the compounds in the training set are then used in the usual manner for predicting the properties of the target and the rest of the compounds of unknown properties in the set. Preliminary results show that the targeted QSPR method yields predictions within the experimental error level for compounds well represented in the database and fairly accurate estimates for complex compounds that are sparsely represented. The cut-off value of the partial correlation coefficient provides an indication of the expected prediction error. Keywords: Quantitative structure-property relationship; QSPR, QS2PR; Property prediction; Process design;.
1. Introduction Modeling and simulation of chemical processes require, in addition to the process model, correlations of physical and thermodynamic properties of the various compounds, often for wide ranges of temperatures, pressures and compositions. Pure component properties are needed to derive the correlations. However, often those properties cannot be measured, or the measurements are expensive and/or unreliable. In the recent years there has been increased interest in the development and use of Quantitative Structure-Property Relationship (QSPR) models [1-7]. The QSPR models are being extensively used for predicting a variety of pure component properties pertaining to chemistry and chemical engineering, environmental engineering and environmental impact assessment, hazard and operability analysis, etc. In the present work we will concentrate on the "most significant common features" QSPR methods, as defined in [1] which we shall call for short QSPRs henceforward. The above QSPRs can be schematically represented by the following equation: y p = f ( x s1 , x s 2 , … x sk ; x p1 , x p 2 … x pm ; β 0 , β 1 … β n )
(1)
where xs1, xs2,… xsk are the molecular structure descriptors of a particular pure compound, xp1, xp2,… xpm are measurable properties of the same compound (such as boiling temperature, melting temperature, toxicity, etc.), β0, β1,… βn are the QSPR parameters and yp is the target property (to be predicted) of the same compound.
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To derive the QSPR, the available data is divided into a "training set" and an "evaluation set". Using the "training set", multiple linear or nonlinear regression, and partial least squares techniques are employed to select the molecular descriptors and/or properties to be included in the RHS of Eq. (1), and to calculate the model parameter values. Model validation is carried out using the "evaluation set". A limitation of the traditional QSPR approach is that if the molecular structure of the target compound belongs to a group that is well represented in the “training set”, the prediction can be expected to be much more accurate than if the target compound belongs to a group which is sparsely represented [e.g. 8]. The structure-property relationships are usually nonlinear, therefore, extrapolation toward a target compound of unmeasured pure component constants can be rather risky and at present the prediction accuracy cannot be assessed. Recently Shacham et al.[9, 10] and Brauner et al. [11] presented a different approach: the Quantitative Structure - Structure Property relationship (QS2PR). This technique enables the derivation of linear property-property correlations based on a structure-structure relationship and provides an estimate of the prediction error. However it can be envisioned that in some cases it will be difficult to apply the QS2PR technique because of the lack of enough predictive compounds for which reliable measured property values exist. In an attempt to overcome the limitations of both the QSPR and QS2PR techniques we have developed a quantitative measure of similarity between molecules and a new "targeted QSPR" (TQSPR) technique, which are described in the next section. 2. The Targeted-QSPR method The TQSPR method attempts to tailor a QSPR to an unknown (target property) of a particular compound (target compound). For its effective use a database of molecular descriptors, xij and physical properties yij for the predictive compounds, where i is the number of the compound and j is the number of the descriptor/property, is required. Molecular descriptors for the target compound (xtj) should also be available. The same set of descriptors is defined for all compounds in the database, and the span of molecular descriptors should reflect the difference between any two compounds in the database. In principle, the database should be as large as possible, as adding more molecular descriptors and more compounds to the database can increase its predictive capability. At the first stage of the targeted QSPR method, a similarity group (cluster, training set) for the target compound is established. The similarity group includes the predictive compounds, identified as structurally similar to the target compound by the partial correlation coefficient, rti, between the vector of the molecular descriptors of the target compound, xt, and that of a potential predictive compound xi, i.e., rti = xt xiT, where xt and xi are row vectors, centered and normalized to a unit length. Absolute rti values close to one ( rti ≈1) indicate high correlation between vectors xt and xi (high level of similarity) between the molecular structures of the target compound and the predictive compound i. The similarity group includes the first p compounds with highest rti values. Another option is to form the similarity group only with compounds for which the rti values exceed a prescribed threshold value. To tailor a QSPR for a property of the target compound (applicable for all members of the similarity group) only members of the group for which data for the particular property are available are considered (N compounds). In view of the limited variability
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of the property values within the similarity group, a linear structure-property relation is assumed of the form: y = β 0 + β 1 x1 + β 2 x 2 … β m x m
(2)
where y is an N vector of the target property values, N is the number of compounds included in the similarity group, x1, x2 … xm are N vectors of predictive molecular descriptors (to be identified via a stepwise regression algorithm), and β 0 , β 1 , β 2 … β m are the corresponding model parameters to be estimated. The signal-to-noise ratio in the partial correlation coefficient (CNRj) is used as a criterion for determining the number of the molecular descriptors that should be included in the model (m). The calculation of CNRj requires specification of error levels for the molecular descriptor data. The error (noise) in the molecular descriptors is assumed to be of the order of the round-off error of the calculated values. For integer data (no. of carbon atoms, for example) the noise level is the computer precision. Addition of new descriptors to the model can continue as long as the CNRj is greater than one for, at least, one of the descriptors which are not yet included. Detailed description of this stopping criterion can be found in Shacham and Brauner[9-11]. It should be noted that if necessary, nonlinear functions of molecular descriptors may also be considered in the RHS of Eq. (2). As in a typical most “significant common features” method [1], a stepwise regression program is used to determine which molecular descriptors should be included in the QSPR to best represent the measured property data of the similarity group and to calculate the QSPR parameter values. The QSPR so obtained can be subsequently used for calculating the estimated value of the corresponding property values for the target compound and for other (p-N) compounds in the group that do not have measured data, i.e. using the equation: y t = β 0 + β 1 xt1 + β 2 xt 2 … β m xtm
(3)
where yt is the estimated property value of the target compound and xt1, xt2, … xtm are the corresponding molecular descriptors values of the target compound. The targeted QSPR method ensures that the most pertinent information available in the data base (as measured values and molecular descriptors) is used for prediction of each property of the structurally similar compounds.
2. Application of the Targeted QSPR method for Property Prediction For practical study of the targeted QSPR method, we used the molecular descriptor and property database of Cholakov et al. [2] and Wakeham et al. [1]. The database contains 260 hydrocarbons, the molecular structure of which is represented by 99 molecular descriptors, and values for five physical properties. The properties included in the database are the normal boiling temperature (NBT), relative liquid density at 20º C ( d 420 ), critical temperature (Tc), critical pressure (Pc) and critical volume (Vc). The list of the hydrocarbons in the database, the sources and quality of the property data are given in the corresponding references [1, 2]. In general, the molecular descriptors include the molar mass along with carbon atom descriptors, descriptors from simulated molecular mechanics (total energy, bond stretch energy, etc.) and some of the most popular topological indices, calculated with unit
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bond lengths and with the bond lengths of the minimized molecular model, obtained by molecular mechanics. A complete list of all molecular descriptors in the database can be found elsewhere [10]. The 99 molecular descriptors in the data base were normalized dividing each descriptor by its maximal absolute value over the 260 database compounds. The stepwise regression program SROV [9] was used for identification of the similarity group, by sorting the compounds in descending order according to their |rti| values. The first p = 50 compounds were included in the similarity group. This number was arbitrarily set. The SROV program was also used for deriving the structureproperty relation (Eq. 3). In the two examples hereunder the practical application of the targeted QSPR method is illustrated. 2.1. Example 1. Prediction of the Properties of n-tetradecane The compound n-tetradecane is a representative of compounds for which accurate experimental data is available for most physical properties, it is densely represented in the database (meaning that there are many similar compounds included) and its properties can be predicted fairly well with existing QSPRs and homologous series techniques. The results of the similarity group selection are displayed in Figure 1. It can be seen that the database contains a large number of compounds with high level of similarity to ntetradecane (|rti| between 0.93195 and 0.99968). The highest correlations are with the immediate neighbors of the target compound in the homologous series, n-pentadecane and n-tridecane. The lowest |rti| is with 1-nonacosene. The similarity group was used to derive QSPRs for the NBT, d 420 , Tc, Pc and Vc for compounds structurally related to n-tetradecane in the form of Eq. (2). Those QSPRs were subsequently used for predicting the properties using Eq. (3). A summary of the QSPRs for the various properties derived for the similarity group of n-tetradecane is shown in Table 1. It can be seen that the QSPRs for the various properties include different molecular descriptors. The linear correlation coefficient R2 values (>0.999 in all the cases) indicate an excellent fit. Only three descriptors were enough for R2>0.999 for prediction of Pc, while for prediction of the other properties four predictors were needed. In Table 1 the property prediction errors obtained with the “targeted” QSPR are compared with experimental errors assigned by DIPPR and with the corresponding prediction errors obtained in previous works [1, 2, 10-11] by applying the QSPR and QS2PR methods to the same data. In general the “targeted” QSPR advocated in this work predicts the properties of ntetradecane better than the traditional QSPRs and with precision comparable to that of the QS2PR [10-11] method (Table 1). However, the errors of both the QS2PR and the “targeted” QSPR (except for Tc) are well within the experimental errors assigned by DIPPR for the target, and hence, when its structure is well represented in the data base, either method can be used. 2.2. Example 2. Prediction of Unmeasured Properties of Members of the Similarity Group of n-tetradecane For three members belonging to the similarity group of n-tetradecane, namely 2,5dimethyldecane, 2,5-dimethyldodecane and 4-methyloctane, there are no experimental
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values for the critical properties and the relative liquid density (except for 4methyloctane). The unknown properties of those compounds can be predicted using the same targeted QSPR that was derived for n-tetradecane. In Table 3 the property values obtained with the TQSPR are compared with measured values (whenever available) and with predictions obtained with the QSPR method of Wakeham et al. [1]. The largest differences between measured and predicted values for 4-methyloctane are: for NBT - 0.4 %; for d 204 - 0.36 %, for Tc - 1.6 %, for Pc - 1.6 % and for Vc - 3.6 %, all within experimental error.
3. Conclusions The partial correlation coefficient between vectors of molecular descriptors has been found to be an efficient and convenient measure for identifying structurally similar compounds and creating a training set of structurally similar compounds for traditional QSPR techniques. The preliminary results obtained with the new targeted QSPR method show that it yields predictions within the experimental error level for compounds that are well represented in the database, and fairly accurate, reliable estimates for complex compounds which are sparsely represented. The cut-off value of the partial correlation coefficient provides an indication for the expected prediction error. Thus, the new method can complement the QS2PR and the traditional QSPR technique for prediction of properties of compounds which are sparsely represented in the molecular descriptor – property database. More research is required in order to determine the relationships between the prediction reliability, the threshold value used for the partial correlation coefficient, the number of compounds included in the similarity group and the accuracy of their property data, and the improvement that might be eventually achieved by inclusion of nonlinear terms in the QSPR model. Another important avenue for future research is the potential for application of the partial correlation coefficient between the vectors of molecular descriptors in computer aided design of molecules, structurally related to a compound with well established useful properties.
Bibliography 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Wakeham, W.A.; Cholakov, G.St. and Stateva, R.P. J. Chem. Eng. Data. 47( 2002) 559. Cholakov, G.St.; Wakeham, W.A. and Stateva, R.P. Fluid Phase Equil. 163 (1999) 21. Lydersen, A.L., Univ. Wisconsin Coll. Eng., Eng. Exp. Stn. Rep. 3, Madison, Wis. (1955). Daubert, T. E. J., Chem. Eng. Data, 41(1996) 942. Boethling, R.S.; Mackay D., eds, Handbook of Property Estimation Methods for Chem., Lewis, Boca Raton, FL, USA (2000). Poling, B.E.; Prausnitz, J. M. and O’Connel, J. P., Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York (2001). Dearden J. C., Environmental Toxicology and Chemistry, 22 (2003) 1696. Yan, X.; Dong, Q. and Hong, X., J. Chem. Eng. Data, 48 (2003) 380. Shacham, M. and Brauner, N. Comp. Chem. Engng. 27 (2003) 701. Shacham, M.; Brauner, N.; Cholakov, G.St. and Stateva R.P. AIChE J. 50 (2004) 2481. Brauner, N.; Shacham, M.; Cholakov, G.St. Stateva, R.P. Chem. Eng. Sci. 60 (2005) 5458.
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Table 1. Summary of structure-property correlations for various properties of n-tetradecane Property
R2
Descriptors
Experiment (DIPPR) NBT
x3,x84,x85,x86 x3,x42, x88, x95
20 4
0.99988 0.99932
d Tc x59, x88, x92,x95 0.99956 Pc x65, x77, x85 0.99946 Vc x72, x86, x95, x98 0.99987 * 8 descriptors[1, 2], ** 4 descriptors[10].
Prediction error, % Targeted QSPR* QSPR
QS2PR**
<1 <1
<0.01 0.09
1.92 0.40
0.05 0.04
< 0.2 < 10 < 10
0.29 1.46 0.547
0.42 0.07 1.04
0.06 0.70 0.10
Table 2. Prediction of properties of members of the n-tetradecane similarity group Properties
NBT 20 4
2,5-dimethyldecane
2,5-dimethyldodecane
4-methyloctane
Targeted QSPR
Published* (QSPR**)
Targeted QSPR
Published* (QSPR**)
Targeted QSPR
Published* (QSPR**)
470.33 0.7502
471.25 (0.7502)
506.76 0.7630
506.75 (0.7646)
417.24 0.7225
415.60 0.7199
d Tc 647.8 (642.2) 683.6 (672.5) 588.1 (589.0) Pc 1.900 (1.878) 1.659 (1.633) 2.337 (2.355) Vc 709 (728) 843 (854) 533 (553) * Published values (without brackets), ** Predicted (inside brackets), 8 descriptors[1],[2] 1.01 Correlation Coefficient
1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0
20
40
60
80
100
120
Compound No.
Figure 1. Partial correlation coefficients in the group of compounds similar to ntetradecane.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Global bounds on optimal solutions in chemical process design U.-U. Hausa , J. Gangadwalab , A. Kienlebc∗ , D. Michaelsa , A. Seidel-Morgensternd , R. Weismantela a
Institut f¨ ur Mathematische Optimierung, Otto-von-Guericke-Universit¨at, Universit¨atsplatz 2, D-39106 Magdeburg, Germany
b Max-Planck-Institut f¨ ur Dynamik komplexer technischer Systeme, Sandtorstr. 1, D-39106 Magdeburg, Germany c
Institut f¨ ur Automatisierungstechnik, Otto-von-Guericke-Universit¨at, Universit¨atsplatz 2, D-39106 Magdeburg, Germany
d
Institut f¨ ur Verfahrenstechnik, Otto-von-Guericke-Universit¨at, Universit¨atsplatz 2, D-39106 Magdeburg, Germany In this paper a new approach for computing global bounds on optimal solutions of mixed-integer nonlinear programs is presented. These type of problems frequently arise in optimal design of chemical processes. The approach is based on a hierarchy of polyhedral relaxations leading to mixed-integer linear programs, which can be solved rigorously. Application is demonstrated for the optimal design of combined reaction distillation processes and for feasibility studies of simulated moving bed chromatographic processes. 1. Introduction The optimal design of chemical processes using mathematical optimization often leads to mixed-integer nonlinear programs (MINLP). Due to nonconvexity MINLP problems are usually difficult to solve. Typically either gradient based local optimization methods are used for this purpose or stochastic optimization methods like simulated annealing or genetic algorithms (see e.g. [8]). However, in both cases no guarantee can be given that the solution found by the algorithm is the global optimum. To overcome this problem, a new global approach is proposed in this paper. It is based on techniques to derive polyhedral approximations of the underlying nonlinear equations in such a way that a mixed-integer linear relaxation of the original problem is obtained, which can be solved rigorously. Since the feasible set of the original nonlinear problem lies within the feasible set of the relaxed problem, the latter provides global lower bounds for the optimal solution of the original problem. The lower bound approaches the true global optimum as the number of grid points of the relaxation is increased. The applicability of the approach is demonstrated for two different challenging fields. The first application is concerned with the optimal design of reaction distillation processes ∗
Author to whom all correspondence should be addressed. Email:
[email protected]
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involving reactor separator recycle systems, reactive distillation columns and side reactor concepts. The second application is concerned with the feasibility of simulated moving bed chromatographic processes. With the new method one can prove whether it is possible to meet given product specifications with a given process configuration and given column efficiency. 2. Methods Optimal design of chemical processes often leads to mixed-integer nonlinear programs, which can be written in the following generic form min f (x, y) s. t. h(x, y) g(x, y) x y
= ≤ ∈ ∈
0, 0, X ⊆ Rn , Y ⊆ Zd ,
(MINLP)
where f : Rn×d → R is a real function, h : Rn×d → Rq and g : Rn×d → Rp are vectors of real functions, X := {x ∈ Rn | Ax ≤ a} ⊆ Rn is a compact polyhedron, and Y := {y ∈ Zd | By ≤ b} ⊆ Zn is the set of integer point lying in a compact polytope. Several approaches for deriving bounds on the model (MINLP) have been suggested in the literature that we briefly survey below. Of major importance are the Generalized Bender Decomposition approach and the outer-approximation technique (see e.g. [3,4] and references therein). The two types of algorithms work in two phases: In the outer phase only the integer variables are manipulated. In each inner iteration a nonlinear subproblem is solved for the continuous variables for fixed values of the integer variables. The inner optimal solution is used to construct a relaxation in form of a second order problem to determine better values for the integer variables. Both approaches differ in the way the second order problem is constructed. Whereas Generalized Bender Decomposition methods use dual information, the outer-approximation approaches construct a mixedinteger linear relaxation for the primal problem. It is worth noting, that only in special cases these approaches yield global results. An important tool for treating general nonlinear functions occurring in model (MINLP) is to resort to convex underestimators for the corresponding functions. This ensures that relaxations of MINLP can be defined that only involve convex functions. Neglecting the integrality requirements yields a convex relaxation for model (MINLP) that is well tractable. Following this approach, interesting global bounds on the optimal value of synthesis problem can be given if one is able to derive good convex underestimators. For special functions convex underestimators have been given (see e.g. [2] and references therein). In this paper, we propose an alternative approach, where integrality requirements for the discrete variables y are explicitly taken into account. Instead of using convex underestimators a hierarchy of polyhedral relaxations is constructed. The construction of these relaxations is based on combinatorial substructures arising from the nonlinearities of the specific application. Each member of this hierarchy is a mixed-integer linear program in an extended space of variables and defines a global bound on the optimal value
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of the overall problem, because it contains all feasible solutions of the original nonlinear model. In the past decades several algorithmic techniques have been proposed to solve such MILPs in practice. Most notably, clever enumeration strategies in combination with advanced preprocessing and cutting plane techniques make instances tractable today that were out of any reach even ten years ago [1].
(a)
f(x)
(b)
Figure 1. For the nonlinear function Φ(x) on a box [l, u] in (a), a polyhedral relaxation for the graph of Φ(x) is given in (b): The domain is subdivided into four subregions. For each subregion a polyhedron is defined enclosing the graph of Φ(x). In order to apply this idea, it is crucial to decompose the function f and each component function of h and g, i. e., to express the function as the sum of functions φ1 . . . , φr . Next, one introduces a new variable for the value of φi . More precisely, the following relaxation procedure is applied as illustrated in Fig. 1: To this end, let (x, y)> ∈ Rn × Zd be a vector of n continuous variables xi ∈ [li , ui ], i = 1, . . . , n, and d integer variables yj ∈ {Lj , . . . , Uj }, j = 1, . . . , d, and consider a nonlinear function φ : Rn × Zd → R restricted Qd Q n d to D := ni=1 [li , ui ] × j=1 {Lj , . . . , Uj } ⊆ R × Z . The main steps of our relaxation procedure includes • a subroutine ANALYZE in which the nonlinear function φ(x, y) is investigated for local and global properties, e. g., local and global extrema, discontinuities, etc., using elementary differential geometry. • a subroutine BINMOD that returns a hierarchy of subdivisions of the domain D. Each subdivision consists of a set of support vectors. A subdivision that is higher in the hierarchy contains all the support vectors of the lower-level ones. • a subroutine POLYPROX that defines an enclosing (not interpolating) polyhedron for the graph of φ(x, y) on every subregion D ν based on the results of the subroutine ANALYZE. • a subroutine COMBINE that, based on combinatorial substructures given by linear (and nonlinear) relations between the variables occurring in φ(x, y), determines valid inequalities involving the corresponding binary variables. The hierarchy of subdivisions provided by subroutine BINMOD gives rise to a hierarchy of nonlinear mixed-integer optimization problems. Each such instance is an extended
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formulation of (MINLP) by introducing additional variables. It attains the following form Pr min k=1 fk φk (x, y) Pr for all j = 1, . . . , q, s. t. k=1 hj,k φk (x, y) = 0, Pr for all j = 1, . . . , p k=1 gj,k φk (x, y) ≤ 0, P νi i i P νi i i ≤ xi ≤ for all i = 1, . . . , n, ι=1 sι βι ι=1 sι+1 βι , (MDEF) P νi i β = 1, for all i = 1, . . . , n, ι=1 ι x ∈ [l, u] ⊆ Rn , y ∈ [L, U ] ∩ Zd , Qn νi β = (β 1 , . . . , β n )> ∈ i=1 {0, 1} Therein, the additional binary variables βιi indicate subintervals [siι , siι+1 ] ⊆ [li , ui ] for the original domain of xi . Such a division into subintervals can be represented as a list of supporting points, S i := {si1 , . . . , siνi +1 } ⊆ [li , ui ], with si1 = li , siνi +1 = ui , and siι Q < siι+1 , for all ι ∈ {1, . . . , νi }. In this manner, the original domain [l, u] is divided into ni=1 νi subboxes. Analytic and geometric properties of the underlying nonlinear functions φk for specific examples give rise to specific lists of supporting points S i , i = 1, . . . , n. In the integer programming community problems of the form (MDEF) involving linear functions only are known as fixed-charge mixed-integer optimization problems that include a large variety of applications (see [7]). In the remainder, focus will be on applications arising in the design of chemical processes. In order to tackle these problems we make use of a variety of new combinatorial relaxations. These arise from nonlinear flow conservations, mixed-integer Knapsack constraints and stable sets in special conflict graphs. 3. Applications 3.1. Optimal design of combined reaction separation processes The first application is concerned with the optimal design of combined reaction distillation processes, which play an important role in chemical industry. Depending on the physical properties of the system, the given constraints on production rate, product specifications, and the given costs different process candidates can be attractive including reactive distillation columns, or nonreactive distillation columns with side- and/or pre-reactors as illustrated in Fig. 2. In principle, the best process configuration can be found by optimizing some suitable superstructures, which include the relevant process alternatives. The objective are minimal total costs comprising investment and operating costs. This will lead to a complex problem of type (MINLP), which could be solved rigorously with a sequence of linear relaxations of type (MDEF) with successive refinement until the global optimum is reached. This however, can be computationally very expensive. Therefore a combined strategy is proposed, where a quick preliminary ranking of relevant process candidates is obtained with available local MINLP optimization methods. Afterwards the results can be checked
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(a)
(b)
(c)
Figure 2. Characteristic process alternatives for combined reaction distillation processes: (a) reactor with nonreactive distillation column and recycle, (b) reactive distillation column, (c) nonreactive distillation column with side reactors.
with global lower bounds obtained for each subproblem through polyhedral relaxation. The ranking of any two process candidates is either proven if the lower bound on the cost function of the second best candidate is greater than the known local solution of the best candidate. If that can not be achieved with successive refinement of the polyhedral relaxation, we may use the polyhedral relaxation to generate new starting values for the local optimization to find better global minima. Application of the procedure has been demonstrated for production of 2,3-dimethylbutene-1 by isomerization of 2,3-dimethylbutene-2. A detailed description of the mathematical formulation is given elsewhere [5]. 3.2. Feasibility of simulated moving bed chromatographic processes The second application is concerned with the feasibility of simulated moving bed (SMB) chromatographic processes. SMB is an advanced technology to separate isomers and in particular enantiomers. For high purity separations with highly efficient columns triangle theory developed by Storti et al. [9] is used for process design. Triangle theory is based on a true moving bed model and assumes thermodynamic equilibrium between the solid and the fluid phase corresponding to an infinite column efficiency. The theory allows to determine suitable values for the flow rate ratios mI , . . . mIV in the four zones of the process, which allow for complete separation of a given mixture with components A and B as illustrated in Fig. 3. Maximum productivity, i.e. maximum feed rate is obtained for values of mII and mIII at the vertex of the triangle (point W in Fig. 3. The triangle represents the feasible region for complete separation. It should be noted that for infinitely efficient columns total separation is always possible for mixtures with different adsorptivities, which can be described with linear or Langmuir isotherms. Or, in other words, in this case the feasible region is always non empty, and, in particular a maximum value for the feed rate different from zero can be found. However, in practice often lower purities are acceptable and cheaper columns with reduced efficiency can be applied. In this case, the question arises whether a given product
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Figure 3. Characteristic process alternatives for combined reaction distillation processes: (a) reactor with nonreactive distillation column and recycle, (b) reactive distillation column, (c) nonreactive distillation column with side reactors.
purity can be achieved with a given column efficiency corresponding to a given number of theoretical stages in our model formulation. To answer this questions polyhedral relaxations are applied to an optimization problem, where we maximize the feed rate for given purity constraints. By means of polyhedral relaxations on upper bound for the optimal feed rate is obtained. If this upper bound is zero, than it is proven that it is not possible to achieve the required purity for any values of mI , . . . mIV . Application has been demonstrated for a process separating a mixture of fructosedextran T9 and fructose-raffinose. A detailed description of the mathematical formulation is given elsewhere [6]. References 1. D. Bertsimas and R. Weismantel. Optimization over Integers. Dynamic Ideas, Belmont, Massachusetts, 2005. 2. M. Tawarmalani and N. V. Sahinidis. Math. Program., 99(3, Ser. A):563–591, 2004. 3. C. A. Floudas. Nonlinear and Mixed-Integer Optimization. Oxford University Press, New York, Oxford, 1995. 4. I. E. Grossmann. Global Optimization in Engineering Design. Kluwer Academic Publishers, 1996. 5. J. Gangadwala, A. Kienle, U.-U. Haus, D. Michaels, and R. Weismantel. in press Ind. Engng. Chem. Res. 6. U.-U. Haus, D. Michaels, A. Seidel-Morgenstern, and R. Weismantel. In preparation. 7. H. Marchand, A. Martin, R. Weismantel, and L. A. Wolsey. Discrete Appl. Math., 123:397–446, 2002. 8. J. Marriott and E. Sorensen. Chem. Engng. Sci., 58:4991–5004, 2003. 9. G. Storti, M. Mazzotti, M. Morbidelli, and S. Carra. AIChE J., 39:471–492, 1993. Acknowledgment. This research was supported by the German Science Foundation (DFG) within the joint research project under grants FOR 468 and KI 417/1.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Stochastic Grey Box Modeling of the enzymatic biochemical reaction network of E. coli mutants Florin Paul Davidescua , Henrik Madsenb , Michael Sch¨ umperlic , Matthias Heinemannc , c a ∗ Sven Panke and Sten Bay Jørgensen . a CAPEC, Department of Chemical Engineering, Technical University of Denmark, Building 227, DK 2800, Kgs, Lyngby, Denmark b
Department of Informatics and Mathematical Modeling, Technical University of Denmark, Building 321, DK 2800, Kgs, Lyngby, Denmark
c
Institute of Process Engineering, Bioprocess Laboratory, ETH Z¨ urich, Sonneggstr. 5, CH 8092, Z¨ urich, Switzerland This paper describes the application of a gray-box stochastic modeling framework for developing stochastic state space models for dynamic systems based on combining first principle models and experimental data. The framework is used to develop reliable predictive models for a biochemical reaction network isolated from E. coli mutants. The modeling purpose is to use the model to identify the bottlenecks in the reaction network to enable optimizing the production of the desired product through genetic manipulation. 1. Introduction There is an increasing interest in producing complex fine chemicals and intermediates in the pharmaceutical industry using biochemical synthesis. Up to now, only one or a few biotransformation steps are involved in complex synthesis problems in industry, although enzymes are widely known as being specific, fast and working under mild conditions. To develop a purely enzymatic synthesis for complex molecules from completely different substrates, large reaction networks are necessary. One way to construct such a functional network is the System of Biotransformations (SBT). The SBT is based on one single organism’s metabolic network containing the synthesis path including cofactor regeneration reactions in an isolated manner. Thereby, the SBT is performed as cell free extract in the production phase, combining the easy handling of a viable culture with the advantages of in vitro biotransformations [5]. The complexity of such large biochemical reaction networks involves a large number of reaction steps with many metabolites and enzymes, each one of them playing different roles as biocatalysts and/or as feed-forward and feed-back regulators. The general goal of this study is to identify the limitations and bottlenecks, to reduce them and to optimize the productivity of the selected reaction network. The workhorse of the de-bottlenecking and optimization process is a model describing the
∗
Corresponding author
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biochemical reaction network with good long term prediction properties. For this particular application, the key product is Dihydroxyacetone phosphate (DHAP ). DHAP is an important precursor for the production of phosphorylated, non natural carbohydrates. Thereby, the DHAP-producing SBT contains all the reactions of the glycolysis, leading to a system of high dynamic and complexity. Therefore, it is not realistic to develop a ”perfect model” from first principle engineering methods. For this reason, in this work the gray-box stochastic model development framework [1] will be used to develop a stochastic state space model. The purpose of this paper is to describe the work-flow driving application of the gray-box stochastic modeling framework for development of a kinetic model for a batch reaction network. 2. Stochastic gray-box modeling background methodology The gray-box stochastic modeling framework, [1] was originally developed for fed-batch cultivations but it can be employed for modeling of complex nonlinear dynamic processes as well. The framework combines different mathematical and statistical tools and assists the model development in a systematic way. First, the model equations are derived from first engineering principle and then completed with diffusion terms/functions to obtain the Stochastic Differential Equations. The diffusion terms accounts for model errors and/or for the un-modeled effects. Formulating the diffusion terms by only having the diagonal terms in the square matrix of the diffusion terms ([1]) it is possible to improve the process model in a systematic way. The measurement equations include the measurements errors as well, thus in this approach it is possible to make a clear distinction between the measurement errors and process noise or model error. In the next step the set of unknown parameters together with the diffusion terms and the variances of the measurements are estimated from experimental data using a maximum likelihood or a maximum a-posteriori method. In the estimation method it will just be mentioned that the solution of the stochastic differential equation system and the innovation terms that appears in the maximum likelihood function is based on an Extended Kalman Filter. The model is un/falsified using different statistic tests. Then the model is reformulated and the iterations continued until the model is un-falsified using available data or all the information contained in the data with respect to the dynamics is exhausted. A workflow diagram of the whole modeling framework is given in figure 1.
Figure 1. Stochastic gray-box modelling framework, from [1]
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3. Experimental data and procedures The experiments have been conducted by ref [5]. In phase I, fed-batch (semi-batch) fermentations of E. coli W3110 tpi are conducted until the optical density (OD600) in the bioreactor reaches a preset value. The broth is centrifuged and the cells are resuspended in SBT-buffer (100 mM HEPES, 0.84 mM KCl, 1 mM ZnSO4 and at pH = 7). The cells are disrupted by high-pressure homogenization. The remaining solids are eliminated by centrifugation/filtration and the liquid extract is recovered. The total protein concentration is determined Bradford and adjusted to the desired concentration by dilution with SBT buffer. The liquid extract contains the enzymes and compounds present in the cell at the time when the fermentation was stopped. In phase II, a volume of 5 ml of SBT extract is used for each experiment. Defined amounts of Hexokinase, (HK) and Lactate − DH as well as AT P and N AD+ , are added. The reactions are initiated by glucose. Samples are collected according to a previously defined time plan. The experiments are terminated after 300 minutes. First, the proteins are removed by precipitation with HCl followed by centrifugation. The samples are analyzed by enzymatic assays. Glucose and glucose-6-phosphate (G6P ) are determined together by addition of both HK and glucose-6-phosphate-dehydrogenase to form N ADP H, which is determined spectrophotometrically. DHAP is determined by addition of glycerol-3-phosphate-dehydrogenase and measuring the N ADH consumption spectrophotometrically. A series of four experiments has been used for the model development; three for parameter estimation and one for model validation.
4. Model development for an SBT isolated from E. coli mutants In the model formulation the first measurement equation was assigned to glucose. The first step in the model development is model formulation. In order to formulate a model the existing biochemical reaction network in E-coli is presented with focus on the reactions around the product of interest (DHAP ) considering the genes which are knocked out. The simplified biochemical reaction network used for model development is depicted in figure 2. The reaction between DHAP and G3P does not take place since the tpi gene i.e. responsible for the expression of the enzyme catalyzing the reaction has been knockedout. For the current version of the model all the reactions from glucose to fructose1,6-biphosphate, (F 16B) were lumped into a single reaction r1 . The second reaction considered is the reaction from F 16B to G3P and DHAP , r2 catalyzed by aldolase. The reactions consuming the G3P down to pyruvate in the central carbon metabolism were all lumped into one single reaction r3 . The reaction producing lactate from pyruvate was included as reaction r4 . The reason to include these two reactions is that it is desirable to account for the consumption-production of co-factors AT P and N AD+ . The model consists of dynamic mass balances for all the species involved in the four reactions plus one for each of the two co-factors. The model equations eq. 1–10 have been completed with the diffusion terms as mentioned above. In this first model formulation it has been considered that the reaction rates r1 − r4 are constant and then estimated together with the model parameters and with the initial values of the states
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= = = = = = = =
−r1 + σ11 · dw r1 − r2 + σ22 · dw r2 + σ33 · dw r2 − r3 + σ44 · dw r3 − r4 + σ55 · dw r4 + σ66 · dw −2 · r1 + 2 · r3 + σ77 · dw −r3 + r4 + σ88 · dw
(1) (2) (3) (4) (5) (6) (7) (8)
Measurements equations: Figure 2: Reaction netyGLC = cGL + e, e ∈ N (0, S11 ) work used for model deyDHAP = cDHAP + e, e ∈ N (0, S22 ) velopment
(9) (10)
The parameters have been estimated using three of the available data sets and gray-box stochastic modeling software CTSM [2]. After the estimation step, significance tests have been performed for the estimates and the parameters estimate correlation matrix has been calculated as well. Several parameters were insignificant or highly correlated, therefore some parameters have been fixed and the remaining parameters reestimated. The fit for one step ahead prediction as well as pure simulation has been plotted in figure 3 and the numerical results for the parameters are given in table 1. The fit for pure simulation data, clearly indicates that the model needs improvement. Inspecting the diffusion terms in table 1 shows that the corresponding σ11 − σ33 are significant, thus the drift terms of these equations are deficient. Following the methodology mentioned above r1 and r2 are included in the state vector one at a time. After including r1 and r2 consecutively, the parameters have been reestimated. At this step in the model development it is necessary to see how we should model the reaction rate r1 . Before the nonparametric methods are applied, it is necessary to reconstruct the states of the model with r1 as extra state. This is done by applying the extended Kalman filter (EKF) with the parameter estimates obtained after r1 was included as a new state. The nonparametric tools e.g. additive models [1,3] are applied in order to identify the shape of the kinetic expression for the reaction rate r1 . Analyzing the reaction network in figure 2, the reaction rate may be a function of glucose, AT P , F 16B concentrations. The graphical results (not shown) indicate that the dependence of r1 versus cGLC appears to be the most significant. In figure 4 this dependence solely, is shown. The same steps have been applied for the second reaction rate, r2 . In this case r2 depends of cG3P and cDHAP as shown in figures 5–6, while the dependence of cF 16B seems to be insignificant (not shown). Considering the shape in figure 4 for the functional dependence of r1 on glucose, then reaction rate r1 can be modeled using Monod kinetics (eq. 11). The parameters are reestimated assuming Monod kinetics (eq. 11) for r1 . The one-step ahead prediction as well
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Table 1 Estimated parameters, standard deviation, t-test, and the in/significance Name cGLC0 cDHAP 0 cAT P 0 r1 r2 σ11 σ22 σ33
Estimate 1.0072E+01 8.5785E-02 1.1619E+01 6.5923E-02 3.9095E-02 4.9197E-01 1.0000E-02 1.0440E-01
Figure 3. cGLC and cDHAP vs. time, exp. data: x, pure simulation: continuous line, one step ahead pred.: dashed
Std. dev. 6.0307E-01 1.2678E-01 7.4897E+01 1.1838E-02 2.5091E-03 3.6057E-02 3.4324E-04 7.4442E-03
Figure 4. r1 vs. cGLC , exp. data: x, local fit: continuous line and 95% conf. intervals: dashed
t-score 16.7004 0.6767 0.1551 5.5688 15.5811 13.6442 29.1343 14.0240
signif.? yes no no yes yes yes yes yes
Figure 5. r2 vs. cG3P , exp. data: x, local fit: continuous line and 95% conf. intervals: dashed
as the pure simulation of the model has improved considerably for the first measurement, see figure 7. r1 = r1max ·
cGLC Ks1 + cGLC
(11)
After modeling reaction rate r1 (eq. 11), reaction rate r2 is included again as a new state and the parameters re-estimated. Using the new set of parameters, states estimation and nonparametric modeling tools are applied again. The individual dependences of r2 on the cG3P , cDHAP and cF 16B seems to be similar with the data obtained before modeling r1 Literature references ([4]), mentions that the reaction rate is related to the equilibrium constant, thus a dependence of the reaction rate on the difference between the forward and the backward reaction was investigated by regressing a dependence on the cF 16B −cG3P ·cDHAP (not shown). Again, the dependence looks like a Monod term but
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Figure 6. r2 vs. cDHAP , exp. data: x, local fit: continuous line and 95% conf. intervals: dashed
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Figure 7. cGLC and cDHAP vs. time, exp. data: x, pure simulation: continuous line, one step ahead pred.: dashed
Figure 8. cGLC and cDHAP vs. time after modeling r1 and r2 , exp. data: x, pure simulation: continuous line, one step ahead pred.: dashed
with this abscissa. The parameters have been reestimated, and the fit has been improved for the second measurement as can be seen in figure 8. 5. Conclusions A gray-box stochastic model for a E. coli extract reaction network is under development. The model development is performed by the application of the gray-box stochastic modeling framework proposed by Kristensen [1]. The current results looks promising and now the focus is in developing and using specific experiments to provide information on different reaction kinetics in the metabolic network. Once the complete reaction network presented in figure 2 is reasonably modeled, productivity optimization will be investigated. REFERENCES 1. Niels Rode Kristensen, Henrik Madsen and Sten Bay Jørgensen, A method for systematic improvement of stochastic gray-box models, Comp. and Chem. Eng., 28 (2004), 1431-1449. 2. Niels Rode Kristensen, Henrik Madsen and Sten Bay Jørgensen, Parameter estimation in stochastic gray-box models, Automatica, 40 (2004), 225-227 . 3. Trevor Hastie and Robert Tibshirani, Bayesian backfitting, Stat. Sci. 15 (2000) 196213. 4. Christophe Chassagnole and others, Dynamic modeling of the central carbon metabolism of Escherichia coli, Biotechnology and Bioengineering, 79, No. 1 (2002) 53-73. 5. Michael Sch¨ umperli, Matthias Heinemann, Anne K¨ ummel, and Sven Panke, in preparation, 2005
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Validated solution of ODEs with parametric uncertainties Youdong Lina and Mark A. Stadtherra a
Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN 46556 USA Abstract We introduce a new methodology for computing a validated enclosure of all solutions of an ODE system with interval-valued parameters and/or initial values. The method is based on a traditional interval approach, but involves a novel use of Taylor models to address the dependency problem of interval arithmetic. Numerical results on a bioreactor kinetics problem, with uncertain initial biomass concentration and uncertain kinetic parameters, demonstrate that this approach provides a very effective way to obtain an enclosure of all possible solutions to a parametric ODE system under uncertain conditions. Keywords: ODE, Validated computing, Parametric uncertainty, Bioreactor kinetics 1. Introduction Initial value problems (IVPs) for ordinary differential equations (ODEs) arise naturally in many applications in process engineering. However, it is often the case that the parameters and/or the initial values are not known with certainty. One common way to represent the uncertainties in such parametric ODEs is to treat the parameters and/or initial values as intervals. Thus, consider the problem of determining a validated enclosure of all solutions of the parametric autonomous IVP, y 0 (t) = f (y, θ),
y(t0 ) = y 0 ∈ Y 0 ,
θ ∈ Θ,
(1)
where t ∈ [t0 , tm ] for some tm > t0 . Here θ is a p-dimensional vector of time-invariant parameters, y is the n-dimensional vector of state variables, and y 0 is the n-dimensional vector of initial values. The interval vectors Θ and Y 0 represent enclosures of the uncertainties in θ and y 0 , respectively. Interval methods [1] (also called validated or verified methods) for ODEs can not only determine a guaranteed error bound on the true solution, but can also verify that a unique solution to the problem exists. Traditional interval methods usually consist of two processes applied at each integration step [1]. In the first process, existence and uniqueness of the solution are proven using the Picard-Lindel¨of operator and the Banach fixed point theorem, and a rough enclosure of the solution is computed. In the second process, a tighter enclosure of the solution is computed. In general, both processes are realized by applying interval Taylor series (ITS) expansions with respect to time, and
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using automatic differentiation to obtain the Taylor coefficients. An excellent review of the traditional interval methods has been given by Nedialkov et al. [2]. For addressing this problem, there are various packages available, including AWA [3], VNODE [4] and COSY VI [5], all of which consider uncertainties in initial values only. In the work described here, we will describe a method for efficiently determining validated solutions of ODEs with parametric uncertainties. The method makes use, in a novel way, of the Taylor model approach that Makino and Berz [6] used to deal with the dependence problem in interval arithmetic, and which they applied in COSY VI [5]. 2. Taylor Models Makino and Berz [6,7] have described a remainder differential algebra (RDA) approach for bounding function ranges and controlling the dependency problem of interval arithmetic. This method expresses a function by a model consisting of a Taylor polynomial, usually a truncated Taylor series, and an interval Taylor remainder bound. Consider a function f : x ∈ X ⊂ Rm → R that is (q + 1) times partially differentiable on X. Based on the Taylor expansion about the point x0 ∈ X, the Taylor model of f (x) then consists of a q-th order polynomial function in (x − x0 ), pf , pf =
q X 1 [(x − x0 ) · 5]i f (x0 ) , i! i=0
(2)
and an interval remainder bound Rf , evaluated here in interval arithmetic, Rf =
1 [(X − x0 ) · 5]q+1 F [x0 + (X − x0 )Ξ] , (q + 1)!
(3)
and is denoted Tf = (pf , Rf ), where Ξ = [0, 1], and [g · 5]k is the operator [g · 5]k =
X j1 +···+jm =k 0≤j1 ,··· ,jm ≤k
k! ∂k jm g1j1 · · · gm . j1 j 1 ! · · · jm ! ∂x1 · · · ∂xjmm
(4)
Arithmetic operations with Taylor models can be done using the RDA operations described by Makino and Berz [6,8], which include addition, multiplication, reciprocal, and intrinsic functions. Therefore, for any function representable in a computer environment, it is possible to compute a Taylor model using RDA operations by simple operator overloading. In performing RDA operations, only the coefficients of pf are stored and operated on. Computation of bounds on Tf over X is denoted by B(Tf ). It has been shown that, compared to other rigorous bounding methods, the Taylor model can be used to obtain sharper bounds for modest to complicated functional dependencies [6,7,9]. 3. Validated Solver for Parametric ODEs The method proposed here uses the traditional two-phase approach, but makes use of Taylor models to deal with the dependency problem arising due to the uncertain quantities (parameters and initial values). In phase 1, the goal is to find a step size hj = tj+1 −tj > 0,
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Algorithm 1 Phase2 (In: Tˆ yj , Aj , V j , hj , Y˜ j , Y j ; Out: T yj+1 , Tˆ yj+1 , Aj+1 , V j+1 ) [k] 1: Z j+1 = hkj F (Y˜ j , Θ) (with Y˜ j from phase 1) k−1 ˆ y + P hi T [i] + Z j+1 2: T U j+1 = T j f ˆ j 3:
Sj = I +
k−1 P
i=1
hij J(f [i] ; Y j , Θ)
i=1
Aj+1 = (m(S j Aj ))−1 ˆ y , RU ) ⇐ T U , with m(RU ) = 0 5: (T j+1 j+1 j+1 j+1 −1 6: V j+1 = (A−1 j+1 S j Aj )V j + Aj+1 RU j+1 ˆ y + Aj+1 V j+1 7: T yj+1 = T j+1 4:
and a rough enclosure Y˜ j of the solution such that existence and uniqueness of the solution can be verified. We apply the traditional interval method to the parametric ODEs by using 0 an ITS with respect to time; that is, for Y j ⊆ Y˜ j , hj and Y˜ j are determined such that
Y˜ j =
k−1 X
0 0 [0, hj ]i F [i] (Y j , Θ) + [0, hj ]k F [k] (Y˜ j , Θ) ⊆ Y˜ j ,
(5)
i=0
where F [i] is the interval extension of f [i] , the i-th Taylor coefficient in terms of y 0 (t) = f(y, θ). Eq.(5) demonstrates that there exists a unique solution y(t; tj , yj , θ) ∈ Y˜ j for all t ∈ [tj , tj+1 ], any y j ∈ Y j , and any θ ∈ Θ. In phase 2, we compute a tighter enclosure Y j+1 ⊆ Y˜ j such that y(tj+1 ; t0 , Y 0 , Θ) ⊆ Y j+1 . This will be done by using an ITS approach to compute a Taylor model T yj+1 of y j+1 in terms of the uncertain quantities (initial values and parameters), and then obtaining the enclosure Y j+1 = B(T y j+1 ). For the Taylor model computations, we begin by representing the interval initial values y 0 ∈ Y 0 by a Taylor model with components Ty0i = (m(Y0i ) + (y0i − m(Y0i )), [0, 0]),
i = 1, · · · , n,
(6)
and the interval parameters θ ∈ Θ by a Taylor model with components Tθi = (m(Θi ) + (θi − m(Θi )), [0, 0]),
i = 1, · · · , p.
(7)
Then, we can determine the Taylor model T f [i] of the ITS coefficients f [i] (y j , θ) by using RDA operations to compute T f [i] = f [i] (T yj , T θ ). The algorithmic procedure of phase 2 is summarized in Algorithm 1. The procedure begins with V 0 = 0, Tˆ y 0 = (m(Y 0 ) + (y 0 − m(Y 0 )), [0, 0]), and A0 = I. J (f [i] ; Y j , Θ) denotes the interval extension of the Jacobian of f [i] over y j ∈ Y j , and θ ∈ Θ.
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Table 1 Bioreactor microbial growth parameters Parameter α k D Si S0
Value 0.5 10.53 0.36 5.7 0.80
Units g S/ g X day−1 g S/l g S/l
Parameter µm KS KI X0
Value [1.19, 1.21] [7.09, 7.11] [0.49, 0.51] [0.82, 0.84]
Units day−1 g S/l (g S/l)−1 g X/l
4. Numerical Experiments In a bioreactor, a simple microbial growth process [10], which involves a single biomass and single substrate, can be described using the following ODE model, X˙ = (µ − αD)X S˙ = D(S i − S) − kµX,
(8) (9)
where X and S are concentrations of biomass and substrate, respectively; α is the process heterogeneity parameter; D and S i are the dilution rate and the influent concentration of substrate, respectively; k is the yield coefficient; and µ is the growth rate, which is dependent on S. We consider two models for µ, the Monod law, µ=
µm S KS + S
(10)
and the Haldane law, µ=
µm S KS + S + K I S 2
(11)
where µm is the maximum growth rate, KS is the saturation parameter, and KI is the inhibition parameter. In this study, the initial value of biomass concentration X0 , and the process kinetic parameters (µm , KS , and KI ) are assumed to be uncertain and given by intervals. Thus, for the Monod law, there are three uncertain quantities, and four for the Haldane law. The values of the initial conditions (X0 , S0 ), the inputs (D and S i ), and parameters (α, k, µm , KS , and KI ) are given in Table 1. We now report experimental results of a C++ implementation of the method described above. This implementation is called VSPODE (Validating Solver for Parametric ODEs). The results for VSPODE were obtained using a k = 17 order interval Taylor series method, and with a q = 5 order Taylor model. For purposes of comparison, as a representative of traditional interval methods, we use the popular VNODE package [4], with a k = 17 order interval Hermite-Obreschkoff QR method. Though, like other available solvers, VNODE is designed to deal with uncertain initial values, it can take interval parameter values as input. However, better performance can be obtained by treating the uncertain parameters as additional state variables with zero time derivatives; thus the parametric uncertainties become uncertainties in the initial values of the extra state variables. All tests were done using a constant step size of h = 0.1, and were performed on a workstation running Linux with an Intel Pentium 4 3.2GHz CPU.
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1.5 ‹S
1.4
VNODE
S
1.3
VSPODE
X/S
1.2 ‹ XVNODE
1.1 1 0.9
X
VSPODE
0.8 0
5
10 t
15
20
15
20
Figure 1. Enclosures for the Monod law 1.5 1.4
SVSPODE ‹ SVNODE
X/S
1.3 1.2 1.1
‹ XVNODE
1 0.9
X
VSPODE
0.8 0
5
10 t
Figure 2. Enclosures for the Haldane law The enclosures computed for t ∈ [0, 20] using VSPODE and VNODE for the Monod law and the Haldane law, are shown in Fig. 1 and Fig. 2, respectively. VSPODE clearly provides a better enclosure, with VNODE failing at t = 9.3 for the Monod law, and at t = 6.6 for the Haldane law. In order to allow VNODE to solve the problem all the way to tm = 20, we divided the intervals into a number of equal-sized sub-boxes and then used VNODE to determine the solution for each sub-box. The final solution enclosure is then the union of all the enclosures resulting from each sub-box. Results showing the final solution enclosures (tm = 20) and their widths, as determined using VSPODE (with no box subdivision) and VNODE with an increasing number of sub-boxes, are given in Table 2 for the Monod law. For example, VNODE-1000 in Table 2 indicates the use of 1000 sub-boxes in VNODE. Even with 1000 sub-boxes, the solution enclosure determined by VNODE is still significantly wider than that obtained from a single calculation with VSPODE, and requires about 200 times more computation time.
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Table 2 Results for the Monod law, showing final enclosures (tm = 20). Method VSPODE
[ [ VNODE–343 [ [ VNODE–512 [ [ VNODE–1000 [ [
Enclosure 0.8386, 0.8450 1.2423, 1.2721 0.8359, 0.8561 1.2309, 1.2814 0.8375, 0.8528 1.2331, 1.2767 0.8380, 0.8502 1.2359, 1.2732
] ] ] ] ] ] ] ]
Width 0.0064 0.0298 0.0202 0.0505 0.0153 0.0436 0.0122 0.0373
CPU time (s) 1.34 68.6 102.8 263.1
5. Concluding Remarks We have described a new method for obtaining validated solutions of initial value problems for ODEs with interval-valued initial conditions and parameters. The dependence of y 0 (t) = f (y, θ) on t is handled using ITS methods, as in VNODE [4]. However, the dependence on the initial state y 0 and the parameter vector θ is handled by using, in a novel way, Taylor models of the form described by Makino and Berz [6,8]. Numerical results on a bioreactor kinetics problem demonstrate that this approach provides a very effective way to obtain an enclosure of all possible solutions to a parametric ODE system under uncertain conditions. Acknowledgment. This work was supported in part by the State of Indiana 21st Century Research and Technology Fund and by the U. S. Department of Energy. References 1. R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966. 2. N. S. Nedialkov, K. R. Jackson and G. F. Corliss, Appl. Math. Comput., 105(1999), 21. 3. R. J. Lohner, Computations of guaranteed enclosures for the solutions of ordinary initial and boundary value problems, In Computational Ordinary Differential Equations, J. Cash and I. Gladwell (eds), Clarendon Press, Oxford, 1992. 4. N. S. Nedialkov, K. R. Jackson and J. D. Pryce, Reliab. Comput., 7(2001), 449. 5. M. Berz and K. Makino, Reliab. Comput. 4(1998), 361. 6. K. Makino and M. Berz, Remainder differential algebras and their applications, In Computational Differentiation: Techniques, Application, and Tools, M. Berz, C. Bishof, G. Corliss and A. Griewank(eds), SIAM, Philadelphia, 1996. 7. K. Makino and M. Berz, Reliab. Comput. 5(1999), 3. 8. K. Makino and M. Berz, Int. J. Pure Appl. Math., 4(2003), 379. 9. A. Neumaier, Reliab. Comput. 9(2002), 43. 10. G. Bastin and D. Dochain, On-line Estimation and Adaptive Control of Bioreactors, Elsevier, Amsterdam, 1990.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Optimal experimental design for ill-posed problems André Bardow a,b a
Institute of Polymers, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland Lehrstuhl für Prozesstechnik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
b
Abstract Modern high-resolution measurement techniques offer the possibility to determine unknown functional dependencies directly from the data. The underlying inverse problems, however, are much more demanding than standard parameter estimation. Still, systematic strategies for experimental design of such ill-posed problems are missing. A new approach is proposed here that in particular achieves the sound integration of the bias-variance trade-off critical to the solution of ill-posed problems. The new design approach is based on the minimization of the expected total error (ETE) between true and estimated function. The ETE design approach is exemplified for the classical example of determination of reaction rates from measured data. Keywords: experimental design, inverse problem, parameter estimation, reaction kinetics, numerical differentiation.
1. Introduction In model-based experimentation, the goal is often to extract an unknown functional relationship from the data. Standard examples are e.g. reaction rates or phase equilibria as function of the state variables. The usual approach is to reduce the problem complexity: first, a model structure (or several candidates) is specified; then the unknown parameters contained are determined from experiments [1]. However, it would often be desirable to avoid the separation of the problem in two parts and to determine the unknown function directly. With the advent of high-resolution measurement techniques, modern process information management systems and advanced mathematical methods (e.g. data mining) this direct route is now becoming increasingly feasible [2]. Still, the identification of unknown functions represents an infinitely dimensional inverse problem. In addition, these problems are generally ill-posed, i.e. the solution is not unique or does not depend continuously on the data [3]. The solution of ill-posed problems for function estimation therefore poses much higher requirements on the data than standard parameter estimation problems where a finite number of parameters are determined in a known model structure. Despite the increased complexity, the systematic generation of optimal experimental conditions for ill-posed problems has received only little attention. Model-based optimal design theory for parameter estimation, pioneered by Box & Lucas [4], is now well established. The approaches available for ill-posed problems are generally direct extensions of these classical design methods [5,6,7]. Since they are set in the maximum likelihood framework they assume unbiased estimates. However, in the solution of illposed problems, bias is systematically introduced to stabilize the problem. The trade-off between variance and bias is then the key element [3]. A sound approach to optimal
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experimental design for ill-posed problems therefore has to incorporate this trade-off. However, none of the approaches currently available includes the bias effect. A new design criterion for ill-posed problems is therefore introduced in this work. The criterion minimizes the statistically expected total error (ETE) between the true and the estimated function. It thus incldues both error contributions: bias and variance. The new criterion is derived next. Estimation of reaction rates from experimental data is then considered as an example application. A discussion of the new approach concludes this paper.
2. Design criterion for ill-posed problems In order to limit the discussion to the essence of the method only linear problems are considered. Nonlinear problems can be treated using proper linearization as in standard design theory [4,5]. Linear ill-posed problems are often obtained from integral equations [3]
g (t ) = ∫ K (t , s; d ) f ( s )ds ,
(1 )
T
where f(t) is the unknown function to be identified from the measured data g(ti). Data is usually available only at discrete points ti and corrupted by measurement errors (assumed here to be Gaussian with zero mean and variance σ2). The kernel function K(t,s;d) is generally known from theory and contains also the design parameters d that can be chosen by the experimenter. It is the goal of experimental design to find the optimal settings for these parameters. For the solution of the inverse problem, direct inversion of Eq. (1) would lead to unstable solutions. Therefore, regularization methods have to be employed. The most common approach is Tikhonov regularization where the estimate for the unknown function f is determined as [3] 2
n ⎞ 1 ⎛ fˆ = arg min ∑ 2 ⎜⎜ g (t i ) − ∫ K (t i , s; d ) f ( s )ds ⎟⎟ + λ Lf i =1 σ i ⎝ T ⎠
2 L2
.
(2 )
Here, the first term is the data error. The second term represents a penalty ensuring smoothness. For the operator L, the identity or the second derivative are frequently used. The regularization parameter λ gives the relative weight to both contributions of the objective.
The goal of a successful experiment should be that the estimate fˆ is as close as possible to the true solution f. The expected value for the total error (ETE) between the Tikhonov estimate and the true function can be computed as [8]
E ⎛⎜ f − fˆ ⎝
⎞ = f − K −1 (λ ) Kf ⎟ L2 ⎠
2
(
2 L2
)
−1
(
where K (λ ) = K K + λL L K . −1
T
T
(
)) T
+ σ 2trace K −1 (λ ) K −1 (λ ) , T
(3 )
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Assuming that an initial guess of the true solution and the measurement error is available it is therefore proposed here to obtain the optimal experimental design from minimizing the expected total error with respect to the design variables d. Thus, the optimal design d* is determined from ETE criterion
min E ⎛⎜ f − fˆ d ,λ ⎝
2 L2
⎞. ⎟ ⎠
(4 )
The first term of the ETE criterion in Eq. (3) reflects the bias introduced by the penalty term whereas the second term summarizes the variance in the estimate. Thus, the biasvariance trade-off is properly incorporated into the new ETE design criterion. The regularization parameter λ integrates naturally as an additional free variable of the design optimization problem (4) and is determined along with the experimental settings. The ETE criterion thus provides a consistent rule to determine λ. Previous approaches had to rely on a priori knowledge [5,6]. Discretization of Eq. (3) is not critical. A simple trapezoidal scheme is usually sufficient since the discretization error is typically much smaller than regularization and data error [8].
3. Example: Identification of reaction rates – Numerical differentiation The specific merits of the new approach are discussed in the light of an example. For this purpose, the determination of reaction rates as function of time from measured concentration data is considered [9]. Such model-based reaction rate measurements typically form the starting point for the identification of constitutive equations [2]. The core of the underlying mathematical problem is the differentiation of experimental data. This by itself is a standard problem in chemical engineering beyond the area of reaction kinetics since often not the measured quantity itself but its derivative is of interest. In practice, the finite difference scheme is often employed to determine the unknown derivative f=dg/dt from the measurements g(ti) [9]. Equidistant measurements with sampling interval dt=ti-ti-1=const. are assumed here. The discretized kernel K is then a lower triangular matrix with all entries identical to dt [6]. In an experiment, the sampling interval dt can be chosen by the experimenter himself. It thus serves as design parameter. It is well known that if the sampling is too coarse the approximation will be poor. However, in the inverse problem, too fine sampling can also lead to an amplification of the error since the measurement noise will corrupt the result and the variance increases [3]. The ETE design criterion (3) is now applied to determine the optimal sampling interval for finite differences. No additional regularization parameter λ is required as the sampling interval itself has a regularizing effect. For the sound incorporation of the bias effect, the first term in the objective (3) is therefore computed using interpolation of the estimated solution on a finer grid. In the example, a first-order reaction, leading to an exponential decay, serves as true function, i.e. f=exp(-10t). Measurement standard deviation is σ=0.01. The ETE design objective (3) is shown as function of the sampling interval dt in Fig. 1. The new criterion shows the expected behavior for the ill-posed problem. The optimal sampling time is found as the trade-off point between bias and variance contribution. Variance dominates the error for small time steps while bias increases for large time steps.
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0
6
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4
-2
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2
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0
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-4
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total error bias variance 0
0.02
0.04
0.06
0.08
-2
E-optimal criterion
expected error
10
10
-4
10 0.1
dt Fig. 1: ETE design objective as function of sampling interval dt. Bias (dotted) and variance (dashed) contributions to the objective are also shown (left axis). The E-optimal design criterion (thin full line) is shown on the right axis.
Criteria proposed previously for the design of ill-posed problems [5,6,7] solely focus on the variance contribution. In these methods, the so-called Fisher information matrix is usually introduced as a variance measure. The Fisher matrix corresponds here to the term inside the trace in Eq. (3). As an example for these design criteria, the E-optimal experimental design criterion [5,6] is plotted on the right axis in Fig. 1. In E-optimal design, the smallest eigenvalue of the Fisher information matrix is maximized [10]. It can be seen that the classical design criteria suggest the use of the maximum sampling time. Thus, these criteria are not able to reflect the specific nature of ill-posed problems. In order to assess the quantitative accuracy of the ETE criterion a simulation study was performed. Simulated measurement data was corrupted with random noise and the finite difference scheme was applied to this data. The average deviation from the true signal was then evaluated and averaged over 10,000 replications. The average error is shown in Fig. 2. It can be seen that the ETE criterion truly captures the behavior found in the actual experiment. The predicted optimal sampling time is slightly larger than the value found in the simulation study which adds to the robustness of the estimate. In summary, it can be concluded that the ETE criterion is able to find the best sampling time with good accuracy. The example of numerical differentiation studied here is well suited to show the specific properties of the new approach. However, it is also special since the design variable, the sampling time, serves at the same time as implicit regularization parameter. The success of the approach therefore shows at the same time that the new method is also able to initialize a regularization parameter. This step was missing in previous approaches [5,6].
4. Discussion and conclusions It could be shown that the new ETE criterion is suitable for the experimental design of ill-posed problems whereas other approaches fail. Still, the new approach requires some
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0
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-1
average error
10
-2
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-3
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-4
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0
0.02
0.04
0.06
0.08
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dt Fig. 2: Error as function of sampling interval dt computed using true solution and simulated measurement data averaged over 10,000 cases.
discussion. In particular, it relies on the assumption that an initial guess for the true solution f – which is actually sought – is available a priori. One may thus wonder about the practicality of the approach. However, it should be noted that this is the standard dilemma in experimental design theory. For nonlinear problems, more relevant in chemical engineering, it already cannot be avoided even in the simpler case of design for parameter estimation [4]. An iterative experiment cycle is thus usually required to find the desired solution [1]. This strategy may also be applied to the ETE approach. Still, even in the initial stage of an analysis, the ETE criterion can be adapted to the level of a priori knowledge available as briefly sketched in the following discussion. Often, the experimenter has at least some qualitative knowledge about the general class of functions the solution should belong to. This is even true for more complex cases than presented in Section 3 (e.g. exponential decay for reaction rates, peak shaped functions for spectra, polynomials for transport coefficients). The criterion may then be used to study the influence of the design variables for the expected function class. This may already give important insight into the proper design. Robust design formulations (e.g. average, min-max design) could then be applied to obtain quantitative design rules [10]. These robust formulations could be of even more importance for nonlinear problems in order to capture the effect of local linearization. In a case when there is really no reasonable assumption available the first term of the ETE criterion (3) may simply be neglected (f=0). The criterion then corresponds to a direct extension of the well-known A-optimal design criterion [10] to ill-posed problems. Such a design is therefore expected to provide at least a reasonable initial experiment. In general, it is an important feature of the formulation that it identifies the individual contributions for bias and variance. Note that the assumed measurement error variance enters the formulation only as relative weight of these two terms (cf. Eq. (3)). A deeper
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problem understanding can therefore be gained by a separate analysis of the dependence of bias and variance on the design variables. In this context, it should be noted that the impact of the design variables on the bias could be approximately analyzed even without assuming any a priori knowledge on functional form for f. After discretization, the bias contribution is given by (cf. Eq. (3))
(I − K (λ )
−1
)
K f ≤ I − K (λ ) −1 K
f ,
(5 )
where I is the identity matrix. The right hand side follows from the submultiplicative property of the matrix norm. Assuming the true solution to be bounded and of order 1 (always possible by proper scaling) an analysis of the bias term could be based on the first matrix norm of the right hand side. Thereby, an upper bound for the bias would be studied. This would thus correspond to standard design theory where a lower bound for the variance from the Cramer-Rao theorem is used [10]. In summary, the expected total error (ETE) design criterion introduced in this work seems to provide the first sound framework for the experimental design of ill-posed problems. As discussed above, the method even yields design guidelines with minimal a priori knowledge. This property underlines the practical utility of the new approach.
Acknowledgements The author thanks Olaf Kahrs, Ernesto Kriesten, Adel Mhamdi and Wolfgang Marquardt (RWTH Aachen) for helpful comments and suggestions on this work. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References 1. S.P. Asprey, S. Macchietto, Statistical tools for optimal dynamic model building, Comp. Chem. Eng., 24 (2000) 1261-1267. 2. W. Marquardt, Model-based experimental analysis of kinetic phenomena in multi-phase reactive systems, Chem. Eng. Res. Des., 83 (2005) 561-573. 3. H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996. 4. G.E.P. Box, H.L. Lucas, Design of experiments in non-linear situations, Biometrika, 46 (1959) 77-90. 5. K. Ito, K. Kunisch, Maximizing robustness in nonlinear illposed inverse problems, SIAM J. Contr. Opt. 33 (1995) 643-666. 6. J. Liu, Optimal experimental designs for linear inverse problems, Inverse Probl. Eng., 9 (2001) 287-314. 7. A. Sylte, E. Ebeltoft, A.A. Grimstad, R. Kulkarni, J.E. Nordtvedt, A.T. Watson, Design of two-phase displacement experiments, Inverse Probl. Eng., 10 (2002) 65-84. 8. J. Weese, A reliable and fast method for the solution of Fredholm integral-equations of the first kind based on Tikhonov regularization, Comput. Phys. Commun. 69 (1992) 99-111. 9. A. Bardow, W. Marquardt, Incremental and simultaneous identification of reaction kinetics: methods and comparison, Chem. Eng. Sci., 59 (2004) 2673-2684. 10. E. Walter, L. Pronzato, Qualitative and quantitative experiment design for phenomenological models - A survey, Automatica, 26 (1990) 195-213.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Dynamic Oil and Gas Production Optimization via Explicit Reservoir Simulation D.I. Gerogiorgis1, M. Georgiadis1.3, G. Bowen2, C.C. Pantelides1,3, E.N. Pistikopoulos1 1
Centre for Process Systems Engineering (CPSE), Imperial College London, SW7 2AZ, UK Schlumberger Cambridge Research Ltd., High Cross, Madingley Road, Cambridge CB3 0EL, UK 3 Process Systems Enterprise (PSE) Ltd., 107a Hammersmith Bridge Road, London W6 9DA, UK 2
Abstract Dynamic oil and gas production systems simulation and optimization is a research trend with a potential to meet the challenges faced by the international oil and gas industry, as has been already demonstrated in a wide variety of publications in the open literature. The complex two-phase flow in reservoirs and production wells governs fuel transport, but is mostly handled by algebraic approximations in modern optimization applications; the true reservoir state variable profiles (initial/boundary conditions) are not known. Integrated modeling and optimization of oil and gas production systems treats oil reservoirs, wells and surface facilities as a single (yet multiscale) system, focusing on computing accurate reservoir and well state variable profiles, useful for optimization. This paper discusses a strategy for interfacing reservoir simulation (ECLIPSE®) with equation-oriented process optimization (gPROMS®) and presents a relevant application. Keywords: oil and gas production, modeling, multiphase flow simulation, optimization.
1. Introduction and Motivation In an era of globalized business operations, large and small oil and gas producers alike strive to foster profitability by improving the agility of exploration endeavors and the efficiency of oil production, storage and transport operations (Economides et al., 1994). Consequently, they all face acute challenges: ever-increasing international production, intensified global competition, price volatility, operational cost reduction policies, aggressive financial goals (market share, revenue, cash flow and profitability) and strict environmental constraints (offshore extraction, low sulphur): all these necessitate a high level of oilfield modeling accuracy, so as to maximize recovery from certified reserves. Straightforward translation of all considerations to explicit mathematical objectives and constraints can yield optimal oilfield network design, planning and operation policies. Therefore, the foregoing goals and constraints should be explicitly incorporated and easily revised if the generality of production optimization algorithms is to be preserved. This paper provides a summary of a strategy towards integration of equation-oriented process modeling and multiphase reservoir computational fluid dynamics (CFD), in order to include the dynamic behavior of reservoirs into oil and gas production models. The problem of fuel production optimization subject to explicit oilfield constraints has attracted significant attention, documented in many petroleum engineering publications. A comprehensive literature review by Kosmidis (2003) classifies previous algorithms in 3 broad categories (simulation, heuristics, and mathematical programming methods) and underlines that most are applied either to simple pipeline networks of modest size, relying on heuristic rules of limited applicability, only suitable for special structures. Reducing the computational burden (focus on natural-flow wells or gas-lift wells only, or reducing well network connectivity discrete variables) is a crucial underlying pattern.
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Dynamic oil and gas production systems simulation and optimization is a research trend which has the clear potential to meet the foregoing challenges of the international oil and gas industry and assist producers in achieving business goals and energy needs. Previous work (Lo, 1992; Fang and Lo,1996; Kosmidis et al., 2004, 2005) has addressed successfully research challenges in this field, using appropriate simplifying correlations (Peaceman, 1977) for two-phase flow of oil and gas in production wells and pipelines. A series of assumptions are adopted to achieve manageable computational complexity: the fundamental one is the steady-state assumption for the reservoir model, based on the enormous timescale difference between different spatial levels (oil and gas reservoir dynamics evolve in the order of weeks, the respective ones of pipeline networks are in the order of minutes, and the production optimization horizon is in the order of days). The decoupling of reservoir simulation from surface facilities optimization is based on these timescale differences among production elements (Peaceman, 1977; Aziz, 1979). While the surface and pipeline facilities are in principle no different from those found in any petrochemical plant, sub-surface elements (reservoirs, wells) induce complexity which must be addressed via a systematic strategy that has not been hitherto proposed. The complex two-phase flow in production wells governs crude oil and gas transport. Despite intensive experimentation and extensive CFD simulations towards improved understanding of flow and phase distribution, commercial optimization applications have not benefited adequately from accurate sub-surface multiphase CFD modeling, and knowledge from field data is not readily implementable in commercial software. model integration can enable the employment of two-phase reservoir CFD simulation, towards enhanced oil or gas production from depleted or gas-rich reserves, respectively. The concept of integrated modeling and optimization of oil and gas production treats oil reservoirs, wells and surface facilities as a single (albeit multiscale) system, and focuses on computing accurate reservoir state variable profiles (as initial/boundary conditions). The upper-level optimization can thus benefit from the low-level reservoir simulation of oil and gas flow, yielding flow control settings and production resource allocations. The components of this system are tightly interconnected (well operation, allocation of wells to headers and manifolds, gas lift allocation, control of unstable gas lift wells). These are only some of the problems that can be addressed via this unified framework. Figure 1 presents the concept of integrated modeling of oil and gas production systems.
Figure 1: Integrated modeling concept for oil and gas production systems optimization: illustration of the hierarchy of levels and production circuit elements (Kosmidis, 2003).
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2. Previous Work and Current Challenges A number of scientific publications address modeling and simulation of oil extraction: they either focus on accurate reservoir simulation, without optimization considerations (Hepguler et al., 1997; Litvak et al., 1997), or on optimal well planning and operations, with reduced (Lo, 1992; Fang and Lo, 1996; Stewart et al., 2001; Wang et al., 2002) or absent (Van den Heever and Grossmann, 2000; Saputelli et al., 2002) reservoir models. Computational Fluid Dynamics (CFD) is a powerful technology, capable of elucidating the dynamic behavior of oil reservoirs towards efficient oilfield operation (Aziz, 1979). The MINLP formulation for oilfield production optimization of Kosmidis (2004) uses detailed well models and serves as a starting point in the case examined in this study. Therein, the nonlinear reservoir behavior, the multiphase flow in pipelines, and surface capacity constraints are all considered (multiphase flow is handled by DAE systems, which in turn comprise ODEs for flow equations and algebraics for phys. properties). The model uses a degrees-of-freedom analysis and well bounding, but most importantly approximates each well model with piecewise linear functions (via data preprocessing). Here, explicit reservoir flow simulation via a dynamic reservoir simulator (ECLIPSE®) is combined with an equation-oriented process optimizer (gPROMS®), towards integrated modeling and optimization of a literature problem (Kosmidis, 2005 – Ex. 2a). An asynchronous fashion is employed: the first step is the calculation of state variable profiles from a detailed description of the production system (reservoir) via ECLIPSE®. This is possible by rigorously simulating the multiphase flow within the reservoir, with real-world physical properties (whose extraction is laborious: Economides et al., 1994). These dynamic state variable profiles (pressure, oil, gas and water saturation, flows) are a lot more accurate than piecewise linear approximations (Kosmidis, 2003), serving as initial conditions for the higher-level dynamic optimization model (within gPROMS®). Crucially, these profiles constitute major sources of uncertainty in simplified models. Considering the oil and gas pressure drop evolution within the reservoir and along the wells, one can solve single-period or multi-period dynamic optimization problems that yield superior optima, because piecewise linear pressure underestimation is avoided. While integrating different levels (sub-surface elements and surface facilities – Fig. 1) is vital, interfacing CFD simulation with MINLP optimization is here pursued in an asynchronous fashion (given the computational burden for CFD nested within MINLP). The concept of integrated modeling and optimization is illustrated in detail in Figure 2:
Figure 2: Integrated modeling and optimization of oil and gas production systems: illustration of the explicit consideration of multiphase flow within reservoirs and wells.
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3. Problem Definition and Model Formulation Dynamic CFD modeling for explicit multiphase flow simulation in reservoirs and wells comprises a large number of conservation laws and constitutive equations for closure: Table 1 presents only the most important ones, which are implemented in ECLIPSE®. The black-oil model (Peaceman, 1977) is adopted in this study, to manage complexity. More complicated, compositional models are widely applied in literature (Aziz, 1979), accounting explicitly for different hydrocarbon real- or pseudo-species concentrations. A black-oil model allows for multiphase simulation via only 3 phases (oil, water, gas). Table 1: Multiphase flow CFD model equations (Nomenclature as in: Kosmidis, 2004). kk S w [ ro ( Po Ugh)] qo (I o ) Oil (1) Po Bo wt Bo Water [
Gas
[
k k rw ( Pw Ugh)] qw P w Bw
k k rg
P g Bg
S w (I w ) wt Bw
( Pg Ugh)] [ Rs
(2)
Sg IS w (I Rs o ) Bo wt Bg
k k ro ( Po Ugh)] q g P o Bo
W w ( x) S
(3)
Total pressure gradient
dP dx
Capillary pressure (oil/gas)
Pcog ( So , S g )
Po Pg
(5)
Capillary pressure (oil/water)
Pcow ( S o , S w )
Po Pw
(6)
Multiphase mixture saturation
So S w S g
1
(7)
Multiphase mixture density
U m ( x) U l ( x) El ( x) U g ( x) E g ( x)
(8)
Multiphase mixture viscosity
P m ( x) Pl ( x) El ( x) P g ( x) E g ( x)
(9)
gU m ( x) sin(T )
Multiphase mixture sup. velocity U m ( x)
A
U g ( x) U l ( x) U sl ( x) U sg ( x) U m ( x) U m ( x)
Multiphase mixture holdup closure E g ( x) El ( x) 1 Drift flux model (gas holdup)
Eg
Choke model (for well & valve i)
qL , i
Choke setting (for well & valve i)
ci
Performance (flow vs. pressure)
q j ,i
(10) (11)
f d (U sl ,U sg , mixture properties)
f c (di , Pi ( xch ), Pi ( xch ), ci , qg ,i , qw,i ) ch
(4)
ch
max(cc , Pi ( x ), Pi ( x ))
f j ( Pwf , j ,i ), i I , j {o, w, g}
(12) (13) (14) (15)
Reduced (1D) multiphase flow balances are solved using a fully implicit formulation and Newton’s method (Kosmidis, 2003), but only for the wells and not for the reservoir. The present paper uses: (a) explicit reservoir and well 3D multiphase flow simulation, (b) elimination of Eq. (15) (performance relations/preprocessing obsolete due to CFD), (c) CFD profiles as initial conditions (asynchronous fashion) for dynamic optimization. The MINLP optimization objective (maximize oil production) and model structure is adopted from the literature (Kosmidis, 2005) via a gPROMS® – SLP implementation. Adopting an SQP strategy can increase robustness as well as computational complexity.
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Reservoir Multiphase Flow Simulation Results Dynamic multiphase flow simulation results (from ECLIPSE®) are presented in Figure 3
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4. Oil Production Optimization Results Table 1: Oil production optimization using reservoir simulation boundary conditions. Example 2a,Kosmidis et al (2005) Oil production (STB/day) Gas production (MSCF/day) Water production (STB/day)
Total capacity 35000 60000 14000
Kosmidis et al (2005) 29317.2 60000 11294.3
This work (%) 30193.7 (+2.9%) 60000 11720.1 (+3.8%)
5. Conclusions and Future Goals The combination of dynamic multiphase CFD simulation and MINLP optimization has the potential to yield improved solutions towards efficiently maximizing oil production. The present paper addresses integrated oilfield modeling and optimization, treating the oil reservoirs, wells and surface facilities as a combined system: most importantly, it stresses the benefit of computing accurate state variable profiles for reservoirs via CFD. Explicit CFD simulations via a dynamic reservoir simulator (ECLIPSE®, Schlumberger) are combined with equation-oriented process optimization software (gPROMS®, PSE): the key idea is to use reduced-order copies of CFD profiles for dynamic optimization. The literature problem solved shows that explicit use of CFD results in optimization yields improved optima at additional cost (CPU cost and cost for efficient separation of the additional water; the percentage difference is due to accurate reservoir simulation). These must be evaluated systematically for larger case studies under various conditions.
Acknowledgements The authors acknowledge financial support as well as a postdoctoral fellowship from the European Union (FP6) under the auspices of a Marie Curie Research Training Network: “Towards Knowledge-Based Processing Systems”/PRISM (MRTN-CT-2004-512233).
References Aziz, K. (1979). Petroleum Reservoir Simulation. Applied Science Publishers, London, U.K. Economides, M. et al. (1994). Petroleum Production Systems, Prentice Hall, NJ, USA. Fang, W.Y., Lo, K.K. (1996). A Generalized Well Management Scheme for Reservoir Simulation, Paper SPE 29124 (www.spe.org). GeoQuest (2000). ECLIPSE 300 Technical Description (2000A), GeoQuest, Schlumberger SCR. Hepguler, G. et al. (1997). Integration of Field Surface and Production Network with a Reservoir Simulator, Paper SPE 38937 (www.spe.org). Kosmidis, V.D. (2003). Integrated Oil and Gas Production Optimization. Ph.D. Thesis, Department of Chemical Engineering, Imperial College London, UK. Kosmidis, V.D., Perkins, J.D., Pistikopoulos, E.N. (2004). Optimization of well oil rate allocations in petroleum fields, Ind. & Eng. Chem. Res. 43: 3513-3527. Kosmidis, V.D., Perkins, J.D., Pistikopoulos, E.N. (2005). A mixed integer optimization formulation for the well scheduling problem on petroleum fields, Comput. & Chem. Eng. 29: 1523-1541. Litvak, M., et al. (1997). Integration of Prudhoe Bay Surface Pipeline Network and Full Field Reservoir Models, Paper SPE 38885 (www.spe.org). Lo, K.K. (1992). Optimum Lift-Gas Allocations Under Multiple Production Constraints, Paper SPE 26017 (www.spe.org). Peaceman, D.W. (1977). Fundamentals of Numerical Reservoir Simulation. Elsevier, NY, USA. Process Systems Enterprise (PSE) Ltd. (2000). gPROMS® Advanced User Guide. London, U.K. Saputelli, L., et al. (2002). A Critical Overview of Artificial Neural Network Applications in the Context of Continuous Oil Field Optimization, Paper SPE 77703 (www.spe.org). Stewart, G. et al. (2001). Field-Wide Production Optimization, Paper SPE 59459 (www.spe.org). Van den Heever, S.A., Grossmann, I.E. (2000). An Iterative Aggregation/Disaggregation Approach for Solution of an MINLP Oilfield Infrastructure Planning Model, Ind. & Eng. Chem. Res. 39: 1955-1971. Wang, P. et al. (2002). Optimization of Production from Mature Fields, 17th WPC, Rio de Janeiro, Brazil.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Multi-scale modelling and optimization of hydrogen storage systems using advanced solid materials Eustathios Kikkinidesa, Michael C. Georgiadisb, Maria Konstantakoua,c, Athanasios Stubosd a
University of Western Macedonia, Department of Engineering and Management of Energy Resources, Kozan 50100, Greece b Process Systems Enterprise Ltd, Academic Office,Thessalonik 57001, Greece c National Center for Scientific Research “DEMOKRITOS”, Institute of Nuclear Technology and Radiation Protection, Athens 15310, Greece
Abstract The aim of the present study is the development of a multi-scale modeling and optimization framework for hydrogen storage in carbon-based nanoporous adsorbents. The outlined methodology is generic and can be easily adapted to the storage of several gases of relevant importance and/or different physisorbing nanoporous materials. The results indicate clearly how operating constraints (e.g. temperature limitations due to safety considerations) can affect the material design in terms of its pore size distribution and how material design constraints (e.g. due to manufacturing limitations) can effect the operation and efficiency of the process.
Keywords: multi-scale modelling; dynamic optimization; hydrogen storage 1. Introduction Environmental and energy problems related to the emission of greenhouse gases and to the depletion of fossil-fuel natural resources, have led to significant research effort on alternative and cleaner fuels (Agrawal et al. 2005). During the coming century, gasoline is expected to be replaced by a cleaner, renewable motor-fuel such as hydrogen while fuel cells should take the place of the internal combustion engine. One of the main barriers towards widespread usage of hydrogen energy in automotive industry is the storage problem. Conventional storage methods such as gas compression and liquefaction are impractical since the former requires very heavy gas tanks and the latter is too expensive to be employed in public vehicles. Storing hydrogen in advanced solid materials, such as carbon-based porous adsorbents and metal hydrides, appears to be a promising, cost effective and safe method of hydrogen storage in the near future. The operation of hydrogen storage tanks packed with these materials presents distinct challenges in process modeling and optimization. In the literature very little attention has been paid on exploring the synergetic benefits between material design and process
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operation/design in a view of deriving a process which, on the one hand, can operate safely and on the other in the most economically attractive way. This work presents an integrated approach that formally exploits the synergistic benefits between material and process design.
2. Microscopic Simulation of Hydrogen Storage in Carbon-Based Materials The grand canonical Monte Carlo method is employed in this work in which the chemical potential (or gas fug acity), volume and temperature of the system are fixed and the simulation calculates the number of particles (gas molecules) in the system and the configurational energy corresponding to a particular choice of n, V and T. The method is discussed in detail in a number of books (Nicholson and Parsonage, 2001). In the present study, the system is considered to be a classical one (i.e. the quantum mechanical character of hydrogen is ignored) (Cracknell, 2001). Hydrogen-hydrogen interactions were modelled using Lennard- Jones potential.
u HH
⎡⎛ σ HH ⎞ 6 ⎛ σ HH ⎞12 ⎤ = 4ε HH ⎢⎜ ⎟ −⎜ ⎟ ⎥ ⎢⎣⎝ r ⎠ ⎝ r ⎠ ⎥⎦
(1)
where u HH is the energy of the (pairwise) interaction between Lennard- Jones sites and ε HH and σ HH are the well depth energy and hard sphere diameter parameters for the interaction, respectively. A two-site Lennard- Jones model is employed, with the interactions summed over all site-site interactions. The parameters for the two-site model were devised in full accordance with similar recent studies (Cracknell, 2001). Pore walls are treated as stacked layers of carbon atoms separated by a distance Δ=0.335 nm, and having a number density ρw=114 atoms/nm3 per layer. The theoretical surface area of this idealiz ed adsorbent is 2620 m 2/g The slit- pore width, H, is defined as the carbon to carbon distance on opposing pore walls (Cracknell, 2001). The simulation does not therefore model any edge effects. The interaction between hydrogen and each wall of the micropore is given by the '10-4-3' potential of Steele (Steele, 1974).
u w ( z ) = 2πρ wε CH σ
2 CH
4 ⎡ 2 ⎛ σ CH ⎞10 ⎛ σ CH ⎞ 4 ⎤ σ CH Δ⎢ ⎜ ⎟ −⎜ ⎟ − ⎥ 3Δ(0.61Δ + z ) 3 ⎥⎦ ⎝ z ⎠ ⎢⎣ 5 ⎝ z ⎠
(2)
The Lennard-Jones parameters for the hydrog en-wall interaction were found from the parameters given in (Cracknell, 2001).
3. Macroscopic modelling of hydrogen storage in the tank A two-dimensional pseudo-homogeneous macroscopic model is developed based on mass, momentum and energy balances assuming a cylindrical bed packed with a carbon-based adsorbent (Delahaye et al. 2002). Due toace sp limitations the details of
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the model are not presented here. The Langmuir isotherm is described by the following equation:
q* =
q s bP q b exp[− ΔH / RT ]P = s 0 1 + bP 1 + b0 exp[− ΔH / RT ]P
(3)
where q s and b0 are parameters depending on the selected material and P the total pressure. Parameter (-ΔH) is the heat of adsorption and in the present study is considered to be identical to the isosteric heat of adsorption obtained from the microscopic model, in accord with the considerations imbedded in the Langmuir isotherm. Proper boundary and initial conditions complement the model.
4. Determination of Optimal Sorption Properties from the Macroscopic Model The process model along with the boundary and initial conditions involve certain parameters that must be optimally selected in order to achieve an economic and safe process performance. Such parameters are the macroscopic Langmuir constants q s and b0 whose values affect the maximum amount of hydrogen that can be stored in the bed for a specific charging time. There are two main issues, which must be taken into consideration when establishing optimal control strategies for this system. The first is to ensure that the maximum process storage efficiency is achieved. The second is to ensure satisfaction of all operating and design constraints. This can be expressed by imposing an upper bound on the average bed temperature in order to account for potential safety concerns. The problem is posed as a dynamic optimization problem and solved using gPROMS dynamic optimization capabilities (Process Systems Enterprise Ltd 2004).
5. Determination of Optimal Pore Size Distribution from the Microscopic Model In this work the micropore range (from 0.5 to 2.0 nm) was subdivided in N equidistant intervals (classes of pores) with 0.1 nm spacing between them. The fraction of the total pore volume associated with each interval, is calculated on the basis of an assumed Particle Size Distribution (PSD) and keeping the total pore volume equal to the measured one. Thus, the amount of gas adsorbed in every class at a certain pressure is evaluated by the simulation, and consequently, a computed isotherm is constructed. This, after comparison to its experimental counterpart, results in the optimum micropore size distribution provided by the best fit. In the present study the “experimental” isotherm that is used to derive the optimal pore size distribution is obtained from the Langmuir equation where the parameters b0 , q s , have been optimally determined from the macroscopic simulations described in section 4.
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The procedure for the determination of the optimum PSD involves the numerical solution of a minimization problem under certain constraints. In practice, the problem consists of minimizing the function: N
qi − ∑ d ij w j
i=1..,M,
j=1…,N
j =1
(4)
for M different pressure values Pi; where qi (gr/m3) is the “experimentally” adsorbed amount determined at pressure Pi from the Langmuir isotherm (eq. 3) with the optimally determined parameters b0 , q s , (section 4). Variable dij is the calculated density of H2 P i, andw j represents the fraction of pores with size Hj . in a pore of width Hj at pressure
6. Integration of microscopic and macroscopic models Τhe macroscopic model determines the optimum isotherm parameters that should be used further in the microscopic model to determine the optimum pore size distribution of the material. On the other hand, both the heat of adsorption, |ΔH|, and the bulk in macroscopic model, depend density of the material, ρs, that are input parametersthe on the pore size distribution, which is determined by the microscopic model. It is therefore clear that the macroscopic model depends on the results of the microscopic model, and particularly on the pore size distribution which determines the values of ΔH different bounds on and ρs. Therefore the following iterative procedure is employed for the temperature. • From the macroscopic model the optimum sorption isotherm parameters are determined, given initial guess of the heat of adsorption, |ΔH(0)|, and bulk density of the material, ρs(0). • From the microscopic model the optimum pore size distribution is determined corresponding to the Langmuir isotherm data based on the above parameters. • From the optimum pore size distribution the “new” values Δ of | H| andρ s are computed. • The above steps are repeated until convergence is achieved. 7. Results 0.6 nm
and Discussion P=20 bar
2.4 nm
0.6 nm
P=100 bar
2.4 nm
Figure 1: Visual representation of H2 physisorption in the graphite nanopores (T=298 K)
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Molecular simulation results regarding hydrogen adsorption on different carbon slit sizes are shown in Figures 1. It is seen that strong confinement effects are present at small pore sizes (0.6 nm) resulting in close packing structures of the adsorbed hydrogen, which once formed, no longer change significantly with pressure. On the other hand at large pore sizes (2.4 nm) there is significant space for the storage of hydrogen molecules resulting in a continuous increase of sorption capacity as pressure increases further. However, even at pressures of 100 bar, there is a lot of empty space in the pore except from regions in the vicinity of the pore walls. Typical results from the iterative process are shown in Figure 2. It is seen that after 4-5 iterations no significant change is observed in the values of ΔH and essentially the iterative procedure has converged. The same is true for the values of the optimized Langmuir parameters, qs and b0, and for the resulting pore size distribution as seen in Figure 2.
Figure 2: Convergence of the heat of adsorption and the Langmuir parameters through the multiscale iterative procedure.
Figure 4 illustrates the resulting densities of H2 stored in the carbon-based optimized materials on a per volume and per weight basis. It is interesting to observe that the two densities show a completely opposite trend as the temperature constraint changes. In particular, as ΔTb (mean aeverage temperature) decreases so does the volumetric density of H2 while its gravimetric density increases. The apparent contradiction is easily explained since as ΔTb decreases so does the density of the optimized material (not presented here). The optimal pore size distributions that result from the optimization procedure are depicted in Figure 5. It is evident that when loose bounds are imposed on the temperature rise in the bed (high ΔTb), the optimum pore size distribution is very narrow and it is limited in sizes of 1 nm or lower. On the other hand, as the temperature bounds are tightened, we observe a shifting of a fraction of pores towards larger sizes, where both the volumetric hydrogen density and the heat of adsorption of the material are significantly lower.
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8. Conculsions The present work presents a multi-scale modeling and optimization framework for hydrogen storage in carbon-based materials. The outlined methodology is generic and can be easily adapted to the storage of several gases of relevant importance (e.g. methane, carbon dioxide) and/or different nanoporous adsorbents (metal-doped carbon nanotubes, zeolites, metal-organic frameworks, etc.). The results indicate clearly the strong interactions between material and process design.
References R. Agrawal, M. Offutt, MP. Ramage,.2005, Hydrogen Economy – An opportunity for Chemical Engineers, AIChE Journal, 51: 6: 1582. R.F. Cracknell, 2001, Molecular Simulation of Hydrogen adsorption in graphic nanofibres. Phys. Chem. Chem. Phys., 3, 2091. A. Delahaye, A. Aoufi, A. Gicquel , I. Pentchev, 2002, Improvement of Hydrogen Storage by Adsorption using 2-D Modelling of Heat Effects, AIChE Journal, 48, 2061. D. Nicholson , NG. Parsonage, 2001, Computer Simulations and the Statistical mechanics of Adsorption. Academic Press, New York. Process Systems Enterprise Ltd. 2004, gPROMS Advanced Users Guide, London, UK. W.A. Steele, 1974, The interaction of Gases with solid surfaces. Pergamon, Oxford.
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Risk analysis and robust design under technological uncertainty R.F. Blanco Guti´erreza∗ , C.C. Pantelidesa† and C.S. Adjimana a
Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London SW7 2AZ, U.K. Technological innovation in process design often leads to increased technological risk arising from incomplete knowledge. We propose a systematic approach to manage this risk using mathematical models that are sufficiently detailed to quantify risk. Global sensitivity analysis is used to determine the complete probability distributions for the key performance indicators of the process, thereby allowing informed decisions to be taken regarding the acceptability of the risk inherent in a given design. It also produces global sensitivity indices which allow the identification of the critical uncertain parameters on which additional R&D needs to be focused if the risk is deemed to be unacceptably high. If the risk is acceptable, then scenario-based approximation is used to handle the residual uncertainty in the critical parameters. Issues regarding the robust and efficient solution of problems involving large numbers of scenarios based on nonlinear models with thousands of variables are considered. The methodology is demonstrated via a case study concerning the design of a catalytic tubular reactor. 1. INTRODUCTION Technological innovation in processes and products almost inevitably implies increased risk with respect to performance, operability and safety. Although this risk can often be reduced by investing time, money and other resources in R&D activities, the increased cost and time spent can significantly reduce the competitive advantage arising from this innovation, e.g. by reducing the probability of achieving a leading market position. Therefore, the potential implications of any residual risk have to be weighed against the potential benefits that may be realised by the deployment of new technology. The use of model-based methodologies for process design and operation can accelerate R&D activities by complementing experimental investigations at the laboratory, pilot plant and industrial plant scales. In principle, instead of searching the, often large, space of possible designs and operations, experimental R&D can be focused on deriving an accurate model (e.g. by identifying the fundamental chemistry associated with a new catalyst). The model can then be used for the relatively rapid and inexpensive consideration and screening of many alternatives. Once one or more promising alternatives are identified, their predicted performance may be verified again experimentally (e.g. using pilot plants). Clearly, the effectiveness of this three-step approach depends crucially on the accuracy of the model derived at the first step. Recent years have witnessed significant advances in this context. It is now practically feasible to use detailed models of experimental apparatus to interpret experimental measurements correctly, estimating multiple model parameters from measurements taken from multiple steady-state and/or dynamic experiments. A posteriori statistical significance analysis can provide estimates of the errors in the parameter estimates. We also have at our disposal model-based techniques for experiment design techniques which can determine the optimal conditions for executing further experiments aiming at achieving maximum model accuracy. Nevertheless, it has to be recognised that, irrespective of the above advances, model uncertainty cannot be fully eliminated, and consequently, a number of important questions need to be addressed: 1. Given a certain level of model accuracy and external disturbances, what is the resulting uncertainty in the key performance indicators (KPIs) of a process or product designed using this model? ∗ Financial
support: CONACyT and CPSE. author. Tel.: +44 (0) 20 7594 6622; fax: +44 (0) 20 7594 6606. E-mail address:
[email protected]
† Corresponding
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2. If the risk associated with this uncertainty is unacceptable, and further R&D is required to resolve some of the inaccuracies in the model, which are the critical model aspects on which such R&D needs to be focused? 3. If the risk is, in principle, acceptable, then what is the best design that can take account of the residual model uncertainty? There is already a large body of research aiming to provide answers to the questions posed above, with particular emphasis on the last one. Work to date has employed different metrics such as flexibility indices [1], trade-offs between flexibility indices and maximum regret [2], expected economic performance [3], and the cost of R&D [4]. Different tools have been proposed for the analysis of feasible regions in the presence of uncertainty, e.g. [5–8], and specific aspects such as technology evolution can be included [9]. The problem can be formulated as a two-stage stochastic optimisation problem (e.g. [10]). This paper aims to complement the above work by providing a quantitative model-based methodology for addressing the first two of the questions posed above. Of course, the use of models for the quantification of technological risk will be successful only if the models can predict the situations that potentially give rise to such risk, e.g. the formation of undesirable by-products through side reactions, or the occurrence of hot spots in reactors through imperfect mixing. Almost always, such models will be more complex than those used for the prediction of nominal performance (e.g. the yield of the main reactor product or the average reactor temperature), and risk-management techniques need to be able to cope with such increased model complexity. The issue of model complexity also affects the practical feasibility of techniques for addressing the last of the three questions above. Most of the work to date reported in the open literature (e.g. scenario-based optimisation) has been applied only to rather small models involving small numbers of uncertain parameters which can be explored using a relatively small number of scenarios. 2. METHODOLOGY The proposed methodology outlined in figure 1 starts by constructing a detailed process model and validating it using techniques of the type outlined in the introduction. This validation process results in optimal estimates of the values of model parameters and also in estimates of the accuracy of these values (e.g. in the form of confidence ellipsoids or joint probability density functions). In the second step, the model, with the nominal values of its parameters, is used to determine an optimal design and operating conditions using standard deterministic optimisation techniques.
Figure 1. Proposed methodology.
Figure 2. Pseudo-dynamic optimisation approach.
2.1. Global sensitivity analysis The third step of the methodology aims to quantify the effects of parametric uncertainty on the process KPIs, including the objective function (e.g. economic performance) and constraints (e.g. relating to product quality and process safety and operability). This task is often performed using local sensitivity analysis based on the partial derivatives of the KPIs with respect to the uncertain parameters. Albeit conceptually simple and computationally inexpensive, this approach has certain important deficiencies. First, local values may fail to capture the KPI variability induced by the model parameters varying over ranges of values. Secondly, most processes have controls which can be used during operation to counteract
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the effects of parameter uncertainty; hence, the effective sensitivity with respect to a certain parameter may be smaller than that implied by the local sensitivity value. Finally, any single measure of sensitivity is unlikely to contain sufficient information for assessing whether the risk inherent in a certain design is acceptable. Consequently, here we adopt a different approach based on global sensitivity analysis (GSA). This involves solving the following optimisation problem for a fixed design d and a sequence of values of the uncertain parameters θ: Φ(d, θ) = max Φ(d, u, x, θ) u∈U
s.t.
f (d, u, x(z), xz (z), xzz (z), θ) = 0 h(d, u, x(z), xz (z), xzz (z), θ) = 0 g(d, u, x(z), xz (z), xzz (z), θ) ≤ 0 y = Y(d, u, x(z), xz (z), xzz (z), θ) yL ≤ y ≤ yU
∀z ∀z ∀z ∀z
∈Ω ∈ Γ(Ω) ∈Ω ∈Ω
(1)
Here Φ represents the objective function (e.g. an economic performance criterion), u a vector of control variables that may be varied over a space U, and x is a vector of state variables which may be distributed over a domain Ω of independent variables z (e.g. spatial position). The model equations f are generally mixed systems of partial differential and algebraic equations involving d, u, x and the latter’s partial derivatives, and subject to boundary conditions h and performance constraints g. The KPIs y are given functions Y of the other variables and are subject to lower and upper bounds. The above optimisation determines the best set of operating conditions for the given design under a certain realisation of the uncertain parameters θ. The latter vary over a given domain Θ with a given probability distribution3 . For the purposes of the GSA, the space Θ is sampled using a low-discrepancy sequence due to Sobol’[11] which has a number of desirable properties. First, for any positive integer k, a sequence of 2k points covers the uncertainty space uniformly. Secondly, and unlike uniform grids, the projection of N sample points onto any parameter axis results in N distinct values of that parameter. One valuable output of the GSA is an estimate of the complete probability distribution of each and every KPI. This provides a good assessment of the “upside” and “downside” inherent in design d and allows a more detailed assessment of the risk than what can be achieved based on aggregate measures such as expected value and variance. If the risk is deemed to be unacceptable, then one may have to go back to step 1 of the methodology to obtain more accurate estimates of the model parameters. Usually, this implies further experimentation, the cost of which may not be trivial. It is, therefore, important to focus this experimentation on those parameters which have the most impact on the process KPIs. Such critical parameters may be identified via global sensitivity indices also computed by GSA. Here we employ the indices proposed by Sobol’[12] which are derived from the “analysis of variances” (ANOVA) decomposition of the nonlinear functions Φ(d, θ) and y(d, θ) defined by the solution of optimisation problem (1). For example, in the case of two parameters θ1 and θ2 , the decomposition is expressed as: Φ(d, θ1 , θ2 ) = Φ0 (d) + Φ1 (d, θ1 ) + Φ2 (d, θ2 ) + Φ12 (d, θ1 , θ2 )
(2)
For a given d, the variances of the functions Φ1 , Φ2 and Φ12 can be calculated from the values of Φ determined during the sampling. A global sensitivity index is then defined as the ratio of the variance of each of these functions to the overall function variance. For example, the first-order global sensitivity index for parameter θ1 in Eq. (2) is defined as: S1Φ (d) =
V arθ1 [Eθ2 (Φ(d, θ1 , θ2 ))] V ar[Φ1 (d, θ1 )] = V ar[Φ(d, θ1 , θ2 ) − Φ0 (d)] V arθ1 ,θ2 (Φ(d, θ1 , θ2 ) − Φ0 (d))
(3)
y Φ P y global sensitivity indices Si and Si are quantities in the range [0, 1] which satisfy PTheΦ first-order S = S = 1. The parameters θ with the largest sensitivity indices in Φ and/or y are flagged as i i i i critical uncertain parameters on which any further experimental R&D effort needs to be focused. This method also allows the calculation of higher-order parameter interactions. For instance, given Eq. (2), Φ is the global sensitivity index of Φ for the interaction between θ1 and θ2 . S12 3 An
estimate of this is produced by the a posteriori statistical significance analysis during the model validation step.
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The variances V ar[.] and expected values E(.) in expressions of type (3) are multidimensional integrals calculated using a technique developed by Sobol’ [12]. In our implementation, the process model is constructed in the gPROMS modelling tool[13] which is also used for the solution of the optimisation problem (1). The GSA is implemented in C++ as a gPROMS-Based Application (gBA) interacting directly with the gPROMS solution engine. The application has been parallelised for execution on distributed computer networks using MPI-based communication to allow the simultaneous evaluation of multiple sampling points. 2.2. Scenario-based optimisation Even if the GSA indicates that the risk associated with a nominal design is acceptable, the design may not be optimal when one considers the variation of the objective function value caused by parameter variability. For some values of the uncertain parameters, it may not even be feasible, violating some of the inequality constraints and bounds in (1). Therefore, we need to determine a new design which takes explicit account of the parameter variability. This is a well-known problem which has been the focus of attention of much of the literature mentioned in the introduction to this section. A standard technique for solving the problem is by postulating a set of scenarios s = 1, ..., N S, each corresponding to a different realisation of the parameters θ[s] , and then determining a design d and controls u[s] , s = 1, ..., N S which maximise some probabilistic measure of performance (e.g. the expected value of Φ). To obtain a good estimate of expected values, it is desirable to use a large number of scenarios. However, this significantly increases computational cost. In addition, with complex models of the type of interest here, numerical convergence (i.e. obtaining a set of variable values that satisfy the equality constraints in (1)) becomes a difficult task, and this can compromise the robustness of the overall algorithm. Here we use a pseudo-dynamic optimisation formulation (cf. figure 2) to solve the scenario-based problem, where smooth transitions between scenarios are achieved via a homotopy/continuation approach. This facilitates the initialisation process by requiring only one set of initial guesses.
Figure 3. Probability distribution for profit from GSA.
Figure 4. Reduction in number of scenarios when no parameter interactions exist.
A further reduction in problem complexity can be achieved in cases where the higher-order sensitivity indices (cf. section 2.1) indicate no significant interactions between parameters. In this case, a reduced set of scenarios can be found by sampling the uncertain parameter space along each parameter domain independently, keeping other parameters fixed at a single set of values (see figure 4). Even with the above reductions in the numbers of uncertain parameters and scenarios, the solution of the scenario-based optimal design problem may remain prohibitively expensive for complex systems. To address this issue, we use a Sample Average Approximation (SAA) approach[14,15] which approximates the optimal design through the solution of a sequence of problems, each involving a much smaller number N of scenarios. Such N -scenario problems are formulated and solved until the average values and standard deviations of the objective function and design variables obtained up to a certain point converge to constant values. If the total number of N -scenario problems solved is M , the computational cost is usually much smaller than what would be required for solving a single problem with M × N scenarios. Moreover, this approach is more amenable to paralellisation as several N -scenario problems can be solved in parallel.
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Table 1 Uncertain parameters and first-order sensitivity indices for the objective function. Parameter Mean Std Dev % S CO heat of formation, ∆Hf (CO) (kJ kmol−1 ) -110440 3.3 0.1822 3.3 0.7332 COCl2 heat of formation, ∆Hf (COCl2 ) (kJ kmol−1 ) -222850 57686.3 3.3 0.0003 Kinetic coeff., Ekr (kPa m3 kmol−1 ) 0.05 3.3 0.0050 Radial heat transfer coeff., kr (kW m−1 K−1 ) 0.05 3.3 0.0077 Axial heat transfer coeff., kz (kW m−1 K−1 ) 0.096 3.3 0.0374 Overall heat transfer coeff., U (kW m−2 K−1 ) 293 0.33 0.0127 Cooling water inlet temp., Tcwin (K) 293 0.33 0.0000 Feed stream inlet temp., Tin (K)
3. CASE STUDY As an illustration of the proposed methodology, we consider the design of an externally cooled catalytic tubular reactor producing phosgene. It is desired to determine the reactor length and diameter which maximise the annualised profit. The available controls include the cooling water flowrate, the feed stream partial pressures for the two reactants and feed stream velocity. The model is steady-state and spatially distributed along the axial and radial dimensions, the spatial discretisation of which results in approximately 5,900 variables. It involves the 8 uncertain parameters listed in table 1. Here these are assumed to be described by independent normal distributions; however, any type of joint probability density function can be used. As described in the methodology (figure 1), an optimal nominal design is first identified based on the mean values shown in table 1. The global sensitivity indices for the eight parameters are then calculated; this requires the evaluation of 17 (=2×8+1) multidimensional integrals, each computed using a Sobol’ sequence of 212 points. The computation requires the solution of 69,632 problems of type (1), with a total CPU time of 55 hours, spread over 32 parallel processes on a Linux cluster of Pentium 4 processors with speeds ranging from 1.79 GHz to 3.39 GHz. The global sensitivity indices for the objective function are shown in the last column of table 1. The heats of formation of CO and COCl2 are clearly the critical parameters in this case. The second-order sensitivity index for these two parameters is S∆Hf (CO) ,∆Hf (COCl2 ) = 0.020, which indicates that there is little interaction between them. The GSA also determines the complete probability distribution for the objective function, as shown in figure 3. Assuming that the risk is acceptable, the final step involves optimisation based on multiple realisations (“scenarios”) of the two critical parameters. All other parameters are fixed at their nominal values, which greatly reduces the number of scenarios that need to be considered. We consider three different approaches. The first samples the space of ∆Hf (CO) and ∆Hf (COCl2 ) using a 6 × 6 uniform grid. The second approach exploits the lack of interaction between the two parameters (as indicated by the low value of the corresponding second-order sensitivity index) to reduce the number of scenarios. Thus, each parameter is sampled independently at six points while keeping the other parameter constant, which results in a total of 11 scenarios (see figure 4). The third approach employs the SAA method solving a sequence of 5-scenario problems. Figure 5 shows the evolution of the cumulative means and standard deviations of the objective function and the optimal reactor length with the number of problems solved. Both the optimal reactor length and the optimal reactor radius (not shown in the figure) converge rapidly to their final values. The convergence of the profit is somewhat slower. The average CPU time per 5-scenario problem is 409 s. Convergence to the optimal design is achieved after about 10 5-scenario problems requiring 3889 CPU s. All three approaches give identical results in terms of the design variables, but show large variations in the expected value of the objective function value and its variation over the set of scenarios studied. These differences arise from the difficulty of obtaining good estimates of the corresponding two-dimensional integrals using relatively small numbers of scenarios. To illustrate this point, the last row of table 2 shows benchmark results evaluated a posteriori by applying 200 scenarios to the optimal reactor design. It can be seen that the values obtained by the SAA approach are nearest the benchmark values, as might be expected in view of the fact that the SAA makes use of a much larger number of scenarios than the other two methods.
196 Table 2 Results of the case study for different methods. Method Total number Profit (USD/yr) of scenarios Mean Std deviation Single optimisation 36 466,000 31,000 Single optimisation, 11 474,000 26,000 independent params SAA 20×5 479,000 10,000 Benchmark 200 477,000 7,000
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Reactor length (m) 1.403 1.403
Reactor radius (m) 0.714 0.714
CPU hours 73 0.9
1.403 1.403 (fixed)
0.714 0.714 (fixed)
2.3 –
4. CONCLUSIONS A systematic methodology has been proposed to manage technological risk arising from incomplete knowledge at the process design stage. It uses mathematical models which are sufficiently detailed to establish a quantitative relationship between the uncertain parameters and the process KPIs. Although the construction of such models is not trivial, it is increasingly being undertaken in industrial practice. An optimisation-based global sensitivity analysis, based on sampling via low-discrepancy sequences, is performed to identify critical parameters affecting the KPIs while exploiting the flexibility afforded by the process control variables. The complete probability distribution of the KPIs is also obtained, thus allowing informed decisions to be made regarding acceptability of the inherent risk. If the risk is deemed to be acceptable, then scenario-based optimisation is employed to determine a design that performs optimally given the variability of the critical parameters. As the number of scenarios may increase exponentially with the number of parameters being considered, the GSA plays a crucial role in eliminating non-critical parameters, and in assessing the extent to which interactions among the critical ones need to be considered. Our results indicate that sample average approximation methods may provide an effective means of handling large numbers of scenarios using nonlinear models involving thousands of variables. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
R.E. Swaney and I.E. Grossmann, AIChE J. 31 (1985) 621. M.G. Ierapetritou and E.N. Pistikopoulos, Comput. Chem. Eng. 18 (1994) 163. F.P. Bernardo, E.N. Pistikopoulos and P.M. Saraiva, Ind. Eng. Chem. Res. 38 (1999) 3056. F.P. Bernardo, P.M. Saraiva and E.N. Pistikopoulos, Comput. Chem. Eng. 24 (2000) 1695. W.C. Rooney and L.T. Biegler, Comput. Chem. Eng. 23 (1999) 1563. W.C. Rooney and L.T. Biegler, AIChE J. 47 (2001) 1794. N.J. Samsatli, M. Sharif, N. Shah, L.G. Papageorgiou, AIChE J. 47 (2001) 2277. V. Goyal and M.G. Ierapetritou, AIChE J. 49 (2003) 1233. L. Cheng and E. Subrahmanian and A.W. Westerberg, Comput. Chem. Eng. 27 (2003) 781. J. Acevedo and E.N. Pistikopoulos, Comput. Chem. Eng. 22 (1998) 647. I.M. Sobol’, USSR Computational Math. and Mathematical Phys. 16 (1976) 236. I.M. Sobol’, Mathematics and Computers in Simulation 55 (2001) 271. gPROMS ModelBuilder v. 2.3.6, Process Systems Enterprise Ltd, www.psenterprise.com (2005). A.J. Kleywegt, A. Shapiro and T. Homem-de-Mello, SIAM J. Optim. 12 (2002) 479. J. Wei and M.J. Realff, Comput. Chem. Eng. 28 (2004) 333.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Network of three catalytic reactors with periodical feed switching for methanol synthesis: bifurcation analysis. Marco Potaa, Lucia Russob, Erasmo Mancusic and Silvestro Crescitellia a
Dipartimento di Ingegneria Chimica, Università “Federico II”, Piazzale Tecchio 80, 80125 Napoli, Italy b Dipartimento di Ingegneria Chimica ed Alimentare,Università di Salerno, Via Ponte Don Melillo, 84084, Fisciano(SA), Italy. c Facoltà di Ingegneria, Università del Sannio, Piazza Roma, 82100, Benevento, Italy.
Abstract In this paper the bifurcation analysis of a network of three catalytic reactors with periodical feed switching for methanol synthesis is carried out and the influence of switch time on the stability and on the performance is addressed. With a methodology based on the construction of discrete maps, the continuation of periodic regimes and the detection of bifurcations are systematically conducted as the switch time is varied. Several complex regimes are found in a wide range of the switch time. Catastrophic transitions from periodic to quasiperiodic and multiperiodic regimes are detected and discussed. These catastrophic bifurcations are found very close to the optimal periodic regime in methanol yield. Keywords: periodically forced react or, network of reactors, methanol synthesis.
1. Introduction Periodically forced catalytic reactors have attracted considerable interest in the last years. Such operation modes possibly overcome thermodynamic and kinetic limitations by changing the feed direction or by periodically changing some operating parameters such as temperature or concentration of the system. Many studies have shown that catalytic reverse flow reactors (RFR), in which the flow direction is periodically reversed, is very efficient to conduct autothermally the purification of industrial off-gas with a low concentration of volatile organic compounds. The RFRs have proven to be cost-effective also for other catalytic processes, in which equilibrium-limited exothermic reactions are carried out (e.g. methanol synthesis (Froment, 1990), ammonia synthesis and oxidation of SO2 (Matros and Bunimovich (1996)). In these processes, reverting the flow direction produces a shaped bell- temperature profile in the reactor close to the optimal temperature distribution which increases yield and selectivity towards the main product. To overcome some intrinsic disadvantage of the RFRs as the washout effect (the loss of reactants immediately upon the flow reversal), Matros (1985) has suggested a network of catalytic reactors in series equipped with a valve system that allows cyclic permutation of the feed position. Van den Bussche and Froment (1996) studied the feasibility of methanol synthesis in a periodically forced network of three catalytic reactors (the STAR configuration) which gives a higher conversion than the reverse flow reactor. Velardi and Barresi 002) (2 analyzed the methanol synthesis in a three reactors network with a periodical changing of the feed position, asoriginally proposed by Matros (1985). Using a brute force analysis they assess that periodic
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regimes and autothermal behavior are possible only for two relatively small ranges of the switch time, although the network allows higher conversions than in RFR. In a successive work Velardi et al. (2004) showed that complex dynamics may arise close to the conditions of maximum conversion. In this framework, nonlinear dynamics tools are proved very successful to predict and completely characterize all regimes as the model parameters are varied. In this paper, we conduct a systematic study of the dynamic behaviour of a periodically forced network of three catalytic reactors for the methanol synthesis through bifurcation analysis. Using the methodology based on the symmetry properties of a periodically forced network of reactors previously established (Russo et al.2002, 2006), bifurcation diagrams are derived with a continuation technique based on the construction of discrete maps (Russo et al.2002). The switch time is chosen as bifurcation parameter for its relevance on both design and control of the system.
2. The mathematical model. In the present work, a network of three fixed-bed reactors is considered. Each fixed-bed reactor is modeled as a heterogeneous system with heat and mass transfer resistance between the gas and the solid phase, axial dispersion in the gas phase, axial heat conduction in the solid phase, and cooling at the reactor wall. A similar model was implemented by Velardi and Barresi (2002) but in the present work we considered a constant value of the total concentration in the gas phase and a pseudo-steady-state hypothesis for mass balance in the solid phase. Within the hypotheses, the mathematical model for the reactors network reads: keff ∂ 2TGi h f ⋅ av ∂TGi ∂T i = −v⋅ G + ⋅ (TSi − TGi ) , (1) 2 ∂t ∂x ε ⋅ ρ ⋅ cP ,G ρ ⋅ cP ,G ∂x h f ⋅ av ⎤ ⎞ ∂TSi λS ⎛ ∂ 2TSi ⎞ 1 nr ⎡⎛ nr = ⋅ (TSi − TGi ) + (2) ⎢⎜ ∑ν i , k R 'k ⎟ ( −ΔH f ,i ) ⎥ ⎜ 2 ⎟− ∑ cP , S i =1 ⎣⎝ k =1 ρ S cP , S ⎝ ∂x ⎠ ρ S cP , S (1 − ε ) ∂t ⎠ ⎦ nr ∂yGi , j ∂ 2 yGi , j ∂yGi , j kG , j kG , k i i i = Deff , j ⋅ − v + ⋅ y − y ⋅ a − y ⋅ ( ySi , k − yGi , k ) ⋅ av (3) ( ) S, j G, j v G, j ∑ 2 ∂t ∂x ε ∂x k =1 ε Nr
kG , j ⋅ cG ⋅ ( ySi , j − yGi , j ) ⋅ aν = ρ S (1 − ε ) ⋅ ∑ν j , k R 'k
(4)
k =1
with the following boundary conditions: Deff , j ∂yGi , j = yGi , j (0, t ) − ⎡⎣1 − f ( t − (i − 1)τ ) ⎤⎦ yG 0, j − ⎡⎣ f ( t − (i − 1)τ ) ⎤⎦ yGi −,1j ( L, t ) v ∂x
(5)
0
keff
ρ ⋅ v ⋅ cP , G λS
∂TSi ∂x
∂TGi ∂x
= TGi (0, t ) − [1 − f (t − (i − 1)τ ] TG 0 − ⎡⎣ f ( t − (i − 1)τ ) ⎤⎦ TGi −1 ( L, t )
= Deff , j 0
(6)
0
∂yGi , j ∂x
= keff L
∂TGi ∂x
= λS L
∂TSi ∂x
=0
(7)
L
The subscript j = 1...nr indicates the j-th chemical species while the superscript i=1,2,3 indicates the i-th reactor of the network. For the nomenclature we refer to that adopted by Velardi and Barresi (2002).
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The function f(t) is a square wave that accounts for the discontinuous feed scheme (Fig.1). The reactors are fed according to the sequence 1-2-3 in the time range [0,τ]; after the first switch, that is in the range [τ,2τ], they are fed according to the sequence 23-1; then, after the second switch (t∈[2τ, 3τ]), the reactors are fed as the sequence 3-12; the next switch brings the feed sequence to the first one, i.e., 1-2-3, and the permutation cycle restarts. It is apparent that the vector field changes discontinuously in time, and it recovers the same form after a time 3τ=T. Indeed,f(t) is a discontinuous periodic function with minimum period T, and the non–autonomous system (1)-(3) is T periodic. f(t) 1 0
τ
2τ
3τ
4τ
5τ
6τ t
Figure 1 – The forcing function f(t) appearing in the boundary conditions of Eqs. (5)-(7).
For methanol synthesis from CO, CO2 and H2 over commercial Cu-Zn-Al catalyst, the model is completed by the kinetic equations given by Graaf et al. (1988), based on three independent reactions: ( A) CO + 2 H 2 R CH 3OH ( B ) CO2 + H 2 R CO + H 2O
(8)
(C ) CO2 + 3H 2 R CH 3OH + H 2O
The adopted reaction rates, reported in Graaf et al. (1988), are written in terms of partial pressure (Velardi and Barresi, 2002). Here we assumed a catalyst efficiency equal to one.With these hypotheses, the model (1)-(7) is a system of 6 partial differential equations and 6 algebraic equations. The numerical simulations are carried out by discretising the system through a collocation method with 8 collocation nodes for each reactor. The time integration of the discretised system of ODEs is performed with LSODE code. The conditions used for the numerical results are the same adopted by Valardi and Barresi (2002)and they are reported in Table I. Reactor length Void fraction Catalyst density Catalyst void fraction Pellets diameter Total pressure YH2in YCOin Fin TGin Table I Conditions adopted in the simulations.
0.5 m 0.4 1750 Kg m-3 0.5 0.0054 m 50 atm 0.935 0.045 32.65 mol m-2 s-1 420 K
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3. Results The switch time is an important operating parameter as proper forcing may induce reactor ignition and different performances, and it can be used as manipulated variable in a control law. Thus, the knowledge of the bifurcation behaviour of the reactor network as the switch time is varied isvalue of as it allows a quick and complete characterization of reactor performances in terms of ignition and switch off. The regime solution diagram shown in Fig. 2 presents the influence of the switch time on network dynamics and it is obtained with the continuation technique described in Russo et al. (2002, 2006). Each point eofdiagram th is representa tive of a T-periodic regime. 280
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S4 0
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tS(sec)
Fig.2. The symmetric T-periodic solution diagram with the switch time, τ, as bifurcation parameter. The solution is reported with the solid temperature at the exit catalyst layer. Solid lines: stable T periodic regimes; dashed lines: unstable T periodic regimes; Fold bifurcations are indicated with the letter S, Flip bifurcations are indicated with the letter F and filled triangles.
The diagram is constituted by two curves:isola an and a mushroom-shaped curve. On the curve with a mushroom shape four fold bifurcations are detected. The two branches, corresponding to 0<τ<τS4 and τ>τS6 , are constituted by non-ignited T-periodic regimes where Tout≈Tin, while the upper branch, delimited by the fold bifurcations S3 and S5, corresponds to ignited T-periodic regimes. Then, a multiplicity is detected in the ranges [τS3, τS4] and [τS5, τS6]. On the T-periodic curve corresponding to the isola delimitated by [τS1, τS2] , different bifurcations leading to complex regimes are detected. These regimes are determined via simulation, and their asymptotic behavior, which varies with the switch time, is reported in Fig.3. In Fig.3, one thousand iterates of the Poincarè map after transients have died out are plotted for each τ-value. Data have been calculated via numerical simulation starting from one regime solution and the bifurcation parameter has been continuously changed. Once the regime solution was reached, this solution was used as initial condition of the subsequent simulation. An aperiodic complex regime (quasiperiodic or chaotic) corresponds in Fig.3 to a vertical line for a fixed parameter value, while a kT-periodic regime corresponds to k points placed on a vertical line. It is apparent from Fig.3 that a quasiperiodic regime exists over a wide range of the bifurcation parameter ([0, 40]). This quasiperiodic regime ar ises from the subcritical NS2 and it coexists with a stable ignited T-periodic regime in almost all range of existence. After a range where only the T-periodic regime exists as ignited regime (40<τ<50), a multiplicity of two 4T-periodic regimes appears. Increasing further the
Network ofT hreeCatalyticRe actors for MethanolSynthesis
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switch time, windows of multiperiodic and quasiperiodic ignited regimes alternate in a wide range of the bifurcation parameter. The transitions from quasiperiodic regimes to multiperiodic ones are characterized by frequency-locking phenomena (Mancusi et al.2004).
Fig.3. The asymptotic behavior for τ∈[0, 110]. The state is represented by the solid temperature at the exit catalyst layer.
It should be noted that the presence of subcritical, and thus catastrophic, Neimark Sacker bifurcations (N-S2 and N-S3 in Fig.2) may lead to plant operating problems. Indeed, when the network operates in a T-periodic regime close to Neimark Sacker bifurcation points N-S2 or N- S3, a disturbance in the bifurcation parameter could lead to jump to a quasi-periodic attractor characterized by larger amplitude oscillations of the vector state. Moreover, even far from the Neimark-Sacker bifurcation, the coexistence of operating T-periodic regime with other complex regimes may induce some problems to the process conduction. Despite of the complexity of the dynamic behaviour of the system, this range of the switch time has a great interest in terms of methanol yield. Indeed, as it is shown in Fig.4, for low switch time (τ=35.969) and in correspondence of out
the T-periodic regime, the methanol yield is maximum ( y CH 3OH = 0.0387 ). However,
the optimal T-periodic regime coexists with a quasiperiodic regime, and then it should be preferred, for safety reasons, a T-periodic regime with a slightly lower yield but in a range of the switch time without multiplicity. For low switch times, this range is very narrow [40,50] and it is embedded in the parameter region where quasiperiodic and multiperiodic regimes can be easily found. It should be noted that, although these complex regimes should be avoided for a safe operation, they are characterized by high average methanol yield. From the other hand, for higher switch time in the range [200,280], although the methanol yield is not maximum, a safe operation can be conducted with a methanol yield comparable with that obtained with a reverse flow reactor (Velardi and Barresi, 2002).
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Fig.4. The asymptotic behaviour for τ∈[0, 300]. The state is represented by the average mole fraction of methanol at the exit.
4. Conclusions In this work a non linear analysis of a network of three catalytic fixed-bed reactors for methanol synthesis has been performed. The bifurcation analysis of the periodic regime is conducted by a continuation technique based on the construction of discrete maps. Using the switch time as bifurcation parameter, two different curves of T-periodic regimes are detected: an isola and a mushroom shaped curve. On the isola catastrophic bifurcations are detected which lead to complex regimes like quasiperiodic and multiperiodic regimes. The complex dynamic behaviour is found only in the range of low switch time values. In this parameter range, despite of the complexity of the dynamics, it was detected the optimal T-periodic regime in yield in methanol. Thus, a suitable control policy should be adopted to operate at low switch times.
References G.F. Froment, 1990, Reverse flow operation of fixed bed catalytic reactor, In YU. Sh. Matros (Ed), Unsteady state processes in catalysis, Utrecht :VPS BV, 57;152. G.H. Graaf, E.J. Stamhuis and A.A.C.Beenackers, Chem. Eng. Sci., 1988: 12, 3185. E.Mancusi, L. Russo, G.Continillo, S.Crescitelli, Comp. Chem. Eng., 2004: 28; 187. Yu. Sh. Matros. Unsteady process in catalytic reactor. Amsterdam: Elsevier, 1985. Yu.Sh. Matros, Bunimovich GA, Catalysis Reviews: Science and Engineering, 1996: 38; 2333. L. Russo, Mancusi E, Maffettone PL, Crescitelli S., Chem. Eng. Sci., 2002: 57; 5065. L. Russo, P. Altimari, E. Mancusi, P.L. Maffettone and S. Crescitell, Chaos Solitons & Fractals 2006 : 28; 682. K.M. Vanden Bussche and Froment G.,1996, the STAR configuration for the methanols synthesis in reversed flow reactors, Cana J of Chem Eng, 1996:74(5); 729. S.Velardi and A.Barresi, Chem.Eng.Sci.,2002: 57; 2995. S. Velardi , Barresi AA, Manca D, Fissore D., Chem Eng. Sci. 2004; 99: 117.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
CFD Model of a Semi-batch Reactor for the Precipitation of Nanoparticles in the Droplets of a Microemulsion Alper A. Öncül,a Björn Niemann,b Kai Sundmacher,b,c Dominique Thévenina a
Otto-von-Guericke-University Magdeburg, Chair of Fluid Dynamics and Technical Flows (LSS/ISUT), Universitätsplatz 2, 39106 Magdeburg, Germany b Max Planck Institute for Dynamics of Complex Technical Sytems, Sandtorstr. 1, 39106 Magdeburg, Germany c Otto-von-Guericke-University Magdeburg,Chair of Process Systems Engineering (SVT/IVT), Universitätsplatz 2, 39106 Magdeburg, Germany
Abstract Precipitation inside the droplets of a microemulsion is a promising technology for the production of nanoparticles with tailored properties, like particle size or shape [1]. In this work, a water-in-oil (w/o)-microemulsion consisting of water, cyclohexane and the non-ionic technical surfactant Marlipal O13/40 is used to synthesise BaSO4 nanoparticles. The reaction is initiated by the mixing of two microemulsions, one containing the first reactant BaCl2 and the other containing the second reactant K2SO4, in semi-batch operation mode in a standard Rushton tank (V = 300 ml). A narrow particle size distribution (Fig. 1) and particle sizes between 4 and 40 nm have been achieved by experiments [2].
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A population balance model (PBM) assuming a discrete two-dimensional Poisson distribution for the dissolved Ba2+ and SO42- ions has been derived [3]. Rate constants for nucleation and growth kinetics proposed by Baldyga et al. [4] for bulk precipitation were fitted to take the slow-down effect of the droplet exchange into account. The obtained results showed a qualitatively good agreement with the experimental data (Fig. 1), but bigger deviations between the model and the experiments were observed for cases with a high concentration difference between the two microemulsions. The reason might be that the precipitation strongly depends on the supersaturation and that especially high concentration differences near the feed lead to a locally varying supersaturation profile, which is responsible for the deviations. To analyse this behaviour the zero-dimensional (homogeneous) PBM with over 600 ODEs is reduced into a set of 4 ODEs and implemented into the Computational Fluid Dynamics (CFD) code FLUENT 6.2 via user-defined scalars and functions. Hence, it became possible to investigate the reaction process for various three-dimensional inhomogeneous hydrodynamic conditions with reasonable computation times. Keywords: Nanoparticles, precipitation, microemulsion, PBM and CFD. 1. Numerical model and calculations The 4 equations of the transformed ODE system are only valid in the case that only one particle can exist in one droplet. This assumption is reasonable due to the fact that the droplet size is around 5 nm and only a few dissolved ions are present inside. Taking into account droplets with several particles would make the model considerably more complex, so that this point is not considered in the present formulation. The total number of droplets without particles (Eq. (1)) consists of a sink term for droplets in which nucleation takes place and a source term for the feeding of new droplets. 0 (t ) nmax +1 nmax +1 (i, j ) dn droplet feed (t ) P2 D (t ) ⋅ rN(i , j ) ⋅ V H 2O (t ) + n droplet =− ∑ (1) ∑ dt i = n +1 j = n +1 crit
crit
The total number of molecules in solid phase (Eq. (2)) is calculated by the addition of dissolved molecules which undergo a nucleation process and the addition of dissolved molecules which contribute to particle growth. dnP (t ) nmax +1 nmax +1 (i, j ) (i, j ) = ∑ ∑ N P ⋅ P2D (t ) ⋅ rN(i, j ) ⋅ VH 2O (t ) dt i =ncrit +1 j =ncrit +1 (2) nmax +1 nmax +1 π ⋅ N A ⋅ ρ BaSO4 2 (i , j ) (i , j ) 0 total + ∑ ∑ P2D (t ) ⋅ rG ⋅ ⋅ d P (t ) ⋅ ndroplet (t ) − ndroplet (t ) 2 ⋅ M BaSO4 i =2 j =2
(
)
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The total number of droplets (Eq. (3)) is only changed by the feeding of new droplets due to the fact that microemulsions are thermodynamically stable and the fusion of two droplets directly leads to the fission into two equally sized droplets. total (t ) dndroplet
dt
feed (t ) = ndroplet
(3)
Due to mass conservation, new Ba2+ can only enter the reactor by the feeding and therefore the total number of Ba2+ in the reactor can be calculated by total dn Ba 2 + (t )
dt
feed droplet (t ) . = n Bafeed 2+ ⋅ n
(4)
Further necessary variables, e.g. the mean numbers of reactant ions in one droplet needed for the determination of the Poisson distribution, are calculated by a few algebraic equations. A quite similar PBM has been recently developed by Singh & Kumar [5] and validated by the results of exact Monte-Carlo computations. The mean particle diameter is calculated by
d P (t ) = 3
6 ⋅ M BaSO4
π ⋅ N A ⋅ ρ BaSO
⋅ 4
total droplet
n
nP (t ) 0 (t ) − ndroplet (t )
(5)
and the kinetic rate approaches depend on the individual supersaturation, S, of each possible ion combination inside one droplet are represented by
(
)
rm( i , j ) = k meff ⋅ S ( i , j ) − 1
αm
(6) , m = N, G. This simplified model has been first applied for the homogeneous reaction process (zero-dimensional analysis) by MATLAB 7.0 before performing the inhomogeneous computations (three-dimensional analysis) in FLUENT. The results obtained from this time-dependent calculation are later compared with those obtained by CFD. The computation takes only a few seconds for the zerodimensional case. As explained previously, a Rushton tank has been used with a 6-bladed impeller rotating at 300 rpm in the clockwise direction (Re = 4,500) and with 18 baffles. The geometry and the grid of the reactor are shown in Fig. 2. There exists a total number of 79,776 volume elements in the whole domain.
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The droplets containing Ba2+ ions are fed into the reactor (where the SO42- ion containing droplets are already exist) during a period of 257.14 s with a constant feed rate. The three-dimensional simulations in the reactor have been performed in two stages by assuming a batch system (i.e. the liquid volume is not changing): Firstly the flow field has been simulated without the reaction process until homogeneous hydrodynamic conditions are achieved and secondly the whole reaction process has been simulated by switching off the flow and the turbulence model (i.e. using frozen-flow condition) while activating chemical reactions. By this way, the computations have been considerably accelerated. The computing times are respectively 24 h and 3 h for the first and the second stage employing a single Pentium-IV Linux PC (2.7 GHz/2 GB memory). In order to simulate the motion of the impeller, the multiple reference frames (MRF) model has initially been applied yielding an approximate flow estimation in a steady state and thereafter the obtained solution has been used as an initial guess for the further unsteady state simulation applying the sliding mesh model (SMM). Unlike the MRF model, the SMM is capable of reflecting the impeller-baffle effects taking the time-dependent location of the impeller into consideration. Therefore the SMM delivers much more accurate results for an unsteady state calculation [6]. During these flow field simulations 2nd-order discretization has been used in space and a 2nd-order implicit time formulation has been chosen for the unsteady solution of the SMM. The standard k-ε approach has been used for the turbulence model since this model should supply a reasonable accuracy and short computing times for such baffled tanks in which no strong, swirling flow occurs [7]. 2. Results Time-periodic flow conditions have been achieved after 100 full rotations of the impeller (i.e. 20 s real time) during the unsteady simulation employing the SMM. The velocity vectors on the middle cross-sectional surface of the reactor and the turbulent kinetic energy contours around the impeller with respect to the final results of the first-stage simulations are shown in Fig. 3.
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The evolution of the mean number of dissolved Ba2+ and SO42- ions per droplet and the mean particle diameter with respect to time according to the secondstage simulation (including only the inhomogeneous precipitation on top of the frozen flow) are represented in Fig. 4 for both zero- and three-dimensional analysis. Referring to this comparison, it can be concluded that an almost homogeneous reaction process is observed in the reactor. This proves that the mixing conditions inside the reactor are almost perfect so that three-dimensional effects are quite negligible in this configuration.
3. Conclusions and outlook The aim of this work was to develop a simplified model describing the precipitation of BaSO4 in a microemulsion droplet population and to implement this model for the three-dimensional, real case analysis of the reaction process in a semi-batch reactor. The initial numerical calculations demonstrate that the inhomogeneous case results agree with the homogeneous case results, which leads to the conclusion that the mixing conditions are almost perfect. The embedding of the PBM formulation within the CFD code allows numerically efficient unsteady flow computations. Next, non-stoichiometric processes will be simulated for various feeding rates and hydrodynamic conditions in order to analyse the influence of different inhomogeneous conditions. Acknowledgement We would like to thank Pro3, the German Process Engineering Expertise Network, for its financial support during this work.
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References 1. B. Niemann, F. Rauscher, D. Adityawarman, A. Voigt and K. Sundmacher, Chem. Eng. Process. (2006), in press. 2. D. Adityawarman, A. Voigt, P. Veit and K. Sundmacher, Chem. Eng. Sci. 60 (2005), 3373. 3. B. Niemann, J. Recksiedler, D. Adityawarman, K. Sundmacher, Annual AIChE Meeting (2005), Cincinnati, OH, USA. 4. J. Baldyga, W. Podgorska and R. Pohorecki, Chem. Eng. Sci. 50 (1995), 1281. 5. R. Singh, S. Kumar, Chem. Eng. Sci. 61 (2006), 192. 6. E.L. Paul, V.A. Atiemo-Obeng and S.M. Kresta, Handbook of Industrial Mixing: Science and Practice, John Wiley, Hoboken N.J. (2004), 292. 7. G. Montante, A. Bakker, Fluent News – Fall (2004), XIII(2), 8.
Nomenclature mean particle diameter [m]
dP
k
eff
effective kinetic rate constant
M BaSO4
molecular weight of BaSO4 [g/mol]
NA
Avogadro number [1/mol]
NP
ion number matrix: securing of an equal amount of both ions for 1 nucleus (nmax × nmax)
total nBa 2+
total number of Ba2+ ions in the reactor (in solid phase and as dissolved ions)
nBafeed 2+
mean number of Ba2+ ions inside one droplet of the feed
ncrit
critical number of molecules needed to form a stable nucleus
n
number of droplets without a particle
total ndroplet
total number of droplets
feed n droplet
feeding flux of droplets [1/s]
nmax
maximum number of dissolved ions of one kind per droplet
nP
total number of BaSO4 molecules in solid phase
nP
mean number of BaSO4 molecules in solid phase per droplet
P2 D
2-dimensional Poisson distribution for dissolved Ba2+ and SO42- ions (nmax × nmax)
Re
Reynolds number
rG
growth rate matrix (nmax × nmax) [m/s]
rN
nucleation rate matrix (nmax × nmax) [1/(m3 s)]
S
supersaturation ratio
V H 2O
total volume of the water inside the droplets [m3]
λ
mean number of dissolved Ba2+ and SO42- ions per droplet
0 droplet
ρ BaSO
4
density of BaSO4 [g/m3]
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Solution of the Population Balance Equation using the Sectional Quadrature Method of Moments (SQMOM) Menwer M. Attarakiha, Hans-Jörg Bartb,* and Naim M. Faqirc a Al-Balqa Applied University, Faculty of Engineering Tech., Chem. Eng. Dept., PO.Box 15008, 11134-Amman, Jordan. b University of Kaiserslautern, Faculty of Mechanical & Process Engng., Institute of Process Engng., POB 3049, D-67653 Kaiserslautern, Germany c Naim M. Faqir, University of Jordan, Faculty of Engng. & Technology, Chemical Engng. Department, 11942 Amman, Jordan *Corresponding author: E-mail:
[email protected], http://www.uni-kl.de/LS-Bart/
Abstract A numerical framework is introduced for solving the population balance equation based on accurately conserving (from theoretical point of view) an unlimited number of moments associated with the particle size distribution. The key idea in this work is based on the concept of primary and secondary particles, where the former is responsible for the distribution reconstruction while the latter one is responsible for different particle interactions such as breakage and coalescence. The numerical method is found to assemble all the advantages and disadvantages of the sectional and moment methods and hence the name: SQMOM. The method is illustrated here by considering pure breakage in a well-stirred vessel; however, it is already extended and tested for particle coalescence (agglomeration) and growth. Keywords: population balance, SQMOM, sectional methods.
1. Introduction Population balance equation (PBE) forms nowadays the cornerstone for modeling dispersed-phase systems arising in many engineering applications such as aerosols dynamics, crystallization, precipitation, liquid-liquid, gas-liquid and combustion processes. The resulting model equations of these processes range from integrodifferential to integro-partial differential equations with no general analytical solutions. Accordingly, there exist in the literature many numerical methods as attempts to solve certain type of the PBEs. These methods ranges from simple finite differences (FDS) or sectional methods using linear grids (in terms of particle diameters) to Galerkin and orthogonal collocations methods on finite elements. An exhaustive review of the available numerical methods is presented by Attarakih et al. (2004a). The quadrature method of moments (QMOM) as first introduced by McGraw (1997) to solve the PBE with pure growth is found very efficient from accuracy and computational cost point of view. Unlike the sectional (finite difference) methods, the QMOM has a drawback of destroying the shape of the distribution and retain information about it only through information stored in its moments. A recent comparison between the QMOM and the finite difference schemes could be found in Attarakih et al. (2006). On the other hand, one limitation of the finite difference schemes is their inability to predict accurately
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integral quantities (low-order moments as a especial case) associated with populations of sharp shapes (Ramakrishna, 2000). So, the objective of this work is whether it is possible to have a finite difference scheme that retains the advantages of the QMOM without destroying the shape of the distribution? The answer to this question is yes, where it is found in this work that all the attempts made previously to increase the accuracy of the finite difference schemes such as the fixed and moving pivot techniques (Ramkrishna, 2000) or the conservative descretization approach of the present authors (Attarakih et al., 2004b) are all merely limited answers to the above question. In this work the fundamental framework to combine the FDS and the QMOM is introduced and thoroughly tested using the available analytical solutions when it is possible. It is found that the new framework as it is called the Sectional Quadrature Method Of Moments (SQMOM) is very accurate in solving the PBEs and furnish a Gauss-like quadrature to evaluate any integral quantity associated with the population density.
2. The Population Balance Equation The population balance equation for a well-stirred continuous vessel of residence time, τ , could be written as: ∂f (d , t ) f = ∂t
feed
τ
−f
− Γ(d ) f (d , t ) +
∫
d max
d
Γ(d ') β (d | d ') f (d ')∂d '
(1)
where f(d,t) is the average number of droplets per unit volume of the vessel at time t. The first term on the left hand side denotes the rate of accumulation of droplets of size d. The term on the right hand side is the net rate of particles as a result of entry and exit events, breakage and coalescence. The source term, is rather complex and for simplicity, only the breakage part is presented (for the complete details see Ramkrishna, 2000). Γ and β are the breakage frequency and the daughter particle distribution respectively.
3. The Sectional Quadrature Method Of Moments (SQMOM) In the finite difference or sectional methods the particle size (here is denoted by the particle diameter, d) is discretized into finite number of sections, Ms. The population in each section is considered to behave like a single particle, and hence it is concentrated at a representative size usually at the middle of the section. In the present framework of descretization, this single particle will be called the primary particle and it will be responsible for the reconstruction of the shape of the distribution. In this way, the greater the number of primary particles (Ms), the more accurate is the reconstruction of the distribution. Unfortunately, large number of primary particles required is to estimate integral quantities of the distribution accurately and hence increasing extensively the computational loads when the population balance equation is coupled for example to a CFD calculations (Marchisio and Fox, 2005). The interaction between primary particles in different sections, due to breakage event for example, results in a new primary particle with no representative size due to the discrete approximation of the distribution. Because of the newly-birthed particle could not conserve any of its low order moments but one (if it is located at the middle of the section), the rest of the low-order moments are predicted with low accuracy and hence the associated integral quantities.
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To overcome this fundamental problem of the sectional methods, secondary particles are generated in each section with positions (abscissas) d <j i > , i = 1, 2,...M s , j = 1, 2,...N q (see Fig.(1) upper panel), where the number of these secondary particles dictates the desired number of low-order moments to be conserved. The population density in each section is partitioned between these particles according to the variation of the population density in this section by assigning weights ( w<j i > , i = 1, 2..M s , j = 1, 2..N q ) to each particle. These secondary particles are exactly equivalent to the number of quadrature points in Gauss-like quadratures or the QMOM. Accordingly, each secondary particle could conserve or reproduce two low-order moments and in general 2Nq moments, where Nq is the number of secondary particles. Secondary particle
Nq
1
w
w
wNq
1
w
d1 d 2 d3 d Nq d1 d 2 d3
di −1/ 2
di +1/ 2
di +3/ 2
Nq
wi = ∑ wm
wi +1
di
di
m =1
di −1/ 2
di +1/ 2
< i +1> d Nq
Primary particle
di +3/ 2
Fig.(1): The concept of primary and secondary particles. In this framework, the particle mechanisms such as breakage and coalescence occur through interactions between the secondary particles. It is obvious from Fig.(1) above (the upper panel) that N q × M s particles are contributing in the breakage and coalescence events. The distribution could be reconstructed from the secondary particles by averaging the total weights of the secondary particles with respect to the section width and locating it at the mean size of the secondary particles as shown in Fig.(1) (the lower panel). In pure mathematical sense, the above presentation is equivalent to applying the QMOM to each section of arbitrary width: [di −1/ 2 , di +1/ 2 ], i = 1, 2,...M s resulting in a set of sectional moments that could be written as:
μr (t ) = ∫
di +1/ 2
di −1/ 2
d r f (d , t )∂d ,
r = 0,1, 2...2 N q − 1
(2)
By applying this set of moment transformations to Eqs.(1) and after some algebraic manipulations one could get: Ms T T d μr (t ) μr , feed − μr< i > = − Br< i > ⎡⎣Γ < i > i w< i > ⎤⎦ + ∑ Cr< i , m > ⎡⎣ Γ < i > i w< i > ⎤⎦ τ dt m =i
(3)
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where: Br< i > = ⎡ (d1 ) r (d 2< i > ) r .... (d N< iq> ) r ⎤ , ⎣ ⎦ < i > < i > < i > Γ = ⎡ Γ(d1 ) Γ(d 2 ) .... Γ(d Nq ) ⎤ , w< i > = ⎡ w1 w2< i > .... wN ⎤ and, ⎣ ⎦ ⎣ ⎦ <m> <m> min( d , d ) min( d , d ) i +1/ 2 1 i +1/ 2 N q ⎡ ⎤ Cr = ⎢ ∫ d r β (d | d1< m > )∂d ... d r β (d | d N< qm > )∂d ⎥ ∫di−1/ 2 ⎣ di−1/ 2 ⎦
Note that each secondary particle in the ith section is characterized by its location (abscissa), d <j i > , and weight, w<j i > . These characterization variables are only function of time and could be calculated by inverting the ith moment problem assuming equal number of secondary particles in each section as follows: Nq
μr = ∑ (d <j i > ) r w<j i >
(4)
j =1
The above 2 N q equations are solved uniquely for the N q abscissas and N q weights using the standard product-difference algorithm as outlined by McGraw (1997). For the special cases of one and two secondary particles an analytical solution could be found. The solution when one secondary particle is used is trivial; however, for two secondary involved (but straight forward) particles ( N q = 2 ) the algebraic manipulations are rather and the result is presented below: 1 1 d1,2 = ψ ( μr<,ir>= 0,1,2,3 ) ± ψ 2 ( μr<,ir>= 0,1,2,3 ) − 4 χ ( μr<,ir>= 0,1,2,3 ) 2 2 1,2
w
=μ
0
⎛ σ ⎜ ⎜ ⎝ d − d1,2
(5)
2
⎞ 1 ⎟ 2 ⎟ ⎠< i > 1 + ⎣⎡σ /[d − d1,2 ]⎦⎤
(6)
where: ψ , χ , σ are functions of the first four moments and d < i > = μ1< i > / μ0< i > The system given by Eqs(3), (5) and (6) is a differential algebraic equation system (DAE) that could be reduced to only a differential system by substituting Eqs.(5) and (6) into (3). Note that it is clear by this combination that the solution of the system (3) guarantees the conservation (reproduction) of 2 N q low order moments ( μ r , r = 0,1,...2 Nq −1 ). Since the number of secondary particles, N q , is unlimited from theoretical point of view, it follows that the discretized PBE given by the system (3) is guaranteed to reproduce an unlimited number of low-order moments (internally consistent with respect to 2 N q moments). This makes the present framework of discretization generally consistent and accurate for solving general PBEs without placing any assumption on the shape and type of the distribution or breakage functions. Accordingly, all the attempts in the literature that are concerned with conserving certain and hence limited number of moments appear to be special cases of the present descretization method by varying the number of primary and secondary particles. For example, when the number of the primary particles equals one the standard QMOM is recovered, while when the number of secondary particles equals one, the standard moving pivot technique is recovered by conserving the total number and volume of the particles in each section (the zero and third moments are conserved).
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4. Numerical Results and Discussion Due to the space limitation, only one example is presented here for the case of particle (droplet) breakage in a well-mixed continuous vessel where the analytical solution for Eq.(1) is available (Attarakih et al., 2004b) using the following set of functions: 3
= 3d 2 e − d , Γ = d 6 , β = 6d 2 / d '3 , f (d , 0) = 0 , d min = 0.001 , d max = 2 and τ = 100s . The sectional moments are evolved in time using the trapezoidal rule with fixed step size of 0.1 second. First, Fig.(2a) compares the convergence of the SQMOM at fixed number of secondary particles by varying the number of primary particles from 2 to 4. It is clear how the first two moments ( μ0 & μ1 ) are over predicted using only two primary and secondary particles. The inaccuracy is attributed to the sharpness of the distribution as it is evolved in time (see Fig. 3a). By doubling the number of primary particles or equivalently the number of sections, the width of each section is decreased resulting in an accurate integration over the sharp distribution as expected where this fact is true for all Gauss-like quadrature methods. On the other hand, by increasing the number of secondary particles from 2 to 3 as seen in Fig.(2b), the same result is almost obtained, which is expected since the accuracy of the quadrature methods is increased by increasing the number of the quadrature points (secondary particles). In Fig.(3a), the average number concentration as predicted using the SQMOM is compared to the analytical solution at different periods of time. It is clear that using 35 primary particles is enough to follow the shape of the number concentration function very accurately. However, since the predicted shape of the distribution is not used in the prediction of any integral property, small number of primary particles is found enough to get an idea about the shape of the distribution. Consequently, the location and weights (Eqs.(5) & (6)) of the secondary particles is used to evaluate any integral over the unknown distribution with the desired accu racy. To get more insight on the convergence proper ties of the SQMOM, the systematic error anal . num. d 30 − d 30 ) based on the mean particle diameter ( d 30 = μ3 / μ0 ) is studied as ( function of the number of primary and secondary particles. It is evident that the order of convergence is increased by increasing both the primary and secondary particles due to the increasing accuracy of evaluating the unclosed integrals in the PBE. The increasing accuracy by increasing the number of secondary particles is reported by many researchers (McGraw, 1997 and Marchisio, 2005). f
feed
12
10
12
(a)
10
SQMOM: Ms = 2, Nq = 2 SQMOM: Ms = 4, Nq = 2 analytical
(b)
8 moments
moments
8
μ0
6
μ0 SQMOM: Ms = 2, Nq = 2 SQMOM: Ms = 2, Nq = 3 analytical
6
4
μ1
4
μ1
2
μ2
2
μ2
0 0
μ3 50
100 150 200 250 300 dimensionless droplet diameter ( )
350
400
0 0
μ3 50
100 150 200 250 300 dimensionless droplet diameter ( )
350
400
Fig.(2): Convergence of the first four moments using the SQMOM: a- By varying the number of primary particles. b- By varying the number of secondary particles.
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The present framework is already extended to particle coalescence and growth where the results are found very accurate and the computational load is dependent on the accuracy and details needed by the user. -1
25
10
(a )
(b)
-2
10
a
time = 100 s
n
20
SQMOM: Nq = 2 SQMOM: Nq = 3
systematic error = | d30 - d30 |
τ = 100 s
3
average number concentration (1/m )
SQMOM: Ms = 35, Nq = 2 analytical
15
10 time = 50 s
5
-3
~ O (1/Ms) 2.8
10
~ O (1/Ms)2.4
-4
10
time = 10 s
0 0
-5
0.5 1 1.5 dimensionless droplet diameter ( - )
2
10
0
10
1
10
2
10
number of primary particles ( Ms )
Fig.(3): a- Comparison between the analytical solution and that predicted by the SQMOM. b – Convergence of the SQMOM in terms of the systematic error in d30.
5. Conclusions The present framework for solving the PBE based on the concept of the primary and secondary particles is found general where all the previous attempts in literature to overcome the problem of internal consistency are merely especial cases of the present framework. In this way, the primary particles are responsible for the distribution reconstruction, while the secondary ones are responsible for breakage, coalescence .. etc. events and carry a lot of information about the distribution. The SQMOM is found extremely accurate and converges very fast by increasing either the number of primary or secondary particles; however, at the expense of the computational load. This computational load is up to the user and the degree of details required about the distribution. Accordingly, the flexibility of the method by its reduction to the standard QMOM when the number of primary particles equals one makes it very attractive from computational point of view. For example, if if Ms = 1 and Nq = 2, only four ODEs are N q = 2, then twenty ODEs are to be solved. to be solved; however, if Ms = 5 and
References M. M. Attarakih, H.-J. Bart, & N. M Faqir (2006). Numerical solution of the bivariate population balance equation for the interacting hydrodynamics and mass transfer in liquid-liquid extraction columns., Chem. Engng. Sci. 61, 113-123. M. M. Attarakih, H.-J. Bart, & N. M Faqir (2004a). Numerical solution of the spatially distributed population balance equation describing the hydrodynamics of interacting liquid-liquid dispersions.Chem. Engng. Sci. 59, 2567-2592. M. M. Attarakih, H.-J. Bart, & N. M Faqir (2004b). Solution of the droplet breakage equation for interacting liquid-liquid dispersions: a conservative discretization approach. Chem. Engng. Sci.,59 , 2547-2565. L. D. Marchisio. (2005). Solution of the population balance equations using the direct quadrature method of moments. J. Aerosol Sci., 36, 43-73. R. McGraw (1997). Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. & Tech.,27, 255-265.
D. Ramkrishna. (2000). Population Balances, Academic Press, San Diego.
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A Global Parametric Programming Optimisation Strategy for Multilevel Problems N.P. Fa´ıscaa , V. Duab , P.M. Saraivac , B. Rustema and E.N. Pistikopoulosa† a
Centre for Process Systems Engineering, Imperial College London, SW7 2AZ, U.K.
b
Centre for Process Systems Engineering, University College London, WC1E 7JE, U.K.
c
Gepsi - PSE Group, Department of Chemical Engineering, University of Coimbra, 2020-290 Coimbra, Portugal Abstract In this paper, we outline the foundations of a general global optimisation strategy for the solution of multilevel hierarchical and general decentralised multilevel problems based on our recent developments in multiparametric programming theory. The core idea is to recast each optimisation subproblem in the multilevel hierarchy as a multiparametric programming problem and then transform the multilevel problem into a single-level optimisation problem. For decentralised systems, where more than one optimisation problem is present at each level of the hierarchy, Nash equilibrium is considered. A three person dynamic optimisation problem is presented to illustrate the mathematical developments. 1. Introduction It is widely recognised that the successful design of large and complex systems involves some type of decomposition of the original problem into smaller and intercommunicating subsystems, typically arranged in a multilevel hierarchy. Such multilevel problems arise commonly in process systems engineering [9,12,13], with bilevel programming problems being the simplest and most studied [8,14]. Bilevel programming problems involve an optimisation hierarchy of two levels, of the following form: min F (x, y) x,y
s.t.
G(x, y) ≤ 0 x∈X y ∈ argmin{f (x, y) : g(x, y) ≤ 0, y ∈ Y }
(1)
where X ⊆ Rnx , Y ⊆ Rny and both are compact convex sets; F and f are real functions: R(nx+ny) → R; G and g are vectorial real functions, G : R(nx+ny) → Rnu and g : R(nx+ny) → Rnl ; nx, ny ∈ N and nu, nl ∈ N ∪ {0}. We also define the rational † Corresponding
author: [email protected]
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reaction set as follows: M(x) = {y ∈ Y : y ∈ argmin{f (x, y) : y ∈ C(x)}};
(2)
where, C(x) = {y ∈ Y : g(x, y) ≤ 0}; Multilevel and decentralised optimisation problems, which typically arise in many engineering [4,9,12,13] and financial applications [2,10], involve a hierarchy of such optimisation levels, as in (1); where each optimisation level (or subproblem) controls a subset of the overall optimisation variables. When more than one subproblem is present at the same hierarchical optimisation level the problem is called a decentralised multilevel problem. Nash equilibrium is often a preferred strategy to coordinate such decentralised systems [13]. Despite their significance, general solution strategies for solving such complex problems are limited, especially due to the multi-layer nature, non-linearities and nonconvexities occur [14]. In addition, the potential presence of logical decisions (which requires the inclusion of binary variables) increases further the complexity of the problem. Therefore, it is widely accepted that a global optimisation approach is needed for the solution of such multilevel optimisation problems [8]. Recently, Pistikopoulos and co-workers [1,5,6,11] have proposed novel solution algorithms, based on parametric programming theory [7], which open the possibility to address general classes of multilevel programming problems. The core idea of this approach is to recast each optimisation subproblem as a multiparametric programming problem, and hence obtain an analytical solution for the rational reaction set for each of the subproblems. These analytical expressions can then be used to compute, through direct comparison the Nash equilibrium between subproblems in the same optimisation level, for decentralised problems. 2. Methodology The proposed approach is illustrated by considering a multiple person dynamic linearquadratic optimisation problem [10], which involves the coordination of a number of controllers within a complex system. Consider the dynamic system represented in Figure 1, where u, v 1 and v 2 are input variables, and x, y 1 and y 2 output variables: u, v 1 , v 2
System
x, y 1 , y 2
Figure 1. Schematic representation of the dynamic system
The discrete dynamic behaviour of this system is described by the following linear state transition model: xt+1 = xt + ut − 2vt1 + vt2 , 1 = yt1 + 2vt1 , yt+1 2 = yt2 + 2vt2 , yt+1
t = 0, 1, 2
(3)
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Parametric Global Optimisation for Decentralised Multilevel Systems with constrains on the input and state variables as follows: −30 ≤ v01 , v11 , v21 , v02 , v12 , v22 ≤ 30, −20 ≤ u0 , u1 , u2 ≤ 20, −10 ≤ x0 , y01 , y02 ≤ 10.
(4)
In Process and Systems Engineering, the performance of the system is in most of the cases optimised regarding just one objective function (e.g. optimal control). But, it is also common to have conflicting goals during in the management of a dynamic process. Since the aim is to optimise the overall performance of the system, suitable cost-functions should be considered. For example, we consider a three-controller system [10]: J1 = min 4x3 + 3y31 + 2y32 + u0 ,u1 ,u2
2 n o X 2 2 2 (ut ) + vt1 − vt2 + 2ut xt + x2t ,
(5)
t=0
J2 = 2min 2x3 + 3y32 + 2 2 v0 ,v1 ,v2
2 n X 2 2 o 2 · ut vt2 + vt1 + 1 + vt2 + 1 ,
(6)
t=0
J3 = 1min x3 + 2y31 − 10y32 + 1 1 v0 ,v1 ,v2
2 n X 2 o 2 −15ut + vt1 − 1 − 2vt1 vt2 + vt2 .
(7)
t=0
Where J1 , J2 and J3 correspond to Controllers 1,2 and 3, respectively. Figure 2 further displays two possible configurations for the control structure of the considered system.P Controller 1 Controller 1
Controller 2
Controller 3
Controller 2
Controller 3 Nash Equilibrium
(a) Three-level controller structure
(b) Multifollower controller structure
Figure 2. Three-controller multilevel problem
The objective then is to derive suitable optimal strategies for the two controller structures. Case (a) of Figure (2) corresponds to a three-level optimisation problem, whereas case (b) refers to a bilevel multifollower optimisation problem. In the following subsections, we briefly describe the developed optimisation strategy for the global solution [6] of these two classes of optimisation problems.
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2.1. Three-level programming problem The steps of the proposed parametric global optimisation strategy for the threelevel programming problem follow directly from the strategy adopted to the bilevel programming problem [6], and can be summarised as follows: Step 1. Recast the lower optimisation problem, J3 , as a multiparametric programming problem, with the control variables belonging to the other two levels being the parameters (x0 , y01 , y02 , ut , vt2 ). Solve the resulting problem using a multiparametric optimisation algorithm [5]; Step 2. Include the rational reaction set, vt1 = f (x0 , y01 , y02 , ut , vt2 ), into the optimisation problem corresponding to Controller 2, J2 ; Step 3. Recast the optimisation problem J2 as a multiparametric programming problem, with the control variables belonging to the upper level being the parameters (x0 , y01 , y02 , ut ), and solve it using a multiparametric optimisation algorithm; Step 4. Include the rational reaction set from the two levels below, vt1 = f (x0 , y01 , y02 , ut , vt2 (ut )), and vt2 = f (x0 , y01 , y02 , ut ), into the optimisation problem corresponding to the leader controller, J1 ; Step 5. Recast the multilevel optimisation problem in a single-level multiparametric programming problem, having as parameters the state-space (x0 , y01 , y02 ), and solve it using a multiparametric optimisation algorithm. If overlapping regions were created at Step 5, the comparison method described in [1] is employed. The result for this problem is listed in Table 1.
Table 1 Solution to the three-level optimisation problem Critical Region 1 Critical Region 2 u0 = 6.84615 − 0.76928x0 u0 = −0.333333 − 1.8519x0 u1 = −20 u1 = −1.33333 + 2.8148x0 u2 = 15.2308 + 0.15388x0 u2 = −2 − 2.4444x0 −10 ≤ x0 ≤ −6.63161 −6.63161 ≤ x0 ≤ 7.36377 Critical Region 3 Critical Region 4 u0 = −1.53333 − 1.6889x0 u0 = −9 − 0.72732x0 u1 = 8.26667 + 1.5111x0 u1 = 20 u2 = −20 u2 = −20 7.36377 ≤ x0 ≤ 7.76466 7.76466 ≤ x0 ≤ 10 v01 = v02 = −2 − 0.5u0 ; v11 = v12 = −2 − 0.5u1 ; v21 = v22 = −2 − 0.5u2
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2.2. Bilevel multifollower programming problem The solution steps for the bilevel multifollower optimisation problem are as follows: Step 1. Recast optimisation subproblems corresponding to Controller 2 and Controller 3 as multiparametric programming problems, with parameters being the set of variables out of their control, (x0 , y01 , y02 , ut , vt2 ) and (x0 , y01 , y02 , ut , vt1 ), respectively. Then solve each one using a multiparametric optimisation algorithm; Step 2. Compute the Nash equilibrium point (see Appendix I), through direct comparison of the two explicit analytical rational reaction sets,
vt1 = f1 (x0 , y01 , y02 , ut , vt2 ) ; vt2 = f2 (x0 , y01 , y02 , ut , vt1 )
(8)
Step 3. Incorporate both expressions into Controller 1, J1 , and formulate a multiparametric optimisation with the state-space (x0 , y01 , y02 ) being the parameter. The unique solution for this problem, in the analysed state space (−10 ≤ x0 , y01 , y02 ≤ 10), is shown in Table 2. Table 2 Solution to multifollower problem Critical Region 1 u0 = 1 − x0 u1 = −8 + x0 u2 = 5 − x0 v01 = v02 = −6 + x0 v11 = v12 = 3 − x0 v21 = v22 = −10 + x0 −10 ≤ x0 ≤ 10
The complexity of this solution procedure clearly depends on the complexity of the underlying parametric programming algorithms, as studied in our previous work [5]. 3. Concluding Remarks A novel global optimisation based strategy has been described for the solution of the hierarchical multilevel and decentralised multilevel systems problems to global optimality. Based on recent developments in parametric programming theory and algorithms [5,11], each subproblem of the optimisation hierarchy is interpreted as a multiparametric programming problem with the variables from the other subproblems being the parameters. The approach has been successfully tested using a three person dynamic problem illustrative example, for the two different optimisation strategies. While the illustrative example involves the same model for the three objective functions (controllers), the proposed optimisation strategy is equally applicable for the case when different models are involved (i.e. all control subproblems are treated in a decentralised fashion). The developed algorithm can address linear and quadratic (controller) objective functions, and a linear model for the dynamic system. Extensions towards general nonlinear models are currently under development.
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4. Acknowledgments Financial support from EPSRC (GR/T02560/01) and Marie Curie European Project PRISM (MRTN-CT-2004-512233) is gratefully acknowledged. 5. Appendix I In this section, the computation of the Nash equilibrium point, using the analytical expressions for the rational reaction sets, is briefly described. Being ut (Controller 1), vt2 (Controller 2) and vt1 (Controller 3) the optimisation variables, the Nash equilibrium for the lower level (u, vt2 ∗, vt1 ∗), Figure 2(b), is reached when [3]: J2 (u, vt2 ∗, vt1 ∗) ≤ J2 (u, vt2 , vt1 ∗) J3 (u, vt2 ∗, vt1 ∗) ≤ J3 (u, vt2 ∗, vt1 )
∀vt2 ∈ Vt2 ∀vt2 ∈ Vt2
(9)
As mentioned before, this equilibrium is easily computed since the expressions for the rational reaction sets are explicitly obtained. Thus, this equilibrium point is equivalent to the solution of the following system: 1 vt = f1 (ut , vt2 ) . (10) vt2 = f2 (ut , vt1 ) REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
J. Acevedo and E.N. Pistikopoulos, Ind. Eng. Chem. Res. 36 (1997) 717. G. Anandalingman, J. Opl. Res. Soc. 39 (1988) 1021. T. Ba¸sar and G.J. Olsder, Dynamic Noncooperative Game Theory, London, 1982. P.A. Clark, Embedded Optimization Problems in Chemical Process Design, Ph.D. Thesis, Carnegie-Mellon University 1983. V.Dua, N.A. Bozinis and E.N. Pistikopoulos, Comput. Chem. Eng. 26 (2002) 715. N.P. Fa´ısca, V. Dua, P.M. Saraiva, B. Rustem and E.N. Pistikopoulos, Submitted for publication in J. Global Optim., May (2005). A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, New York, 1983. C. Floudas, Deterministic Global Optimization, Dordrecht, 2000. M. Morari, Y. Arkun and G. Stephanopoulos, AIChE Journal 26 (1980) 220. P. Nie, L.Chen and M. Fukushima, Eur. J. Opl. Res. 169 (2006) 310. E.N. Pistikopoulos, V. Dua, N.A. Bozinis, A. Bemporad and M. Morari, Comput. Chem. Eng. 24 (2000) 183. G. Stephanopoulos and C. Ng, Journal of Process Control 10 (2000) 97. A.N.Venkat, J.B. Rawlings and S.J. Wright Proceedings of AIChE 2005. L.N. Vicente and P.H. Calamai, Journal of Global Optimization 5 (1994) 291.
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Modelling deammonification in biofilm systems: Sensitivity and identifiability analysis as a basis for the design of experiments for parameter estimation Doris Brockmann,a Karl-Heinz Rosenwinkel,a Eberhard Morgenroth,b,c a
Institute of Water Quality and Waste Management, University of Hanover, 30167 Hanover, Germany b Department of Civil and Environmental Engineering and cDepartment of Animal Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Abstract A procedure for selecting parameters for identifiability analysis and parameter estimation and for the design of experiments as a basis for parameter estimation was described for a model of deammonification in biofilm systems. A larger number of parameters were identifiable using data from batch experiments than from continuous reactor operation. However, not all sensitive parameters were identifiable from experimental data due to a large degree of parameter correlation. Keywords: deammonification, experimental design
anammox,
biofilm
modelling,
identifiability,
1. Introduction For the treatment of wastewater with high ammonium concentrations and low organic carbon to nitrogen ratios (C/N ratio) classical biological nitrogen elimination becomes cost-intensive. Thus, new biological nitrogen elimination processes, like deammonification (Helmer et al., 1999a) and CANON (Strous, 2000), attract increasing interest. These processes are a combination of partial oxidation of ammonium to nitrite and anaerobic ammonium oxidation (anammox). Anaerobic ammonium oxidation converts ammonium and nitrite directly to dinitrogen gas. Both steps of the deammonification can be combined in biofilms or granules into a single-stage operation (Helmer et al., 1999b; Sliekers et al., 2002). Modelling and simulation of the deammonification can help to optimise the process. For the application of the model it is essential to estimate a set of model parameters that fit the experimental data. To be able to estimate a unique set of parameters it is necessary that the model parameters are not correlated and that the experimental data has sufficient information content with respect of the parameters that are estimated. The ability to obtain a unique parameter set that is able to describe the behaviour of a system is called “identifiability” (Petersen, 2000). Structural/theoretical identifiability is based on the model structure and deals with the ssibility po to obtain a unique value for each parameter from a set of noise-free (perfect) data (Dochain et al., 1995). In contrast, practical identifiability deals with the question whether the experimental and noise corrupted data available are informative enough for giving accurate parameter values (Vanrolleghem et al., 1995). The design of an experiment, e.g. mode of reactor operation, operation conditions, measured variables and measuring intervals, has an influence on the identifiability of the parameters (Holmberg, 1982; Dochain and
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Vanrolleghem, 2001). Hence, designing an experiment which provides as much information as possible to determine and estimate the regarded parameters can help to reduce the experimental effort. Van Hulle (2005), for example, performed an experimental design to determine the nitrite inhibition constant for the anammox bacteria from ammonium and nitrite concentration profiles in the bulk liquid of a biofilm reactor. In this paper, a procedure for selecting parameters for identifiability analysis and following parameter estimation as well as the design of experiments for the estimation of parameters are presented.
2. Materials and methods 2.1. Mathematical model The model included growth and inactivation of aerobic ammonium oxidisers, XNH, aerobic nitrite oxidisers, XNO, and anaerobic ammonium oxidisers, XAN. The growth processes were described with Monod kinetics for substrate utilisation. Ammonium consumption for biomass growth was neglected due to the low production of autotrophic biomass. Inactivation processes were defined for all three organism groups which reduced the amount of active biomass and formed inactive or inert biomass. Default values for the parameters were taken from Hao et al. (2002). The affinity constant for oxygen of the nitrite oxidisers was modified to obtain a better fit to measured data in steady state. The model describing the kinetics was integrated into the biofilm compartment of AQUASIM (Reichert, 1998) which assumes a one-dimensional biofilm structure. The reactor set-up of the simulated continuously operated reactor was based on layout and operation parameters of a lab-scale moving-bed biofilm reactor operated for deammonification (Hippen et al., 2001). The layout of the batch reactor was according to the set-up used by Helmer et al. (2001) to investigate the conversion processes in the lab-scale moving-bed plant for deammonification. The biofilm thickness in the model was set to 450 μm as biofilm thicknesses of 420 to 510 μm were measured on the moving-bed carriers of the investigated lab-scale moving-bed reactor (Tromm, 1999). 2.2. Sensitivity and identifiability analysis In the following, the used procedure for parameter selection and identifiability analysis is briefly described. The procedure is presented in more detail by Brockmann (2006). 2.2.1. Selection of parameters for identifiability analysis and subsequent parameter estimation The selection of parameters for identifiability analysis and parameter estimation was based on results of a regional steady state sensitivity analysis. The regional sensitivity analysis was carried out based on a factorial design as described by Box et al. (1978). A high sensitivity is one requirement for an identifiable parameter the parameters with the largest impact on the model output were selected for identifiability analysis and subsequent parameter estimation. 2.2.2. Collinearity as measure for non-identifiability of parameters The second requirement for an identifiable parameter is that a shift in the model output caused by a shift in the parameter may not be approximately compensated by appropriate changes in other parameters. The “compensability” was quantified by the calculation of the collinearity index γk defined by Brun et al. (2001). The collinearity ~ index was calculated from the scaled sensitivity matrix S .
Modelling Deammonification in Biofilm Systems γk =
1 min
β =1
~ Sβ
223
(1)
The local sensitivity functions sj were calculated for default parameter values taken from Hao et al. (2002). 2.3. Design of experiments Different experimental layouts, operation conditions and measurements were tested and evaluated regarding identifiability of the kinetic parameters. The aim was to determine the experiment providing the most information on the selected parameters. This experiment should then be used to estimate the selected and identifiable parameters. On the one hand, continuous reactor operation (almost in steady state) was studied and compared to the results obtained for a batch experiment. On the other hand, six different experimental layouts for a batch experiment were evaluated.
3. Results and discussion 3.1. Parameters selected for identifiability analysis and parameter estimation Regional steady state sensitivities were calculated to determine the parameters with the largest impact on the model output. The growth and inactivation rates for all three groups of microorganisms had a very large influence on the process (data not shown). All three affinity constants for oxygen had a distinct effect as well. The affinity constants for nitrite had the smallest impact on the output and were therefore not selected for parameter estimation. In total, nine parameters were selected for parameter estimation: the growth and inactivation rates for all three groups of microorganisms and all three affinity constants for oxygen. 3.2. Parameter identifiability: Continuous reactor operation versus batch experiment Parameter identifiability was studied for continuous reactor operation and a batch experiment carried out for a dissolved oxygen concentration (DO) of 0.7 mg O2/L and an ammonium influent and initial concentration, respectively, of 150 mg NH4-N/L. Collinearity indices were calculated using Eq. (1) for all possible combinations of ammonium (NH4+), nitrite (NO2-) and nitrate (NO3-) measurements. Parameter subsets with a collinearity index below a threshold of 15 were considered identifiable. From continuous reactor operation parameter subsets with a maximum of four parameters were identifiable from NH4+, NO2- or NO3- measurements (Figure 1 a). Combining measurements of the different chemical compounds did not enhance parameter identifiability as parameter subsets with a maximum of only three parameters were identifiable from combined measurements (Figure 1 b). The “good” information a certain extent by the “poor” obtained from NH4+ measurements were compensated to information derived from NO2- and NO3- measurements. From the batch experiment parameter subsets of size 6 could be identified from a combination of all measurements (NH4/NO2/NO3) (Figure 1 c and d). In this case, combining measurements of the different chemical compounds improved the parameter identifiability. The nine selected parameters were, however, not uniquely identifiable based on measurements from continuous reactor operation or from the batch experiment. Even combining the measurements from batch and continuous experiments did not enhance parameter identifiability. The “poor” information derived from continuous experiments compensated to a certain degree the “good” information obtained from batch experiments.
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Figure 1. Collinearity indices for all parameter subsets for continuous reactor operation and batch experiment at DO 0.7 mg/L
3.3. Experimental design for batch experiments For the experimental design six different batch experiments were investigated regarding parameter identifiability (Table 1). Besides three different dissolved oxygen concentrations in the bulk liquid (No. 1-3), the addition of nitrite at the start of the experiment (No. 4), the pulse injection of ammonium during the experiment (No. 5) and switching off the aeration at the halftime of the experiment (No. 6) were analysed concerning identifiability. Table 1 summarises the number of identifiable subsets of parameters for NO3measurements and all possible measurement combinations. These measurements and measurement combinations provide significantly more information for the identifiability analysis and the estimation of parameters compared to NH4+ and NO2- measurements. The numbers of identifiable parameter subsets are given for parameter subsets of size 5 or 6 parameters. Except for experiment No. 4, a maximum of six parameters was identifiable from the studied experimental designs based on a combination of all three measurements. Although variations of the experimental design enhanced the identifiability of the parameters for some measurement combinations not all nine selected parameters could be identified from the experimental data for any of the experimental designs. For each group of microorganisms high collinearity was observed between the growth rate and the affinity constant for oxygen. Due to the correlation between the parameters of these three subsets of size 2 only six of the nine selected parameters were identifiable.
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Table 1. Number of identifiable parameter subsets of size 5 (left) or 6 (right) parameters for the investigated experimental designs No.
Experimental design
1
DO 0.7
2
NO3
NH4/NO2
NH4/NO3
NO2/NO3
NH4/NO2/NO3
/;/
/;/
/;/
16 ; /
38 ; 8
DO 2
/;/
/;/
/;/
/;/
36 ; 8
3
DO 5
/;/
/;/
/;/
/;/
36 ; 8
4
DO 0.7, NH4 + NO2
/;/
/;/
/;/
/;/
36 ; /
5
DO 2, NH4 pulse
43 ; 10
10 ; /
6;/
24 ; /
36 ; 8
6
DO 5 off after 3 h
8;/
/;/
/;/
8;/
36 ; 8
4. Conclusions Batch experiments provide significantly more information for the identifiability analysis and the estimation of parameters compared to continuous reactor operation at steady state. Careful selection of measurements measurement or combinations as well as the experimental design improves the identifiability of the selected parameters. Nevertheless, due to high correlations among some of the selected parameters not all of them may be identifiable from the data. To solve the parameter identifiability problems either more parameter values have to be assumed based on values from literature or parameter combinations need to be defined for highly correlated parameters.
Acknowledgements
This study was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) under project No. RO 1221/8-1.
References Box, G.E.P.; Hunter, W.G.; Hunter, J.S. (1978): Statistics for experimenters: An introduction to design, data analysis and model building. Wiley series in probability and mathematical statistics. John Wiley & Sons, Inc. Brockmann, D. (2006): Modelling nitrification and deammonification: Sensitivity analysis, identifiability analysis and design of experiments. Faculty of Civil Engineering and Geodetic Science. University of Hanover. Ph.D. thesis. Brun, R.; Reichert, P.; Kunsch, H.R. (2001): Practical identifiability analysis of large environmental simulation models. Water Resources Research 37 (4), 1015-1030. Dochain, D.; Vanrolleghem, P.A. (2001): Dynamical Modelling and Estimation in Wastewater Treatment Processes. IWA Publishing. Dochain, D.; Vanrolleghem, P.A.; van Daele, M. (1995): Structural identifiability of biokinetic models of activated sludge respiration. Water Research 29 (11), 2571-2578. Hao, X.; Heijnen, J.J.; van Loosdrecht, M.C.M. (2002): Sensitivity analysis of a biofilm model describing a one-stage completely autotrophic nitrogen removal (CANON) process. Biotechnology and Bioengineering 77 (3), 266-277. Helmer, C.; Kunst, S.; Juretschko, S.; Schmid, M.; Schleifer, K.-H.; Wagner, M. (1999a): Nitrogen loss in a nitrifying biofilm system. Water Science and Technology 39 (7), 13-21. Helmer, C.; Tromm, C.; Hippen, A.; Rosenwinkel, K.-H.; Seyfried, C.F.; Kunst, S. (1999b): Einstufige biologische Stickstoffelimination durch Nitritation und anaerobe AmmoniumOxidation im Biofilm. gwf Wasser Abwasser 140 (9), 622-632.
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Helmer, C.; Tromm, C.; Hippen, A.; Rosenwinkel, K.-H.; Seyfried , C.F.; Kunst, S. (2001): Single stage biological nitrogen removal by nitritation and anaerobic ammonium oxidation in biofilm systems. Water Science and Technology 43 (1), 311-320. Hippen, A.; Helmer, C.; Kunst, S.; Rosenwinkel, K.-H.; Seyfried, C.F. (2001): Six years' practical experience with aerobic/anoxic deammonification in biofilm systems. Water Science and Technology 44 (2-3), 39-48. Holmberg, A. (1982): On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities. Mathematical Biosciences 62, 23-43. Petersen, B. (2000): Calibration, identifiability and optimal experimental design of activated sludge models. Faculty of Agricultural and Applied Biological Sciences. Gent University. Ph.D. thesis. Reichert, P. (1998): AQUASIM 2.0 - User Manual, Computer program for the identification and simulation of aquatic systems. Dübendorf, CH, Swiss Federal Institute for Environmental Science and Technology (EAWAG). Sliekers, A.O.; Derwort, N.; Campos Gomez, L.;J.Strous, M.; Kuenen, J.G.; Jetten, M.S.M. (2002): Completely autotrophic nitrogen rem oval over nitrite in one single reactor. Water Research 36, 2475-2482. Strous, M. (2000): Microbiology of anaerobic ammonium oxidation. Department of Biotechnology. Delft University of Technology. Ph.D. thesis. Tromm, C. (1999): Batchversuche zur Identifikation von Stickstoffumsetzungen im Biofilm. Institut für Siedlungswasserwirtschaft und Abfalltechnik. Universität Hannover. Master thesis. Van Hulle, S.W.H. (2005): Modelling, simulation and optimisation of autotrophic nitrogen removal processes. Faculty of Agricultural and Applied Biological Sciences. Ghent University. Ph.D. thesis. Vanrolleghem, P.A.; van Daele, M.; Dochain, D. (1995): Practical identifiability of a biokinetic model of activated sludge respiration. Water Research 29 (11), 2561-2570.
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The Combined-Continuum-and-Discrete-Model (CCDM) for simulation of liquid-particle flows Kevin F. Malone, Bao H. Xu, Michael Fairweather Institute of Particle Science & Engineering, University of Leeds, Leeds LS2 9JT,UK
Abstract The Combined-Continuum-and-Discrete-Model (CCDM) is a technique that can simulate microscale behaviour of fluid-particle systems. Previous studies have focused on gas-solids flows; however, the technique is equally applicable to liquid-solid systems providing the model is expanded to account for complex fluid-particle interaction forces and changes to interparticle contact behaviour caused by the liquid medium. In this work, liquid-fluidized beds have been simulated using CCDM. Results indicate that modifications to account for the effect of the liquid have little impact on macroscopic system qualities such as minimum fluidization velocity and bed expansion, but a significant improvement in terms of the microscale particle mixing behaviour produced by the model. Keywords: liquid fluidization; multiphase flow; computer simulation; particle contacts
1. Introduction Unit operations involving solid particulate materials submersed in liquids are common in industrial processes. Examples include crystallisation, sedime ntation, filtration, hydrotransport, and liquid fluidization. Knowledge of the behaviour of liquid-particle systems is clearly of interest to those working in these areas. While some useful information can be obtained from experimental measurements, examination of microscale motion cannot be realised by experimental methods due to the complexity of these systems. Fortunately, computer simulation techniques may be able to provide a solution. The Combined-Continuum-and-Discrete-Model, or CCDM, is a technique that can simulate microscale behaviour of fluid-particle systems. Previous CCDM–type studies of multiphase systems have focused on gas-solids flows, in particular the behaviour of gas fluidized beds [1, 2]. However, the technique is equally applicable to many industrial liquid-solids flows. Here we discuss the simulation of liquid-fluidized beds using CCDM. Examination of liquid fluidization allows evaluation of CCDM’s usefulness for more general liquidsolids flows. In addition, increases in the number of applications that make use of liquid-fluidized beds in recent years gives an incentive to better understand the behaviour of these systems. For liquid-particle systems more complex fluid-particle interactions, as well as the effect of the more viscous fluid on particle collisions, must be accounted for in the model formulation. Comparison of results obtained using the original, or ‘gas’, CCDM, and the modified, or ‘liquid’, CCDM are presented here.
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2. Methodology CCDM uses a combination of the Discrete Element Method (DEM) for predicting the particle motion, and Computational Fluid Dynamics (CFD) for modelling the behaviour of the continuum fluid. In DEM [3] simulations, the trajectoriesd anrotations of individual particles are evaluated based on Newton’s second law of motion, using a numerical time stepping scheme. Contact forces are calculated at each time step using appropriate contact laws, and resolved into their normal and tangential components. The key assumption in DEM is that disturbances cannot propagate from any particle further than its immediate neighbours, providing a sufficiently small time step is used. For calculation of the continuum fluid flow, the locally-averaged [4] continuity and Navier-Stokes equations are solved using the SIMPLE method [5] to give the fluid velocity and pressure. This CFD calculation for the fluid is combined with the DEM model of the particles’ behaviour by carefully applying Newton’s third law of motion to the fluid-particle interaction force. This ensures the two sets of equations, which are solved on different length scales, are correctly coupled. More details of the CCDM model formulation as applied in gas-solids systems are given in [2]. The modifications to the CCDM which are necessary to correctly simulate liquidsolid systems are described below. 2.1. Fluid-particle interaction forces In liquid-particle systems, high fluid viscosity and small density difference between the phases means certain fluid-particle interactions that are negligible in gas-particle systems must be considered. In the ‘gas’ CCDM, only the steady-state drag force is considered. In the ‘liquid’ CCDM, we consider the added-mass, the Magnus (spin) lift, and the pressure gradient forces, in addition to the steady-state drag force. The overall fluid-particle interaction force is therefore:
(
2 ⎡ 3 − χ +1 + Cm ( u f - u p ) ⎢ 4d C D 0 u f − u p ( u f − u p ) ε p π fd = ρ f d p3 ⎢ ⎢ 6 ⎛ du f ⎞ d ⎢ + Ca ( u f − u p ) + ⎜ −g⎟ dt dt ⎢⎣ ⎝ ⎠
)⎤⎥⎥
⎥ ⎥ ⎥⎦
(1.1)
CD0, the steady state drag coefficient, and the exponent χ are functions of the particle Reynolds number, Re, as given in [6]. Cm, the Magnus lift force coefficient, is also a function of Re, and is calculated as described in [7]. Ca is the added-mass coefficient, taken to be 0.5. The final term on the right-hand side is the pressure gradient force [8]. 2.2. Particle-particle and particle-wall contacts In liquid-particle systems, interparticle collisions differ significantly from those in gasparticle systems due to the effect of hydrodynamic lubrication forces between the
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229
particle surfaces which depend on the fluid density and viscosity. To account for this in the ‘liquid’ CCDM, each particle’s coefficient of restitution was taken to be a function of the particle Stokes number, based on the relation given in [9]: ⎛ St ⎞ eliquid = egas ⎜1 − c ⎟ St ⎠ ⎝
(1.2)
where egas is the particle coefficient of restitution in air, and Stc is the critical impact Stokes number, below which rebound does not occur. In this work, Stc was set equal to 10 [9]. St, the particle Stokes number, is given by: St =
mv Re ρ p = 2 6πμ r 9 ρf
(1.3)
3. Simulation conditions Solid phase Particle shape
Fluid phase Spherical
Fluid
water
Number of particles
1600
Viscosity, μ
1.00× 10-3 kgm-1s-1
Particle diameter, d
5.00×10-3 m
Density, ρf
1.00× 103 kgm-3
Particle density, ρp
2,750 kgm-3
Bed width
2.00×10-1 m
Spring constant, kn
1.50× 106 Nm-1
Bed height
1.00 m
Sliding friction, γ
0.3
Bed thickness
5.00×10-3 m
Dry damping coeff’t, η
1.10 kgs-1
Cell width
1.00×10-2 m
Time step, Δt
5.00× 10-7 s
Cell height
1.00×10-2 m
Table 1. Parameters used in the simulations.
An initial packing was generated by allowing randomly distributed particles to fall under the influence of gravity, without fluid effects. This packing was used in the fluidized bed simulations with both the original and modified CCDM models. A uniform fluid inlet velocity across the base of the bed was used in all cases.
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4. Results and discussion
Pressure drop/bed weight ΔP/mg
1.10 1.05 1.00 0.95 0.90 0.85 0.80 0
0.05
0.1
0.15
0.2
0.25
0.3
Liquid velocity, U (m/s) Original CCDM
Modified CCDM
Figure 1: Relationship between liquid velocity and pressure drop for both CCDM models
Figure 1 shows the evolution of the bed pressure drop as the liquid velocity is increased. The curve reaches a plateau at the minimum fluidization velocity, Umf. In the cases shown Umf = 0.13 ms-1 for the original CCDM and 0.15 ms-1 for the modified CCDM. These values are higher than predicted from the Richardson-Zaki equation, which gives Umf as 0.08 ms-1. -0.1 -0.2
ln U
-0.3 -0.4 -0.5 -0.6
y = 2.2168x R2 = 0.9976 y = 2.2669x R2 = 0.998
-0.7 -0.8 -0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
ln ε Original code
Modified code
Fitted line (Original)
Fitted line (Modified)
Figure 2: Richardson-Zaki bed expansion, both models – gradients give the value of exponent n
Figure 2 shows the relationship between voidage and liquid velocity, U. Both fitted lines have similar gradients: 2.27 for the original CCDM; 2.22 for the modified CCDM. These values are close to the theoretical value of n = 2.4 for systems with a terminal Re > 500 (in this system terminal Re = 2800). While Figures 1 and 2 suggest that the modifications have not had a great impact on the macroscopic system behaviour, since the differences in Umf and bed expansion between the two versions of CCDM are small, results from the modified CCDM exhibit quite significant differences in terms of particle-scale mixing and flow behaviour. Figures 3
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231
and 4 show snapshot images of fluidized beds simulated with the original (Fig. 3) and modified CCDM (Fig. 4). t=0.75s
t=1.50s
t=2.25s
t=3.00s
t=3.75s
t=4.50s
Figure 3: Snapshot images of bed fluidized at 0.4 ms-1, as simulated with original CCDM model. (Particles coloured according to initial position to allow visual observation of mixing).
t=0.75s
t=1.50s
t=2.25s
t=3.00s
t=3.75s
t=4.50s
Figure 4: Snapshot images of bed fluidized at 0.4 ms-1, as simulated with modified CCDM model. (Particles coloured according to initial position to allow visual observation of mixing).
In Figure 3, the uneven and unsteady nature of the bed surface is apparent, and a certain degree of mixing is exhibited, whereasFigure in 4 the bed has a smooth surface and is not as well-mixed. The animations from which these snapshots are taken show distinct differences in the flow behaviour of the two beds: the simulation with the original CCDM exhibits bubbling behaviour akin to that observed in a gas-fluidized bed; while the modified CCDM produces a smoother fluidization with less bubbling, as is commonly observed in liquid fluidized beds. Figure 5, which shows distributions of particles’ axial (a) and radial (b) component velocities, supports this finding. There is a noticeable difference in both plots. Results from the original CCDM exhibit greater deviation from the mean value (close to zero in all cases), indicative of greater mixing; while distributions from the modified CCDM are more tightly grouped around the mean, as expected from the lesser degree of mixing observed in Figure 4.
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0.35 0.3
0.25
0.25
0.20
0.2 0.15 0.15 0.10
0.1
0.05
0.00 -0.112
0.05
-0.080
-0.048
-0.016
0.016
Original Modified
0.048
0.080
0.112
0 -0.112
-0.080
-0.048
-0.016
0.016
0.048
0.080
0.112
Original Modified
a b Figure 5: Individual particle component velocity distributions for fluidized beds, velocity 0.4 ms-1 (a) Axial velocity distribution; (b) Radial velocity distribution.
5. Conclusions The Combined-Continuum-and-Discrete-Model (CCDM) has been applied to simulate liquid-fluidized beds. Inclusion of additional fluid-particle interaction forces and revision of the way interparticle contacts are treated resulted in similar values to the original CCDM in terms of macroscopic bed properties, but better results in terms of the particle-scale mixing behaviour. Further studies are being performed in order to determine the relative sensitivity of the model to each of the individual fluid-particle interaction forces, and of the revised contact mechanics.
Acknowledgments The authors would like to thank Nexia Solutions Ltd. and the Engineering and Physical Sciences Research Council for financial support to Mr. Malone in the form of CASE studentship award number GR/P03711/01.
References Y. Tsuji, T. Kawaguchi, & T. Tanaka. Powder Technol. , 77(1) 1993. p. 79. B.H. Xu & A.B. Yu. Chem. Eng. Sci., 52(16) 1997. p. 2785. P.A. Cundall & O.D.L. Strack. Geotechnique, 29(1) 1979. p. 47. T.B. Anderson & R. Jackson. I&EC Fundam., 6(4) 1967. p. 527. S.V. Patankar, Numerical heat transfer and fluid flow. Hemisphere, London, 1980. R. Di Felice. Int. J. Multiph. Flow, 20(1) 1994. p. 153. Y. Tsuji, Y. Morikawa, & O. Mizuno. J. Fluids Eng.-Trans. ASME, 107(4) 1985. p. 484. 8. L.S. Fan & C. Zhu, Principles of gas-solid flows. Cambridge University Press, Cambridge; New York, 1998. 9. G.G. Joseph, R. Zenit, M.L. Hunt, & A.M. Rosenwinkel. J. Fluid Mech. , 433 2001. p. 329.
1. 2. 3. 4. 5. 6. 7.
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Implementation of efficient logic-based techniques in the MINLP process synthesizer MIPSYN Marcel Ropotar, Zdravko Kravanja University of Maribor, Faculty of Chemistry and Chemical Engineering, P.O. Box 219, 2000 Maribor/Slovenija
Abstract The main aim of the research is to implement the most advanced modeling and solution techniques in the automated process synthesizer MIPSYN. In particular, different modeling formulations are studied, rooted in disjunctive programming and convex hull representation. Alternative modeling is proposed for logical interconnection nodes and alternative outer approximation formulation. Initial research indicates that they could be efficient for solving large-combinatorial process network problems. Keywords: disjunctive programming, outer-approximations, MINLP, process synthesis, process synthesizer.
1. Introduction Over the last couple of decades significant advances have been achieved in modeling and mathematical programming techniques (see e.g. Grossmann and Kravanja, 1997; Biegler and Grossmann, 2004). Recent developments in logic-based optimization (e.g. Grossmann and Biegler, 2004) are regarded as one of the most important achievements for effectively modeling and solving discrete-continuous synthesis problems. Although several general-purpose MINLP solvers (see www.gamsworld.org/minlp/solvers.html), including the logic-based solver LOGMIP (Vecchietti and Grossmann, 1997), have been developed, almost no automated synthesis environment, based on recent advanced techniques, and specializing in the synthesis of process flowsheets, has been developed so far. This paper reports on the experience gained in developing such a synthesis environment, and experiences gained when solving process network problems using up to several hundred discrete variables. Different formulations for logical interconnection nodes are applied and the following representations of outer approximations (OA) for the Outer Approximation/Equality Relaxation algorithm are compared: h(x l ) + ∇ x h(x l )(x − x f ) ≤ M (1 − y ) (1) Big-M formulation:
(
)
Convex hull representation: ∇ x h(x l )x ≤ ∇ x h(x l ) x l − h(x l ) y T
(
)
(2)
(3) An alternative formulation: ∇ x h(x l )x ≤ ∇ x h(x l )x f + ∇ x h(x ) (x l − x f ) − h(x l ) y Unlike convex hull representation, where the continuous variables x are usually forced into zero values when the corresponding disjunctives are false, in the new formulation the variables are forced into arbitrarily-forced values, xf. We report our experience in the selection of different xf and implementation of different formulations in the MINLP process synthesizer MIPSYN (Mixed-Integer Process SYNthesizer), the successor of PROSYN-MINLP (Kravanja and Grossmann, 1994). l T
2. An alternative convex-hull representation An efficient way of formulating discrete/continuous nonlinear problems in the area of process synthesis is to use Generalized disjunctive programming (GDP) (e.g. Türkay
M. Ropotar and Z. Kravanja
234
and Grossmann, 1996). One of the most important features of GDP is that NLPs are solved only in the reduced space of global and currently selected alternatives. The other important feature is that, before the first outer approximation disjunctive program (OADP) is solved, outer approximations (linearizations) are derived for the whole problem. Both features significantly improve efficiency when solving (OADP) problems. The conventional (OADP) is given in the following form: min Z = ∑ ck + α k
α ≥ f (x l )+ ∇ x f (x l ) ( x − x l )⎫⎪ T
s.t.
( )
⎬, l =1,..., L
( ) ( x − x ) ≤ 0 ⎪⎭
g xl + ∇ x g x
l T
l
A g ( x) ≤ b g ⎤ ⎡Yik ⎥ ⎡¬Yik ⎢ ⎤ γ c = ⎥ ⎢ ⎢ i i ⎥ c 0 = ⎥ ⎢ i ⎢ LO UP ⎥ x x x ∨ ≤ ≤ ⎥ ⎢ B ik x = 0⎥ k ∈ SD, i ∈ Dk , ⎢ ik ⎥ ⎢ ⎢ A ( x) ≤ bik ⎥ ⎥ ⎣⎢ ⎢ ⎦⎥ T T l l l l ⎢⎣∇ x hik x x ≤ ∇ x hik x x − hik x , l ∈ Lik ⎥⎦ Ω(Y ) = true
( )
( )
(OADP)
( )
x ∈ R n , c ∈ R m , Y ∈{true, false}
m
where qualitative logical and discrete decisions are represented by disjunctives (i ∈ Dk, k ∈ SD) and propositional logical constraints Ω(Y), whilst continuous quantitative decisions by (non)linear (in)equality constraints, which can be global (g(x) ≤ 0, A g (x ) ≤ b g ) or belong to local representations of alternatives ( hik (x ) ≤ 0 , Aik (x ) ≤ b ik ). Note that when an alternative is not selected, its linearizations do not apply, and x is set to zero. Türkay and Grossmann (1996) developed convex-hull OAs for variables x that take zero or nonzero values by disaggregating vector x into sub vectors of zero xZ and nonzero xNZ variables. Here, an alternative and more general OADP is proposed, where vector x can be set to any value xf when the alternative is not selected: min Z = ∑ ck + α k
α ≥ f (x l )+ ∇ x f (x l ) ( x − x l )⎫⎪ T
s.t.
⎬, l =1,..., L T g x l + ∇ x g x l ( x − x l ) ≤ 0 ⎭⎪ A g ( x) ≤ b g
( )
( )
⎡Yik ⎢ ⎢ci = γ i ⎢ LO UP ⎢x ≤ x ≤ x ik ⎢ A ( x) ≤ bik ⎢ ⎢⎣∇ x hik x l T x ≤ ∇ x hik x l T x l − hik x l
( )
( )
(A-OADP)
⎤ ⎤ ⎡¬Yik ⎥ ⎥ ⎢ c 0 = ⎥ ⎥ ⎢ i ⎥ ⎥ ⎢ f x x ∨ = ⎥ k ∈ SD, i ∈ I , l ∈ Lik ⎥ ⎢ LO ik ⎥ ⎥ ⎢ A ( x − x ) ≤ bik ⎥ ⎥ ⎢ ⎥⎦ ⎢⎣∇ x hik x l T x ≤ ∇ x hik x l T x f ⎥⎦
( )
( )
( )
Ω(Y ) = true x ∈ R n , c ∈ R m , Y ∈{true, false}
m
Note that, auxiliary linear inequalities ( ∇ x hik (x l ) x ≤ ∇ x hik (x l ) x f ) are applied in order to preserve the feasibility of OAs in MILP when an alternative is not selected and the T
T
Implementation of Efficient Logic-Based Techniques in the MINLP Process
235
corresponding x is set to xf. By replacing Yik in (A-OADP) with binary variable yik, the following alternative convex-hull formulation for OAs can be derived at:
( )
T
(
( ))
( )
T
( )
T
∇ x hik x l x ≤ ∇ x hik x l x l − hik x l yik + ∇ x hik x l x f (1 − yik )
which can finally take the form:
( )
∇ x hik x l
T
( )
x ≤ ∇ x hik x l
T
(
( )
T
( ))
x f + ∇ x hik x l ( x l − x f ) − hik x l yik
(4)
(5)
In addition, in order to set x to xf when an alternative is not selected, the following constraints should be applied: x ≤ x f + ( x UP − x f ) y ik
(6)
x ≥ x f + ( x LO − x f ) yik (7)
The key feature of the alternative OAs (eq. 5) is that they preserve feasibility, even in the presence of nonconvexities when alternatives are not selected and x is set to xf. This enables the use of variables with nonzero lower bounds, directly without additional logical constraints on the variables. Note that when xf is equal to the lower bounds (xf = xLO), inequality (7) becomes redundant and can be omitted from the formulation. Similarly, ineq. (6) can be omitted when xf is equal to xUP. This reduces the size of the MILP problem. An interesting feature of the proposed formulation of OAs (ineq. 5) is that nonzero xf can be chosen, such that linearization coefficients at y become zero, and the mixed-integer OAs become pure-continuous constraints that are much easier to solve, especially when the number of binary variables is very high. However, forcing x to a nonzero xf, transforms pure-continuous linear constraints Aik (x ) ≤ bik into mixed-
integer constraints Aik (x − x LO yik ) ≤ bik . It is then obvious that the selection of xf and, especially the selection of the most suitable OA and modeling representation, may not be a straightforward task and may significantly influence the efficiency of the search. The earliest experience indicates that the best efficiency is achieved when xf is set to xLO. A procedure for a systematic selection of the most suitable xf is under way. Until recently only big-M formulation of OAs and big-M representation of logical interconnection nodes (single-choice mixers and splitters) were used in MIPSYN to solve MINLP synthesis problems. Now, OAs and logical interconnected nodes are also represented by the conventional convex-hull and the alternative convex-hull formulations.
3. Examples Three synthesis problems of different sizes and complexities are solved using all three OAs and modeling representations, in order to test and compare their efficiencies. The first numerical example is a network synthesis example with a simple model but very large-scale combinatorics with 400 binary variables. The second example is the synthesis of heat exchanger network (HEN) comprising different types of exchangers. The model exhibits moderate complexity and high combinatorics (249 binary variables). The last, alil chloride example, is the synthesis of a reactor/separator network in to an overall heat integrated process scheme, with a complex model and smaller-size combinatorics (32 binary variables).
M. Ropotar and Z. Kravanja
236 3.1. Network synthesis problem
Fig. 1 shows a superstructure comprising a sequence of exclusive-or alternatives. This model consists of a linear objective function, nonlinear design equations, formulation for single-choice splitters and mixers and exclusive-or logical constraints (detailed formulations will be given in the extended paper). The objective is to minimize total cost at the fixed demand of the final outflow. The problem was solved by using all three OA and modeling representations. Solution statistics until the 3rd major MINLP iteration is reported in Table 1. x 11 x1
z11 x12
z12 x3
x1i
x2
x21
z21 x22
z22
x2i
z1i xi+1 z2i
x1N
z1N 5
x 2N
z2N
Figure 1: Superstructure of the network synthesis problem. Table 1: Solution statistics of the network synthesis problem. Best NLP n/a
BigM
Integrality gap, % n/a
No. of eq./ No.of var. 3802/1801
No.of nodes n/a
CPU for 3 it., sec. n/a
Nodes/CPU for 3 it. n/a
Convex-hull
183.870
0.868
3402/1801
319
15.071
21.2
ACH (xf = xLO)
183.870
0.868
2202/1801
293
4.274
68.6
f
UP
ACH (x = x )
183.870
0.868
3402/1801
2264
46.209
49.0
ACH (xf = x1)
183.870
0.868
3402/1801
341
24.142
14.1
As can be seen in Table 1, it was impossible with big-M formulation to solve the problem in a reasonable time, whilst both convex-hull representations enable the solving of this high combinatorial problem very quickly. Also it can be seen that for the alternative convex-hull formulation (ACH) the selection of xf is very important and that the best efficiency of the search is achieved when xf = xLO. Note that with the same integrality gap and somewhat smaller number of constraints, the alternative formulation with xf = xLO could solve the problem in only a quarter of the CPU time needed to solve the problem using the conventional convex-hull formulation. 3.2. HEN synthesis problem Each match in a stage-wise superstructure is comprised of a double pipe, a plate and frame, a shell and tube exchanger, and a by-pass when the match is rejected. The model is described in detail by Soršak and Kravanja (2002). Consideration of different types of exchangers enables the simultaneous selection of exchanger types; however, it significantly increases the number of binary variables. In this example of 4 hot and 5 cold process streams and 4 stages, the problem originally had 320 binary variables. By prescreening alternatives the number was reduced to 249. Table 2 shows statistics for three different representations. With respect to integrality gap, CPU time and the number of nodes, both convex-hull representations outperform the big-M one whilst the efficiency of the alternative convex-hull formulation is slightly better than the conventional formulation one. Also, with big-M, a slightly inferior solution was obtained than with the convex-hull representations.
Implementation of Efficient Logic-Based Techniques in the MINLP Process
237
Table 2: Solution statistics for the HEN synthesis problem.
BigM Convex-hull f
LO
ACH (x = x )
Best NLP 821.00
Integrality gap, % 31.321
No. of eq./ No.of var. 8414/5595
No.of nodes 18950
CPU for 8 it., sec. 86.050
Nodes/CPU for 8 it. 220.2
818.69
7.465
6814/5595
4817
29.779
161.8
818.69
7.465
5534/5595
4065
28.207
144.0
3.3. Alil chloride example Details of the alil chloride problem are given by Iršič-Bedenik et al. (2004). The reactor/separator superstructure comprises a series of basic reactor substructure elements with side streams and intermediate separators at different locations. In each element a recycle reactor (a recycle stream around a PFR) and a CSTR are embedded in parallel arrangement so as to enable a different feeding, recycling and bypassing. In addition, each PFR consists of a train of several alternative elements. The corresponding DAE system is modeled by the orthogonal collocation on finite elements. Simultaneous heat integration was performed by a multi-utility configuration model (Duran and Grossmann, 1986). The overall model is highly nonlinear and nonconvex. 32 binary variables were assigned to discrete decisions. The objective is to maximize the net present value at a fixed production for alil chloride. The solution statistics of all three OA and modeling representations is given in Table 3. Table 3: Solution statistics of alil chloride problem. Integrality gap, %
BigM OAs
Best NLP k$/a 83.709
0.348
Convex-hull
83.679
0
86.245
0
f
LO
ACH (x = x )
No. of eq./ No.of var.
No.of nodes
CPU for 7 it., sec.
Nodes/CPU for 7 it.
2046/10426
568
66.027
8.6
4408/10426
53
10.567
5.0
3903/10426
9
5.866
1.5
When logical constraints (x ≤ xUPy) are imposed on all continuous variables presented as alternatives, integrality gaps of both convex-hull approaches are decreased practically to zero which significantly facilitates the efficiencies of the first couple of MILPs. However, in the tighter MILP representations, the effects of nonconvexities become more severe, causing a significant increase in the number of nodes in the subsequent MILPs. It is interesting to note that, due to the presence of nonconvexities, even with the zero integrality gaps the efficiencies of the convex-hull representations do not improve. In order to decrease the troublesome effect of nonconvexities a special convex test (Kravanja and Grossmann, 1994) was applied and violating OAs were temporarily dropped out of the master MILPs. Statistics of solutions for both convex-hull representations are now significantly improved (Table 3), especially in the case of alternative convex-hull representation, where the best solution was found and the least computational effort was needed to obtain it.
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4. Conclusion The main aim of this research is oriented towards the development of an advanced and robust synthesizer shell, capable of solving large-scale applications in different engineering domains. The performances of different OA and modeling representations are summarized in Table 4. Both convex-hull representations usually outperform the big-M one. The earliest high performance solutions with alternative representation, indicates that the alternative convex-hull representation could be more efficient in solving high combinatorial problems than the conventional one and has the smallest problem size. On the other side it exhibits the strongest sensitivity to the effects of nonconvexities and the model formulation is probably the most complicated. It should be noted that so far the research has been focused only to the OA algorithm. The application of the alternative convex-hull formulation with other MINLP techniques is under way. Table 4: Performance of different OA and modeling representations.
Easiness of modeling Problem size Effect of nonconvexities Nodes/sec of CPU time
Big-M The most easy From the smallest to the largest
Convex-hull Moderate
Alternative xf= xLO The most complicated
The largest
Moderate
The smallest
Moderate
The strongest
The largest
Moderate
The smallest or moderate
References Biegler L.T. and I. E. Grossmann (2004). Retrospective on optimization, Computers and Chemical Engineering, 28, 1169-1192. Duran, M.A. and I.E. Grossmann (1986). Simultaneous optimization and heat integration of chemical processes, AIChE J. 32, 123-138. Grossmann, I. E. and Z. Kravanja, Mixed-integer nonlinear programming: a survey of algorithms and applications. V: Biegler, L. T. (ed.). Large-scale optimization with applications. Part 2, Optimal design and control, (The IMA volumes in mathematics and its applications, Vol. 93). New York: Springer-Verlag, 73-100 (1997). Grossmann, I. E. and L. T. Biegler (2004). Part II. Future perspective on optimization, Computers and Chemical Engineering, 28, 1193-1218. Iršič Bedenik N., B. Pahor and Z. Kravanja (2004). An integrated strategy for the hierarchical multilevel MINLP synthesis of overall process flowsheets using the combined synthesis/analysis approach, Computers and Chemical Engineering, 28, 693-706. Kravanja Z. and I.E. Grossmann (1994). New Developments and Capabilities in PROSYN – an Automated Topology and Parameter Synthesizer, Computers and Chemical Engineering, 18, 1097-1114. Soršak A. and Z. Kravanja (2002). Simultaneous MINLP synthesis of heat exchanger networks comprising different exchanger types. Computers and Chemical Engineering, 26, 599-615. Türkay M. and I. E. Grossmann (1996). A logic based outer-aproximation algorithm for MINLP optimization of process flowsheets. Computers and Chemical Engineering, 20, 959-978. Vecchietti A. and I.E. Grossmann (1997). LOGMIP: a disjunctive 0-1 nonlinear optimizer for process systems models, Computers and Chemical Engineering, 21, 427-432. www.gamsworld.org/minlp/solvers.html, GAMS MINLP solvers (2005).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Calculation of three-phase bubble columns Dierk Wiemanna, Dieter Mewes Institute of Process Engineering, University of Hannover, Hannover, Germany a present: Bayer Technology Services GmbH, Uerdingen, Germany
Abstract The scope of this work is the numerical calculation of the three-dimensional, timedependent velocity and concentration fields in cylindrical bubble columns with twophase gas-liquid and three-phase gas-liquid-solid flow. Therefore all phases are described by an Eulerian approach. In particular the local interfacial area density and the interphase transfer terms for mass and momentum are calculated based on a population balance equation approach. The proposed approach enables an effective way to couple population balance and computational fluid dynamics. For three-phase gas-liquid-solid flow heavy particles with diameters in the range of 100 µm are considered as catalyst for a heterogeneous chemical reaction. The solids phase viscosity and pressure are described based on the granular flow theory. The influence of particles on bubble coalescence has been investigated to extend the model. From the calculation the threedimensional, time-dependent velocity and concentration fields are obtained. Keywords: bubble column, population balance equation, CFD, mass transfer,
1. Introduction The flow pattern in bubble columns is strongly influenced by the superficial gas velocity. The homogeneous flow regime arises for low superficial gas velocities. In this flow regime the integrated volume fraction of gas and the interfacial area density increase almost linearly with the superficial gas velocity. However for technical applications the heterogeneous flow regime is of more importance. In this flow regime increasing coalescence of small bubbles lead to the formation of larger ones. These large bubbles rise up much faster than the small ones thus a large amount of gas is entrapped with them. The liquid flow pattern is characterized by large scale vortices, which cause a large degree of backmixing. If mass transfer occurs between the gas and the liquid phase backmixing influences the local concentration difference. The dimension of bubble column reactors is widely based on empirical models for the interfacial area, the phase velocities and backmixing (Deckwer [1], Nigam and Schumpe [2]). The scale-up of these models is however limited to the experimental dimensions since the reactor geometry has a strong influence on these parameters. In contrast computational fluid dynamic methods enable a physical based prediction of the flow field independent of the column dimension. For the description of bubbly flow the Euler-multi-fluid model and the Euler-Lagrange approach are commonly used for the calculation of large-scale flow fields. Other approaches such as direct numerical simulations are restricted to detailed investigations of small numbers of bubbles. The description of bubbly flow requires the knowledge of the interfacial area since mass, momentum and energy transport are proportional to it. Therefore the population balance equation is often used to calculate the interfacial area in dependence of the flow field. In this work the multi-fluid model is coupled with a population balance equation
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approach according to Lehr et al. [3]. From the numerical solution of the population balance equation a bi-modal bubble size distribution is obtained for the heterogeneous flow regime, thus the gas phase can be divided into one fraction containing small bubbles and a second fraction containing large bubbles. Using the self-similarity of the calculated bubble size distributions a transport equation for the mean bubble diameter of the small and large bubble fraction is derived. Both bubble fractions are coupled by bubble coalescence and break-up, thus the volume fraction and the bubble size vary throughout the flow field.
2. Modeling bubbly flow In this section the model is described briefly. A detailed description can be found in [4] to [6]. The calculation considers three Eulerian phases: the liquid phase, a gas phase representing small bubbles and a gas phase representing large bubbles. For three phase gas-liquid-solid flow an additional Eulerian phase arises for the solid phase. For each phase the momentum transport equation ∂ (α i ρ i uG i ) + ∇(α i (ρ i uG i uG i )) = ∂t
( (
))
G G G G G − α i ∇p i + ∇ α i η i Δu i + ( Δu i ) T + Fmass + α i ρ i g + Fij
, i=g1, g2, l
(1)
is solved independent of the physical phase distribution. The inIn the above equation all phases share the same bulk pressure. The temporal and convective changes of momentum on the left hand side of eq. (1) are balanced by several forces on the right hand side. These forces are due to the bulk pressure gradient, shear, secondary fluxes due to mass transfer, gravitational forces and interphase momentum transfer. The index l refers to the liquid phase, g1 and g2 refer to the small and the large bubble phase. For the multi-fluid approach in particular modeling of the interphase momentum transfer is important. The most important interphase force is due to interphase drag. The drag force per unit volume is calculated to G G G G 3 α G Fil = C D ρ l i u i − u l (u i − u l ) . 4 di
(2)
based on the drag on a single sphere. In eq. (2) the drag coefficient is calculated following Clift et al. [7] ⎡ 24 2 8 ⎫⎤ ⎧ C D = max ⎢ (1 + 0.1 Re 0.75 ); min ⎨max(0.44, Eo1 / 2 ), ⎬⎥ 3 3 ⎭⎦ ⎩ ⎣ Re
(3)
in dependence of the Reynolds- and Eotvos-number Re i =
G G ui − u l di νl
(4)
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Calculation of Three-Phase Bubble Columns Eo =
g (ρ l − ρ i )d i2 . σ
(5)
In eq. (5) the surface tension between the liquid and gas phase is σ. The Sauter-bubble diameter di is calculated from a transport equation for the mean bubble volume (Lehr et al. [3]). In addition secondary fluxes of momentum occur due to mass flux between the phases G G ⎧⎪ − M i→ j u i , Fmass = ⎨ G ⎪⎩ M i→ j u i ,
phase i phase j
.
(6)
In eq.(5) the mass flux from phase i to phase j is labeled M i → j . For the gas phase also
secondary fluxes due to coalescence and break-up of bubbles are considered. The bulk mass balance equation for each phase is ⎧⎪ ∂ (α i ρ i ) G + ∇(α i ρ i u i ) = ⎨ ∂t ⎪⎩
−M i→l , i = 1,2 +M , i = l ; j = 1,2
(7)
j→l
considering mass transfer from the gaseous to the liquid phase. For multi-component flow with n species in addition to the bulk mass balance a species mass balance equation ⎧⎪ ∂ (α i ζ Ai ρ i ) G + ∇(α i ζ Ai ρ i u i ) − ∇(D Ai ∇(ρ i ζ Ai )) = ⎨ ∂t ⎪⎩
− ζ Ai M i→l , i = 1,2 +ζ M , i = l ; j = 1,2 Aj
j→l
(8)
is solved for (n-1) species. In eq. (8) one of these (n-1) species is named A. The species represent the tracer substance or the transferring component. For the calculation of the mass transfer rate the phase equilibrium at the gas-liquid interface is described following Henry’s law. The mass transfer across a turbulent air-water surface is investigated by Law and Khoo [8]. The experimental results indicate that the mass transfer rate is correlated to the turbulence near the surface. However the authors emphasize that for the implementation into a multi-fluid model the dependency between the near surface turbulence and the bulk phase turbulence needs further investigations. Therefore in this work the mass transfer coefficient is calculated in dependence of a Sherwood-number. The mass transfer rate is calculated
i ,l = m
(
cl β l ρ A,pl − ρ A,l c l − c A ,l
)
(9)
with the bulk molar concentration of the liquid phase cl and the bulk molar concentration of the transferred component cA,l. The Sherwood-number is calculated according to Brauer [12].
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In bubbly flow turbulent velocity fluctuations arise in the liquid phase. These fluctuations are caused due to the shear flow but also the presence of the bubbles induce turbulence. So far no general accepted model for the description of the turbulence exists. In this work the turbulence in the liquid is described by the k-ε model with additional source terms accounting for bubble induced turbulence following the proposal of Lopez de Bertodano et al. [9]. For three-phase flow the solids phase is considered by an Eulerian-phase. The momentum balance is written to G ∂ (αsρs uG s ) + ∇(αsρs uG s uG s ) = −αs∇p − ∇ps + ∇⎛⎜ αs τs ⎞⎟ − Fd + ρsgG . ⎝ ⎠ ∂t
(10)
In addition to the bulk pressure p the solids pressure ps arises. The solids pressure describes the additional pressure due to interactions between the solids. Inter-particle collisions are considered by the shear stress tensor τ. For the calculation of the solids pressure and the shear stress tensor the theory of granular flow is applied [10]. Collisions between solids and bubbles lead to momentum transfer between the gas and solid phases. Based on the assumption of elastic collisions the momentum transfer term Fd,g,s = 3,519α i α s ρ s
1 ρ ⎛d d Bi + i ⎜⎜ P ρ s ⎜ d Bi ⎝
2
⎞ ⎟ d ⎟⎟ P ⎠
u 2rel
(11)
arises in the gas and solid phases. In eq. (11) the solids diameter is dp, the diameter of the bubble is dBi and the relative velocity between solids and bubbles is urel. For the description of the local bubble size and the local interfacial area density a simplified solution of the population balance equation is used. This model enables the prediction of the volume fraction and bubble size for homogeneous and heterogeneous bubbly flow. The resulting transport equations for the volume fraction and bubble size are coupled with the balance equation for mass and momentum. Thus the flow field is calculated in dependence of the local bubble size. In this work three-phase gas-liquidflow with small but heavy particles is considered. The solids represent the catalyst for a heterogeneous chemical reaction. The resulting set of equations is solved with the code CFX-5.7 using the method of finite volumes. The flow domain is discretized using a block-structured grid with hexahedral volumes. The edge length of the grid is 1 cm. Near the wall region a finer grid is used. The flow field in bubble columns strongly varies with time and space. For the temporal resolution time steps in the order of 0.01s to 0.05s are made. These time steps provide the calculation of the large-scale velocity fluctuations in the flow field. The convective terms are discretized with second order accuracy to reduce numerical diffusion.
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Calculation of Three-Phase Bubble Columns
3. Results 3.1. Experimental investigation of coalescence in three-phase flow
Massenanteil solids massFeststoff fraction
The transport equation for the mean bubble volume describes the local bubble diameter in dependence of bubble-break-up and coalescence processes. For the case of small particles in the order of 100µm the collision between particles and bubbles does not cause bubble break-up. However bubble coalescence can be affected by the presence of solids. From two-phase gas-liquid flow it is known, that coalescence arises if the relative velocity between the bubbles perpendicular to their surface is smaller than a certain critical velocity. To determine the influence of solids loading on this critical velocity binary collisions between bubbles are analyzed. The liquid phase is de-ionized water, the gas phase is air and the solids are glass spheres with a mean diameter of 78.11µm. In fig. 1 several sequences for different solids loadings are depicted. From these images the relative velocity perpendicular to the bubbles surfaces is determined. For high velocities the bubbles bounce, whereas for small values coalescence occurs.
Koaleszenz 0%
3%
2 1 0ms
1 10mm
10mm
2 0ms
10mm
3
3 1
1 60ms
80ms
1
1
1
20ms 2
40ms 2
60ms 2
1
1
1
1
10%
2
20ms
1 0ms 2
2
2
2
2
2
10ms
20ms
30ms
88ms
100ms
3 70ms
3
40ms
3 80ms
3
50ms
Figure 1: Bubble coalescence due to binary collisions for different solids loadings In fig. 2 the critical velocity is shown in dependence of the solids mass fraction. For two-phase gas-liquid flow the critical velocity is 0.095 m/s. In case of ten per cent solids mass fraction the critical velocity decreases to 0.06 m/s. Thus coalescence is hindered due to the presence of the glass spheres.
3.2. Numerical calculated flow fields From the numerical calculation the three-dimensional, time-dependent flow fields are obtained. The liquid is water, the gas phase is air. For the solid phase glass spheres of 100µm diameter are assumed. The overall solids loading is 0.114. In fig.3 the instantaneous flow fields of the solid and liquid phase are shown. The streamlines of the
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solid and the liquid phase are colored with the volume fraction and the axial velocity. The solids motion is similar to the liquids, thus the solids are transported upwards in the core region of the column and transported downwards near the column wall. In contrast to the volume fraction of gas, high volume fractions of solids are calculated near the wall, whereas low volume fractions are calculated in the core region.
kritsche Geschwindigkeit ukrit
10 cm/ s 9 8 7 6 0 5 0.00
0.02
0.04
0.06 0.08 0.10 Massenanteil Feststoff = Masse Feststoff Masse Feststoff+Wasser
Figure 2: Influence of solids loading on the critical velocity
Calculation of Three-Phase Bubble Columns
245
The model is extended to consider a heterogenous chemical reaction for which the solids represent the catalyst. As example the hydrogenation of anthraquinone has been chosen. For that purpose the gas phase is assumed as hydrogen, the liquid is a solution, which contains a certain amount of anthraquinone. The solid particles represent the palladium catalyst. The model includes the absorption of the gas phase, the transport of the absorbed hydrogen and the anthraquinone to the solids surface and the chemical reaction at the surface. For the Euler model the chemical reaction is represented by a quasi-homogeneous reaction rate. The reaction rate depends on the volume fraction of solids, the solids density and the molar concentration of anthraquinone in the liquid phase. The chemical reaction rate is calculated to rv = k r α s ρ s c Anthr ηs
(12)
In eq. (12) the constant is kr=0.0014 m³/(kg s) [11] and the solids efficiency is set to ηs=1. The reaction rate is introduced in the mass balance equation thus an additional source term arises for the liquid phase, whereas for the gaseous phases sink terms arise. In fig. 4 the calculated volume fraction of the gas phase and the mass fractions of the absorbed hydrogen, the anthraquinone and the resulting hydroanthraquinone are shown. In accordance with the decrease of anthraquinone the mass fraction of hydroanthraquinone increases along the column height.
4. Conclusion The three-dimensional, time-dependent flow fields for three-phase gas-liquid-solid flow in cylindrical bubble columns are calculated using an Eulerian model. In particular the balance equations for mass and momentum are coupled with a transport equation for the mean bubble volume. For a heterogeneous chemical reaction the solid phase is
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considered as catalyst. The calculated flow fields are similar to those calculated for twophase gas-liquid flow.
Notation a CD c D d F g H j k
interfacial area density, m-1 drag coefficient, molar concentration, mol m-3 column diameter, m bubble diameter, m force, kg m s-2 gravitational acceleration, m2 s-1 Henry- coefficient, kg m-1 s-2 superficial velocity, m s-1 turbulent kinetic energy per unit mass, m2 s-2 mass flux density, kg m-2 s-1
M i→ j
mass flux from phase i to j, kg m-3 s-1
n
molar flux density, mol m-2 s-1 pressure, kg m-1 s-2 time, s velocity, m s-1 coordinate, m
m
p t u x
Greek letters α volume fraction, β mass transfer coefficient, m s-1 ε turbulent kinetic energy dissipation rate, m2 s-3 η dynamic viscosity, kg m-1 s-1 μ molecular weight, kg mol-1 ρ density, kg m-3 ν kinematic viscosity, m2 s-1 ξ mass fraction, σ surface tension, kg s-2 Subscripts l liquid g1 small bubble fraction g2 large bubble fraction s solid
Acknowledgements The authors gratefully acknowledge the financial support of the German Research Foundation (DFG). The calculations are performed using the IBM pSeries 690 super computer of the North German Supercomputing Combine (HLRN) in Hannover and Berlin.
References [1] Deckwer, W.-D. (1992). Bubble Column Reactors, John Wiley Sons, New York [2] Nigam, K.D.P., Schumpe, A. (1996). Three-phase sparge reactors. Gordon and Breach,
Calculation of Three-Phase Bubble Columns
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Amsterdam. [3] Lehr, F., Millies, M., Mewes, D. (2002). Bubble-size distributions and flow fields in bubble columns. AIChE J., 11, 2426-2443. [4] Wiemann, D. (2005), Numerisches Berechnen der Strömungs- und Konzentrationsfelder in n zwei- und dreiphasig betriebenen Blasensäulen, Thesis, Univ. of Hannover, Cuvillier-Verlag, Göttingen. [5] Wiemann, D., Mewes, D. (2005). Prediction of backmixing and mass transfer in ubble columns using a multifluid model. Ind. & Eng. Chem. Res., 44, 4959-4967. [6] Wiemann, D., Mewes, D. (2005). Calculation of flow fields in two- and three-phase bubble columns considering mass transfer. Chem. Eng. Sci., 60, 6085-6093. [7] Clift, R., Grace, J.R., Weber, M.E. (1978). Bubbles, Drops and Particles, Academic Press, New York, San Francisco, London. [8] Law, C.N.S., & Khoo, B.C. (2002). Transport across a turbulent air-water interface. AIChE J., 48, 1856- 1868. [9] Lopez de Bertodano, M., Lahey, R.T., Jones, O.C. (1994). Development of a k- model for bubbly two-phase flow. J. Fluids. Eng., 1, 128-134. [10] C.K.K. Lun, F.B. Savage, D.J. Jeffrey, N. Chepurniy: Kinetic theories of granular flow and slighly inelastic particles in a general flow-field; J. Fluid Mech. 140 (1984), 223/256. [11] E. Santacesaria, M. Di Serio, A. Russo, U. Leone, R. Velotti: Kinetic and catalytic aspects in the hydrogen peroxide production via anthraquinone; Chem. Eng. Sci. 54 (1999), 13-14, 2799/2806. [12] H. Brauer: Particle/Fluid Transport processes; Progress in Chem. Eng. 17 (1979), 61/99.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A framework for model-based design of parallel experiments in dynamic systems F. Galvanina, M. Baroloa, F. Bezzoa and S. Macchiettoa,b a
DIPIC – Dipartimento di Principi e Impianti di Ingegneria Chimica, Università di Padova, via Marzolo 9, I-35131, Padova, Italy ( [email protected]) b Department of Chemical Engineering, Imperial College London, South Kensington Campus, SW7 2AZ London, UK ([email protected])
Abstract Advanced model-based experiment design techniques are essential for rapid development, refinement and statistical assessment of deterministic process models. One objective of experiment design is to devise experiments yielding the most informative data for use in the estimation of the model parameters. Current techniques assume the multiple experiments are designed in a sequential manner. The concept of model-based design of parallel experiments design is presented in this paper. A novel approach, viable for sequential, parallel and sequential-parallel design is proposed and evaluated through an illustrative case study. Keywords: model-based experiment design, dynamic modelling, parameter estimation, model validation.
1. Introduction Model-based experiment design techniques allow selecting conditions for the next experiment that are “best”, in the sense of having the maximum information content about the underlying process. Typically, it is desired to establish the most appropriate model structure and the best values of the parameters, so as to provide the best fit to experimental data. Based on earlier work of Espie and Macchietto [1] and Zullo [2], Asprey and Macchietto [3] proposed a general systematic procedure to support the development and statistical verification of dynamic process models for both linear and non-linear dynamic systems described by differential and algebraic equations (DAEs). According to this approach and assuming that no model discrimination is required beforehand, three consecutive steps are needed to determine model parameters: 1. the design of a new set of experiments, based on current knowledge (model structure and parameters, and statistics from prior experiments); 2. the execution of the designed experiment and collection of new data; 3. the estimation of new model parameters and statistical assessment. The sequential iteration of steps 1, 2 and 3 typically leads to a progressive reduction in the uncertainty region of model parameters, thanks to the new information obtained from the experimental data. The procedure has been successfully demonstrated in several applications, such as crystallisation processes [4], mammalian cell cultures [5] and biofuels production [6]. A similar procedure for optimum experimental design was developed by Bauer et al. [7], who assessed it on an industrial reactive system. They also adopted a sequential approach. There are a number of research and industrial applications where it is possible to envisage the simultaneous execution of several experiments in parallel rather than
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sequentially. Miniaturisation allows the definition of array of modules (e.g. microreactors for chemical or biochemical reactions) in which several experimental conditions can be simultaneously set up to carry out parallel experiments. Clear advantages in terms of elapsed time saving are presently offset by the lack of a systematic procedure for model-based design of parallel experiments. In this work, the possibility of advancing the current techniques to tackle the design of parallel experiments is discussed. Furthermore, a new approach based on a statistical analysis of the variance-covariance matrix of the parameters to be estimated is developed and assessed. It is shown that this can also be applied to develop hybrid sequential-parallel experiment design strategies. Parallel and sequential-parallel techniques are compared to a standard sequential approach and potential advantages/disadvantages are highlighted. The applicability of the new experiment design methods to dynamic systems and their performance are illustrated via an illustrative case study.
2. The methodology Let us consider a process described by the set of DAEs of the form: f ( x(t ), x (t ), y (t ), u (t ), q, θ ) = 0 ,
(1)
where x(t) and y(t) are vectors of the differential and algebraic variables, u(t) and q are vectors of the time-varying and time-invariant control variables, and θ is the set of Nθ unknown model parameters to be estimated. Here it is assumed for simplicity the all the M differential variables x can be measured (the case where only a subset is measured being a trivial extension). Model-based experiment design for parameter precision aims at determining the optimal vector ϕ of experimental conditions (initial conditions x0, control variables u and q and the times when measurements are sampled) required to maximise the expected information content from the measured data generated by these experiments, i.e. to minimise the confidence ellipsoid of the parameters to be estimated. This means that some measure ψ of the variance-covariance matrix Vθ of the parameters has to be minimised. If we take into account a number Nexp of experiments, the matrix Vθ is the inverse of the Nθ × Nθ information matrix Hθ [8]: −1
−1
⎡ Nexp ⎡ Nexp M M −1 ⎤ −1 ⎤ Vθ (θ , ϕ ) = Hθ (θ , ϕ ) = ⎢ ∑ Hθ* |k + ( Σθ ) ⎥ = ⎢ ∑ ∑∑ σ ij |k Qi|k Q j |k + ( Σθ ) ⎥ , ⎢⎣ k =1 i =1 j =1 ⎣⎢ k =1 ⎦⎥ ⎦⎥ −1
(2)
where H*θ |k is the information matrix after the k-th experiment, σij is the ij-th element of the inverse of the estimated variance-covariance matrix of the residuals Σ=cov(xi, xj), Qi is the i-state matrix of the sensitivity coefficients at each of the nsp sampling points: ⎡ ∂x ⎤ Qi = ⎢ il ⎥ ⎣ ∂θ m ⎦
l = 1,..., nsp
m = 1,..., Nθ ,
(3)
and Σθ is an approximate variance-covariance matrix of the parameters. Prior information on the parameters can be ignored by dropping the dependency of equation (2) on Σθ [9]. A common choice for the measure ψ is the E-optimality criterion [10], which aims at minimising the largest eigenvalue λ1 of matrix Vθ. Note that the definition of matrix Vθ and the E-optimality criterion are quite general and do not
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depend on whether the experiments are run sequentially or simultaneously. If a sequential approach is considered, the information matrix is defined as: Hθ =
N exp −1
∑ Hθ k =1
* i |k
+ Hθ* | Nexp (θ , ϕ ) = K + Hθ* | Nexp (θ , ϕ ) ,
(4)
where K is a constant matrix defined by the previous (Nexp−1) experiments. In the above information matrix, only the vector ϕ of the experimental conditions for the new experiment, Nexp, is available for optimisation. On the other hand, Nexp new experiments can be designed simultaneously. In this case, the information matrix becomes: N exp
Hθ = ∑ Hθ* , k (θ , ϕ k ) .
(5)
k =1
Here, all vectors ϕk , one for each experiment k are optimized simultaneously, using, as before, the largest eigenvalue λ1 of the overall matrix Vθ (E-optimality) as objective function to be minimised. It is noted that, as the inversion of Hθ is a nonlinear operation, the optimum Vθ thus obtained will not be the same as the sum of the Vθ obtained by optimizing each individual experiment Nexp times. In other words, the Nexp new optimal experiments will normally be distinct. The main drawback of this approach is that a much larger optimisation problem needs solving. An alternative method is also proposed here. According to this novel approach each experiment is designed a-priori to deliver a vector of experimental conditions producing information which is totally different (orthogonal) from the other ones. In mathematical terms, that means that the information matrix Hθ is split into its singular values identified by its Nθ eigenvalues λi : the new optimisation criterion, called SV-optimality, aims at maximising the information linked to the Nexp largest singular values of Vθ. Thus, the overall optimisation problem is split into Nexp separate optimisation problems, where the k-th measure ψk is defined as:
ψ k = λk (Vθ )
k = 1,..., N exp ≤ Nθ
λ1 > λ2 > ... > λN
exp
.
(6)
The obvious advantage of SV-optimality is that it is easier to solve Nexp small optimisation problems rather than a single large one. The second potential advantage is that we do not design the experiments to maximise the information content of the overall set, but each experiment is designed to maximise a specific component of the available information. Note that this approach can also be applied for sequential experiment design: the first experiment will aim at minimising the largest eigenvalue of the variance-covariance matrix, the second will minimise the second largest eigenvalue, and so on.
3. Case study The methodology discussed in the previous section is applied to a biomass fermentation process that appeared in several papers on the subject [1,3,8]. Assuming Monod-type kinetics for biomass growth and substrate consumption, the system is described by the following set of DAEs: d x1 = ( y − u1 − θ 4 ) x1 , dt
d x2 yx = − 1 + u1 ( u2 − x2 ) , dt θ3
y=
θ1 x2
θ 2 + x2
,
(7)
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where x1 is the biomass concentration (g/L), x2 is the substrate concentration (g/L), u1 is the dilution factor (h-1), and u2 is the substrate concentration in the feed (g/L). The experimental condition that characterise a particular experiment are the initial biomass concentration x10 (range 1-10 g/L), the dilution factor u1 (range 0.05-0.20 h-1), and the substrate concentration in the feed u2 (range 5-35 g/L). The initial substrate concentration x20 is set to 0 g/L. Both x1 and x2 can be measured during the experiment. The objective is to design a set of experiments to yield the best possible information for the estimation of the four parameters θi. The total duration of a single experiment is set equal to 40 h. It is assumed that each experimental run involves 5 sampling intervals. A piecewise-constant profile over 5 switching intervals is assumed for both controls. A total of 15 variables are optimised in each experiment. The elapsed time between any two sampling points is allowed to be between 1 and 20 h and the duration of each control interval between 2 and 20 h. “Experimental data” are obtained by simulation with θ =[0.310, 0.180, 0.550, 0.050]T as the “true” parameters and by adding multivariate normally distributed noise with a mean of zero; two possible M×M covariance matrix Σ of the simulated measurements error will be considered: ⎡ 0.01 0 ⎤ ΣA = ⎢ 0.05⎥⎦ ⎣ 0
0 ⎤ ⎡0.05 ΣB = ⎢ . 0.08⎥⎦ ⎣ 0
(8)
The initial guess for the parameters’ values is set to θ =[0.313, 0.202, 0.551, 0.050]T. 3.1. Proposed experiment designs and results Different experiment design approaches are compared assuming that we wish to design the same number of new experiments. Initially, the following designs are implemented: 1. D1: sequential experiment design (E-optimality), 2 experiments 2. D2: parallel experiment design (E-optimality), 2 experiments 3. D3: sequential experiment design (SV-optimality), 2 experiments 4. D4: parallel experiment design (SV-optimality), 2 experiments Each design is applied first assuming “clean” measurements (Case A: matrix ΣΑ) and then noisy ones (case B: matrix ΣΒ). Results, in terms of the a-posteriori statistics obtained after the optimally designed experiments were executed and model parameters re-estimated with the new data, are summarised in Table 1. In all cases, the model responses with the estimated parameters give a statistically good fit of the data derived from the designed experiments, as expressed by the χ2 value, which is in all cases less than χ2ref based on a Student distribution. It should be noted that the χ2 values for the different cases cannot be compared to each other, since each represents the capability of the model to fit the data from the experiments of that specific design. Here, the different designs could be assessed by comparing the estimated parameter values to the true ones. However, in “real life”, this test is not possible since the true values are of course not known. Therefore, the best approach is to evaluate the accuracy of the design by observing for each parameter either the interval of estimation confidence or the t-value statistics. For a set of experiments to produce a reliable parameter estimation the t-value must be greater than a computed reference value derived from a Student distribution (t-test). 3.1.1. Case A – Clean measurements All designs provide statistically sound results (all t-values are above the reference threshold). Note, that from this point of view, parallel design is a sensible alternative to save time since the experimental session requires half the time as either D1 or D3 (but, of course, double equipment is needed). One drawback of design D2 is that, as
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previously stated, it requires the solution of a larger optimisation problem (30 variables) and, therefore, it may be more upset by convergence issues and, more importantly, by a larger number of local minima. This issue is overcome by design D4. Table 1. Comparison of sequential and parallel approaches for model-based experiment design (two experiments). Superscript * indicates t-values failing the t-test Design
Param. estimate
Conf. interval (95%)
t-value (tref=1.75)
χ2 (χ2ref = 26.30)
D1-A
θ = [0.305, 0.164,
[±0.0110, ±0.0518, ±0.0243, ±0.0101]T
[27.87, 3.17, 22.29, 4.52]T
21.46
θ = [0.299, 0.145,
[±0.0137, ±0.0582, ±0.0474, ±0.0097]T
[21.80, 2.50, 10.79, 4.32]T
19.17
θ = [0.305, 0.163,
[±0.0107, ±0.0520, ±0.0221, ±0.0096]T
[28.43, 3.14, 24.60, 4.82]T
21.63
θ = [0.305, 0.269,
[±0.0134, ±0.1431, ±0.0384, ±0.0120]T
[22.80, 1.88, 13.58, 3.41]T
15.35
θ = [0.300, 0.185,
[±0.0390, ±0.1202, ±0.1138, ±0.0387]T
[7.69, 1.54*, 4.60, 0.98*]T
22.19
[±0.0443, ±1.283, ±0.0769, ±0.0182]T
[7.22, 0.93*, 6.16, 1.73*]T
17.12
[±0.026, ±0.1084, ±0.0564, ±0.0188]T
[11.20, 1.40*, 9.10, 2.15]T
20.48
[±0.0278, ±0.1122, ±0.0627, ±0.0287]T
[10.78, 1.17*, 8.55, 1.53*]T
22.80
0.541, 0.046]T D2-A
0.512, 0.042]T D3-A
0.542, 0.046]T D4-A
0.521, 0.041]T D1-B
T
0.523, 0.038] D2-B
θ = [0.320, 1.189, T
0.474, 0.032] D3-B
θ = [0.292, 0.151, T
0.513, 0.040] D4-B
θ = [0.300, 0.132, T
0.536, 0.044]
The best parameter estimation in terms of confidence interval and t-values is obtained by means of design methods D1 e D3, i.e. the two sequential ones. This is as expected, since the second experiment is designed using the information content from the first experiment. It is interesting to note that approach D3 performs slightly better than D1. In particular, D3 produces a more confident estimation of parameter θ3, hinting that some of the information content related to that parameter belong to a different direction in the variance-covariance matrix. Although less precise, a similar behaviour can be detected by comparing D2 and D4. D4 is less precise as far as the estimation of parameters θ2 and θ4 is concerned. Nonetheless, a better estimation of θ3 is obtained. 3.1.2. Case B – Noisy Measurements These results are rather more interesting. First of all, no design is capable of providing a full set of reliable parameters (D2 produces a particularly bad θ2 estimation). More experiments are needed. In this case SV-optimality is a better criterion. Both designs D3 and D4 are sensibly more performing. Design D3 is the only one providing a statistically sound estimation of three parameters. However, what is surprising is that D4 is overall a better design than D1. Exploiting the information related to λ2 is more important than having the chance to design the second experiment by using the information of the first experiment. Once again, it can be seen that SV-optimality leads to a good estimation of parameter θ3, while E-optimality provide a better estimation of parameter θ2. This confirms the hypothesis that the direction identified by the second eigenvalue contains some valuable information related to the third parameter.
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In view of the above results, it seems reasonable to design a set of 3 experiments aiming first at extracting most of the information related to the first eigenvalue (indeed, the most informative) and then at maximising the information related to the next two largest eigenvalues. Two more design formulations are thus considered: 5. D5: sequential experiment design (E-optimality), 3 experiments 6. D6: sequential-parallel experiment design (E+SV-optimality), 1+(2 parallel) experiments Results are summarised in Table 2 (from the same initial conditions as before). Design D5 shows that three sequential experiments are still insufficient to reliably estimate all parameters: the estimate of parameter θ2 is nearly acceptable, but that of θ4 is not. On the contrary, the results from design D6 are fully satisfactory. Not only is it possible to obtain (in a shorter time period) a statistically precise estimation of the entire set θ (particularly of θ3), but all parameters are better estimated than in D5. This seems to confirm that valuable information is related to the smaller eigenvalues and that a proper exploitation of such information can produce more effective experimental designs. Table 2. Comparison of sequential and sequential-parallel approaches for model-based experiment design (three experiments). Superscript * indicates t-values failing the t-test Design
Param. estimation
Conf. interval (95%)
t-value (tref=1.70)
χ2 (χ2ref = 38.85)
D5-B
θ = [0.305, 0.189,
[±0.0297, ±0.1118, ±0.0920, ±0.0307]T
[10.28, 1.69*, 5.79, 1.34*]T
29.78
[±0.0105, ±0.0364, ±0.0237, ±0.0080]T
[13.87, 2.11, 10.85, 2.61]T
27.54
T
0.532, 0.041] D6-B
θ = [0.298, 0.158, 0.528, 0.043]T
4. Final remarks A novel procedure based on the decomposition of the variance-covariance matrix has been suggested, which is applicable to the model-based design of both sequential and parallel experiments. Preliminary results on an illustrative application demonstrate the promising potential of this new approach. Future work will assess the applicability of the methods to larger applications and the development of a systematic procedure to help determine the best approach to use for model-based experiment design, whether sequential, parallel, or mixed sequential-parallel.
References [1] D. Espie and S. Macchietto, AIChE J., 35 (1989) 22. [2] L. Zullo, PhD Thesis, The University of London, 1991. [3] S.P Asprey and S. Macchietto, Comput. chem. Engng., 24 (2000) 1261. [4] B.H. Chen, S. Bermingham, A.H. Neumann, H.J.M. Kramer and S.P. Asprey, S.P, Ind. Eng. Chem. Res., 43 (2004) 4889. [5] F.R. Sidoli, A. Manthalaris and S.P. Asprey, Ind. Eng. Chem. Res., 44 (2005) 868. [6] G. Franceschini and S. Macchietto (L. Puigjaner and A. Espuna Eds), ESCAPE –15, CACE Series 20A, Elsevier, Amsterdam, The Netherlands, (2005) 349. [7] I. Bauer, H.G. Bock, S. Körkel and J.P. Schlöder, J. Comput. Appl. Mathem., 120 (2000) 1. [8] S.P. Asprey and S. Macchietto, J. Proc. Control, 12 (2002) 545. [9] G.E.P. Box and H.L. Lucas, Biometrika, 46 (1959) 77. [10] J. Kiefer and J. Wolfowitz, Ann. Math. Stat., 30 (1959) 271.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
OPEN CHEMASIM™: Breaking Paradigms in Process Simulation Hans Hassea, Bernd Besslingb, Roger Böttcherc a
Institute of Thermodynamics and Thermal Process Engineering (ITT), University of Stuttgart, D-70550 Stuttgart, Germany Chemical Engineering, BASF AG, D 67056 Ludwigshafen, Germany c Corporate Engineering, BASF AG, D 67056 Ludwigshafen, Germany Email: [email protected] Internet: http://chemasim.itt.uni-stuttgart.de b
Abstract Since September 2005 OPEN CHEMASIM™, a process simulator with features similar to commercial programs is available to the academic community as an Open Source Code. The code was developed by BASF for over 30 years as an in-house tool, and has now been transformed into an Open Source Code for academic use. For the distribution, the internet platform http://chemasim.itt.uni-stuttgart.de was set up at ITT, University of Stuttgart. Academic institutions may use OPEN CHEMASIM™ freely for teaching and in research as long as the results are published unrestrictedly. The code can be distributed to students, e.g., for project work. The present paper reports on this unprecedented project in process simulation. Features of the OPEN CHEMASIM™ program are briefly described and it is explained how the OPEN CHEMASIM™ as an academic noncommercial project works. Keywords: Process Simulation, Open Source, OPEN CHEMASIM™, Software, BASF.
1. Introduction It is well known, that if the same problem is solved with different codes, even for only moderately complex problems, the solutions often differ outside the numerical uncertainty [1]. It is therefore highly desirable that, it can be tracked in the code what really was done. This is only possible with Open Source Codes. Furthermore, as in principle an unlimited number of people can actively participate in debugging an Open Source Code, these codes will in the long run generally be more reliable than undisclosed codes. More fundamentally, it can be argued that black box simulations are unacceptable for any scientific purpose. One of the most essential requirements of scientific work is repeatability and, more stringent, traceability. Reports on scientific experiments or simulations must put other scientists in a position as to be able to repeat the described experiments or simulations and to trace what has been done in all relevant aspects. Of course this ideal can not always be reached, but it is scientific practice to try to come close to it. Using a commercial program often does not even allow repeating the simulations as the program version with which the simulations were made may no longer be available by the time the repeatability is to be checked. Scientifically more important is the fact that in simulations with black box programs it is generally not fully traceable what has been done. Questions that can arise at any point in a scientific discussion may, hence, not be clarified. Open Source Codes do not have that problem as, at least in principle, everything can be traced down to the roots. Of course, with rising complexity of their studies, scientists often have no choice: they need to use powerful commercial software even if it is only poorly documented. But if there is a choice, from a scientific standpoint, it is surely more attractive to use an Open Source Code.
2. OPEN CHEMASIM™ OPEN CHEMASIM™ breaks many paradigms in process simulation: it is neither a commercial product by a software company nor is it a shareware or commercial product created by an academic institution. It started with the decision of BASF to share its in-house process simulator CHEMASIM with the academic community – in a non-commercial way. CHEMASIM has a long history within BASF [2–5]. In the early seventies process developers at BASF, as in many other chemical companies, realized that process simulation was a key to success in their business. As there were no commercial process simulators then, BASF started developing their own simulator, called CHEMASIM (German: CHEMie Anlagen SIMulation). Since then, CHEMASIM was continuously improved
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by a highly motivated and skilled group of mathematicians and software engineers at BASF who always worked closely together with the engineers who applied the tool. Over the years CHEMASIM steadily grew, and became more powerful and versatile. Since the eighties CHEMASIM had to compete with commercial process simulators, and the question arose if the development of the in-house tool was to be supported further. Up to now, CHEMASIM was successful in that competition. But the fact remains that BASF is no software company and that commercial programs get better: it is a race, and the question is how long it is going to continue as it did now for more than 20 years. OPEN CHEMASIM™ is a completely unexpected solution to that puzzle. BASF has decided in 2005 to open their process simulator to the academic community, not only the executable objects but also the source code. CHEMASIM became OPEN CHEMASIM™. Never before has the academic community had access to a similar process simulation source code. There are no limitations; OPEN CHEMASIM™ users can use all parts of the software; they can add their own software. They can freely use OPEN CHEMASIM™ for teaching and academic research. Also BASF will continue to use CHEMASIM in the future, hopefully in a new active “Verbund” with academic partners.
3. OPEN CHEMASIM™ Program Features OPEN CHEMASIM™ is a package of several programs originally created to meet the requirements of chemical engineers working in the field of process design and development in the chemical industry. The program CHEMASIM is the heart of OPEN CHEMASIM™: it is a process simulator suited for simulations of large chemical plants. In CHEMASIM basically, mass and energy balances are solved based on equilibrium thermodynamics. CHEMASIM contains models of all important unit operations like reactors, distillation, absorption, and extraction columns, evaporators, condensers etc. The most important thermodynamic fluid property models are implemented. The features provided by CHEMASIM are similar to those of the well-known commercial process simulators. The main focus of CHEMASIM is the simulation of processes with extremely non-ideal multicomponent mixtures; e.g., three phase distillations with chemical reactions can routinely be handled. CHEMASIM also allows automatic parameter variation and optimization. The philosophy behind CHEMASIM has always been to solve the given process simulation problem equation-oriented, simultaneously, i.e., the full set of equations describing the problem is set up and solved numerically, unlike in many commercial process simulators which are based on solving subproblems representing different units and iterating to find the solution for the entire flow sheet [6]. The equation oriented approach is especially advantageous for simulations of complex processes with many recycles [7]. Like in all process simulators, the basic mathematical task in CHEMASIM is solving a large set of non-linear equations f(x) = 0 (1), where f : IRn → IRn contains all equations describing the flowsheet, i.e. mass- and energy balances, equilibrium and stoichiometric equations, reaction balances [5]. x ∈ IRn is the vector of the variables for which values are found by CHEMASIM. n is typically of the order of 103 – 104. In CHEMASIM, first the structure of the flowsheet is defined, i.e., the user input of streams, process units, reactions and specifications is translated into the form of the function f. Due to a dynamic allocation of storage the absolute size of the problem in CHEMASIM is not fixed or bounded. The next step is to initialise all the variables x, which may be done by user estimates or with information from old solutions, either of the entire problem or of subproblems. Finally the set of equations (1) is solved by a Newton method using a relaxation technique based on a Gauss algorithm. Figure 1 shows this structure in a flowchart. Building up the structure of the flowsheet and solving the equations is done simultaneously in CHEMASIM, cf. inner loop over all units in Figure 1. This is one of the reasons for the fast response of CHEMASIM. Especially the thermodynamics part of CHEMASIM contains many features developed over the years to keep the iterations in the calculation robust und reliable.
OPEN CHEMASIMTM: Breaking Paradigms in Process Simulation
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user input
defining structure f
creating initial values x
last solution
loop over all units ν
unit: fν ( xk ), fν ' ( xk )
Thermodynamics Reaction balances and further more
creating and inversion of block structure up to unit ν: ⎛ f 1 ' ( x k ) a12 … … f1 ( xk ) ⎞ ⎜ ⎟ … ⎜ ⎟ ⎜ a aν 2 fν ' ( x k ) fν ( x k ) ⎟⎠ ⎝ ν1
solving the linear system ∂f(xk)/∂x Δxk = – f(xk)
relaxation and up date:
xk+1 = xk + ωk Δxk
accuracy reached
output
Figure 1: Structure of CHEMASIM. CHEMASIM has been shown to be able to solve problems for which commercial process simulators fail. A recently published example is the simulation of a process for trioxane production from aqueous formaldehyde [8, 9]. The process has three major recycle streams and is characterized by the oligomerization reactions of formaldehyde with water that lead to complex multicomponent mixtures. Meaningful simulations of that process can only be performed by explicitly accounting for at least 10 chemical reactions on every stage of the process.
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CHEMASIM is a FORTRAN 90 program. An example for a CHEMASIM output is presented in Figure 2. CHEMASIM was developed in Germany and was, up to 2005, mainly used in BASF AG in Ludwigshafen, Germany. The CHEMASIM input/output and documentation is therefore presently in German. However, OPEN CHEMASIM™ is set up as an international project; the OPEN CHEMASIM™ language is English. We hope that an English version of the program will be made available through OPEN CHEMASIM™ before long. Together with CHEMASIM some other programs are supplied in OPEN CHEMASIM™. They mainly help creating reliable fluid property models (data fitting, setting up the fluid property data file, visualization, evaluation, calculations of azeotropic points, distillation and residue curves, miscibility gaps).
Figure 2: Example for a CHEMASIM output (distillation column with profiles).
4. The OPEN CHEMASIM™ Project OPEN CHEMASIM™ is more than just a downloadable process simulation software. The idea behind OPEN CHEMASIM™ is to create a living system in which CHEMASIM will continue to develop and grow. OPEN CHEMASIM™ users agree to share their own contributions to CHEMASIM with the OPEN CHEMASIM™ community, in a non-commercial way. The system is organized like a star, cf. Figure 3.
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User A User B
User F Open Chemasim Forum
User C
User E User D
Figure 3: The OPEN CHEMASIM™ Star
The OPEN CHEMASIM™ forum allows users to exchange ideas and experiences, to download software and upload their own versions. The forum is moderated by ITT, University of Stuttgart. ITT also supplies a master version of OPEN CHEMASIM™ (executable object and source code), and will try to keep track of new developments in the forum, test them and continuously improve the master version. Note that only non-profit academic institutions or natural persons working at a university or non-profit research institution can become OPEN CHEMASIM™ users, not companies. The source code is only made available to academic institutions, not to individual natural persons. Users must agree to use OPEN CHEMASIM™ only for academic research and teaching, any commercial activity is excluded. The use of OPEN CHEMASIM™ by academic users in joint projects with industrial partners is allowed only if the results are published unrestrictedly. Standard Users get access to the OPEN CHEMASIM™ executable objects and installation routines, Developers get full access, which includes the source code. It is legal to make copies of CHEMASIM for scientific or educational use within the working group, also for use by students. Working groups at academic institutions only need to register once either as Standard User or Developer. OPEN CHEMASIM™ users do not have to pay for the software itself. There is only a one-off registration fee. There are no annual fees. The money from the registration fee is used for keeping the OPEN CHEMASIM™ project alive, to run the server, to handle registration, to support users getting started, and last but not least to supply up-to-date master copies of the OPEN CHEMASIM™ source code and the executable objects. The fee is presently 490 € for Standard Users, and 990 € for Developers.
5. Current State and Outlook OPEN CHEMASIM™ was presented to the academic community first in a German chemical engineering national conference in September 2005 [10]. By the time this paper is written, it is too early for predictions on the further evolution of the project. More information will be available in June 2006 when the ESCAPE 16 conference will be held. Presently, for the reasons given above, registrations come only from Germany. Besides chemical engineers also mathematicians and software engineers have shown their interest in OPEN CHEMASIM™. Dechema has declared its willingness to supply fluid property data by linking the DETHERM data bank to OPEN CHEMASIM™. The executable version will be distributed to chemical engineering students for project work in several German universities in 2006.
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In the near future an English version of OPEN CHEMASIM™ should be made available to enlarge the potential user group. If the academic OPEN CHEMASIM™ project turns out to be successful, ways should be discussed how to open the project also to interested industrial parties.
Acknowledgements Many people in BASF have contributed to CHEMASIM over the years. Their work is gratefully acknowledged. Thanks are also due to all those who have supported the idea of OPEN CHEMASIM™ and have helped to make it real. Special thanks are due to C. Adami, C. Grote, B. Hegner, F. Güttner, A. Klein, A. Polt, H. Schoenmakers (from BASF), and M. Bidner, O. Ryll, J. Vrabec, F. Schaal (from ITT).
References [1] F. Y. Dupradeau, J. Rochette: Bugs in Computational Chemistry Software and their Consequences: the Importance of the Source Code. J. Mol. Model (2003) 271 – 272. [2] B. Hegner, D. Hesse, D. Wolf: Möglichkeiten der Berechnung bei heteroazeotroper Destillation, Chem. Ing. Techn. 45 (1973) 942-945. [3] U. Block, B. Hegner: Aufstellung und Anwendung eines Rechenmodells für die Dreiphasen-Rektifikation, AIChE J. 22 (1976) 582-589. [4] D. Gärtner, B. Hegner, M. Molzahn, R. Schmidt: Simulation des gekoppelten Stoff- und Wärmeüberganges in Gegenstromapparaten, Chem. Ing. Techn. 51 (1979) 322 – 323. [5] B. Hegner: Steady state simulation, Dechema-Monographie 115, p. 85-98, Dechema, Frankfurt (1988). [6] B. Hegner, H. Schoenmakers: Die Berechnung gekoppelter thermischer Trennungen mit simultanem Lösungsalgorithmus, Chem. Ing. Techn., 56 (1984) 229. [7] B. Hegner, H. Schoenmakers: CHEMASIM – experience with BASF’s simultaneous process simulator, Inst. Chem. Eng. Symp. Ser. 92 (1985) 365-375. [8] T. Grützner, H. Hasse, N. Lang, M. Siegert, E. Ströfer: Development of a New Distillation Based Process for the Production of Trioxane, AIChE Spring National Meeting, 11.-13.04.2005, Atlanta, GA, USA. [9] T. Grützner: Entwicklung eines neuen destillationsbasierten Verfahrens zur Herstellung von Trioxan, PhD Dissertation, Universität Stuttgart (2006). [10] H. Hasse: Open Source Prozesssimulation – Eine Chance für die Verfahrenstechnik, GVC/Dechema-Jahrestagungen, 06.-08.09.2005, Wiesbaden.
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Simulation of the population balance for droplet breakage in a liquid-liquid stirred tank reactor using H-matrix methods Jürgen Koch,a Wolfgang Hackbusch,b Kai Sundmachera,c a
Max-Planck-Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, Magdeburg D-39106, Germany b Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig D-04103, Germany c Otto-von-Guericke-University Magdeburg, Process Systems Engingeering, Universtiätsplatz 2, Magdeburg D-39106, Germany
Abstract In population balance equations particle breakage is often described by a Volterra integral operator. Naive discretisation of this operator leads to quadratic complexity. In this paper an efficient numerical treatment for Galerkin discretisation of the integral operator is suggested, based on the idea of H-matrices, which leads to linear complexity. Keywords: H-matrices, fast methods.
1. Introduction Emulsions are mixtures of two immiscible liquids (e.g. water and oil) and a surfactant, needed to stabilise the liquid-liquid interface. One of the liquids, i.e. the disperse phase usually establishes a population of droplets and so the emulsion can be considered as a disperse system. Such systems are often applied directly in consumer designed products or they appear as intermediates in chemical processes. Furthermore, well characterised emulsions can be used as a structured reaction medium to manufacture solid particles in the sub micrometer range [6]. This approach allows to control the particle properties by droplet size and other emulsion properties. For the particle synthesis two similar emulsions, each containing dissolved educts in water droplets, are mixed in a stirred tank. By droplet coalescence and breakage events the reaction is initialised leading subsequently to the formation of solid particles in the droplets. The correct understanding of the droplet population behaviour is thus of practical importance. Commonly for the description of the system a population balance equation is used, where the coalescence and breakage events are characterised by Volterra integral operators of the first kind. After a naive discretisation these operators have usually a complexity of O(n2) for storage and matrix-vector-multiplication if n denotes the problem size. In this work, for the operator being responsible for the breakage of droplets, introduced in [1] from Coulaloglou and Tavalarides, a Galerkin discretisation will be presented using the ideas of the H-matrices, compare Hackbusch [4]. After some modifications this discretisation leads to a complexity of O(n).
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2. General model In literature population balance equations can be found in arbitrary detailed versions. The balance equation according to [1] has a non-local character and can be written in the general form
wF x, t wt
Fin x, t Fout x, t Fco x, t Fco x, t Fbr x, t Fbr x, t . (1)
The individuals of the droplet population are described by the number density function F. This function specifies the number of droplets of the size (volume) x at the time t in the tank reactor. The range of F in the first variable is given by [xmin, xmax] [0, xmax], where xmin stands for the smallest and xmax for the biggest possible droplet size. The terms Fin and Fout denote the inflow and the outflow of liquid and droplets, respectively. The operator Fco+ and Fco name the source and sink terms of the coalescence and the operator Fbr+ and F br the source and sink terms of breakage. The interest is directed towards the operators representing the breakage phenomenon. In detail they read as
Fbr x, t
K lin >F ., t @ x
Fbr x, t
K diag >F ., t @ x
³
xmax
x
Q x M x, y E br y F y, t dy
and
E br x F x, t .
(2) (3)
Here Ebr is the breakage frequency, Q(x) represents the number of daughter droplets formed from a droplet with size x, and M(x,y) is the distribution of daughter droplets generated from the breakage of a droplet of size y. The discretisation of the diagonal operator Fbr is trivial henceforth we focus on the integral operator Fbr+.
3. Kernel function for droplet breakage The equation (2) can be rewritten with the help of the kernel function N
K lin >F @ x
³
xmax
x
Q x M x, y E br y F y dy
³
xmax
x
N x, y F ( y )dy
(4)
where the time dependency is omitted. In the explicit construction of the kernel function it is assumed, that a breaking droplet forms always two daughter droplets, i.e. Q(x){2. The breakage frequency Ebr represents as
E br x c1 x
2 9
5 ª º exp « c 2 x 9 » ¬ ¼
(5)
where the constants c1 and c2 depend on the stirrer diameter, its revolution speed, the density of the liquids and their surface tension, see [1], and for the a orders of magnitude holds c1a1 and c2a106. Further on the daughter droplet distribution M is based on a Gaussian distribution and can be written as
M x, y
ª c3 2 x y 2 º exp « c 4 » y y2 ¬ ¼
(6)
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where the constants c3 and c4 are chosen in the way that 99.6% of the broken droplets are located in the interval [0, y], compare again [1]. The order of magnitude is c3~ c4~1.
4. Numerical approach For the discretisation of the integral operator (4) a Galerkin scheme is used. The Galerkin system matrix K has for i, j =1,...,n the entries
K ij
³
supp bi
bi x ³
supp b j > x , xmax @
b j y N x, y dydx ,
(7)
where bi and bj denote the corresponding basis functions and n the degrees of freedom. Assuming piecewise constant basis functions with supp(bi)=[xi,xi+1] for i=1,...,n. The entries of K can be written
K ij
° 0 for xi d x j 1 ° xi1 x j1 ® ³x ³x N x, y dydx for x j ! xi 1 , i,j=1,...,n. ° xii1 xjj1 °³ ³ N x, y dydx for xi x j ¯ xi x
(8)
Though the resulting matrix is an upper triangular with complexity O(n2). First we assume the integral operator (4) has a fixed lower boundary xmin (Fredholm operator) and the kernel N is separable, i.e. it can be written in the form
N x, y
¦
k l 1
) l x
(9)
where k is called the separation rank. Then the entries of the corresponding system matrix K would have the following form
K ij
xi 1
³ ³ xi
x j 1
xj
N x, y dydx
k
xi 1
l 1
xi
¦ ³
) l x dx ³
x j 1
xj
In this case the matrix Knun can be written as a low rank matrix with help of two matrices Snuk and Tkun in the following way
Sil
³
xi 1
xi
) l x dx and Tlj
³
x j 1
xj
In the format K=ST only 2kn entries have to be stored. A matrix-vector-multiplication of K with a vector x can be done in two steps. First x has to be multiplied with T, to get the vector Tx. That requires a2kn operations. Next vector Tx is multiplied with S. That requires also a2kn operations, see Figure 1.
K x S T x Figure 1. Multiplication scheme of a low rank matrix.
S
Tx
y
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For a Volterra operator and separable kernel, the matrices S and T are created as in the Fredholm case. For multiplication the following algorithm is used, where the entries of the vectors x and y are denoted by xj and yi. Algorithm 1: for (l = 1 to k) begin sum = 0.0; j = n; for (i=n-1 down to 1) begin while (j>i) do begin sum = sum + Tlj xj; end; yi = yi + Sil * sum; end; end Note, that the diagonal elements of K have to be added separately. We call this kind of low rank matrix Volterra low rank matrix. The overall complexity is also O(kn).
5. Construction of special H-matrices The kernel function N from (9) is not separable, but it could be approximated by a separable kernel expansion, e.g. by Lagrange interpolation in the first variable
N x, y
¦
k l 0
N xl , y Ll x
where the xl denote the interpolation knots and k denotes the polynomial degree. Usually this will not deliver satisfying results over the whole integration interval. In the concept of H-matrices the matrix K is split up blockwise with the help of a so-called block cluster tree for a class of kernel functions, see [4,3]. Each matrix block is connected to an integration area [a,b]u[c,d], where [a,b],[c,d] [xmin, xmax], and on each area a different kernel approximation is used. This procedure leads to low rank matrix blocks, with the advantages mentioned above. However not on every block this procedure is reasonable, thus some (small) blocks are represented by a full matrix, for details see again [4,3]. In [5] a modification of H-matrices is introduced taking into account the properties of the kernel (9) and of the Volterra character of the integral operator (4). The resulting modified H-matrix is sketched out in Fig. 2. On each matrix block, with corresponding integration area [a,b]u[c,d], the interpolation error (in x or y) can be estimated by
N 3 k >N @ f ,>a ,b @ d 1 / k N
1 f ,C
2 k 1
(10)
independent of the interval [c,d]. Here 3k denotes the Lagrange interpolation operator of degree k and /k the so-called Lebesgue constant, see [2]. The definition of the number ||N ||f,C can be found in [5]. Important in (10) is the factor 1/2k1, it allows to decrease the error with increasing degree k. The relative error made by using the H-matrix instead of the full matrix for the system matrix K (refer to system (8)) is given in Table 1 in the spectral norm. The error decreases by the factor a0.25 every time k is increased by 1. The complexity of the H-matrix sketched out in Fig. 2 is of O(kn), i.e. O(n) for fixed k, instead of formerly O(n2). The amount of storage for k=12 is pointed out in Table 2.
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Due to the quadratic behaviour of the full system matrix e.g. for n=8192 it needs a140 times more memory than the H-matrix with linear behaviour.
In the Tables 3 and 4 the times for matrix creation and matrix-vector-multiplication are compared. Here the H-matrix method is a100 times faster for n = 8192. Table 1. Relative errors for increasing k in spectral norm (n=1024). k
7
rel. error
8
4.9710
-4
9
1.1010
-4
10
2.4110
-5
5.1210
11 -6
1.1910
12 -6
2.7510-7
Table 2. Used memory in [MB] for k=12. n
Full matrix H-Matrix
521 2.00 0.28
1024 4.00 0.51
2048 32.00 0.95
4096 128.00 1.85
8192 512.00 3.64
Table 3. Time for matrix creation in [sec] for k=12. n
Full matrix H-Matrix
521 -1
8.8510 6.6710-2
1024
2048
4096
+0
+1
+1
3.5710 1.3210-1
1.4110 2.6410-1
6.1210 5.4110-1
8192 2.5210+2 1.0610+0
Table 4. Time for matrix-vector-multiplication in [sec] for k=12. n
Full matrix H-Matrix
521 -3
4.8110 3.4910-4
1024
2048
4096
-2
-2
-1
1.9110 1.5310-3
8.0210 3.2010-3
3.5110 6.0210-3
8192 1.4310+0 1.0210-2
As a first illustrated example the numerical solution of a reduced population balance equation
wF x, t wt
Fbr x, t Fbr x, t ,
(11)
was calculated, compare (2) and (3), using the full system matrix as well as a H-matrix with k=12. For the constants of the kernel (4) the following values were used c1 = 2.4, c2 = 4.5, c3 = 0.35 and c4 = 4.7106. A normalised initial population F(x,0)=F0(x) was chosen and the development of the population for different times was evaluated with a simple Euler method, for n=1024.
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Results can be found in the diagrams in Figure 3 and Figure 4 for different time steps t1 and t2. Here the graphs produced with the help of the full matrix and the H-matrix are nearly identical and therefore almost indistinguishable. F
F Initial population Full matrix H−matrix
1
Initial population Full matrix H−matrix
1
0.8
0.8
F0
0.6
F0
0.6
F2
0.4
0.4
F1 0.2
0.2
0
0
0.2
0.4
0.6
Figure 3. Population: F(x,t1) = F1(x).
0.8
x
0
0
0.2
0.4
0.6
0.8
x
Figure 4. Population: F(x,t2) = F2(x).
References 1. C. A. Coulaloglou and L. L. Tavalarides, 1977, Description of interaction processes in agitated liquid-liquid dispersions, Chem. Engng. Sci., 32, 1289-1297 2. P. J. Davis, 1963, Interpolation and Approximation, Blaisdell Publishing Co. Ginn and Co. 3. L. Grasedyck and W. Hackbusch, 2003, Construction and Arithmetics of H-matrices, Computing, 62, 89-108 4. W. Hackbusch, 1999, A sparse matrix arithmetic based on H-matrices, Part I: Introduction to H-matrices, Computing, 70, 295-334 5. J. Koch, 2005, Effiziente Behandlung von Integraloperatoren bei populationsdynamischen Modellen, Otto-von-Guericke-Universität Magdeburg 6. C. Y. Tai, M. H. Lee and Y. C. Wu, 2001, Control of zirconia particle size by using twoemulsion precipitation technique, Chem. Engng. Sci., 56, 2389-2398
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Simultaneous Dynamic Validation/Identification of Mechanistic Process Models and Reconciliation of Industrial Process Data Pablo A Rolandia, José A Romagnolib a
Process Systems Enterprise Ltd., 107a Hammersmith Bridge Road, London W6, UK Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA, 70803 USA
b
Abstract Process models are subject to parametric uncertainty and raw process-instrumentation data are corrupted by systematic and random errors. In this work, we present a framework for dynamic parameter estimation and data reconciliation aiming at integrating model-centric support tools and industrial process operations. A realistic case-study for the rectification of the mass balance of an industrial continuous pulping system is presented. The incentive for gross-error estimation during model-based production accounting and inventory analysis is demonstrated.
1. Introduction An assumption implicit in the execution of any hybrid data-driven/model-based activity is that both the mechanistic model and experimental data describe the behaviour of the process system accurately. In the case of industrial manufacturing systems, these conditions are rarely met. Physical, chemical, and biochemical process phenomena are complex and, therefore, difficult to model conceptually and mathematically. Thermodynamic and transport properties and reaction rates are difficult to characterise experimentally and, hence, subject to parametric uncertainty. Modelling industrial process systems aggravates these problems since site-specific operating conditions render theoretical and/or empirical modelling of some natural phenomena virtually intractable. Concurrently, plant data is abundant and readily available in industrial process systems. However, raw process-instrumentation data are corrupted by systematic and random errors undermining the solution performance of any hybrid datadriven/model-based activity making use of experimental data pools. Joint parameter estimation and data reconciliation techniques provide a framework for simultaneous dynamic validation/identification of mechanistic process models and reconciliation of industrial process data.
2. Problem definition In general terms, the parameter-estimation problem can be stated as finding the optimal estimate of the vector of parametric variables T , subject to the constraints imposed by the fundamental principles of conservation (i.e. the mathematical model of the process). Conventionally, optimality implies maximising the likelihood of predicting the experimental set of measurements or, alternatively, minimising a measure of the distance between experimental measurements and predicted values. Similarly, simultaneous data reconciliation and gross-error estimation can be stated as finding an optimal vector of random measurement errors İ and systematic errors E so that the
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corrected measurements satisfy the fundamental principles of conservation imposed by the mechanistic process model. Within the scope of data reconciliation, optimality is usually associated to minimising a measure of the error between experimental measurements and predicted values. In this work, we shall adopt a notation that eases the declaration of a given processengineering problem in terms of the conventions of state-of-the-art high-level symbolic languages. Thus, we define a dynamic estimation problem (DEP) as follows:
min M ~z t , z t , V t
T , E ,Z ,J
F x t , x t , y t , u t , p, T , E
0, t [0, t f ]
I x 0 , x 0 , y 0 , u 0 , p, T , E 0 V t V ~z t , z t , Z , J , t [0, t f ]
(1)
T min d T d T max E min d E d E max Z min d Z d Z max J min d J d J max For each measuring device, the reconciled (corrected) measurement z , the raw measurement ~ z , the measurement error İ and the measurement bias E are given by:
z
~z İ E
(2)
In the error-in-variables measured (EVM) method for parameter estimation/data reconciliation, all measured process variables are assumed to contain systematic and random errors. In order to engage in the solution of this type of DEP using commercial general-purpose process-engineering software tools, a problem formulation consistent with the characteristics of Eq. (1) is needed. In this work, we shall assume that
İ ip
E ip , where ip indicates input process variables; according to this, Eq. (2)
f
becomes:
z ip
~z ip E ip
(3)
Here, z can be uniquely determined from ~ z since, structurally, the number of unknowns in Eq. (3) is one. Consequently, the overall number of decision variables to be estimated is N (i.e. the dimension of the vector E ). Because in plant-wide industrial applications the number of measuring devices to be reconciled is generally in the order of a few hundreds, this approach renders large-scale EVM DEP solvable with available advanced process modelling (APM) tools. From a physical perspective, ip
İ ip
f
ip
E ip denotes sensors with good precision and poor calibration (this
situation is the most common in industrial manufacturing plants). Defining a dynamic estimation problem requires selecting a subset of measurements from the experimental data set that is consistent with the purpose of the particular estimation experiment. In the case of industrial process systems where raw plant data is
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abundant, this process is rarely a trivial task. For example, process measurements are available at sampling periods which are orders of magnitude smaller than the characteristic time constant of the process system, leading to a phenomenon of data over-sampling. As a consequence, raw plant data is not adequate for populating this experimental data subset, and it is advisable to perform some data pre-processing and conditioning in order to improve the solution performance of the dynamic estimation experiment. In this work, the methodology for reconstruction of process trajectories (RPT) proposed by Rolandi & Romagnoli (2006) has been used to reduce the number of observations in the experimental data subset and simultaneously smooth high-frequency temporal fluctuations in process variables. Even though the details of this technique are out of the scope of this contribution, RPT has improved the accuracy, efficiency and robustness of industrial DEP.
3. Methodology A successful and meaningful definition of parameter estimation/data reconciliation problems requires a painless integration of empirical data and large-scale dynamic models. State-of-the-art commercial process-engineering tools lack support mechanisms for manipulating plant data seamlessly and incorporating this information in the formulation of hybrid data-driven/model-based problems. This has precluded the routine validation of plant-wide mechanistic models, as well as the widespread use of advanced model-centric technologies (MCTs) such as joint parameter estimation and data reconciliation techniques. In a companion paper (Romagnoli & Rolandi, 2006), the authors proposed a novel architecture for process-engineering software development and introduced the notion of the so-called Problem Definition Component (PDC). This software object supports a novel methodology for definition of parameter estimation/data reconciliation problems which is based on the refinement of instances of process-engineering data models. In this paradigm, Data Model Templates (DMTs) determine what information (predominantly the model’s structure and control system’s objects) is available to the user and how this information can be manipulated by the end-user. On the other hand, Data Model Definitions (DMDs) represent valid model-based activities and associated experimental process data. DMDs are generated by the user as a series of refinements of the original DMTs according to particularities of the conceptual definition of a given process-engineering problem. This definition process is regulated entirely by the nominated PDC. Due to space constraints, the Problem Definition Environment (PDE) of the System for Support of Process Operations (SYSS-PRO) software prototype will not be shown in this work. Two data models are needed in order to fully describe the mathematical definition of a dynamic estimation problem. These structures are the so-called Process Data Object (PDO) and the Dynamic Estimation Problem data models (DEP). In brief: x PDO model: it contains data representing raw experimental process data in a form suitable for combined discrete/continuous process modelling; not only does this structure support data pre-processing, conditioning and reconstruction techniques, but it also maps process instrumentation from which data was retrieved to the corresponding input/output variables of the process and model variables. x DEP model: it contains data determining structural and numerical information of general dynamic estimation problems; this structure is given by a series of control (input), measured (output) and parametric (decision) process variables which maps into the corresponding model variables and process-instrumentation objects
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(devices); this allows, for instance, a better characterisation of the objective function, selection of process operating parameters and/or measurement biases to estimate, determination of forcing input conditions, etc.; it also keeps information on upper and lower bounds and initial guesses of decision variables.
4. Case-study With the exception of the contribution by Özyurt & Pike (2004), dynamic parameter estimation and data reconciliation of industrial process systems represented by largescale mathematical process models are rare in the open literature, and the solution of this kind of problems still poses several challenges to the research community. In this work, a large-scale mechanistic model of the continuous pulping system of a world-class industrial pulp and paper mill is used to illustrate viability of the proposed framework. Overall, the implementation of the resulting large-scale mathematical model gives rise to approximately 1.5104 differential-algebraic equations; among these, there are 1.4104 algebraic equations, 9.7102 ordinary differential equation, and 3.2102 structural degrees-of-freedom. Concurrently, there are approximately 3.6102 statuses within the state transition network. gPROMS was used as the modelling and solution engine (MSE). The goal of this case-study is to reconcile historian process data focussing on the closure of the general mass balance of the continuous pulping system. We will also aim at demonstrating that the abundance of plant data in today’s industrial manufacturing systems can be readily exploited to solve realistic model-based parameter estimation/data reconciliation problems. 4.1. Problem specification In this case-study, process-instrumentation data obtained from the historian throughout 24hr of operation is used. A set of 26 input process variables is reconstructed in order to force the behaviour of the continuous process system according to experimental process conditions. Among these, 21 are controlled variables and 5 are disturbances. A combined implicit/explicit state initialisation procedure is used to determine the initial state of the process; the details of this technique are outside the realms of this manuscript. Two parametric process variables are subject to estimation. The wood chip impregnation factor is a measure of the flowrate of steam condensate bounded to the interstitial space between wood chips before entering to the chip meter in the feed line. Changes in wood handling operations and operating conditions of the chip bin affect the impregnation of wood, changing the free-liquor pattern flow and affecting the extent of the pulping reactions and closure of the overall mass balance. The pre-multiplier of the fundamental kinetic model also determines the extent of the pulping reactions, accommodating for seasonal wood-composition fluctuations and inadequate wood handling operations. In this case-study, three flow-measurement devices are rectified: the overall white liquor addition; the wash filtrate addition to the digester’s bottom; and the black liquor extraction from the upper screens of the digester. Data from eight sensors are used for the purpose of estimation; three of them are output measured process variables and five are input measured process variables. A weighted least-squares objective function minimising the difference between the model predictions and experimental observations is used in this case-study. The weights of each sensor are proportional to the expected value of the measured time series; hence, relative deviations contribute equally to the magnitude of the objective function irrespectively of the nature of the measured process variable.
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The potential for model-based joint parameter estimation/data reconciliation of a largescale complex industrial process system is demonstrated in this case-study. The problem results in the estimation of five parametric process variables (three of them are measurement biases) from an experimental data pool of eight measured variables and twenty-six control variables. The challenge of this joint parameter estimation/data reconciliation case-study lies on the combination of large amounts of process data and a large-scale mechanistic process model to solve an involved process-engineering problem of interest to mill personnel.
0.349 0.342 0.334 0.327 0.319 0.142 0.162 0.182 0.202 0.222 3 impregnation factor [m /kg]
liquor addition bias [m3/min]
kinetic pre-multiplier [adim]
4.2. Analysis of results Figures 1 to 3 show the confidence regions for the wood chip impregnation factor and the kinetic pre-multiplier, the white liquor and wash filtrate addition biases, and the upper extraction flow bias and kinetic pre-multiplier, respectively. 0.194 0.184 0.174 0.164 0.154 0.144 0.065 0.069 0.073 0.077 0.081 filtrate addition bias [m3/min]
Figure 2. Confidence region. 15
0.16 0.11 0.06 0.01 -0.04 0.142 0.162 0.182 0.202 0.222 3 impregnation factor [m /kg]
Figure 3. Confidence region.
relative error [%]
3
upper extraction bias [m /min]
Figure 1. Confidence region.
10 5 0 -5 0
4
8
12 16 time [hr]
20
24
Figure 4. Mass balance closure.
Figure 4 shows three different trajectories corresponding to the closure of the general mass balance according to different degrees of awareness on the status of the process system. For instance, the conservation of mass in the continuous pulping area on the basis of the measured volumetric flowrates is inaccurate by approximately 11.3%; this discrepancy is due to the trivial fact that mass flowrates and not volumetric flowrates should be used in this calculation; unfortunately, this information is rarely available from industrial process instrumentation. Hence, accurate inventory analysis is virtually impossible without the aid of model-based software tools. In light of these facts, the mechanistic model of the continuous pulping area is used to examine the fulfilment of the principles of conservation. In effect, when the calculated mass flowrates are used, the closure of the general mass balance can be verified, on average, by a reasonable 3.5%. However, joint parameter estimation/data reconciliation enables us to approach this problem from a different perspective. Indeed, the mechanistic process model could be used to attain a more accurate compliance of experimental plant data with the
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fundamental laws of conservation provided that the plant/model mismatch was attributable not only to the mechanistic process model but also to experimental plant data. When gross errors are corrected, the overall conservation of mass is verified within a 0.7% error. These results substantiate the idea that gross-error detection and estimation is more critical to advanced industrial process data management systems than the conventional reconciliation of random errors. Since production accounting and inventory analysis are based on the information arising from process instrumentation, systematic errors introduce a bias in these calculations. From a practical viewpoint, it would be reasonable to estimate those biases which have a strong impact on inventory analysis, or whose quantification is vital for other operational purposes (e.g. inferential soft-sensing). In the case of an industrial continuous pulping system, a cost analysis reveals that the most significant sources of revenue and expenses are likely to be the production of pulp, the cost of chip consumption and the cost of evaporation of weak black liquor. The cost of evaporation of weak black liquor can be partially reconciled from the estimate of the bias of the upper-screen extraction flow meter. The estimated 6.4% error in this process measurement is associated to a material stream which accounts for nearly 32% of the overall weak black-liquor extraction flow from the continuous cooking digester at this nominal production level (~3.1m3/min). Additionally, the treatment of the black liquor in the evaporation area comprises approximately 56% of the variable costs of operation of the continuous pulping area (~ 88US$/min). Hence, a 6.4% measurement error on such a critical process stream is equivalent to a production miscalculation of approximately 0.50 million US$ per year, or an inventory error of roughly 32 thousands cubic meters per year.
5. Conclusions The ability to manipulate plant data seamlessly and to define dynamic estimation problems in the industrial workplace is critical to the success of advanced MCTs. In this paper we described a novel software architecture aiming at this goal, and we presented two process-engineering data models enabling this paradigm shift. A prototype estimation/reconciliation environment was built to ease the manipulation of these data models while defining joint parameter estimation/data reconciliation problems of industrial relevance. A large-scale process model of an industrial continuous pulping system was used. The accuracy of the process model was improved and processinstrumentation data was rectified by joint parameter estimation/data reconciliation techniques. Also, the closure of mass balances was improved drastically, and grosserrors estimation was found to be critical for accurate production accounting, inventory analysis and soft-sensing of industrial process systems. This provided an economic incentive for applying the proposed framework for joint parameter estimation/data reconciliation supporting the advanced operation of industrial process systems.
References Özyurt, D.B., Pike, R.W. (2004). Theory and practice of simultaneous data reconciliation and gross error detection for chemical processes. Computers & Chemical Engineering, 28, 381402. Rolandi, P.A. and Romagnoli, J.A. (2006). Integrated model-centric framework for support of manufacturing operations. Part ii: The simulation environment. Computers and Chemical Engineering, submitted for publication. Romagnoli, J.A., Rolandi, P.A (2006). Model-centric technologies for support of manufacturing operations, PSE 2006/ESCAPE 16.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A model discrimination based approach to the determination of operating regimes for chemical reactors Aidong Yang, Elaine Martin, Gary Montague and Julian Morris Centre for Process Analytics and Control Technology School of Chemical Engineering and Advanced Materials, University of Newcastle, Merz Court, Newcastle upon Tyne, NE1 7RU, UK
Abstract The operation of a chemical process that involves the interplay between chemical reaction(s) and transport phenomena can materialise in the occurrence of a number of different operating regimes. When developing a new process, to achieve successful scale-up, the operating regime which yields ideal performance in smaller scale experiments should be identified and retained in the full scale realization. In the past, experimental procedures have been proposed for identifying operating regimes based on the qualitative trends of the response of a process to the change in operating conditions. In this work, a quantitative approach is proposed, in which the problem of determining operating regimes is formulated as one of model discrimination. The proposed approach makes use of hybrid models to handle missing mechanistic knowledge and an optimal experimental design technique was applied to generate the most discriminative data. A simulated case study on the nitration of toluene demonstrates that, compared with existing qualitative methods, this approach has the potential to achieve sharper discrimination, impose fewer requirements on experimental facilities, and complement existing methods. Keywords: chemical reactor, operating regime, model discrimination, experimental design.
1. Introduction The operation of a chemical process that involves the interplay between chemical reaction(s) and transport phenomena often materialises in the occurrence of a number of different operating regimes. When developing a new process, the operating regime which yields ideal performance (in terms of yield, selectivity, etc.) at a smaller scale should be identified and retained in the full scale realization, in order to achieve successful scale-up (Bourne, 2003). Since operating regimes often relate to different rate processes, a particular regime can be characterized in terms of a specific range for a dimensionless number which denotes the ratio of two characteristic times. However, it may not be possible to calculate such a dimensionless number when a new process is being developed since very little is known about the process at this stage, especially with respect to the chemical reaction kinetics. Previously it has been proposed that an experimental procedure specific to the type of process being investigated is adopted, to determine qualitatively the operating regimes. Furthermore, specific types of equipment such as constant interfacial area cells may be required to support the determination process (Atherton, 1993; Bourne, 2003).
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In contrast to the existing qualitative approaches, this paper proposes a model-based quantitative approach for determining the operating regimes, based on the model discrimination technique. Model discrimination is a task whereby a limited number of appropriate models are selected from a larger number of candidate models (e.g. Kitrell, 1970; Stewart, et al, 1998; Jarosch et al, 2002). A typical approach is to fit all the candidate models using available experimental data and then assess the fitness of these candidate models according to a certain statistical criterion (Verheijin, 2003). To obtain the experimental data that realises the most efficient discrimination, optimal experimental design techniques can be applied (Atkinson & Denov, 1992). The primary goal in this research is not to develop or identify the best model for a chemical process but to apply model discrimination as a tool to identify the true operating regime of a chemical process, where each candidate model represents one of the possible regimes. Since the knowledge about a process is incomplete during the early stages of process development, such a model is likely to be more “inexact” than those utilised, say, for the purpose of making predictions. However it is hypothesised that such models will be sufficient to enable differentiation between different operating regimes. In Section 2, the proposed numerical approach to regime determination via model discrimination is presented. In Section 3, a case study on the process of toluene nitration is described and results and discussions are presented. Some concluding remarks are given in Section 4.
2. Determination of operating regime via model discrimination The goal of optimal experimental design for model discrimination is to generate data that maximise the distance between candidate models. Since the distance between two models and that between a model and the data are dependent on the values of model parameters, a sequential procedure should be adopted which alternates between the estimation of parameters and the design/execution of the experiments. As the number of observations increases, the best model (with its associated parameterization) among the candidate models will approach the “true” model (Atkinson & Denov, 1992). This procedure requires to be customized in two aspects for the purpose of determining operating regimes. The first aspect is the preparation of the candidate models. For the determination of the operating regime of a chemical process that involves the interplay of the chemical reaction and the transport phenomena, it is assumed, in this study, that the overall model structure reflecting conservations and the model of transport phenomena are available, but the knowledge about chemical reaction kinetics is incomplete, as may occur in practice. To allow for a candidate model to participate in the discrimination procedure, a black-box or grey-box approximation needs to be adopted to replace the missing mechanistic knowledge. Consequently, each candidate model is deemed to be hybrid, and it is the black-box part where the parameters are to be estimated in the parameter estimation stage. The second aspect is the examination of the stopping criterion. A statistical test, Bartlett’s chi-square test of homogeneity of variances (Verheijin, 2003), is applied to gradually eliminate the unfavourable models and to terminate the discrimination procedure when no more models can be eliminated. The statistic is defined as:
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Determination of Operating Regimes for Chemical Reactors M
T =
∑ (n − m =1
2 / s m2 ) p m ) ln( s tot
⎡ ⎢ M 1 1 ⎢∑ 1+ − 3 ( M − 1) ⎢ m = 1 ( n − p m ) ⎢⎣
⎤ ⎥ 1 ⎥ M ( n − p m ) ⎥⎥ ∑ m =1 ⎦
~ χ
2
M −1
,
2
where M is the number of models, n is the total number of data points, sm is the 2
residual sum of squares computed for model m, stot is the average residual sum of squares of all models, and pm is the number of parameters in model m. The hypothesis of homogeneous variances is rejected when the computed value of T is larger than the 95% quantile of the χ M −1 distribution. Applying the above customization defines the procedure actually adopted in this study, which is summarised in Figure 1. 2
Start Prepare models and identify conditions affecting chemical kinetic and transport phenomena Conduct preliminary experiments, estimate parameters, and rank the residual sum of squares Design experiment to maximize the distance between two best-fitted models
No
First design?
Distance between two best-fitted models significantly enlarged? No
Yes
Yes Conduct newly designed experiment, estimate parameters of each model with all existing observations, and rank their residual sum of squares
Stop with indistinguishable operation regimes
Perform test of homogeneity of variances Yes
Positive? No Remove the worst-fitted model
No Only one model left?
Yes
Stop with the identified true operation regime
Figure 1. Procedure for operating regime determination via model discrimination.
3. Case study: Nitration of Toluene Nitration of toluene is a liquid-liquid reaction process that takes place (as considered here) in a stirred batch reactor, which involves mass transfer between the organic phase and the aqueous phase. Depending on the operating conditions, the reactor may be operated under one of several regimes. In this case study, four regimes are considered, namely a very slow reaction, a slow reaction, a slow-fast transition, and a fast reaction. Detailed characterization and mathematical modelling of these regimes can be found in Bourne (2003), Doraiswamy & Sharma (1984), and Zaldivar, et al (1995&1996). The models are not presented here due to the space limitation but were implemented in gPROMS (PSE, 2004) to undertake dynamic simulations.
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Under isothermal conditions, the apparent reaction rate constant of nitration, k, is a function of the composition of the mixed acid in which the nitration takes place. For the study of operating regime determination, the mechanistic model of k is assumed unknown, thus represents missing knowledge. For formulating the problem as one of model discrimination, a fourth-order polynomial black-box model was adopted for modelling k. This model was then used with the other parts of the model that corresponds to a specific operating regime, thereby forming a complete candidate model that was comparable with those representing other regimes. The overall structure of the simulated study was such that (1) each of the four possible regimes was taken once to be the true one, resulting in four study groups (A-D); (2) for each group, the complete rigorous model corresponding to the true regime was used to generate the simulation data; (3) the person determining the operating regime would be unaware of which regime was the true one and thus perform the determination by discriminating between the four candidate models applying the strategy defined in Section 2 - this strategy was implemented in gPROMS; (4) the distance between two model candidates is computed by the time-integration of the difference between their predictions on the concentration of toluene in the organic phase during a batch; and (5) the operation condition used to increase the distance between candidate models was the concentration of H2SO4 in the mixed acid. At the beginning of the study, each of the four groups was initiated by undertaking three preliminary (simulated) experiments to attain an initial estimate of the parameters of the black-box model. The discrimination procedure was then performed for each group and the results are shown in Table 1. The numbering of the models (M-) and the corresponding residual sums of squares (S-) are as follows: very slow reaction - 1, slow reaction - 2, slow-to-fast transition - 3, and fast reaction - 4. The ranges of the amount of H2SO4 (in kg) applicable to these operating regimes were set as: Regime 1: 0.4-0.6; Regime 2: 0.7 - 0.9; Regime 3: 1.3-1.5; Regime 4: 1.8-2.0. Table 1. Results of operating regime determination via model discrimination True regime
Number of experiments
Group of test A
Ranking of the residual sum of squares
1
3
S1(2.15)S2(2.75)S3(2.7 5) S4(11.5) S1(2.18)S2(33.4) S3(33.6)S4(230) S4(1.54)S1(15.1) S3(29.6) S2(33.2) S3(30.6)S2(34.4) S4(199)S1(9.28e3) S3(31.4)S2(35.4) S3(7.81)S4(39.0) S1(236)S2(6.07e3) S3(8.30)S4(1.45e3) S2(9.31e3) S1(4.30e4) S3(5.03)S4(6.66) S1(49.4)S2(7.84e4) S3(6.90)S4(6.88) S1(1.03e4) S2(1.68e5)
3+1 B
2
3 3+1
C
3
3+2 3 3+1
D
4
3 3+1
Results of Optimal Exp. Design Optimal amount of H2SO4 (kg) 0.6
Maximum Distance between two best fitted models 27.3
Remaining model(s) through Bartlett’s Test
M1 0.9
3.71e3
0.9
1.585
M2, M3
0.9 1.3
1.241 6.00e2
M2, M3 M3
1.8
6.84e-2
1.8
2.26e-2
M3, M4
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The results of the case study demonstrate the advantages of the proposed method. If the slow-to-fast transition regime (No. 3) were not considered, the other three operating regimes would have been able to be successfully distinguished from one another and correctly identified as the true regime. It should be noted that no special requirement on the experimental equipment is imposed to obtain this result. In comparison, for the existing qualitative method for regime determination, the differentiation between the slow-reaction and the fast-reaction regimes requires the use of a constant interfacial area cell to maintain the interfacial area between the two phases (Atherton, 1993; Bourne, 2003). Additionally, the quantitative method has been able to correctly identify the transition regime as the true one. The handling of the transition regime by the existing qualitative method has not been reported, but it would be difficult for that method to distinguish between the slow reaction and the transition regime, whichever is the true regime, because their qualitative responses to a change in agitation and other conditions are similar. There was one case (cf. Group D in Table 1) in which the proposed method failed but the existing qualitative method that makes use of a constant interfacial area cell may succeed. If such a reactor is used, it may be possible to exclusively identify the fastreaction regime as the true regime by distinguishing it from the transition regime. This is because the transition regime is sensitive to a change in the phase volume ratio while the fast-reaction regime is not. The use of a constant interfacial area cell allows for separating the influence of the phase volume ratio from that of the interfacial area, and therefore makes the distinction between the two possible. This analysis suggests that, in this particular case, a constant interfacial area cell should be applied to cope with the difficulty faced by the proposed approach.
4. Concluding remarks A quantitative approach to the determination of operating regimes based on model discrimination has been proposed, which is supported by techniques including hybrid modelling, optimal experimental design, and a statistical test to assess the homogeneity of the variances. A case study on a toluene nitration process demonstrated that, by leveraging mathematical models of possible operating regimes where black-box regressions are used to represent missing knowledge on chemical kinetics, the proposed approach has the potential to achieve sharper discrimination and require less experimental facilities in most cases. This is in contrast with existing qualitative methods where special experimental facilities are necessary. There are two concerns pertaining to the effectiveness of the proposed approach. One is that the measurements of chemical process variables are often corrupted with noises. The result presented in the case study has not considered measurement errors which may make a difference on the estimate of the parameters of the black-box model, hence on the distances between different model candidates. However, a preliminary numerical study indicated that a major influence is most likely to occur only in the cases where the distance between the compared model candidates is insignificant, which are indeed also the cases where the proposed approach tends not to work well. Another concern is that, whilst the reported case study has assumed an unknown expression for the reaction rate constant, in some other cases the entire chemical kinetics may be unclear (e.g. information such as the order of the reaction is not available at all). Under such a situation, this approach will still be applicable, simply by employing a black-box model to represent the entire chemical kinetics (i.e. one that
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correlate the intrinsic reaction rate with the influential variables). A case study addressing this situation will be undertaken as future work. This case study has shown particularly the difficulty of distinguishing the transition regime from its neighbours. It is suspected that the inter-model distance of these regimes might have been reduced due to the use of the black-box regression model embedded in every model being compared. This may contribute to the difficulties in the effective discrimination of these regimes. Research that addresses this issue is ongoing.
Acknowledgements The authors acknowledge the financial support of the EPSRC award GR/R64407/01 “Vertical Integration of Product Development and Manufacturing”.
References Atherton, J. H., 1993. Methods for study of reaction mechanisms in Liquid-liquid and liquid-solid reaction systems and their relevance to the development of fine chemical processes. Trans. Inst. Chem. Eng. A, 71, 111. Atherton, J. H., 1999. Chemical Aspects of Scale-up. In: W. Hoyle (Ed), Pilot Plants and Scaleup of Chemical Process II. The Royal Society of Chemistry, Cambridge, UK. Atkinson, A.C., Donev, A.N., 1992. Optimum Experimental Designs, Oxford Univ. Press, NY. Bourne J.R., 2003. Mixing and selectivity of chemical reactions. Organic Process Research and Development, 7 (4), 471-508. Doraiswamy, L.K., Sharma, M.M., 1984. Heterogeneous Reactions: Analysis, Examples, and reactor Design. Volume 2: Fluid-Fluid-Solid Reactions. John Wiley & Sons, New York. Jarosch, K., Solh, T., de Lasa, H. I., 2002. Modelling the catalytic steam reforming of methane: discrimination between kinetic expressions using sequentially designed experiments. Chemical Engineering Science, 57 (16), 3439-3451. Kittrell, J. R., 1970. Mathematical modelling of chemical reactions. Adv. Chem. Eng., 8, 97-183. PSE, 2004. gPROMS Advanced User Guide, Process Systems Enterprise Ltd., 23.02.2004. Stewart, W. E., Shon, Y., Box, G. E. P., 1998. Discrimination and Goodness of Fit of Multiresponse Mechanistic Models. AIChE Journal, 44 (6), 1404-1412. Verheijen, P. J. T., 2003. Model Selection: an overview of practices in chemical engineering. Computer-Aided Chemical Engineering, 16, 85–104. Westerterp K.R., van Swaaij, W.P.M., Beenackpers, A.A.C.M., 1990. Chemical reactor Design and operation. John Wiley & Sons, New York. Zaldivar, J.M., Molga, E., Alós, M.A., Hernández, H., Westerterp, K.R., 1995. Aromatic nitrations by mixed acid: slow liquid–liquid reaction regime. Chem. Eng. Process. 34, 543– 559. Zaldivar, J.M., Molga, E., Alós, M.A.,Hernández, H., Westerterp, K.R., 1996. Aromatic nitrations by mixed acid: fast liquid–liquid reactions. Chem. Eng. Process. 35, 91–105.
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A performance comparison of some high breakdown robust estimators for nonlinear parameter estimation Eduardo L.T. Concei¸c˜aoa and Ant´onio A.T.G. Portugala a CEM group, Department of Chemical Engineering, University of Coimbra, P´olo II, Pinhal de Marrocos, 3030–290 COIMBRA, Portugal
While the inevitable occurrence of departures from the assumptions made beforehand can damage least squares reliability, robust estimators will resist them. A number of alternative robust regression estimators have been suggested in the literature over the last three decades, but little is known about their small-sample performance in the context of nonlinear regression models. A simulation study comparing four such estimators together with the usual least squares estimator is presented. It is found that the MM- and τ -estimators are quite efficient when the proportion of outliers in data is not too large. Keywords: high breakdown point, robust regression, nonlinear regression, Monte Carlo 1. INTRODUCTION In the nonlinear regression model, one observes the response variable y obeying the model yi = f (xi , θ) + ei ,
i = 1, . . . , n,
(1)
where x is a vector of explanatory variables, θ is a vector of unknown true parameters to be estimated, and e is the measurement error. Define the residuals corresponding to θ as ri (θ) = yi − f (xi , θ). It is common to consider the errors ei as independent and identically distributed random variables with zero mean and variance σe2 , which follow a specified type of distributions. The goal of each possible estimator is to draw reliable estimates of the parameters from data and additionally protect against departures from statistical model assumptions made beforehand because in practice it is very unlikely that the model assumptions hold perfectly. They may include the presence of outlying observations and other departures from the imposed model distribution. Of course, it is recognized that neither classical least squares (LS) nor, more generally, maximum likelihood methodology is satisfactory as far as the robustness requirement is concerned, since it depends heavily on the assertion that the actual error process follows exactly the distribution assumed. For this reason, a vast amount of literature in robust alternative techniques was developed over the last 30 years. A measure of robustness frequently used in the literature is the breakdown point (BP) which is, roughly speaking, the smallest proportion of contaminated data which leads to
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unreliable model parameters. Thus, a regression estimator with high breakdown point (HBP) is capable of handling multiple outliers, even if they are grouped. Besides, it is also important to know the uncertainty (bias and sampling variability) of point estimates on “clean data”. This is assessed by the statistical efficiency criterion calculated as the ratio of the mean squared error (MSE) in a least squares estimate to the actual MSE of a (robust) estimate, computed at the Gaussian (normal) distribution. Unfortunately, HBP estimators tend to have low efficiency. Rousseeuw [1] proposed the first high breakdown regression estimator, the least median of squares (LMS), but its very low asymptotic efficiency is a well known drawback. The same author [1] suggested the least trimmed squares (LTS) estimator which is more efficient than the LMS estimator. Since then, several methods have been proposed which combine good asymptotic efficiency with HBP. Among them are the three-stage MM-estimator starting with initial HBP regression estimates of Yohai [2] and the τ -estimator of Yohai and Zamar [3]. Little is known about the small-sample properties of these estimators in the context of nonlinear regression models. Thus, the main purpose of this article is to investigate their small-sample performance by means of a Monte Carlo simulation study based on real data sets. The simulation design considers the effects of proportion of outliers in data and different error distributions. Another goal is to compare the use of the LMS and LTS estimators as the initial HBP estimator in the MM-estimator. The remainder of the paper is organized as follows. Section 2 defines the LMS, LTS, MM-, and τ -estimates. In Section 3 we summarize the basic aspects of the simulation study. The different estimators are then compared in Section 4. 2. DEFINITIONS OF ROBUST ESTIMATORS Least median of squares (LMS) Rousseeuw [1] proposed the first regression estimate with the highest possible BP of 1/2, by minimizing the median of squared errors, that is θˆLMS = arg min med ri2 (θ), θ
(2)
i
where θˆ is an estimate of θ and med denotes the median. Least trimmed squares (LTS) The LTS estimate is defined as [1] θˆLTS = arg min
h X
θ
2 r(i) (θ),
n/2 6 h 6 n,
(3)
i=1 2 (θ) is the ith squared residual sorted from smallest to largest and h is the number where r(i) of these terms which are included in the summation called the coverage of the estimator. Therefore, the n − h “trimmed” observations that correspond to the largest residuals do not directly affect the estimator. Let α = 1 − h/n be the amount of trimming called the trimming proportion, with 0 6 α 6 1/2. The maximal BP for LTS equals 1/2 and is obtained by choosing α close to 1/2. However, one may expect a tradeoff between a high value for α and a loss in efficiency. Thus, the choice of α (or equivalently h) determines the overall performance of
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the LTS estimator, and some effort is required to tune this parameter. Hence, it has been suggested that lower values for α (the most commonly suggested values are 0.25 and 0.1) will give a good compromise between robustness and efficiency. Note that the objective function of the LTS estimator is nonconvex and not differentiable the same happening for the LMS estimator. Consequently, these optimization problems cannot be solved by standard derivative-based methods. The MM-estimator This method proposed by Yohai [2] involves the following steps as suggested by Stromberg [4]: 1. Compute an initial HBP estimate θˆHBP (we use LMS as well as LTS) and obtain the corresponding residuals ri (θˆHBP ). 2. Next, calculate the robust residual scale estimate sn given by the solution of the following equation à ! n 1X ri (θˆHBP ) ρ0 (4) = b with ρ0 (u) = ρ(u/k0 ) and b/ρ0 (∞) = 0.5, n i=1 sn where k0 = 0.212 and ρ is the Hampel loss function defined as follows 2 u for |u| < a 2¡ ¢ a for a 6 |u| < b a |u| − 2 " ¶2 # µ ρ(u) = c − |u| 2 ab − a2 + (c − b) a2 1 − for b 6 |u| 6 c c−b 2 for |u| > c, ab − a2 + (c − b) a2
(5)
where a = 1.5, b = 3.5, and c = 8. (Scale estimators measure dispersion and are used to standardize residuals.) 3. Obtain the LS estimate θˆLS . Then find the M-estimates [5] θˆ0 and θˆ1 to minimize µ ¶ n X ri (θ) Q(θ) = with ρ1 (u) = ρ(u/k1 ), ρ1 (6) sn i=1 which satisfies Q(θˆ0 ) 6 Q(θˆHBP ) and Q(θˆ1 ) 6 Q(θˆLS ), respectively, where k1 is chosen as 0.9014 to achieve 95% asymptotic efficiency at the Gaussian distribution. This means that a local minimum can be used. The final MM-estimate is then θˆMM = min(θˆ0 , θˆ1 ). The basic idea is that this estimate inherits the HBP of the initial estimate and simultaneously improves the efficiency with the M-estimator at step 3. τ -estimator Yohai and Zamar [3] proposed another HBP estimator with high efficiency. The τ -estimates are defined by minimizing a robust scale of the residuals given by v u n ¶ µ u1 X ri (θ) τ (θ, sn ) = sn t (7a) ρ1 n i=1 sn subject to the constraint µ ¶ n ri (θ) 1X = b, ρ0 n i=1 sn
(7b)
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where b/ρ0 (∞) = 0.5 and sn is an M-estimator of scale implicitly defined by equation (7b). The choice of ρ0 regulates the robustness, whereas the choice for ρ1 can be tuned to give good asymptotic efficiency under the Gaussian model. Yohai and Zamar [3] and Tabatabai and Argyros [6] used the ρ function ´ ( 2³ u u2 u4 1 − for |u| 6 c + 2 4 c 3c ρc (u) = 22 (8) c for |u| > c. 6 They recommended ρ0 = ρc0 with c0 = 1.56 and ρ1 = ρc1 with c1 = 1.608. In this case, the τ -estimator’s BP is 0.5 and its asymptotic efficiency at the Gaussian distribution is 95%. Note that the above ρ function does not have a continuous second derivative, which might result in outcomes far from optimality using standard optimization algorithms. 3. SIMULATION STUDY Description of the test model: oxidation of propylene We consider for the oxidation of propylene the model that involves rate constants with Arrhenius temperature dependence analyzed in Watts [7] −rC3 H6 =
ka kr c0.5 O 2 c C 3 H6 , 0.5 ka cO2 + nkr cC3 H6
(9)
where rC3 H6 denotes the rate of propylene disappearance, ka and kr denote the rate constants of adsorption of oxygen and oxidation of propylene, respectively, c denotes concentration, and n = (moles oxygen required)/(mole propylene reacted) is the stoichiometric number. To reduce correlation between kinetic parameters in the Arrhenius expression for a rate reaction we used the reparametrization reported in Lohmann et al. [8] resulting in θ = (ln ka (350 ◦ C), ln ka (390 ◦ C), ln kr (350 ◦ C), ln kr (390 ◦ C)) as the vector of parameters to be estimated. Experimental Simulation is conducted to compare the small-sample behavior of the estimates described in the former section. More precisely, we compare LS, LMS, LTS(α) for α = 0.1, 0.25, and 0.5, τ -, and MM-estimators starting with three different HBP initial estimates—LMS, LTS(0.25), and LTS(0.5). Each sample contains 66 observations (xi , yi ) in which xi is taken from the experimental data. We have taken the set of LS estimates of the experimental data as the true parameters to generate predictions of the measured quantities yi according to model (1). The error terms ei are generated from five different distributions: Gaussian N(0, σe2 ), Cauchy, Skew-Normal [9], and two “scale” contaminated Gaussians 0.9N(0, σe2 ) + 0.1N(0, (2σe )2 ) and 0.7N(0, σe2 ) + 0.3N(0, (5σe )2 )— denoted as CN(0.1, 2) and CN(0.3, 5), respectively. Two proportions of outliers in data are considered in each simulated data set, namely 10% (small contamination) and 30% (high contamination). Certain observations, chosen at random, are modified to be “bad” data points by shifting upwards the response variable by 5σe for 10% bad data and 10σe for 30% qP n bad data. Here, σe may be estimated from the data as σ ˆe = ri (θˆLS )/(n − np ) with i=1
np being the number of parameters in model (1). The number of Monte Carlo replications
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Gaussian
30% outliers CN(0.1, 2)
30% outliers CN(0.3, 5)
30% outliers 30% outliers Cauchy Skew−Normal
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MM-LMS MM-LTS(0.50) MM-LTS(0.25) LS
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MM-LMS MM-LTS(0.50) MM-LTS(0.25) LS
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0% outliers CN(0.1, 2)
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MM-LMS MM-LTS(0.50) MM-LTS(0.25) LS
40 60 80
120
40
60
80 100 30 40 50 60 70
50 100 150 200
40 60 80 100
100 med(|θˆLS − θ|)/ med(|θˆ − θ|)
Figure 1. Efficiencies of the competing estimates for ln ka (390 ◦ C). The efficiency criterion has been normalized by its value for the LS estimate obtained only under Gaussian error (contamination 0%). Each circle shows the value of efficiency, whereas the darker line segment shows the bootstrap [15] percentile 95% confidence interval obtained with 999 bootstrap replications.
is 100. We used the criterion med(|θˆ − θ|) [10], a robust analog of the MSE, to evaluate the performance of an estimator. Computing the estimates The major computational difficulty with the estimates considered in this paper is that they cannot be calculated by standard optimization algorithms. We therefore adopted the improved version by Lee et al. [11] of the differential evolution (DE) algorithm proposed by Storn and Price [12] for all the regression estimators. This method is a stochastic global search heuristic that applies to bound constrained problems. Note that if the univariate scale estimator is computed from (7b), then by plugging sn into (7a) a τ -estimate can be obtained by solving an unconstrained minimization
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problem. A convenient procedure to obtain the solution for both the scale estimators (4) and (7b) is the algorithm of Brent [13] that does not require derivatives. The final stage of the MM-estimator uses the L-BFGS-B algorithm [14]. 4. MONTE CARLO RESULTS Fig. 1 displays results concerning the performance of the robust estimators. For shortness, we only report the simulation results for the ln ka (390 ◦ C) parameter. As expected, for departures from Gaussian distributed data (especially for CN(0.3, 5) and Cauchy), we clearly see the advantage of the robust methods over the classical. This becomes even more visible for the outlier contamination scenarios. Generally, no significant difference could be found between the MM-estimates computed with the LMS estimator and when using the LTS estimator, except when the outlier proportion is 30% for which LTS(0.25) is worst. Furthermore, we also note that for (uncontaminated) Gaussian distributed errors the loss in efficiency of both τ - and MM-estimates with respect to least squares is rather small or barely distinguishable. For null or small contamination levels, we can observe that the τ - and MM-estimates show an overall best behavior, albeit quite close to the LTS(0.1) estimator. On the other hand we can see that, in general terms, the LMS and LTS(0.5) estimates are the worst, followed by LTS(0.25). For high contamination, essentially these estimators behave the opposite way compared to small fractions of contamination. Note that among HBP estimates, LTS(0.1) and MM- clearly lose. These simulation results support the use of the MM- or τ -estimator as a valuable alternative to the existing classical methods in the practical applications for which the proportion of outliers in data is not too large. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
P.J. Rousseeuw, J. Am. Stat. Assoc. 79 (1984) 871. V.J. Yohai, Ann. Stat. 15 (1987) 642. V.J. Yohai and R.H. Zamar, J. Am. Stat. Assoc. 83 (1988) 406. A.J. Stromberg, J. Am. Stat. Assoc. 88 (1993) 237. P.J. Huber, Robust Statistics, Wiley, New York, 1981. M.A. Tabatabai and I.K. Argyros, Appl. Math. Comput. 58 (1993) 85. D.G. Watts, Can. J. Chem. Eng. 72 (1994) 701. T. Lohmann, H.G. Bock, and J.P Schl˘oder, Ind. Eng. Chem. Res. 31 (1992) 54. A. Azzalini, Scand. J. Stat. 12 (1985) 171. J. You, Comput. Stat. Data Anal. 30 (1999) 205. M.H. Lee, C. Han, and K.S. Chang, Ind. Eng. Chem. Res. 38 (1999) 4825. R. Storn and K. Price, J. Glob. Optim. 11 (1997) 341. R.P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, NJ, 1973; reissued by Dover Publications, Mineaola, NY, 2002. 14. C. Zhu, R.H. Byrd, P. Lu, and J. Nocedal, ACM Trans. Math. Softw. 23 (1997) 550. 15. R. Wehrens, H. Putter, and L.M.C. Buydens, Chemom. Intell. Lab. Syst. 54 (2000) 35.
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Equivalent dynamic solution of an industrial HDPE slurry reactor Sushant Nigam,a Kannan M. Moudgalya,a Amiya K. Panib a DepartmentofChemicalEngineering,IITBombay,Powai,Mumbai400076,India b DepartmentofMathematics,IITBombay,Powai,Mumbai400076,India Abstract A discontinuous industrial High Density Polyethylene (HDPE) reactor system, described by 15 Differential Algebraic Equations (DAEs) and 12 DAEs, has been numerically solved in this work. This system is characterised by frequent switches between these two sets of DAEs, making the numerical solution expensive. Using an extension of Filippov’s regularisation approach, we obtain an equivalent, continuous, dynamic model of the discontinuous system. The equivalent dynamic solution is several orders of magnitude faster than the discontinuous solution.
1. Introduction Discontinuous systems are especially common in man made technologies. An example of this is the sliding mode controller. This approach is present in process industry as well. Moudgalya and Jaguste [1] have reported a realistic discontinuous slurry process to produce HDPE. Moudgalya et al. [2] have reported that such discontinuities can be used to understand fundamental mechanisms, to improve mixing, to enhance heat transfer coefficients and to produce stable two phase flows. Numerical solution of these systems, however, is expensive owing to frequent switches. This is especially true when DAEs are present, thanks to frequent initialisations. Moudgalya and Ryali [3] show that these systems exhibit discontinuity sticking, with small step sizes being the solution. Finally, in optimisation processes, designed to tune model parameters [1], a complete simulation has to be carried out for every objective function evaluation. All of the above observations point to the necessity of an efficient integration procedure for such discontinuous processes. In this report, we present the equivalent dynamic solution to the HDPE reactor system studied by [1] and compare its performance with the discontinuous solution.
2. Discontinuous HDPE Model In an industrial slurry process, studied in detail by [1], ethylene is polymerised into HDPE in a well mixed reactor in the presence of hydrogen and a Ziegler Natta catalyst dispersed in a diluent, a schematic of which is shown in Fig. 1. The slurry level in this figure is given by ML4/ρ + ML5, where, the variables are as defined in [1]. First define the switching function φ as follows: ϕ=
ML4 + ML5 − Vd ρ
We obtain the following slurry model when φ > 0 (Fig. 1(a)):
(1)
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(a) Slurry model
(b) Gas model
Figure 1. A schematic of HDPE reactor, described by slurry and gas models
dM1 = F −L −r 1 1 1 dt dM 2 = F −L −r 2 2 2 dt dML3 =F −L 3 3 dt dML4 28r2 = F −L + 4 4 dt 1000 dML5 =F −L 5 5 dt ML1 L1 = ML4 L4 + ML5 + L5 ρ ρ
ML2 L2 = ML4 L4 + ML5 + L5 ρ ρ ML3 L3 = ML4 L4 + ML5 + L5 ρ ρ ML4 L4 = ML4 L4 + ML5 + L5 ρ ρ L4 P − Pout + L = 0.2268C 5 v ρ gs
(2) (3) (4) (5) (6) (7)
(8)
(9)
(10)
(11)
When φ< 0 (Fig. 1(b)), the system is modelled by dM1 = F −G −r 1 1 1 dt dM 2 = F −G −r 2 2 2 dt
(12) (13)
Equivalent Dynamic Solution of an Industrial HDPE Slurry Reactor dML3 =F 3 dt dML4 28r2 =F + 4 dt 1000 dML5 =F 5 dt MG1 G = 1 MG2 G2
(G + G ) 1
2
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(14) (15) (16) (17)
RT P − Pout = 0.2268C v P gg
(18)
The following equations are applicable, whatever the value of φ is: ML + MG = M 1
1
(19)
1
ML2 + MG2 = M 2
(20)
α1MG1
(21)
ML1 = RT ML4 ( MG1 + MG2 ) + ML5 P ρ ηα 2 MG2
RT ( MG1 + MG2 ) P
( MG + MG ) 1
2
=
ML2 ML4 + ML5 ρ
RT ML4 + + ML = V 5 P ρ
(22)
(23)
In the above model, the subscripts 1 to 5, respectively, denote hydrogen, ethylene, catalyst, polymer and diluent. F, L and G, respectively, stand for inflow rate of feed and outflow rates of slurry and gases. M denotes the total holdup in the reactor, while, ML and MG, respectively, denote the holdup in the slurry and the gas phases. All the symbols used above have the same meaning as in [1], except for gs and gg, which now denote the densities of the slurry and the gas streams flowing out of the reactor through the control valve. A proportional integral controller is used to control this system. For further details, the reader is referred to [1]. This system is solved with the help of DASSL [4] using the procedure of [1]. Pressure in the reactor, ethylene concentration in the gas phase and the slurry level, as functions of time, are shown in Fig. 2. After initial swings, the slurry level in the reactor settles down at the level of the dip tube. As gas and slurry leave the reactor alternatively, the applicable model also keeps changing [1].
Figure 2. Pressure, ethylene and slurry level in HDPE reactor
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A mathematical idealisation of this is referred to as sliding. The procedure to handle ODE systems in sliding mode is well known [5].
3. Equivalent Dynamics In this section, we first state the results available for a class of discontinuous, index-1 DAE systems, while in sliding mode [6,7], and then apply them to the HDPE reactor under discussion. Consider a discontinuous system described by the following equations: If ϕ( y )>0 , dy + = f ( y, z , z ) 1 2 dt
(24)
0 = g ( y, z ) 1 + 0 = h ( y, z , z ) 1
(25) (26)
2
Else if ϕ( y )<0 dy − = f ( y, z , z ) 1 3 dt
(27)
0 = g ( y, z )
(28)
1
− 0 = h ( y, z , z ) 1
Here,
(29)
3
j k z (t ) ∈ R , z ( t ) ∈ R 1
2
and
z (t ) ∈ R 3
l
with
j + k +l = m .
We also have,
f + :R n + j + k → R n , f − :R n + j +l → R n , g:R n + j → R j , h + :R n + j + k → R k and h + :R n + j + k → R k . Note that in the region where ϕ( y ) >0 , the system consists of nψ differential equations and j+k algebraic equations while in the region ϕ( y )<0 we have a
system with n differential equations and j+l algebraic equations. In the HDPE system, Eq. (2) to (6) are denoted by Eq. (24); Eq. (7) to (11) are denoted by Eq. (26); Eq. (12) to (16) are denoted by Eq. (27); Eq. (17) and (18) are denoted by Eq. (29). Finally, the vector g that appears in Eq. (25) and (28) stands for Eq. (19) to (23). If the system reaches sliding mode, the equivalent dynamics can be described by [6,7], dy + − = αf ( y , z , z ) + (1 − α) f ( y , z , z ) 1 2 1 3 dt
0 = g ( y, z ) 1 + 0 = h ( y, z , z ) 1
2
− 0 = h ( y, z , z ) 1
3
Where, α is calculated as follows:
(30) (31) (32) (33)
Equivalent Dynamic Solution of an Industrial HDPE Slurry Reactor α=
( Δϕ, f − ) ( Δϕ, f − ) −( Δϕ, f + )
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(34)
We now apply this approach to the HDPE reactor system [8]: 1 Δϕ=(0,0,0, ,1) ρ
28r2 1 + ( Δϕ, f ) = ( F4 − L4 + ) + F5 − L5 1000 ρ 28r2 1 − ( Δϕ, f ) = ( F4 + ) + F5 ρ 1000
(35) (36) (37)
Substituting these in Eq. (34), we obtain F + 28r2 /1000+ρF5 α= 4 L4 +ρL5
(38)
In order to complete the description of the sliding model, we need to calculate only Eq. (30). We obtain the following: dM1 = F1 − G1 − r1 + α (G1 − L1 ) dt dM 2 = F − G − r + α (G − L ) 2 2 2 2 2 dt dML3 = F − αL 3 3 dt dML4 28r2 =F + − αL 4 1000 4 dt dML5 = F − αL 5 5 dt
(39) (40) (41) (42) (43)
The equivalent dynamic system is solved by DASSL, using the same integration parameters as in the discontinuous system. Unlike before, however, there are no discontinuities now. We get results identical to the ones in Fig. 2. The comparison of two approaches, namely, integration of discontinuous and the equivalent dynamic models was carried out in a Mac PowerBook G4, with a 667 MHz processor, 512 MB RAM, operating with Mac OS X, version 10.2.8. The time taken by the equivalent dynamic approach for the dassl parameter of dtout is taken as the metric for comparison. The number of times the discontinuous method required for different dtout values is shown in Fig. 3. Note that we are justified in choosing a fixed value of dtout for the equivalent dynamic system, as there are no discontinuities in it, whereas, one has to choose small values for the discontinuous system, owing to discontinuity sticking. The equivalent dynamic model is several orders of magnitude more efficient than the discontinuous model. As a matter of fact, one can even argue that it is not fair to compare the two models.
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Figure 3. Ratio of time taken by discontinuous model for different dtout values to time taken by equivalent dynamic model with dtout=0.01hr
4. Conclusion Using the extension of Filippov’s regularisation procedure, an equivalent dynamic model of a discontinuous HDPE reactor system, modelled by 15 DAEs and 12 DAEs is obtained. Numerical solution of the equivalent dynamic system is identical to the one obtained through the integration of the discontinuous system. As the proposed method converts a discontinuous system into a continuous one on the sliding surface, it is several orders of magnitude more efficient than the one involving discontinuous models. This is of immense value, especially, when DAEs are to be integrated as a part of an optimisation process.
References [1] K. M. Moudgalya and S. Jaguste, Chemical Engineering Science, 56 (2001) 3611–3621. [2] K. M. Moudgalya, S. K. Singh, K. P. Madhavan and G. Jain, Chemical Engineering Science, 58 (2003) 3973–3983. [3] K. M. Moudgalya and V. Ryali, Chemical Engineering Science, 56 (2001) 3595–3609. [4] K. E. Brenan, S. C. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, SIAM, Philadelphia, 1996. [5] A. F. Filippov, Differential Equations with Discontinuous Right-hand Sides, Kluwer Academic Publishers, Dordrecht, 1988. [6] J. Agrawal, K. M. Moudgalya and A. K. Pani, In SAFEPROCESS 2003, 5th IFAC symposium on fault detection, supervision and safety of technical processes, Washington, D. C., 2003. pp. 795-800. [7] J. Agrawal, K. M. Moudgalya and A. K. Pani, Chemical Engineering Science, in print, 2006. [8] S. Nigam, Equivalent dynamics and efficient integration algorithms for a class of discontinuous, differential algebraic equations in sliding mode, B.Tech Project Report, I.I.T. Bombay, April 2005.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Dynamical and stationary analysis of an electrolyte diode and comparison with experiments Zdeněk Slouka,a Michal Přibyl,a Jiří Lindner,a Dalimil Šnita,a Miloš Mareka a
Institute of ChemicalTechnology, Department of Chemical Engineering, Technická 5, 166 28 Praha 6, Czech Republic
Abstract Stationary and non-stationary behavior of an electrolyte diode system is studied. The system consists of a microcapillary connecting two reservoirs with a strong acid and strong base. If an external electric field is imposed on the capillary, diode like currentvoltage characteristics is observed. A thin layer of electrolyte with extremely large pH gradient is formed in the closed regime of the diode. Transient behavior of the electrolyte diode is studied experimentally and numerically. Observed short-time overshoots of electric current in the current-voltage characteristics are explained based on numerical analysis of the reaction-transport processes in the capillary. Keywords: electrolyte diode, electrophoresis, microchip, transients, mathematical modeling
1. Introduction After connecting DC voltage to a microfluidic device consisting of two reservoirs (one contains a strong acid, HCl, and the other one strong base, KOH) separated by a permeation layer (Fig. 1), complex nonlinear behavior can be observed. Such microdevice is called electrolyte diode due to the similarity of the current-voltage characteristics with a semiconductor diode. When DC voltage is applied, two qualitatively different types of behavior of the electrolyte diode can be observed. If the positively charged electrode (anode) is placed in the KOH reservoir and the cathode is placed in the HCl reservoir, the system is in the open mode. The capillary then contains an electrolyte with high concentrations of potassium and chloride ions (KCl) and is characterized by a high level of electrolyte conductivity. If the electric field polarity is reversed, the electrolyte diode is in the closed mode. In the zone of the microcapillary where hydrogen and hydroxyl ions accumulate, water is formed. The narrow zone of water formation is characterized by a low value of electric conductivity. Hence the observed electric current response to the applied difference of electric potential is much lower than in the open mode. In this work, a positive difference of electric potential is related to the open mode and a negative difference to the closed mode. The difference is defined as a difference of electric potential between the alkaline and the acid reservoirs. Studies on electrolyte diodes were reported, for example, by Hegedüs et al. (1995, 1996, 1998), Merkin et al. (2000), Lindner et al. (2002), and Šnita et al. (2001). The authors dealt with experimental realizations and/or numerical stationary analyses of such systems. Here, we compare experimental and numerical results of a dynamical analysis of the electrolyte diode carried out in a capillary microchip. The interactions of ions with an externally applied electric field are important in many microfluidic applications: capillary electroseparation, electrokinetic dosing,
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sample addressing etc. The results obtained for the studied system can be relevant for more complex systems like microfluidic ionic gates and transistors. Fig. 1 Scheme of the electrolyte diode system.
2. Mathematical model and numerical analysis The mathematical model of the electrolyte diode consists of four mass balances of ionic components ∂ci ∂c z c D F ∂Φ ⎞ ∂ ⎛ = − ⎜ − Di i − i i i ⎟ +ν i r , (1) ∂t ∂x ⎝ ∂x RT ∂x ⎠ where, the index i can represent potassium, chloride, hydrogen, and hydroxyl ions. Poisson equation of electrostatics is used for the evaluation of the electric potential distribution ∂ 2Φ ε 2 = − F ∑ z i ci . (2) ∂x i The symbols ci, Di, zi, νi, F, R, T, Φ, ε denote ionic concentration, ionic diffusivity, ionic charge number, stoichiometric coefficient of the ion i in the chemical reaction, Faraday’s constant, molar gas constant, temperature, electric potential and permittivity of the environment, respectively. The reaction rate of water formation r is defined as r = −k (K w − cH + cOH − ) , (3) where symbols k and Kw represent the kinetic and the equilibrium constants of water formation, respectively. Eqs. (1) and (2) together with Dirichlet boundary conditions form the studied mathematical model. Cf. also the Model A section in Lindner et al. (2002) for detail explanation and for the discussion of the used parameter values. Large moving gradients of electric potential and ionic concentrations bring problems in numerical analysis of the electrolyte diode. Hence both the stationary and the dynamical solvers use a mesh adaptation technique (Přibyl et. al., 2005). The sudden changes of the difference of electric potential, when the non-stationary behavior is studied, are modeled as a fast linear decrease or increase of electric potential on one boundary. The duration of the linear change was set to 0.1 second. Fig. 2 Dependence of the position of the zone with low electric conductivity on the imposed difference of electric potential. The zone is depicted by a light (yellow) stripe.
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Fig. 3 Opening the electrolyte diode. Time courses of electric current density obtained from numerical analysis (A) and experiments (C). Corresponding currentvoltage characteristics (B,D).
3. Experimental arrangement The electrolyte diode is constructed from two polystyrene plates. The plates are manufactured by a micro-milling machine. Two channels forming the flow-through reservoirs of the acid and the base are drilled in the upper plate. The connecting microcapillary is milled into the bottom plate. A dried and shrunk piece of the polyvinylalcohol gel (PVA, Fluka) is inserted into the microcapillary. Then the plates are pressed together and connected by thermosetting in a furnace (15 minutes, 90 °C). When the polystyrene device is cooled, a water electrolyte is supplied into the gel that swells and tightly fills the microcapillary. Finally, the measuring and the source electrodes are inserted into the microchip. To obtain the current-voltage characteristics of the electrolyte diode, the imposed potential difference is increased or decreased in specified time intervals and the current response is recorded.
4. Results and discussion 4.1. Steady state behavior If the electrolyte diode is in the closed mode, the narrow zone of low electrolyte conductivity is characterized by a large gradient of pH. This gradient can be detected by a pH sensitive dye, for example, thymol blue (see Fig. 2). Differences of electric potential were applied on the capillary boundaries and the positions of the large pH gradients were observed. If the applied voltage is negative, the gradient shifts closer to the alkaline reservoir due to difference between the diffusivities of hydrogen and hydroxyl ions (in water DH = 9.31×10-9 m2s-1, DOH = 5.28×10-9 m2s-1). However, the used color indicator brings ionic impurity into the electrolyte diode system. Hence the experimentally observed results are affected by the electromigration transport of thymol blue ions. Numerical results of the stationary analysis of the electrolyte diode are described in Lindner et al. (2002).
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4.2. Numerical and experimental analysis of electrolyte diode transient behavior The dynamical analysis starts in the steady state when no difference of electric potential is applied. After every 300 seconds, the applied difference is switched to another value. The transient response of the system is recorded. First, the transition of the electrolyte diode to the closed mode and its consecutive opening was studied. The following sequence of electric potential differences was used: 0, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10 Volts. The observed transient behavior of the electrolyte diode expressed as the computed/measured electric current density is plotted in Fig. 3. Figs. 3A, B were obtained by numerical analysis of the model equations and the results in Figs. 3C, D were recorded during the experiment. The time courses of the observed electric current are shown in Figs. 3 A, C. The transition of the electrolyte diode into the closed mode at time 300 s is accompanied by a short-time overshoot of electric current density. Then the dependencies are characterized by low values of electric current density when a negative voltage is applied (closed mode). In the open mode, the observed electric current density is linearly proportional to the applied potential difference (see Figs. 3 B, D). The currentvoltage characteristics obtained by numerical and experimental analyses are in good qualitative agreement. However the absolute values of electric current density differ because of a decrease of ionic diffusivities in the PVA gel with respect to the values for dilute water solutions (considered in the simulations). The decrease leads to the reduction of both the diffusion and the electromigration fluxes of ions and thus to lower values of electric current density. The transition of the electrolyte diode to the open mode and its consecutive closing was studied in the next step (Fig. 4). The following sequence of the difference of electric potential was used: 0, 10, 8, 6, 4, 2, 0, -2, -4, -6, -8, -10 Volts. Two significant overshoots of electric current density are observed (Figs. 4A, C). The overshoots occur when: (i) the diode is switched to the open mode at time 300 s and (ii) the imposed electric potential difference varies from 0 V to -2 V at time 2100 s. The corresponding current-voltage characteristics are plotted in Figs. 4B, D.
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Fig. 5 Electric current density overshoot – switch to the closed mode. Spatial profiles of potassium concentration (A), electric potential (B) and the total ionic concentration (C). The profiles are plotted in 300 s (solid line), 300.07 s (dashed line) and 302 s (dotted line) after the start of numerical simulation.
The two experimentally obtained current-voltage dependencies somewhat differ (compare Figs. 3D, 4D). This is caused by the concentration polarization of the source electrodes that leads to the decrease of the voltage actually imposed on the microcapillary.
4.3. Analysis of transient overshoots The current density overshoot in Figs. 3A, C at time of 300 s can be explained by the analysis of spatio-temporal distributions of model variables obtained from the numerical analysis. Distributions of potassium ions, electric potential and the total ionic concentration (the sum of concentrations of all ions) before the decrease of voltage difference from 0 Volts to -10 Volts are depicted by solid lines in Figs. 5A-C, respectively. When the potential difference is increased, the electric current density grows according to Ohm’s law (dashed line). Consequently, chloride and potassium ions start to migrate out of the system and thus hydrogen and hydroxyl ions become dominant. Their neutralization in the microcapillary leads to the formation of the zone with extremely low electric conductivity where almost no ions are located. Hence the electric potential drop is concentrated within the thin zone (dotted line). The observed value of electric current density returns close to zero and the electrolyte diode approaches the closed mode. The nature of the current density overshoot in Figs. 4A, C at the time 300 s can be explained as follows: The increase of voltage difference from 0 Volts to 10 Volts is accompanied by a growing electromigration transport of the ionic components, which result in an increase of the electric current density (dashed line in Fig. 6). Potassium and chloride ions are transported into the microcapillary whereas hydrogen and hydroxyl ions remain close to their reservoirs. The microcapillary then behaves as a globally electroneutral system. Hence, for example, the concentration of potassium cations migrating from the KOH reservoir cannot exceed the concentration of chloride anions everywhere in the microcapillary except in the regions close to the reservoirs. This reflects the fact that hydrogen and hydroxyl ions in the centre of the microcapillary vanish and their concentration is close to 10-7 kmol m-3 that is characteristic for water. In the early stage after the voltage shift, it results in a decrease of the total ionic concentration, an increase of the electric resistance and a decrease of the electric current density (dotted line in Fig. 6). As chloride and potassium ions further migrate into the microcapillary, the total ionic concentration again increases (dash-dotted line in Fig. 6), which is accompanied by the observed increase of the electric current density.
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Fig. 6 Electric current density overshoot – switch to the open mode. Spatial profiles of potassium concentration (A), pH value (B) and the total ionic concentration (C). The profiles are plotted in 300 s (solid line), 300.07 s (dashed line), 300.3 s (dotted line) and 350 s (dash-dotted line) after the start of numerical simulation.
5. Conclusions Numerical analysis and experiments carried out on the electrolyte diode system reveal a complex transient behavior. Detailed numerical analysis of the reaction-transport processes in the electrolyte diode explains the short-time overshoots in the currentvoltage characteristics when the system is switched into the open and/or the closed mode. The obtained experimental results are in a good qualitative agreement with the results of modeling, which supports correctness of the used mathematical description. The discrepancies between the modeling results and experiments are primarily caused by the use of approximate values of ionic diffusivities in the model. The spatially onedimensional approximation of the real three-dimensional system that can contain, e.g., spatial heterogeneities in the porous structure is the other reason for the differences.
6. Acknowledgement The authors thank to the Czech Science Foundation for supporting the project no. 104/04/1442 and to the Ministry of Education, Youth, and Sport of the Czech Republic for supporting the project no. MSM 6046137306.
References L. Hegedüs, Z. Noszticzius, A. Papp, A.P. Schubert, M. Wittmann, 1995, ACH - Models in Chemistry, 132, 1-2, 207-224. L. Hegedüs, N. Kirschner, M. Wittmann, Z. Noszticzius, 1996, Progress in Colloid and Polymer Science, 102, 101-109. L. Hegedüs, N. Kirschner, M. Wittmann, Z. Noszticzius, 1998, Journal of Physical Chemistry A, 102, 6491-6497. J. H. Merkin, P.L. Simon, Z. Noszticzius, 2000, Journal of Mathematical Chemistry, 28, 1-3, 4358. J. Lindner, D. Šnita, M. Marek, 2002, Physical Chemistry Chemical Physics, 4, 1348-1354. M.Přibyl, D. Šnita, M. Marek, 2005, Chemical Engineering Journal, 105, 99-109. D. Šnita, M. Pačes, J. Lindner, J. Kosek and M. Marek, 2001, Faraday Discussions, 120, 53-66.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Stability Analysis of Differential-Algebraic Equations in AUTO_DAE Bianca C. von Clausbruch, Evaristo C. Biscaia, Jr., Príamo A. Meloa a
Programa de Engenharia Química/COPPE, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Bloco G, Sala G116, Ilha do Fundão, Rio de Janeiro – RJ, CEP 21945-970, Brasil.
Abstract A new computational package (AUTO_DAE) to study the stability of index-1 differentialalgebraic equations (DAEs) is presented. The characterization of the characteristic values of these systems is also presented and a discussion on the stability theorems for ordinary differential equations is performed for the differential-algebraic case. AUTO_DAE is based on the open source continuation and bifurcation computational package AUTO (Doedel et al., 1997), thoroughly used to investigate the behavior of ODEs. Prior to steady-state non linear analysis, AUTO_DAE performs a structural characterization of the DAEs in order to recognize the algebraic equations presented in the model. The characteristic values of the DAE system are evaluated using a standard routine to solve the generalized characteristic value problem. Reliability and robustness of the new code are demonstrated through the analysis of non linear chemical engineering problems.
Keywords: DAE; Stability; Continuation; Bifurcation; Software.
1. Introduction When a process is investigated, model based steady-state and dynamic analysis is usually performed before control and optimization techniques are implemented. Many process models may present non linear responses such as multiplicity of stationary solutions and self-sustained periodic oscillations. In order to study those modes of responses appropriately, one may use the well-known concepts underlying the bifurcation theory as well as methods of parametric continuation. The theory of nonlinear dynamical systems described by ordinary differential equations (ODEs) is very well developed and there are an uncountable number of academic and scientific books, articles and computational packages about this subject. Nevertheless, the theory for systems governed by differential algebraic equations (DAEs) is not so well developed and is modestly discussed in the open literature, and the main results available are usually obtained for simple cases, such as those described by index-1 DAE systems. Besides, open and public domain computational codes for nonlinear analysis of DAEs are almost nonexistent in the literature (see, for instance, Hyanek et al., 1995; Kienle, et al., 1995; Ochs et al., 1996; Mangold et al., 2000). In this work, a new, open source computational package to study the stability behavior of process models described by index-1 differential-algebraic equations is presented. The new code, AUTO_DAE, is based on the well-known bifurcation and continuation package AUTO (Doedel et al., 1997). Prior to steady-state nonlinear analysis, AUTO_DAE performs a structural characterization of the DAEs in order to recognize the algebraic equations presented in the model. The characteristic values of the DAE system are evaluated using a standard routine to solve the generalized characteristic value problem. Reliability and robustness of the new code are demonstrated through the analysis of nonlinear chemical engineering problems.
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2. Review of Theory 2.1. Stability of DAEs The problem of stability and bifurcation analysis of DAE systems has also received attention in other communities than the chemical engineering community (e.g. Reich, 1995; Beardmore and Song, 1998; Chen and Aihara, 2001). A tutorial discussion regarding this subject is presented by Beardmore and Song (1998). When approaching the problem, a few fundamental questions may be posed: Can the methods of nonlinear analysis developed for ODEs be directly applied to DAEs? Is it possible to define the Lyapunov stability for DAEs? If the DAE system has a given parameter, λ, can one find the bifurcation structure of the solutions as λ is varied? Unfortunately, preliminary answers to these questions are generally negative. Nevertheless, if certain regularization conditions, given by Reich (1995), are satisfied, it can be shown that the stability of equilibrium points of DAE system can be analyzed using classical linear algebra and spectral theory concepts. Furthermore, in the neighborhood of the equilibrium point, the DAE system possesses a linearization that is a vector field whose flow is equivalent to that of the DAE system, and whose dimension is equal to that of the local manifold (Beardmore and Song, 1998). These results are particularly important when one is interested in applying classical ODEs theory in the study of DAEs and shows, additionally, that a linearization is possible. The determination of the stability for a DAE system is performed slightly different from the purely differential case. Consider the semi-explicit DAE system below dx = f (x( t ), y ( t ), λ) , (1) dt 0 = g (x( t ), y ( t ), λ ) where f∈ℜn, g∈ℜm are nonlinear functions, and x∈ℜn, y∈ℜm are differential and algebraic state variables, respectively. The notation may be simplified even more as follows d~ x B = F(~ x ( t ), λ) , (2) dt x = [x y ]T ∈ℜn+m. where B∈ℜn+m×ℜn+m has rank(B)
Definition (Matrix pencil). Let A and B be a pair of matrices n×n. The set of all matrices of the form A-μB, with μ∈C, is called matrix pencil (C is the set of complex numbers). The characteristic values of the matrix pencil are elements of the set μ(A,B) defined by
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μ( A, B) = {μ ∈ C / det( A − μB) = 0} . If μ∈μ(A,B) and Av = μBv , with v≠0, then v is called characteristic vector of A-μB.
As discussed by Beardmore and Song (1998), considering that the DAE system satisfies the regularization conditions given by Reich (1995), which guarantees the existence of the linearization, then it may be said that if all pairs (μ,v)∈C × Cn+m satisfying ( A - μB) v = 0 , (5) * x is linearly stable. For vector fields linear stability are such that Re(μ)<0, then ~ implies in local asymptotic stability (Seydel, 1994), which is also valid for regular DAE systems. The calculation of saddle-node bifurcation points, also known as limit points, may be carried out using the limit points theorems for ODEs, as they are also equilibrium points of Equation (2). It should be pointed out, though, that the stability of the equilibrium point should be determined as indicated by Equation (4). As far as Hopf bifurcation points are concerned, Reich (1995) presents a version of the Hopf theorem applied to DAE systems, where the regularization conditions assure the linearization of the system. Reich uses the regularization conditions to justify the Hopf theorems for index-1 DAE systems (Beardmore and Song, 1998).
2.2. Parametric Continuation of Index-1 DAEs AUTO (Doedel et al., 1997) is a continuation package able perform bifurcation analysis of algebraic systems of the form f (x, λ) = 0 , (6) n n where f∈ℜ is a vector of nonlinear functions, x∈ℜ is a vector of state variables and λ∈ℜ is a real parameter of the system. Systems of ODEs of the form dx = f (x( t ), λ) , (7) dt may also be investigated in AUTO. In order to carry out the continuation of stationary curves, AUTO uses a pseudo-arc length with multiple step predictor-corrector technique (Kubiček and Marek, 1996), as long as the user supplies the code with an initial steady-state, and chooses one or more continuation parameters of the mathematical model. It is possible, then, to detect the occurrence of special points, such as limit points (LP) and Hopf bifurcation (HB) points. Besides, continuation of periodic orbits is also possible, as well as the continuation of LPs and HBs in two or three parameters. A strong limitation of AUTO is that, in principle, DAE systems may not be treated directly. Furthermore, AUTO is designed to work with low dimensional problems. Is should be pointed out, though, that literature shows that AUTO may be easily enhanced to work with systems up to a few hundred equations (Hyanek et al., 1995; Melo et al., 2003). Besides, linear algebra methods of sparse matrices are available in the literature, as discussed by Mangold et al., 2000.
3. Stability Analysis in AUTO_DAE In order to perform the continuation of stationary points of DAEs in AUTO, the calculation of characteristic values must be performed as indicated by Equation (4). Notice that, at steady-state, Equation (2) reduces to Equation (6), indicating that the continuation algorithm does not require any change. For the solution of the generalized characteristic value problem, the routine rgg.f of Eispack (Netlib, 2004) was chosen. This routine uses the QZ algorithm of Moler and Stewart (1973). This algorithm does
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not make any inversion of matrix B or of any submatrix of B, and is a generalization of the QR algorithm (Golub and van Loan, 1996). It is important to notice that a fundamental step toward the determination of the stability of equilibrium points of a DAE system is the calculation of matrix B. In order to do that, a structural characterization routine has been added to AUTO. The vector of functions F (cf. Equation (2)) is implemented in AUTO with the aid of another vector, XPRIME, as follows Fi = RHSi − XPRIME i for i = 1,..., n , (8) Fi = RHSi for i = n + 1,..., n + m where RHS stands for the right hand side of the equation. Notice that, as presented in Equation (8), the system possesses n differential equations and m algebraic equations. The structural characterization routine tests the existence of derivatives of the state variables in the equations and characterizes the structure of the DAE system. As a result, matrix B is generated. The major modifications of AUTO described above have resulted in a new code for the analysis of steady-state stability of index-1 DAE system called AUTO_DAE. In order to show the reliability and robustness of the new code, two chemical engineering systems that may be described by DAEs are treated below.
4. Examples 4.1. The CSTR with A→B Reaction The first example is the well-know CSTR with an exothermic first order reaction, A → B, described by Uppal, Ray and Poore (1974). This is a benchmark example for many nonlinear studies as it presents a multitude of nonlinear responses, including multiplicity of steady-state solutions and limit cycle behavior. Mass and energy balances performed on the reactor lead to the following dimensionless mathematical model dx1 = − x1 + Da (1 − x1 ) exp( x 2 ) , (9) dt dx 2 = − x 2 + BDa (1 − x1 ) exp(x 2 ) − β x 2 , (10) dt where x1 is the conversion of species A, x2 is the dimensionless reactor temperature, Da is the Damköhler number, B is the adiabatic temperature rise, and β is the dimensionless heat transfer coefficient. As presented, the model is described by ODEs and, thus, may be directly implemented in AUTO. In order to test the new version of the code, AUTO_DAE, the model was rewritten in such a way to force the appearance of a differential-algebraic structure. In the index-1 formulation of the model one may write dx1 = − x1 + Da (1 − x1 ) x 3 , (11) dt dx 2 = − x 2 + BDa (1 − x1 ) x 3 − βx 2 , (12) dt 0 = x 3 − exp( x 2 ) . (13) Figure 1 presents a typical bifurcation diagram for this system as the Damköhler number is varied. Other parameters were kept constant during the calculation (B=14 and β=2). Saddle-node (limit point) as well as Hopf bifurcations are observed. Stable branches are given by solid lines and unstable branches by dotted lines; Hopf
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bifurcation points are represented by the symbol (■). Three bifurcation diagrams were built: two for the implementation of Equations (9)-(10) in AUTO and in AUTO_DAE, and another for the implementation of Equations (11)-(13) in AUTO_DAE. In all three cases, the bifurcation diagrams are rigorously identical to that presented in Figure 1. 1.0 0.8
x1
0.6 0.4 0.2 0.0 0.00
0.05
0.10
0.15
0.20
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Figure 1 – Bifurcation diagram for Example 1. 4.2. The Evaporative Cooling Reactor The second example presented regards the evaporative cooling CSTR, also known as boiling liquid reactor. In this class of reactors, the reaction heat for exothermic reactions is removed by partial vaporization of the liquid phase (Figure 2a). A first order, exothermic reaction A → B is processed in the vessel. By assuming the quasisteady state hypothesis for the dynamics of the cooling jacket, the reactor mathematical model is given below as presented by Zavala (1997) dx1 = I − Ol , (14) dt dx 2 x = I − O l 2 − kx 2 , (15) dt x1 dx 3 1 = {ICp A ( x I − x 3 ) − Δh r kx1 + O g [Cp l ( x cond − x 3 ) − Δh vap ] , dt x1Cp l
(16)
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where x1 and x2 represent the total number of mols and the number of mols of species A in the liquid phase, respectively, x3 is the reactor temperature, Ol is the exit molar flowrate, and Og is the gas phase molar flowrate (Figura 2a). For this system, there was no need to create auxiliary algebraic variables, as Equations (17)-(18) guarantee the index-1 differential-algebraic structure for the system. It should be noticed, however, that if Equations (17)-(18) were inserted into Equations (14)-(16), then the system would become purely differential. All parameters of the evaporative cooling reactor used for the calculations presented here may be found in Zavala (1997), and are omitted here due to space reasoning. In order to test the new code, three bifurcation diagrams were built: two for the implementation of Equations (14)-(16) in AUTO and in AUTO_DAE, and another for the implementation of Equations (14)-(18) in AUTO_DAE. In all three cases, the bifurcation diagrams are rigorously identical to that presented in Figure 2b. A discontinuity in the state equation used for the calculation of vapor pressure of species A is responsible for the unusual behavior observed in Figure 2b.
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4. Conclusions A new code, AUTO_DAE, is presented to perform stability analysis of process models described by differential-algebraic equations. Tests with chemical engineering problems have demonstrated the reliability and robustness of the new code.
5. Acknowledgments Authors would like to thank CNPQ (Conselho Nacional de Desenvolvimento Científico e Tecnológico – Brasil) for providing scholarship to B.C.C. and supporting this research. Comments from ESCAPE reviewers are also acknowledged.
References Chen, L.; Aihara, K., Stability and bifurcation analysis of differential-difference-algebraic equations. IEEE Trans. Circ. Syst. I. v. 48(3), p. 308-326, 2001. Beardmore, R. E.; Song, Y. H., Differential-algebraic equations: a tutorial review. Int. J. Bif. Chaos. v. 8(7), p. 1399-1411, 1998. Doedel, E.; Champneys, A. R.; Fairgrieve, T. F.; Kuznetsov, Y. A.; Sandstede, B.; Wang, X. AUTO 97: User’s Guide, Concordia University, Montreal, 1997. Golub, G.H.; van Loan, C.F. Matrix Computations. London: The Johns Hopkins University Press, 1996. Hyanek, I.; Zacca, J. J.; Teymour, F.; Ray W. H. Dynamics and stability of polymerization process flow sheets. Ind. Chem. Res. v.34(11), p.3872-3877, 1995. Kienle, A.; Lauschke, G.; Gehrke, V.; Gilles, E.D. On the dynamics of the circulation loop reactor – numerical methods and analysis. Chem. Eng. Sci., v.50(15), p. 2361-2375, 1995. Kubiček, M.; Marek, M. Computational Methods in Bifurcation Theory and Dissipative Structures, New York: Springer-Verlag, 1986. Mangold, M.; Kienle, A.; Gilles, E.D.; Mohl, K.D. Nonlinear computation in DIVA - methods and applications. Chem. Eng. Sci., v. 55(2), p. 441-454, 2000 . Melo, P. A.; Biscaia Jr., E. C.; Pinto, J.C. The bifurcation behavior of continuous free-radical solution loop polymerization reactors. Chem. Eng. Sci., v. 58(13), p. 2805-2821, 2003. Moler, C.B.; Stewart, G.W., An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal., v.10(2), p. 241-256, 1973. Netlib. Netlib Repository. URL: http://www.netlib.org, last accessed in 03/15/2004. Ochs, S.; Rosendorf, P.; Hyanek, I.; Zhang, X.M.; Ray, W. H. Dynamic flowsheet modeling of polymerization processes using POLYRED. Comp. and Chem. Eng., v. 20(6-7), p. 657-663, 1996. Reich, S. On the local qualitative behavior of DAEs. Circ. Syst. Sig. Proc., v. 14(4), p. 427-443, 1994. Seydel, R. Practical bifurcation and stability analysis – from equilibrium to chaos. New York: SpringerVerlag, 1994. Uppal A.; Ray W.H.; Poore, A. B. On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci., v. 29(4), p. 967-985, 1974. Zavala, M.S. Some issues in the dynamic modeling and simulation of large-scale systems in chemical engineering. M.Sc. Thesis, University of Wisconsin at Madison, Madison, Wisconsin, USA, 1997.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Application of particulate models for industrial processes George Skillas,a Catrin Becker,a Marcel Verduyn,a Johannes Vorholza a
Process Technology & Engineering, Degussa AG 1024-319, D-63457 Hanau/Wolfgang, Germany Abstract
Two approaches for modeling and simulating gas phase processes with varying degrees of detail and computational effort are presented. The first example involves a single reactor used in the production of nanoscale particles, the second example deals with an entire spray granulation process producing particles in the millimeter range. In both cases particle size is an important measure for product quality. The aim of the work is to predict particle size distribution (PSD) and flows as a function of process and operation parameters. The reactor is simulated by means of computational fluid dynamics (CFD). Hereby, the computed flow and chemical species fields are used as input for a simulation of particle inception, growth and aggregation. Due to the complexity of the system quantitative results are prone to errors. However, the method is robust and the results for the particle field help understanding and explaining different aspects of the reactor and the aggregate particles produced. In spray granulation processes the PSD depends, very often in a complex manner, on process and/or material parameters. Moreover, process dynamics can also play an important role. Furthermore, the interaction and interconnection of different unit operations (granulators, mills, classifiers) is in the scope of interest, when the application of sophisticated models (as in the first case) is restricted or impossible due to computational limits. In this second example, a literature model for growth and abrasion (Heinrich et al. (2002)) is extended to breakage and implemented into Aspen Custom Modeler. In order to ensure a high flexibility of the model, the process is divided into several unit operations which can be arbitrarily combined within a flowsheet. The model is parameterized using plant measurements combined with suitable estimates and applied to industrial granulation processes. It can be utilized for process conception as well as for process optimization. Keywords: spray granulation, nanoparticles, particle dynamics, particle size distribution, cfd.
1. Carbon black reactor modelling 1.1. Introduction Most carbon black is produced by the furnace black process, Kühner 2000. In this process natural gas and preheated air are burnt in a combustion chamber and then feedstock, i.e. liquid aromatic oil, is injected into the hot gases. The oil decomposes mainly to carbon and hydrogen while carbon black is formed. When the desired particle size and morphology are achieved reactions are stopped by quenching with water. Subsequently the gas is led through a heat exchanger to preheat the process air. Baghouse filters separate gas and carbon black agglomerates. In this paper we will discuss
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the part of the process described above, where oil is injected and carbon black particles are formed. 1.2. The Model For the results presented here Navier-Stokes equations for turbulent flow, energy and chemical composition, coupled with a monodisperse particle model which included inception aggregation and surface growth were solved. Turbulence is computed by the realizable k-ε model. Convective and radiative heat losses are accounted for. Radiation is modelled with the discrete ordinates model. We consider CO2, H2O and carbon black absorption to compute the absorption coefficient. The particle model is monodisperse. The aggregate number N, volume V and area A concentration are transported through the computational domain. Model aggregates consist of monodisperse primary particles. These are modelled as hard spheres i.e. no overlap is allowed. Nucleation, oxidation and growth rates are modelled by Arrhenius expressions each with two parameters (Ri, Ti), while the particle collision coefficient is computed by taking into account expressions for both free molecular and continuum regimes. The Arrhenius form of the nucleation rate was derived by simplifying the classical nucleation theory expression, see Skillas et al. 2005 for details. Atomisation of the liquid oil with detailed droplet break-up mechanisms is not considered in the calculation, instead experimental data is used for droplet diameter and velocity to partially account for these limitations. We considered species mixing and not reacting to be rate limiting, i.e. the composition of the species in the gas is determined under the assumption that the chemical reactions are so fast that chemical equilibrium is achieved locally. The geometry of the simulated reactor can be seen in Figure 1. The computational domain is three-dimensional starting at the end of the combustion chamber and encompasses a narrow tube and the reactor tunnel.
Figure 1: Reactor geometry and discretisation of the computational domain.
1.3. Model application and discussion Mass flow data calculated by simulation and measured mass ow data, taken from Skillas et al. 2005, are shown in Figure 2. Inlet gas oxygen concentration is shown as a λ value for natural gas combustion in the combustion chamber. The trends are in qualitative agreement with experiment for both λ values. The slope value is over-predicted by the model, nevertheless steepness is similar to experiment. The pronounced difference between calculated and experimental value at an oil flow of 430 kg/h, found for the SSA as well Skillas et al. 2005, can be attributed to the performance of the chemical model. The CFD chemical model is assuming equilibrium. Comparison of reactor gas chromatography measurements with equilibrium calculations show a consistent underprediction of CO2 concentration and an over-prediction of CO, i.e. at equilibrium conditions more carbon is consumed by oxygen than in reality. The more O2 present, the lower the calculated carbon black mass compared to reality, leading in the extreme to differences like the one at 430 kg/h. In part this effect is compensated for in the particle
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model through the fit parameters. Considering that this prediction is an extrapolation of the fit the performance is good.
Figure 2: Carbon black mass flow produced by, vs. oil mass ow injected into, the reactor for simulation and experiment. The set of four model parameters Rn, Tn, Rg and Tg are adjusted by comparing the measured and calculated data for the two rhs pairs of points. The two left hand side (lhs) pairs are predicted with the model using the best parameter set found.
As mentioned above, residual oxygen burns oil vapour, raising the temperature in the vicinity of the oil droplets, c.f. Figure 3a. Temperature reaches a peak in this region and approaches the adiabatic flame temperature in the order of 2300 °C, as the chemical model assumes chemical equilibrium in each computational cell. Temperatures in the vicinity of the walls are accurate within 100 °C, when compared with experimental data
Figure 3: From left to right (a) reactor temperature in K and (b) aggregate number concentration in #/mg. The upper portion is a cut through an oil nozzle containing plane, while the lower portion shows a cut through a plane between two oil nozzles. The oil injection region can be seen.
obtained by pyrometry measurements. The low temperature region is caused by energy needed for oil evaporation. Number concentration data are shown in Figure 3b. Particles form on the envelope of the oil droplet evaporation region, where high temperatures needed for particle formation are present. Due to high velocities (low residence time) in the core flow aggregates at the end of the computational domain consist on average of only 7 primary particles (c.f. Figure 4a), while the aggregate concentration drops about an order of magnitude.
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Figure 4: From left to right (a) primary particle diameter in m and (b) axial velocity in m/s. The upper portion is a cut through an oil nozzle containing plane, while the lower portion shows a cut through a plane between two oil nozzles. The core flow residence time is short, resulting in particles with an diameter of about 10 nm. A recirculation zone can be seen, near the reactor walls.
The inhomogeneity in the Nψfield, due to oil injection almost vanishes after about 40% of the reactor tunnel length. To understand the primary particles per aggregate field one has to consider flow velocities, c.f. Figure 4b. Recirculation zones resulting in longer residence times, which promote aggregation, exist in both planes. The lowest aggregate number concentration occurs in the recirculation region and it is lowest between oil nozzles. This is consistent with high temperatures and concentration of growth species locally available, which in reality are needed to promote aggregation through neck-forming between primary particles. The shape of the fields in Figures 3-4 does not vary with process conditions, while the actual numbers follow empirical laws. For example, decreasing the reactor temperature lowers aggregate concentration N and increases primary particle diameter dp. Good agreement between simulation and experiments is found at conditions where the equilibrium assumption holds.
2. Modelling a plant producing particulate products 2.1. Introduction The particle size distribution (PSD) of particulate products, as e.g. granulates, is an important measure for the product quality as it influences properties like solubility, dust formation or dispersibility. In turn, the PSD is influenced by process parameters as for example the hydrodynamic conditions inside a granulator as well as by the design of the whole plant. Especially in the case, where the influence of variations in the interconnection of different unit operations (granulators, mills, classifiers) is in the scope of interest, the application of sophisticated models as discussed in the previous sections is restricted or impossible due to computational limits. Instead, it is desirable to work with models of “moderate” accuracy that, on the other hand, enable the modeling of a larger network of unit operations or even a whole plant. 2.2. The model In the present work a literature model of a spray granulation process (Heinrich et al. (2002)) is extended and implemented in Aspen Custom Modeler (ACM), a part of the industrial standard simulator Aspen Engineering Suite (AES). In order to ensure a high flexibility of the model, the process is divided into several unit operations (granulator, separators as sieves or cyclones, and mills) which can be arbitrarily combined on a
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flowsheet. The utilization of ACM has the additional advantage, that thermophysical property databases (Aspen Properties) can be accessed and/or that user developed unit operations can be combined with “off-the-shelf”-models for “standard” unit operations available in Aspen Plus or Aspen Dynamics. In a spray granulation process a fluid product, e.g. a suspension, is sprayed onto particles in a fluidized bed. A Growth part of the drops is deposited on the particles, the solvent evaporates in the hot, unsaturated fluidization air, and di the solid matter remains on the Nuclei particle, resulting in an onion-like Evaporation growth. At the same time the particle Abrasion Breakage is also subjected to mechanical stress causing abrasion or even breakage resulting in a diameter decrease or the birth of new smaller particles (figure Figure 5: Granulator unit 5). For the granulator unit the population balance account considering growth, abrasion and breakage is solved. The expressions for effective growth and abrasion are based on a model by Heinrich et al. (2002), in addition, breakage terms are introduced. Mills and separators are modeled using short cut methods, e.g. by fixed PSD of the bed spectrum and separation functions. 2.3. Application of the model to industrial spray granulation processes The model is applied for the simulation of the start up behavior of a newly built spray granulation process which was built up by the unit operations shown in figure 5. In the first step the model parameters are determined from plant measurements of an already existing (but smaller and partly designed differently) plant for the same product or estimated by suitable assumptions. After that, the parameters are adopted for the simulation of the new plant. By the simulation of the start up procedure an instability of the new process, i.e. process runaway by accumulation of fines in recycle streams and product quality decrease, was detected, which was confirmed during the start up of the plant. This situation is shown qualitatively in figure 6 on the left hand side. The flexibility in combination of the unit operations enabled the simulation of different scenarios for the stabilization of the process which finally resulted in the introduction of corrective measures that led to stable behavior (figure 6 (rhs)). Another application of the model is the examination of the influence of the change of process parameters (e.g. grinding intensity, suspension flow, or variations in the interconnection of separators) on the product quality, for example the particle size distribution. Whereas in the case when stability problems are considered, qualitatively correct trends might be sufficient, the accuracy of the results needs to be higher, when the PSD is predicted. Typical results (made anonymous) are shown in figure 8. In this case the above mentioned parameters are again fitted to the results of plant measurements. In the second step the model is extrapolated to different operating situations of the same plant (figure 7). Although there are still some quantitative deficiencies the agreement between simulation and experiment can be considered as good.
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References S. Heinrich et al., 2002, Analysis of the start-up process in continuous fluidized bed spray granulation by population balance modelling, Chem. Eng. Sci. 57, 4369-4390 G. Kühner, 2000, What is carbon black?, Degussa AG, Inorganic Chemical Products Division, P.O. Box 110533, D-60000 Frankfurt 11, Germany. G. Skillas et al., 2005, Simulation of particulates in a carbon black reactor, J. Nanop. Res. 7, 1527.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Optimization of Operating Conditions for Ferrichrome Production in a Membrane Bioreactor Using Ustilago maydis A. Drewsb, H. Arellano-Garciaa, M. Wendtc, M. Kraumeb, G. Wozny a a
Department of Process Dynamics and Operation, Berlin University of Technology, Sekr. KWT-9, Str. des 17. Juni 135, Berlin 10623, Germany b Department of Chemical Engineering, Berlin University of Technology, Sekr. MA 5-7, Str. des 17. Juni 135, Berlin 10623, Germany c Infraserv GmbH, Knapsack KG, Chemiepark Knapsack, D-50351 Hürth, Germany
Abstract In this work, a continuous siderophore production system using the phytopathogenic fungus Ustilago maydis is considered. A hybrid process, specifically, a microfiltration membrane bioreactor is employed which is deemed to be advantageous since cells are retained in the vessel while possibly inhibitory products are continuously withdrawn from the system. Accordingly, the process is operated at high cell density thereby increasing productivity. Preliminary analysis and studies for steady state optimization result in the existence of steady-state points for the maximum production rate. Thus, the optimization focus is divided into two operation stages: the startup and the steady-state operation. The former operation implies a dynamic optimization where the optimal policies are determined in order to meet the previously established optimal steady-state. Furthermore, the startup period is characterized through a switching operation mode from batch to continuous. However, the overall aim of the optimization is the maximization of the product total amount per time while minimizing the startup period. To keep the production costs at a convenient level, different constraints are included in the optimization problem such as a glucose waste limit, a lower bound for the outlet product concentration, and also technical constraints which involve upper bounds for the biomass concentration in the reactor. For the simulation and computation of the sensitivities, we propose a new multiple-time-scaling-approach to solve the resulting optimization problem which possesses strong nonlinear properties. Keywords: Ustilago maydis, membrane bioreactor, high cell density, model-based optimization.
1. Introduction Ferrichromes belong to a group of iron chelating agents called siderophores. Possible clinical applications of these compounds include elevating the potency of antibiotics, and anti-cancer drugs. Ustilago maydis, a host specific phytopathogenic fungus (smut fungus), can be used for the production of ferrichrome. In liquid culture, U. maydis grows yeast-like in oval shaped single cells (length approx. 10μm, see Fig. 1). A membrane bioreactor (MBR) is defined as a hybrid process since it combines a common bioreactor and membrane separation units for biomass retention offering process intensification and new possibilities for bioprocesses and wastewater treatment.
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MBR combine the benefits of high biomass concentrations to increase volumetric productivity with the possibility to run a continuous process at controlled biomass retention and thereby controlled biomass growth rate. With the organism used in this work, ferrichrome productivity was increased by a factor of 12 in comparison to repeated batches (Drews and Kraume, 2005). By decoupling of hydraulic and biomass retention times, MBRs offer an additional degree of freedom for process control in comparison to chemostats. Therefore, they can be particularly advantageous in fermentations where inhibitory metabolites occur either as the desired product or as unwanted byproducts since the productive cells are retained in the vessel while soluble products are continuously withdrawn. Although several practical experiences and data are already available for design and operation of MBR processes there is still considerable optimization potential. Here, e.g., to optimize the productivity in MBR, optimization under product inhibition and process constraints is required. Product inhibition means that the product concentration has a strong impact on the growth rate with a decreasing effect at either too high or too low concentrations and an optimum somewhere in between. Thus, the potential for optimization is mainly caused by this inhibitory effect. However, due to process constraints, mostly derived from technical and cost restrictions, it is not always possible to keep the concentrations of the product and other inhibitory components exactly at their optimum level. Consequently the optimization potential is mainly exploited by finding an operation policy, where the compliance of those process constraints is always assured while keeping the concentrations of the inhibitory components as close, as soon and long as possible to the optimum productivity level over the entire time horizon. This makes model-based optimization strategies a challenging task.
2. Experimental Procedure Ustilago maydis (strains urbs- and wild type FB1) was stored in glycerol stocks (25 %) at –80 °C. After a 3-day inoculation on potato-dextrose-agar cells were transferred into shaking flasks containing a modified Grimm-Allen-medium with glucose as the main carbon source. Flasks were shaken for approx. 24 h at 100 min-1 and at 27 °C until they were transferred into a 5 L glass fermentor. In MBR runs, this was equipped with an external ceramic tubular membrane module for biomass retention. Temperature was controlled at 27 °C, pH at 7.2, and pO2 at 40 %. Fig. 1b shows the experimental setup. To study the effects of glucose and ammonia limitation, respectively, and the effects of short- and long-term limitation, different batch, fed-batch, chemostat and MBR cultivations were carried out. The biomass concentration was determined by turbidity
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measurements at 600 nm calibrated against dry weight measurements of washed cells. For determination of substrates and nutrients concentrations, commercial test kits were used. The concentration of ferrichromes in the supernatant was measured by the CASmethod (Schwyn and Neilands, 1987) whereby unchelated siderophores can be detected. All samples were membrane filtered before analyses.
3. Modeling of MBR A model was developed to describe the considered MBR process. The model approach includes mass balances and kinetics. At the given conditions (pH 7.2 and T = 27 °C), the concentrations of glucose cC, ammonia cN and ferrichrome cP were identified as the main influencing parameters for growth of Ustilago maydis. Thus, the following expression is derived. For glucose, a simple Monod term was chosen while for ammonia and ferrichrome inhibition was observed at higher concentrations:
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For each time point, the degree of freedom is four. In case of open loop control, it is recommendable to use the trajectories of the input variables Vin , VB , cC,in and cN,in as control or decision variables in the optimization task. For the dynamic optimization, in particular, the initial concentrations of all components, particularly the substrates, may be used as additional decision variables.
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4. Steady state and dynamic optimization 4.1. Formulation of the optimization problems The general aim of the optimization is the optimal increase of the ferrichrome productivity. Thus, the objective function is defined as the maximization of the total amount of produced ferrichrome divided by the time. The equality constraints are the model equations. Some economic and constructional aspects give rise to further constraints. Since there is an upper bound for the flux through the membrane, and for the membrane area to be coupled to a given reactor volume, the hydraulic residence time τ is restricted (liquid hold-up in the membrane module < reactor volume). With a typical value of 20 L m-2h-1 for filtration of biological suspensions and a typical tubular module packing density of 450 m2m-3, the minimum residence time τmin is estimated to be 0.11 h. Furthermore, through increased viscosity and uptake rates, high cell densities result in decreasing oxygen transfer rates and thereby increased operating costs. A value of 200 gL-1 is chosen as the maximum allowable cB (dry weight). Overall operating costs are also influenced by raw material utilization and product purification. In order not to waste an excess of raw materials, the maximum allowable cC is also restricted. Additionally, an upper bound for cN of 1.5 gL-1 is required to avoid poisoning of the biomass. Moreover, because too low product concentrations cause difficulties and excessive operating costs in product recovery in downstream separation units, e.g., adsorption processes, a lower bound for cP is considered. It should be noted, that the bounds for glucose and ferrichrome are only valid when there is an outflow, consequently there must be an inflow. In batch periods with no inand outflow, a violation of those bounds is feasible. In contrast, the bounds of cN and cB should never be violated (hard constraints). Concerning the bounds of the decision variables, the upper bound of the volumetric inflow rate is determined by τmin. The lower bounds of all decision variables can be set to zero. The formulation of the optimization problem above corresponds to the dynamic optimization. For the steady state optimization, the objective function will be simplified to the maximization of the ferrichrome production rate, and for the constraints, all time dependent variables such as the derivatives of the state variables will be omitted. 4.2. Solution Strategy The steady state problem is relatively small and is solved by the simultaneous approach. Contrary to that, for the dynamic optimization problem, trajectories of the decision variables within a large time horizon need to be computed which requires a discretisation of the time horizon into a large number of time intervals leading to a large scale optimization problem. In this work, the sequential approach is applied. Due to the rather expected non-monotonic behavior of state variables within time intervals (i.e. maxima of glucose and ammonium, and minima of product concentrations) where the feed parameters are constant, the orthogonal collocation method in finite elements (5points) is used as a discretization approach to guarantee robustness and efficiency at each simulation step (Arellano-Garcia et al., 2005). In addition, a step-size control is integrated to assure convergence at each time interval. 4.2.1. Decomposition approach An inherent characteristic of the MBR process is that a batch phase is required at the beginning when the initial condition of the inhibitory product is very small, i.e. far from its peak which is optimal for the growth of biomass. Furthermore, since the volumetric flow rates are set to zero during the batch period, all the other continuous decision variables have no impact on the process either. On the other hand, the initial
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concentrations of the substrates, and the switching time point from the batch to the continuous phase are crucial decision parameters for the objective function and the constraints in the optimization problem. Thus, the original optimization problem is decomposed into two sub-problems. The first one minimizes the batch time subject to end conditions concerning concentrations bounds, which are supplied by the optimizer of the continuous phase where initial concentrations are used as decision parameters as well. This interaction procedure is finished when the end concentrations of the batch period are equal to the corresponding optimized values of the continuous period. For MBR, however, the reformulation of the subproblems is physically equivalent to the original problem formulation. 4.2.2. An approach to multiple time-scaling Due to the large time horizon of the MBR process, the trajectories of the decision variables are discretized into time intervals where the values are piecewise constant. It should be noted, that the size of these time intervals is independent of those used for simulation. The constraints are to be met at all time points in certain or even all periods. However, to solve the optimization problem in the continuous phase, a step size control algorithm is required in the simulation stage to assure robust convergence. Furthermore, it must be guaranteed that a collocation point of the larger time interval, which is used for the decision variables and constraint calculation, is matched with the end of a small time interval. By this means, an accurate sensitivity calculation can be performed which is required for the NLP solver in the optimization layer. This leads to a novel approach of multiple time-scaling where the step size control (inside one large time interval) is carried out by a 3-layer system (Fig. 2). At the beginning of computing the trajectories, the step length is first set up in Layer 1, where it is equal to the time interval for the decision variables. In case of convergence, the sensitivities concerning the constraint values at all collocation points are calculated simultaneously. If not, the step size control goes one layer down and so on forth. In Fig. 2, DT or DTs represents the general term for the interval length while the subscripts u and c indicate the correspondence either to the decision interval (ii) or to the interval between the current collocation points, respectively. However, it can also be seen that the end of one interval in Layer 3 is always limited to the next collocation point.
Figure 2. Layer system for the multiple time scaling
5. Computation results The numerical results begin with preliminary steady state studies where the upper bound of glucose and biomass is considered. For this purpose, steady state optimization is carried out with different upper bounds for glucose concentration over different values of the residence time. These results allow two main conclusions: i) only when the glucose concentration is lower than approx. 5gL-1, significant differences can be observed due to the fact that KS,C = 0.7 gL-1; ii) if the residence time is lower than 2 hours, a major impact on optimal production and growth rate is identified. Due to the
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glucose feed concentration cC,in [gL-1]
fact that cB is always at its upper bound and cN at its optimal value, the optimal value of cP can actually be described as a function of cC and τ. Apart from cP, the dependent state variables remain at a level for maximum allowable µ and, thus, the optimal steady state production rate can never be exceeded during the dynamics of the entire process. Therefore, the dynamic optimization problem is equivalent to a startup problem. 70 60 50 40 30 20 10
batch growth
steady state
continuous growth
0 0
50
100
150
200
250
300
time [h]
Figure 3. Optimal trajectories of the volumetric feed rate and of the glucose feed concentration.
Due to the fact that the maximum possible cB results from the steady state optimization, and that the growth rate is only limited by the constraints of cC and τ, it is obvious, that the production rate has reached its maximum value at the steady state point. Thus, the entire optimal operation strategy can be roughly divided into three periods: batch growth, continuous growth and steady state period. The most significant trajectories of the decision variables are illustrated in Fig. 3. The results show that the feed flow rate rapidly increases at the beginning of the continuous growth period until τmin is reached in order to keep cP as long as possible at its lowest possible point, which is the closest point to its peak in the growth rate function. From that state on, the feed flow rate keeps close to its end value in the steady state period, while the feed concentrations of the substrates approach to their steady state end values relatively continuously in order to compensate the increasing consumption by the growing biomass.
6. Concluding remarks In this work, model-based optimization is applied for ferrichrome production in an MBR using Ustilago maydis. A model is used which was validated with experimental data. For the batch period, instead of the feed conditions, the initial conditions of the feeding substrates and the time point to switch to the continuous period are critical parameters for optimality. Thus, a novel developed decomposition method has been applied for matching the batch period with the following continuous growth period. For the numerical optimization of the growth period, it is necessary to assure robustness in convergence and a reliable sensitivity calculation for the constrained variables at different selected time points. For this purpose, a novel efficient approach to multiple time scaling strategy is proposed.
References Arellano-Garcia, H., Barz, T., Wendt, M., and Wozny, G., 2005, Real-Time Feasibility of Nonlinear Predictive Control for Semi-batch Reactors, Computer-Aided Chemical Engineering, 20A, 967-973. Drews, A. and Kraume, M., 2005, Process Improvement by Application of Membrane Bioreactors, Trans IChemE, Chem. Eng. Res. Des. 83(A3), 276-284. Schwyn, B. and Neilands, J.B., 1987, Universal Chemical Assay for the Detection and Determination of Siderophores, Anal. Biochem. 160, 47-56.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Modelling and Simulation of MSF Desalination Process using gPROMS and Neural Network based Physical Property Correlation M.S. Tanvir and I.M. Mujtaba* School of Engineering , Design and Technology University of Bradford, Bradford BD7 1DP, Great Britain
Abstract Multi Stage Flash (MSF) desalination plants are a sustainable source of fresh water in arid regions. Modelling plays an important role in simulation, optimisation and control of MSF processes. In this work an MSF process model is developed using gPROMS modelling tool. Accurate estimation of Temperature Elevation (TE) due to salinity is important in developing reliable process model. Here, instead of using empirical correlations from literature, a Neural Network based correlation is used to determine the TE. This correlation is embedded in the gPROMS based process model. We obtained a good agreement between the results reported by Rosso et. al. (1996) and those predicted by our model. Effects of seawater temperature (Tseawater) and steam temperature (Tsteam) on the performance of the MSF process are also studied and reported. Keywords: Desalination, MSF, modelling, gPROMS, NN based correlation.
1. Introduction The technique of turning seawater into fresh water is called desalination. Multi-Stage Flash (MSF) distillation process (Fig. 1) has been used for many years and is the largest sector in the desalination industry (El-Dessouky and Hisham, 2002). An MSF process consists of three main sections: brine heater, recovery section with NR stages (flash chambers) and a rejection section with NJ stages. Seawater enters into the last stage of the rejection stages and passes through a series of tubes to remove heat from the stages. Before the rejection section seawater is partly discharged to the sea to balance the heat. The other part is mixed with the recycled brine form the last stage of the rejection section and fed before the last stage of the recovery section. Seawater is flowing through the tubes in different stages to recover heat from the stages and the brine heater raises the seawater temperature to the maximum attainable temperature (also known as Top Brine Temperature, TBT). After that it enters into the first flashing stage and produce flashing vapour. This process continues until the last stage of the rejection
*
all correspondences to Dr. I.M. Mujtaba. Email: [email protected]
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section. The concentrated brine from last stage is partly discharged to the sea and the remaining is recycled as mentioned before. A typical MSF process model includes mass and energy balances, the geometry of the stages and physical properties which are functions of temperature and salinity. Adequate knowledge of the total heat transfer area, the length of the flash chamber, control of the corrosion and scale formation are needed for modelling, design and scale up of MSF processes. These parameters are dependent on/ inter-related with TBT (Spiegler and Liard, 1980). Several correlations for estimating the TE exist in the literature (Bromley et al., 1974). However in this work Neural Network based correlation is used to determine the TE. The NN based correlations can be easily adapted to the new plant data and give more accurate predictions of TE compared to the empirical ones (Tanvir and Mujtaba, 2006).
2. MSF Process Model Models for each unit operation (such as flash chamber, brine heater, splitter and mixer) are developed separately and connected via a high level modelling language using gPROMS. gPROMS is a general Process Modelling System which is capable of performing simulation, optimisation and parameter estimation of highly complex processes. It is chosen in this work because it is reliable and requires less programming knowledge (as in FORTRAN and C). Recovery Stages
CW
Reject Stages
WR
Seawater F
Ws
Steam
BN B0 Stage:
NR
NR +1
NR +NJ
Recycle Brine (R)
TFj CR/CS
CR//CS
TDj-1
TFj+1
Dj TDj
BD Blowdown
1
WR/WS
Dj-1
DN Distillate (Fresh water)
WSt eam
WR/WS
Bj-1 TBj-1 CBj-1
Bj TBj CBj
Fig.1 A Typical MSF Process and Stage j
The steady state model equations (based on Rosso et al., 1996) are given in Fig. 2 (most symbols except few are defined in the original reference). For a total number of stages NS = NR+NJ, the total number of equations (TNE) is: 25NS+27. The total number of variables (TNV) is: 18NS+16. Therefore, the degrees of freedom (D.F. = TNV-TNE) is: 7NS + 11. All physical property correlations shown in Fig. 2 except for TE (temperature elevation due to salinity) are taken from Hellal et al. (1986), Rosso et al. (1996) and Hussain et al. (2003). The NN based correlation for the estimation of TE is described by Tanvir and Mujtaba (2006), which was developed based on Bromley (Bromley et al., 1974) data.
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Here we considered 3-layer NN architecture with 4 neurons in the hidden layer and 1 neuron (TE) in the output layer (Fig. 3). The correlation is shown in Table 1 with the weights and biases. The detailed training, validation and testing of the network is described in Tanvir and Mujtaba (2006) and the predictions of TE by NN based correlation and experimental data are compared by Tanvir and Mujtaba (2006). Stage Model Mass Balance in the flash chamber: B j −1 = B j + V j B j −1 CBi −1 = B j CBj Mass Balance for the distillate tray: D j = D j -1 + V j Enthalpy balance on flash brine: B j = (hBj −1 − hvj ) /(hBj − hvj ) B j −1 hvj = f (TSj ) hBj = f (CBj , TBj ) Overall Enthalpy Balance: WR S Rj (TFj − TFj +1 ) = D j −1 S Dj −1 (TDj −1 − T *) + B j −1 S Bj −1 (TBj −1 − T *) − D j S Dj (TDj − T *) − B j S Bj (TBj − T *) (replace WR for WS rejection stage) Heat transfer equation: WR S Rj (TFj − TFj +1 ) = U j A j X (replace WR for WS rejection stage) X = {(TDj − TFj +1 ) − (TDj − TFj )} / ln {(TDj − TFj +1 ) /(TDj − TFj )} U j = f (WR , TFj , TFj +1 , TDj , D ij , D oj , Lij , f ji ) (replace WR for WS rejection stage) S Rj = f (TFj +1 , TFj , CR ) (replace CR for C S rejection stage) S Dj = f (TDj ) S Bj = f (TBj , CBj ) Distillate and flashing brine temperature correlation: TBj = TDj +TE j + EX j + Δ j Distillate and flashed steam temperatures correlation: TS j = TD j +Δ j TE j = f (TDj , CBj ) Δ j = f (TDj ) EX j = f ( H j , w j , TBj ) Brine Heater Model C B 0 = CR B0 S RH (TB 0 − TF 1 ) = Wsteam λS λS = f (Tsteam ) WR S RH (TB 0 − TF 1 ) = U H AH Y Y = {(Tsteam − TF 1 ) − (Tsteam − TB 0 )} / ln {(Tsteam − TF 1 ) /(Tsteam − TB 0 )} S RH = f (TBO , TF 1 ) U H = f (WR , TB 0 , TF 1 , Tsteam , DHi , DHo , f Hi ) Splitters Model BD = BNS − R CW = WS − F Makeup Mixers Models WR = R + F RCBNS + FCS = WR C R WR hW = RhR + FhF hW = f (TFm , CR ) hF = f (TFNR +1 , CF ) hR = f (TBNS , CBNS ) B0 = WR
Note: T* is reference temperature = 0oC
Fig. 2 MSF Process Model 1st LAYER
2nd LAYER
3rd LAYER
2
b1
•
All the inputs and targets are scaled so that they fall within a specified range.
•
50% data are used for training, 25% for validation and 25% for test.
•
Levenberg-Marquardt backpropagation algorithm is used for training to determine the weights and biases of the multi-layered network.
1
2
w11
3
w11 3
2
2
w21
3
2
b1
w12
2
w22 w31
2
b2
1
2
b3
3
w13
2 w32
3 2
2
w42
w41
2
b4
3
w14
4
Input LAYER
Hidden LAYER
Output LAYER
OUTPUT LAYER
INPUT LAYER
1
2
2
w12
Fig.3 Neural Network Architecture for TE Estimation
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Table 1. The NN based Correlation for TE (Tanvir and Mujtaba, 2006) TE j = TEscaleup std _ TE + mean _ TE
TEscaleup = a13 =w113 a12 + w123 a22 + w133 a32 +w143 a42 +b13
a12 =tanh (w112 xscaleup + w122 BPTscaleup +b12 )
a22 = tanh (w212 xscaleup + w222 BPTscaleup +b22 )
a32 =tanh (w312 xscaleup + w322 BPTscaleup +b32 )
a42 = tanh (w412 xscaleup + w422 BPTscaleup +b42 )
xscaleup = ( x − mean _ x ) / std _ x
BPTscaleup = ( BPT − mean _ BPT ) / std _ BPT
std _ x = 2.169 mean_x = 4.037
std _ BPT = 21.02 mean _ BPT = 91.549
std _ TE = 0.352 mean _ TE = 0.606
2nd layer: 2
w212 =0.213 w312 = 0.514 w412 = -0.580 w122 =1.396 w222 =0.087 2 2 2 2 2 w =-0.174 w42 =0.225 b1 = 2.448 b2 = -0.829 b3 = 0.409 b4 =-.2.398 3rd layer: w113 = 0.005 w123 =6.364 w133 = 0.466 w143 = -1.797 b13 = 2.312 w11 =0.917 2 32
Note
BPT = TDj , x (wt%) = CBj (wt/wt) × 100
3. Results In this work we have considered the case reported by Rosso et al. (1996). There are total of 16 stages with 13 recovery and 3 rejection stages. The specifications (satisfying the degrees of freedom) are same as those used in Rosso et al. and are shown in Table 2. The simulation results are presented in Table 3. The results (shown in plain) are in good agreement with those reported by Rosso et al. (shown in italic). The salinity and brine temperature ranges in this work are 6.29-6.82 wt% and 40-90oC. Note the NN based correlation for estimating TE was developed with salinity range 0.19-7.23 wt% and 60120 oC. Despite the temperature range of this work being slightly outside the range of 60-120oC range, the simulation results are quite close those reported by Rosso et al. (1994) even in the temperature range 35-60oC. Having satisfied with the model presented in this work, we have carried out further simulation to study the sensitivity of seawater temperature ( Tseawater ) and steam temperature ( Tsteam ) on the total amount of fresh water produced ( DNS ), Gained Output Ratio (GOR), Top Brine Temperature (TBT) and final bottom brine temperature (BBT). The results are summarised in Table 4. With the increase of Tseawater both TBT and BBT increase for a given Tsteam = 97 C (Cases 1-3). As the terminal temperature difference decreases, for a given design of the plant (heat transfer area, etc.) the amount of heat removal decreases. This consequently reduces the amount of distillate produced per stage, thus reducing the total amount of freshwater. The corresponding reduction of the steam flow rate (Wsteam) keeps the GOR almost constant. This simulation clearly shows that due to seasonal variation, more freshwater will be produced during winter (Case 1) than in summer (Case 3). For a given seawater temperature, Tseawater = 45 C, with the increase of Tsteam , the terminal temperature difference increases. For a given design of the plant (heat transfer area, etc.) the amount of heat removal therefore increases. This consequently increases the amount of distillate produced per stage and the total amount of freshwater (Cases 4-6). A corresponding increase of GOR is thus noticed. Note, to maintain the supply of
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freshwater in summer at the winter level, there has to be an increase in Tsteam from 97C to 116.5C (compare Case 1 and Case 5). To sustain the high temperature operation this might have a knock-on effect on the capital investment. Table 2. Constant Parameters and Input Data
Aj / AH Brine heater Recovery stage Rejection stage
3530 3995 3530
WS 8
1.131*10 kg/h
D ij / DHi 0.022 0.022 0.024
Tsteam
Tseawater
o
o
97 C
D oj / DHo
35 C
w j / L j / LH
f ji / f Hi -4
0.0244 0.0244 0.0254
1.86*10 1.4 *10-4 2.33*10-5
CS
R
5.7 wt%
12.2 12.2 10.7
Hj 0.457 0.457
CW 6
6.35*10 kg/h
5.62*106 kg/h
Table 3: Summary of the Simulation Results F kg/h 5.68*10
BD kg/hr 6
5.68*106
6
WR kg/hr 7
Wsteam kg/hr 7
CR wt/wt
4.75*10
1.203*10
1.188*10
6.29*10-2
4.75*106
1.203*107
1.188*107
6.29*10-2
Stage Profiles (Brine heater stage j =0) Stage
Bj kg/h
0
1.203E+07 1.203E+07 1.197E+07 1.197E+07 1.191E+07 1.191E+07 . 1.131E+07 1.131E+07 1.110E+07 1.110E+07
1 2 . 12
16
Dj kg/h
CBj wt/wt
TFj oC
TDj oC
TBj oC
57238.2 59403.0 115214.2 118730.0 . 715074.6 719700.0
6.29E-02 6.29E-02 6.32E-02 6.32 E-02 6.35E-02 6.36 E-02 . 6.69E-02 6.69E-02
83.79 83.33 80.87 80.41 . 49.31 49.27
86.15 85.75 83.28 82.87 . 51.97 51.93
90.01 89.74 86.15 86.79 84.35 84.01 . 53.15 53.24
930882.7 934410.0
6.82E-02 6.82 E-02
38.19 38.07
39.84 39.98
41.24 41.51
4. Conclusions Here, gPROMS modelling tool has been used to model an MSF process. A Neural Network based correlation developed earlier (Tanvir and Mujtaba, 2006) for estimating TE is embedded within the gPROMS environment. The simulation results using the new model are in good agreement with the published results. NN based correlation predicts TE very well even slightly outside the range of training. The model is then used to study the sensitivity of two important operating parameters: the seawater temperature which is subject to seasonal variation, and the steam temperature in the brine heater which controls TBT of the process (indirectly controlling the design of the process). The
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results show that the steam temperature plays an important role to maintain the production rate of freshwater at different seasons. However, this may be at the expense of costly design. Table 4. Effect of Tseawater and Tsteam on DNS , GOR, TBT , BBT Case
Tseawater
DNS
Wsteam
GOR
TBT
BBT
1 2 3
23 35 45
1.09E+06 9.31E+05 7.88E+05
1.41E+05 1.19E+05 1.02E+05
7.73 7.82 7.72
88.6 90.1 91.0
30.3 41.2 50.2
4 5 6
111.0 116.5 121.0
1.01E+06 1.09E+06 1.16E+06
1.21E+05 1.29E+05 1.35E+05
8.29 8.48 8.64
103.8 108.8 112.9
51.5 52.0 52.5
Tsteam
Gained Output Ratio (GOR)=Total Fresh water produced/Amount of Steam Needed = DNS / Wsteam
Nomenclature Bj
brine flow leaving stage j, kg/h
BD
Blow down mass flow rate, kg/h
CBj CW CS
Dj
Brine concentration, wt/wt Rejected seawater flow rate, kg/h Seawater salt concentration, wt/wt Seawater salinity in the recovery stages, wt/wt Distillate flow from stage j, kg/h
F
Make-up seawater flow rate, kg/h
CR
R TBj
Recycle stream flow rate, kg/h Temperature of flashing brine
TDj
leaving stage j, oC Temperature of distillate leaving stage j, oC Seawater temperature leaving
TFj Ws WR
stage j, oC Seawater mass flow rate, kg/h Seawater flow in the recovery section, kg/h
Reference L.A. Bromley et al., (1974), AIChE, 20, 326. H.T. El-Dessouky, and H. M.Ettouney, (2002), Fundamentals of salt water desalination, Amsterdam, Elsevier Science Ltd. gPROMS, (2005), Introductory User Guide, Process System Enterprise Ltd (PSE), http://www.psenterprise.com/gproms/ A. M. Hellal et al. (1986), Computer and Chemical Engineering , 10, 327-342. A. Hussain et al., (1993), Desalination, 92, 21-41. M. Rosso et al. (1996), Desalination, 108, 365-374. K.S. Spiegler and A.D.K. Liard, (1980) , Principle of desalination, New York, Academic press. M.S. Tanvir and I.M. Mujtaba, (2006), Neural network based correlations for estimating temperature elevation of seawater in MSF desalination process, in press , Deslaination.
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A New Operation Mode for Reactive Batch Distillation in Middle Vessel Columns: Start-up and Operation Irisay Carmona, Harvey Arellano-Garcia, Günter Wozny Department of Process Dynamics and Operation, Berlin University of Technology, Sekr. KWT-9, Str. Des 17. Juni 135, Berlin 10623, Germany
Abstract The most outstanding feature of batch distillation is its flexibility in design and operation. In particular, when chemical reactions and physical separations have some overlapping operating conditions the combination of these tasks can offer significant benefits. In this work, we propose a new operation mode for reactive batch distillation in middle vessel columns. The special feature of this novel operation mode lies on the fact that depending on the characteristics of the reaction mixture, the reaction will also take place either along the upper column or along the lower column. The performance of the novel operation mode is demonstrated through rigorous simulation of two industrial reactive batch distillation processes. Furthermore, since the performance of the start-up procedure has a large impact on the entire reactive batch distillation process, we also focus on the description of the dynamic behavior of the start-up operation starting from the cold and empty batch columns as the initial state. The proposed model includes both equation and variable discontinuity. Thus, we propose in this work a modeling approach to describe the state transitions during the start-up phase. To show its efficiency, the developed approach has been applied to the start-up operation simulation of different batch distillation column configurations with overlapping chemical reactions. Due to the reliable initial state, the developed models are employed for model based optimization and control. Keywords: start-up modeling, reactive batch distillation, middle vessel column.
1. Introduction Batch distillation is used in chemical, food, and pharmaceutical industries where flexibility is needed. This operational flexibility of batch distillation columns makes them particularly suitable for smaller amounts of products with high added value, multiproduct or multi-purpose operations. Moreover, when chemical reactions and thermal separation have some overlapping operating conditions, the combination of these tasks can, in particular, offer considerable benefits. These benefits may involve: avoidance of reaction equilibrium restrictions, higher conversion, selectivity, yield, removal of side reactions and recycling streams, circumvention of non-reactive azeotropes. Although there is an intensive literature on batch distillation, relatively little has been published on reactive batch distillation. In the conventional reactive batch distillation column, the feed is charge into a large reboiler or reactor at the bottom of the rectifying column. The order of appearance of the components in the distillate is determined by the phase equilibrium characteristics of the mixture. At the end of the separation the content in the column will drain down into the still and a residual bottom fraction may be a remaining product. This is particularly appropriate when one of the reaction products has a lower boiling point than the other products and reactants. When all reaction products have
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higher boiling temperatures, the inverted reactive batch distillation will be more suitable. There are, however, disadvantages associated with the use of a conventional reactive batch rectifier or batch stripper. A combination of these configurations is described as the middle-vessel batch reactive column. The feed mixture is loaded into the middle vessel, where the reaction take place, between the two separation sections, and the products are simultaneously obtained from the top and the bottom of the column reducing the way of separation and pushing the reaction further to the product side. Due to the two product streams, a middle-vessel batch reactive column shows superior performance to a typical or inverted batch distillation process, where distillate cuts of the light and intermediate products are taken sequentially. On the other hand, there are, in fact, some reaction systems (transesterifications or esterifications) where none of these batch column configurations are either suitable or can be used economically to improve the rate of conversion, yield and efficiency. Complications usually arise due to the existence of azeotropes or due to the risk of cracking occurrence (e.g. in esters). To overcome these difficulties, a new operation mode for reactive batch distillation in middle vessel columns is proposed with which the efficiency of reactive batch distillation can be improved, e.g., less batch time and energy consumption, less off-spec products, and more product amount.
2. A New Operation Mode for Reactive Batch Distillation For equilibrium-limited reactions such as transesterifications, it has been demonstrated that conducting reaction and distillation simultaneously allows reaction to overcome the limitations of distillation and to bring reactions to complete conversion of the reactants because the products are continuously removed from the reaction zone. Furthermore, an excess of one educt can be used to shift the reaction towards the product side resulting in a semibatch operation mode. To illustrate the new operation mode, we firstly consider a generic reversible reaction of the form B+C<->A+D representing A and D the components with the lowest and highest boiling point, respectively. In the conventional reactive (or middle vessel) batch distillation column the relative volatilities of the reactants and the products should be such that the products can be removed from the reactor i.e. the reaction zone (middle vessel) easily while the reactants remain in the reactor or they will be shifted back to the middle vessel through thermal separation both in the upper column and the lower column such that they (B, C) meet again in the middle vessel (Fig. 1). In this way, the conversion and selectivity can be increased.
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However, for the industrially relevant and for this work exemplary selected transesterification reaction, which can be described as follows: educt alcohol (B) + educt ester (C) <-> product alcohol (A) + product ester (D) isopropanol + methyl myristate <-> methanol + isopropyl myristate, the middle vessel configuration depicted in Fig. 1 can not be used so easily. Both alcohols could, indeed, be separated thermally in the upper column, but in the lower column is not possible to separate the esters due to the risk of cracking. For other reaction systems could be the problem that azeotropes appear. To overcome these problems, the reaction will be displaced to the entire lower column such that the increase of the column temperature will be limited by the chemical equilibrium. For this purpose, the complete middle vessel configuration will be operated in semibatch mode where the excess of educt alcohol (heated steam) will be supplied to the reboiler of the lower column, and the most volatile components (both alcohols) will be evaporated out of the lower column towards the upper column through a vapor bypass configuration during the batch (Fig. 2). Generally speaking, the special feature of the novel operation model lies on the fact that depending on the characteristics of the reaction mixture, the reaction could take place not only in the middle vessel (or reactor) but also along the upper column or along the lower column. In the former case, the reaction will be enhanced by adding one of the educts into the top of the upper column, and a liquid by-pass configuration connecting both columns. In the latter, as stated before, the educt will be supplied to the reboiler of the lower column (Fig. 2). Another interesting feature of the new operation mode is the possibility of performing those reaction/distillation systems with a low tonnage, which are not suitable to be operated continuously, but at the same time too large charge for a batch process. For this purpose, the flow rates of the educt streams in the new configuration (for instance: C into the middle vessel and B to reboiler, see Fig. 2) will change periodically from continuous to discontinuos mode. Concerning this point, the production of dimethyl ether will be presented as a further case study. To simulate the new operation mode in both case studies, a dynamic rigorous detailed model is used. Comparisons with their conventional operation clearly indicate the improvement by the new operation mode.
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3. A Modelling Approach for the Start-up of Reactive Batch Columns An inherent characteristic of batch distillation is that a batch column will frequently be started up from a cold and empty state. Moreover, the amount, composition as well as nature of the components in the initial charge may be variable from batch to batch. Furthermore, due to the overlapping reaction the states variables will constantly change from the very beginning i.e. the use of conventional modeling strategies like a pseudowarm state or total reflux will not provide a reliable initial state (but may be a model consistent initial state). Therefore, modelling and simulation of the reactive batch distillation start-up operation from the cold and empty batch columns as the initial state plays an important role in the optimal design and operation of such processes. Owing to the dynamic nature, initialization of such a system is a challenging problem. Thus, we propose in this work a hybrid modeling approach to describe the state transitions during the start-up phase. The proposed model includes both equation and variable discontinuity. To show its efficiency, the developed approach has been applied to the start-up operation simulation of different batch distillation column configurations with overlapping chemical reactions (conventional, inverted, middle vessel batch configurations). A detailed tray-by-tray model for each operation mode has been developed. The total equation system consists of mass balance, energy balance, vapourliquid equilibrium relations and tray hydraulics. The tray hydraulics is related to the geometry of the trays and essential for computing the pressure drop and hold-up of each tray. The reaction kinetics is added to the model to depict the reaction both in the middle-vessel and the corresponding batch column. During the start-up phase each tray will be described from a non-equilibrium phase, in which only mass and energy transfer are taking place, to an equilibrium phase in which the vapour-liquid equilibrium is held (Wang et al., 2003). The switching point between these two phases is decided by the relationship of bubble point temperature at the operating pressure. The equilibrium state is attained tray by tray either from bottom to top of the column or vice versa, whereas the liquid hold-up of each tray is mainly filled due to the reflux flow. Figure 3 shows the state transition of the trays in the upper and lower batch columns during start-up. Accordingly, at certain time point, a tray may be at the state of empty (EM), liquid accumulation (LA1 or LA2) or vapour-liquid equilibrium (VLE). The new operation mode involves both start-up approaches: upwards and downwards (Fig. 3). However, the outcomes evidently show the efficiency of the start-up approach to describe the dynamic behaviour during the start-up from the cold and empty state. From this issue, some useful start-up heuristics/strategies could be derived and compared. Start-up upwards …
… …
…
EM
j
EM
LA1 VLE
LA1 VLE VLE
…
Bottom
LA1
… …
…
…
…
…
…
…
J-1
EM
…
…
…
…
LA1 LA2
j
EM
LA1
…
…
LA1 LA2 VLE …
J+1
…
EM
LA1
…
LA2 VLE
Bottom
…
…
EM
LA1 VLE
Top
EM
… J-1
J+1
Start-up downwards
…
Tray
Tray
Top
Time
Figure 3. State transition of the trays during the start-up.
Time
…
…
… …
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In Fig. 4 the simulated temperature and product concentration profiles along both column sections are presented. It shows the changes of the state variables at different time points during the batch including the start-up phase. In this work, different start-up strategies have been investigated for the transesterification process operated with the new operation mode. The strategies differ in the time point when to start up both batch columns. They are summarized as follows: a) after the VLE is achieved in the middle vessel, the upper column will get started and the downstream valve to the lower column will not be opened until the whole upper column has been stabilized; b) both the reflux ratio valve of the upper column and the downstream valve will be opened simultaneously; c) shortly after the vapor from the middle vessel (or reactor) arrive at the bottom of the upper column, the downstream valve to the lower column will then be opened. However, with the proposed simulation approach from the cold and empty state, all strategies were squarely evaluated. It could be proven that depending on the reaction system and the purity requirements, one start-up strategy can perform better than the other. For the slightly endothermic tranesterification is the strategie a) according to the results to be presented the most appropriate. Figure 4 shows also that the product alcohol is enhanced by the reaction taking place along the lower column. The conversion of product ester increases towards the reboiler of the lower column.
4. Operational Aspects of the New Operation Mode In this work, rigourous simulation comprising the start-up phase is applied to conduct a dynamic analysis and to develope rough optimal policies for both batch process case studies complying with the equality constraints of the detailed model equations and the production constraints, which include product purity specifications and the physical restrictions considering all operational decision variables. For the new operation mode, a wide variety of degrees of freedom such as reflux ratio (upper column), downstream flow rate from middle vessel to the lower column, feed rate into the lower column, heat supply both in the middle vessel (or reactor) and in the lower column reboiler, and/or the boil up rate were considered. Computation results of applying this operation mode to two industrial reactive batch processes show significant improvements of operation efficiency in comparison to the conventional operation modes on industrial site.
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Figure 5 shows the developed operating policies for the transesterification process as well as the corresponding composition profiles. A slow increase of the reflux ratio in the first 4 hours is allowed, since a large amount of product alcohol results from the increase of the downstream flow rate. However, the increase of the reflux ratio in the upper column is required in order to fulfill the distillate purity requirements. The drastic decrease of the downstream flow rate can be explained by the time delay between the fill up of the lower column during the start-up and the feed supply of educt alcohol. Further on, the educt alcohol is also restricted to a certain amount. Figure 5 also shows the instantaneous values of the distillate composition as well as the time-dependent changes of the lower column reboiler concentration. However, high purity specifications of both product alcohol and product ester can be achieved.
5. Concluding Remarks A new operation mode for reactive batch distillation in middle vessel columns is proposed. Its performance is studied through simulation of two industrial reactive batch distillation processes. Comparisons between the operation modes with respect to the total batch time, the total amount of required energy and the profit of a batch run provide evidence of the advantage using the new operation mode. However, with the proposed operation mode, the benefits of reactive distillation (driving reactions to the product side, azeotrope breaking, and the potential reduction of costs, etc) can be added to those of the batch operation and its flexibility. Furthermore, an efficient modelling approach for the start-up of reactive batch columns starting from the cold and empty state is also proposed. Based on the reaction system, different start-up strategies are derived. Moreover, due to the reliable initial state, the developed models are currently employed for model based optimization and control. Conceptual designs concerning the application of the new operation mode to other relevant complex reaction systems will be introduced.
References Mutjaba, I.M., 2004, Batch Distillation: Design and Operation, Series on Chemical Engineering, Vol.3, Imperial college Press. Sorensen, E., Skogestad, S., 1996, Optimal startup procedures for bath distillation, C&CE(20), 1257-1262. Wang, L., P. Li, G. Wozny, S. Wang, 2003, C&CE (27), 10, 1485-1497.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Towards a novel optimisation algorithm with simultaneous knowledge acquisition for distributed computing environments Siyu Yang, Antonis Kokossis, Patrick Linke Centre for Process and Information Systems Engineering, University of Surrey, Guildford, GU2 7XH, U.K.
Abstract This paper reports on research into novel optimisation schemes for large-scale distributed computing environments that will enable data analysis and knowledge acquisition in the course of optimisation. The scheme incorporates concepts from the Simulated Annealing search strategy in order to ensure robustness. In contrast to Simulated Annealing, which is a sequential optimisation algorithm, the proposed optimisation scheme consists of a number of solution pools, each of which is associated with a system temperature which defines solution quality within the pool. The solutions in these pools are generated by performing constant temperature Markov processes on existing solutions in these pools. As the individual Markov processes are independent they can be completed in large-scale distributed computing environments, constantly producing new solutions which are stored in a central database. During the optimisation, the solutions are regularly reassigned to pools according to their performance relative to the other solutions that have been generated such that the solution quality improves towards the pool associated with the lowest temperature. This final pool accumulates the set of optimal solutions during the optimisation. The solutions of all pools are stored in a central database from which knowledge about the importance of individual solution features can be extracted in the context of the systems performance. Keywords: Optimisation, distributed computing, knowledge acquisition.
1. Introduction Current optimisation methods are of limited use for decision-support in complex systems due to two main short-comings. Firstly, they require long computational times to identify optimal solutions to complex problems. The algorithms are not easily parallelised for use in large-scale distributed computing environments as transitions from initial towards optimal solutions are largely sequential. Distributed environments become increasingly available with the advent of Grid Technologies and new generations of optimisation methods are required that can exploit the vast available distributed computing resources effectively. Secondly, the results obtained from optimisation runs are often difficult to interpret by the user in the context of the decisions to be taken. This is particularly true for stochastic optimisation methods, which tend to be very robust in addressing complex optimisation problems, where important solution features are often blurred by features not strongly impacting on the systems performance. An optimisation algorithm that could exploit large-scale distributed systems and provide the user with optimal solutions alongside insights into
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the importance of individual solution features would be very attractive for decisionsupport. We have devised a novel optimisation algorithm with the aim of addressing the above shortcomings. The algorithm incorporates concepts from the stochastic optimisation strategy Simulated Annealing to enable robust optimisation, whilst doing away with the inherently sequential nature associated with this meta-heuristic-based search scheme. This sequential nature of the search has hampered previous efforts of parallelising Simulated Annealing algorithms. As a result, past developments allow only minor distribution of computations and the number of processors that can be utilised in optimisation is severely limited. The number of processors that can be deployed depends upon the length of the homogenous Markov chain to be executed at a given system temperature which is also an essential parameter to influence the performance of optimisation (Leite and Topping, 1999). The limited ability of the algorithm to exploit vast distributed computing resources presents a major deficit that prevents the exploitation of advances in computing infrastructures in the form Grids (Antonopoulos et al., 2005). The proposed new algorithm will enable the full exploitation of such resources. Besides allowing the large-scale distribution of the optimal search, the algorithm enables the analysis of information generated during the optimal search as all intermediate and optimal solutions are stored in a central database. It is therefore possible to device information mining schemes that allow the acquisition of knowledge about the individual solution features in the context of the solution performance and to identify those solution features that strongly impact on the systems performance. The following sections outline the new optimisation scheme. An application to a global optimisation test problem is presented to illustrate the performance of the algorithm. The development of information analysis schemes that allow efficient knowledge acquisition for a number of process systems engineering problems is the focus of current research and will be reported separately.
2. Distributed optimisation algorithm development 2.1. Architecture The architecture of the novel optimisation algorithm is shown in Figure 1. The algorithm features a number of pools, each of which is associated with a systems temperature that controls the distribution of solution quality. The highest temperature pool (T1) accepts almost all possible solutions to the problem, whereas the lowest temperature pool (TN) only admits solution of the highest quality. The algorithm is initialised by assigning a number of feasible solutions to the problem to the highest temperature pool (T1). Agents randomly select pools and solutions and perform Markov processes at the corresponding pool temperature for each selected solution. All solutions visited during the Markov process are returned into the corresponding pool. As Markov processes are random operations, some solutions generated at a high temperature will be of high quality and would warrant membership of a lower temperature pool. This calls for a dynamic update of pool memberships based on the solution quality distributions defined by the pool temperatures. This is realised through periodic distribution of solutions amongst the pools. As the search progresses, more and more solutions will penetrate the lower temperature (high performance) pools and the algorithm is terminated as a sufficient number of solutions that warrant membership of the lowest temperature pool has been generated.
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Initial solutions
Pool 1 (T1)
Initial solution
Agents to execute Markov process
Group of solutions Pool 2 (T2) Pool 3 (T3) Pool 4 (T4)
Distribution of solutions according to acceptance criteria
CONTINUE
Pool 5 (T5)
Pool N (TN)
Enough solutions
No
Yes
Figure 1 Novel optimisation algorithm 2.2. Pools and Agents The concept of pools and agents allows the massive parallelisation of optimisation experiments as the agents will be able to constantly generate solutions at different pool temperatures which are stored in a central solution database that also stores information about the pool associated with a solution. In contrast to existing stochastic optimisation methods, there is no direct link to a solution from a previous iteration. This absence of successive transitions, which has hampered previous attempts to parallelise Simulated Annealing, enables massive parallelisation of the optimisation. The periodic distribution of solutions among pools can be performed in parallel to the execution of the Markov processes so that the idle times of the algorithm would be minimal. 2.3. Acceptance and termination criteria The distribution of solutions among different pools requires acceptance criteria to decide on the membership of a solution in a given pool. For a minimisation problem, we accept a solution into a pool at temperature T if: ⎛ Cur (So) − Min (So) ⎞ Exp⎜ ⎟ ≥ Rand (1) T ⎝ ⎠ where Cur(So) is the objective function value of a candidate solution So to be distributed, Min(So) is the current best solution in the pool and Rand is a random number (0 < Rand < 1). The acceptance criterion resembles the Metropolis criterion
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employed in Simulated Annealing (Metropolis et al 1953) and has been implemented in the first instance. The average solution quality and the quality distribution improves from the highest temperature to the lowest temperature pools. The lowest temperature pool therefore contains only the best solutions with the lowest distribution of solution quality. The more solutions are present in the lowest temperature pool, the higher will be the probability that the optimal solution has been found. The search is terminated once a specified number of solutions have entered the lowest temperature pool. 2.4. Prototype implementation We have set up a small prototype system to test our algorithm. An SQL2000 database was set up to store the pools on our research center’s central server. The agents, capable of obtaining an initital solution from a pool, executing a Markov process at the pool temperature, and returning a set of solutions into the pool, as well as the solution redistribution algorithm were coded in fortran 95 with fortransql library. The agents executed their Markov processes on a 731MHz Intel Pentium III processor. The PC and the server communicated via our local area network.
3. Illustrative example We have tested the algorithm on five well-studied nonconvex nonlinear test problems given by Floudas et al. (1999). For lack of space, we can report on only one problem here: ⎧ (0.0039 × x7 + 0.0039 × x8) × (495 × x4 + 385 × x5 + 315 × x6) ⎫ min ⎨ ⎬ x10 ⎩ ⎭ subject to - 0.5 × x9 × x4 × (0.8 × x7 + 0.33333333 3333333 × x8) + x1 = 0 - 0.5 × x9 × x5 × (0.8 × x7 + 0.33333333 3333333 × x8) + x2 = 0 - 0.5 × x9 × x6 × (0.8 × x7 + 0.33333333 3333333 × x8) + x3 = 0 x10 - x7 - ( x8 - x9 ) ≥ 0 x1 - 8.46527343 75 × x10 ≥ 0 x2 - 9.65006510 416667 × x10 ≥ 0 x3 - 8.87167968 75 × x10 ≥ 0 0.5 × x1 × x9 - 2.2 × (8.4652734 375 × x10) 1.33333333 0.5 × x2 × x9 - 2.2 × (9.6500651 0416667 × x10)
≥0
1.33333333 333333
0.5 × x3 × x9 - 2.2 × (8.8716796 875 × x10) 1.33333333 x4 - 0.01117717 47883801 × x7 ≥ 0.2 x5 - 0.01376553 60411427 × x7 ≥ 0.2 x6 - 0.01556638 72253648 × x7 ≥ 0.2 x4 - 0.01117717 47883801 × x8 ≥ 0.2 x5 - 0.01376553 60411427 × x8 ≥ 0.2 x6 - 0.01556638 72253648 × x8 ≥ 0.2
333333
333333
≥0
≥0
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Table 1. Effect of algorithmic parameters Poolnum and Markov on solution quality Poolnum = 100
Markov = 100
Markov
st
Av_Obj
Poolnum
st
Av_Obj
10
8.4E-05
-47.7063
10
2.01E-01
-46.4726
50
1.48E-04
-47.7061
50
1.52E-04
-47.7058
100
1.16E-04
-47.7061
100
1.16E-04
-47.706
500
2.81E-04
-47.7055
500
9.57E-05
-47.7061
Table 2. Comparison of the new optimisation algorithm with Simulated Annealing Simulated Annealing
Novel optimisation algorithm
CPU (sec)
Av_Obj
St
Markov
Poolnum
CPU (sec)
Av_Obj
St
1017.8
-47.7006
1.39E-03
500
100
102.0
-47.7045
5.86E-04
We studied the importance of the two key algorithmic parameters, the length of the Markov chains (Markov) and the number of pools employed (poolnum). The searches were terminated after at least ten solutions have penetrated the lowest temperature pool. The average objectives (Av-Obj) and the standard deviations (st) over all solutions in the lowest temperature pool are reported in Table 1. The performance of the algorithm clearly improves with the number of pools present as a result of a better equilibration of the system during cooling. However, the performance appears independent of the length of the individual Markov processes and very good performances were observed for the shortest chains studied. This behaviour has been observed for all problems studied so far and suggests that massive parallelisation of the algorithm is indeed possible. We also solved the problem using conventional Simulated Annealing to establish a basis for comparison. The Simulated Annealing implementation employed a perturbation framework identical to the one used in our new algorithm. We developed targeting curves with increasing Markov chain lengths for sets of ten runs per case. The performance improved with the Markov chain length but the quality of the solutions did not match those obtained using of the new algorithm, even for extremely long chains. Table 2 compares the performance of new algorithm with that of Simulated Annealing for the case of the longest Markov chains studied (1000). It can be seen that the new algorithm outperforms Simulated Annealing in terms of solution quality and offers massive savings (90%) in CPU time for the case of Markov = 500 and poolnum = 100. Similar observations were made for different combinations of these two parameters. The presented algorithm showed similar behaviour when applied to four other test problem. Most importantly, the performance was observed to be independent of the Markov parameter, which indicates the high potential for massive parallelisation. Detailed results from these tests will be published separately.
4. Conclusions We have presented a new optimisation method that is suitable for large-scale distributed computing environments. The algorithm carries the strengths of stochastic optimisation
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methods such as Simulated Annealing in terms of global optimisation capabilities. A comparison with Simulated Annealing indicates that the new algorithm is also highly computationally efficient. In the absence of sequential searches, the algorithm is, in principle, not limited by the number of processors it can exploit. The algorithm will be applicable to a wide range of optimisation problems in operations as well as in design. As the solutions of all pools are stored in a database, knowledge about the importance of individual solution features can be extracted in the context of the systems performance. This is the focus of current research. We are also in the process of setting up a distributed test bed to evaluate the algorithm further. Applications to typical process and product design problems as well as problems in process operations will be the focus of future activities.
References N. Antonopoulos, P. Linke and A. Kokossis. Chemical Engineering Communications, 28(2004) 2391 J.P.B. Leite and B.H.V. Topping, Computers and Structure Jounal 73 (1999) 545 N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, Chemical Physics, 21(1953) 1087 C.A. Floudas, P.M. Pardalos, C.S. Adjiman, W.R. Esposito, Z. Gumus, S.T. Harding, J.L. Klepeis, C.A. Meyer, and C.A. Schweiger, Handbook of Test Problems in Local and Global Optimization, Kluwer Academic Publishers, 1999.
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Floating Index of Inequality Constrained DAE Systems Domingos Fabiano de S. Souza,a Roberta C. Vieira,b Evaristo C. Biscaia Jr.a a
Programa de Engenharia Química, PEQ/COPPE/UFRJ, Universidade Federal do Rio de Janeiro, CP 68.502, CEP 21.945-970, Rio de Janeiro, Brazil b PETROBRAS Petróleo Brasileiro S.A, Av. República do Chile 65 room 802 CEP 20.035-900, Rio de Janeiro, Brazil.
Abstract Problems of dynamic optimisation with inequality path constraints are common in industrial plants. These constraints describe conditions of the process when it operates with extreme values of the variables, based on safety and/or economics restraints. Normally, during the optimal trajectory some of the inequality constraints are activated, and those remain active during a certain period of time. This behaviour can produce a change in the differential index of the DAE system, leading to the so-called floating index phenomena (Feehery and Barton, 1998). This contribution is motivated by the high computational costs typically associated with each of the steps for the resolution of the floating index problem. The proposed new method unifies the advantages of special regularisation functions with numerical codes which integrate higher index DAE systems, avoiding the reinitialisation and index reduction steps. All the inequality constraints are described by appropriate continuous functions and the resulting DAE system can be numerically integrated directly using numerical code such as PSIDE (Lioen et al., 1998). This new procedure has been applied to two typical example: optimal control problem of index two with a state variable inequality constraint (Jacobson and Lele, 1969) and state constrained Van der Pol oscillator of index one. The main advantage of the new method is that the DAE system can be integrated continuously, preventing the restart of the numerical integration every time an inequality constraint is violated. The obtained results are identical with those obtained elsewhere encouraging new developments and extensions. Keywords: dynamic optimisation, regularisation functions, floating index DAEs.
1. Main Text Chemical processes models are limited by constraints that represent safety conditions, chemical or physical equilibrium or economical constraints. These constraints are generally represented by inequality equations and can be applied to control or state variables. During the dynamic simulation, the exact time when an inequality restriction is activated is normally unknown. After the constraint is activated, a new equation (or information) must be included into the mathematical model, and this equation must be satisfied until the constraint is no longer active. A possible consequence of this fact is that the differential index of the differential-algebraic equation (DAE) system representing the mathematical model of the process can change during the dynamic simulation. This behaviour characterizes the so-called floating index DAE system. Methods of resolution of dynamic optimisation problems with inequality constraints (in the state variables) can be classified in two groups, according to the level of adopted
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discretisation: total discretisation (or simultaneous approach) and partial discretisation (or sequential approach). In the first group, the dynamic system is totally discretised resulting in an algebraic system which, along with the equality and inequality constraints, is annexed to the code of non-linear programming (NLP). An advantage of this approach is the ease of manipulation of the inequality restrictions (Cuthrell and Biegler,1987 and Longsdon and Biegler,1989). However, its spectrum of application limited to a family of particularly simple and relatively small problems. For the second group, only the control variable is discretised. The resulting system of equations can be solved by techniques of dynamic programming or with non-linear programming (NLP) strategies. The main characteristic of this technique is that at each iteration of the NLP code a numerical integration of the dynamic system must be performed. Within the sequential approach, there are two different ways to handle the inequality constraints. (a) approximate methods. In this context, the inequalities constraints are evaluated in the neighbourhood of the feasible region by: (i) introduction of square slack variable, converting inequality constraint to equality (Jacobson and Lele, 1969, Bryson and Ho, 1975); (ii) measuring the degree of violation of the constraint over the entire trajectory by max operator or square max operator (Vassiliadis et al. 1994); (iii) dislocating the limit of the constraint inside of an error defined previously - smooth approximation (Goh And Teo, 1988); (iv) discretising the inequality constraints on a finite number of points and satisfying at the end of the segments (Chen and Vassiliadis, 2004). (b) direct methods. A second context consists of manipulating directly the inequalities and identifying the events (Park and Barton, 1994 and Guyou and Petzold, 2002) of activation and deactivation of the restriction. In this approach, the following steps are needed for the numerical resolution: (i) detection of activation/deactivation of constraints; (ii) index determination (and frequently index reduction); (iii) model switching; and (iv) determination of consistent initial conditions to restart integration (Feehery and Barton, 1998). In both methods, every time an inequality constraint is reached, a new DAE system must be built, a new set of consistent initial conditions must be determined and an index reduction method must be applied in order to restart the numerical integration (Majer et al., 1995, Park and Barton, 1996, Guiyou and Petzold, 2002). The result of the activation and deactivation of the restrictions can be the change in the differential index of the system during the optimisation process and integration. The numerical effort associated to each of those steps increases the computational cost. In this work, all the inequality constraints are described by appropriate continuous functions and the resulting DAE system can be integrated continuously. The new method allies the advantages of special regularisation functions with numerical codes that integrate higher index DAE systems, avoiding the reinitialisation and index reduction steps every time one inequality constraint is violated. This new procedure has been applied to typical example with inequality state constrained. The code PSIDE (Lioen et al., 1998) has been used for numerical integration. The obtained results are identical with obtained elsewhere encouraging new developments and extensions.
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2. Numerical Example Two examples are presented to illustrate the proposed methodology: (i) an optimal control problem with a state variable inequality constraint (Jacobson and Lele, 1969); and (ii) state constrained Van der Pol oscillator (Vassiliadis et al. 1994). Example 1 - Optimal Control Problem with a State Variable Inequality Constraint (Index Two) This problem was originally presented by Jacobson and Lele (1969) and consists in minimize the state variable y3 at final time (tfinal=1) through manipulation of control variable u(t), restricted between lower and upper bounds of -3.0 and 15, respectively. The dynamic system equations are presented in Table 1. Table 1 – Set of Equations of Illustrative Example1. dy1 = y 2 , with y 1 (0) = 0 dt
(1)
dy 2 = - y 2 + u , with y 2 (0) = − 1 dt
(2)
dy 3 = y 12 + y 22 + 0.005u 2 , with y 3 (0) = 0 dt
(3)
y 2 (t ) - 8(t - 0.5) + 0.5 ≤ 0
(4)
2
The main idea of the proposed methodology is to smooth, during the numerical resolution, the transition between the constrained condition to the unconstrained condition. This procedure needs both: (a) the selection of the regularization function and (b) determination of the conditions that describe the feasible and infeasible region. The use of regularization functions in the automatic initialisation of algebraicdifferential systems has been proposed by Vieira and Biscaia Jr. (2000). The authors have established some criteria to guide the selection of those functions and their parameters. The chosen function for the present work is shown in Equation (5), where ξ is a parameter defined by the user (usually 0 < ξ << 1). ⎛ ⎛ g(y, t ) ⎞ ⎜ ⎜⎜ ⎟ ⎜ ξ ⎟⎠ 1 ⎜ ⎝ λ(g, ξ) = ⋅ 1 − 2 2 ⎜ ⎛ g(y, t ) ⎞ ⎜ ⎟⎟ 1 + ⎜⎜ ⎜ ⎝ ξ ⎠ ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(5)
The determination of the set equations that describe the feasible and infeasible region is guided by an analysis of the behaviour of the inequality constraint before and after activation. In the illustrative example, the inequality is converted into a new algebraic
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equation with a new algebraic variable y*, which is equal to the state variable y2 when the inequality constraint is inactive and by 8(t – 0.5)2 - 05 when the inequality constraint is active. In the present example, the sum of the conditions that characterize the feasible region and infeasible is represented by equation:
[
]
y* = λ[g(y, t ), ξ ] ⋅ y 2 + [1 − λ[g(y, t ), ξ ]] ⋅ 8(t - 0.5)2 − 0.5
(6)
where g(y,t) is the inequality constraint and λ[g(y, t ), ξ ] is the regularization function that presents the following property: ⎧1 para arg < 0 λ(arg, ξ) ≅ ⎨ ⎩0 para arg ≥ 0
(7)
The state variable y2 is replaced by the new state variable y* in Equations (1) and (3). Then, the new dynamic model system is rebuilt with the Equations (1) to (3) and Equation (6). It should be pointed out that the computer code PSIDE (Lioen et al., 1998) has been used to perform the numerical integration of the correspondent DAE system. This code can deal with fully implicit DAE systems of index up to 3, and its selection eliminates the need of index reduction. In this example, the index of the system is equal to 2 during the activation of the inequality constraint. The profiles obtained for the objective function and the unconstrained and constrained state variables are presented in Figures 1 and 2, respectively. 0.16 0.14 0.12 0.10
y3(t)
0.08 0.06 0.04 0.02 0.00 -0.02 0.0
0.2
0.4
0.6
0.8
1.0
Time
Figure 1 – Objective function profile for example 1.
0.2
1.6 1.2
-0.2
0.8
-0.4
0.4
y*(t)
y2 (t)
0.0
-0.6
0.0
-0.8
-0.4 -0.8
-1.0 0.0
0.2
0.4
0.6
Time
0.8
1.0
inequality constraint 0.0
0.2
0.4
0.6
0.8
1.0
Time
Figure 2 - Unconstrained and constrained state variables profiles in example 1.
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Example 2 - State Contrainted Van der Pol Oscillator (Index One) This problem was presented by Vassiliadis et al. (1994) and consists of minimizing the state variable y3 at final time (tfinal=5) through manipulation of control variable u(t), restricted between its lower and upper bounds of -0.3 and 1.0, respectively. Table2 – Set of Equations of Illustrative Example 2.
(
)
dy1 = 1 - y 22 ⋅ y1 - y 2 + u , with y1 (0) = 0 dt
(8)
dy 2 = y1 , with y 2 (0) = 1 dt
(9)
dy 3 = y12 + y 22 + u 2 , with y 3 (0) = 0 dt
(10)
y1 (t ) ≥ - 0.4
(11)
In this example, the feasible region is limited by Equation (11). When this constraint is active, the time derivative of the variable y1 (represented by the right hand side of Equation 8) must be null, what leads to an additional constraint for the control variable. Hence, two new algebraic equations are added to the original system in order to represent the restrictions on the state variable, y* = λ[g(y, t ), ξ ]⋅ y1 + [1 − λ[g(y, t ), ξ ]]⋅ (- 0.4 )
and on the control variable.
[
(12)
(
u * = λ[g(y, t ), ξ ] ⋅ u + [1 − λ[g(y, t ), ξ ]] ⋅ y 2 + 0.4 ⋅ 1 - y 22
)]
(13)
The results obtained for state variables are presented in the Figure 3. 3.2
y3
2.8
y2 *
y
State Variables
2.4 2.0 1.6 1.2 0.8 0.4 0.0 -0.4 0
1
2
3
4
5
Time
Figure 3 - State variables profiles in example 2.
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3. Conclusions In this contribution, a novel strategy for resolution of floating index DAE has been presented. In the two examples presented, the index of the DAE systems changed when the constraints became active, characterizing the floating index behaviour. The smooth switch between models via a regularisation function has been effective and suitable. Numerical results previously presented for those systems have been reproduced and the simulation effort has been greatly reduced, since the steps of reinitialization and index reduction have been completely eliminated of the simulation. The regularization function used to change the value of the weight λ from 0 to 1 (or vice-versa) is continuous up to the first derivative. If a higher degree of continuity is required, alternative formulations have been tested by the authors. The reported function, Equation (5), has been considered the most suitable after a cost benefit analysis. Additional examples have been studied by the authors, and the results obtained have always been encouraging, Unfortunately, due to space limitations, it has not been possible to present additional numerical results, or even to extend the discussion concerning the examples presented. The methodology proposed in this work for handling the inequality constraints depends on the particular model being solved. However, this “taylor-made” characteristic does not compromise its utilization, due to the lack of extensive algebraic manipulation (such as differentiations) and to the simplicity of final formulation of the problem.
References A. E. Bryson, Y-C. Ho, 1975, Applied Optimal Control, Hemisphere Publishing Corporation, Washington D. C.. C. J. Goh, K. L. Teo, 1988, Control Parameterization: A Unified Approach to Optimal Control Problems with General Constraints, Automatica, v. 24, pp. 3-18. C. Majer, W. Marquardt, E. D. Gilles, 1995, Reinitialization of DAE’s After Discontinuities, Computers and Chemical Engineering, v. 19, Suppl., pp. 507-512. D. H. Jacobson, M. M. A. Lele, 1969, Transformation Technique for Optimal Control Problems with a State Variable Inequality Constraint, IEEE Trans. Autom. Control, v. 14, 5, pp.457-464. J. E. Cuthrell, L. T. Biegler, 1987, On the Optimization of Differential-Algebraic Process Systems, AIChE J., v. 8, pp. 1257-1270. J. S. Logsdon, L. T. Biegler, 1989, Accurate Solution of Differential-Algebraic Optimization Problems, Ind. Eng. Chem. Res., v. 28, pp. 1628-1639. M. Guiyou, L. R. Petzold, 2002, Efficient Integration Over Discontinuities for DifferentialAlgebraic Systems, Mathematics and Computers in Simulation, v. 43, pp. 65-79. R. C. Vieira, E. C. Biscaia Jr., 2000, Direct Methods for Consistent Initialisation of DAE Systems, Computers and Chemical Engineering, v. 25, pp. 1299-1311. T. Park, P. I. Barton, 1996, State Event Location in Differential-Algebraic Models, ACM Transactions on Modeling and Computer Simulation, v. 6, 2, pp. 137-165. T. W. C. C. Chen, V. S. Vassiliadis, 2004, Inequality path constraints in optimal control: a finite iteration ε-convergent scheme based on pointwise discretisation, Journal of Process Control, v. 15, pp. 353-362. V. S. Vassiliadis, R. W. H. Sargent, C. C. Pantelides, 1994, Solution of a Class of Multistage Dynamic Optimization Problems. 1. Problems with Path Constraints, Process Design and Control by Ind. Eng. Chem. Res., v. 33, pp. 2123-2133. W. F. Feehery, P. I. Barton, 1998, Dynamic Optimization With State Variable Path Constraints, Computers and Chemical Engineering, v. 22, 9, pp.1241-1256. W. M. Lioen, J. J. B. De Swart, W. A. Van Der Veen, 1998, PSIDE users guide, Report MASR9834, CWI, Amsterdam, Holanda. URL: http://www.cwi.nl/cwi/projects/PSIDE.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Predictive modeling of ionic permselectivity of porous media Libor Seda, Juraj Kosek ICT Prague, Department of Chemical Engineering, 166 28 Prague 6, Czech Republic
Abstract Transport and separation processes in ionic systems located in the porous medium are investigated. The software for the modeling of the combined electroosmotic, migration, diffusion and pressure driven flow in the spatially 2D or 3D porous medium has been developed. This software allows to determine the mapping from the parametric space of the porous structure and distribution of fixed charge into the space of application properties such as ionic permselectivity or perfusion flow. Software capabilities are illustrated on case studies of systems where the Debye length is either comparable or negligible with respect to the characteristic pore size. The concept of the reconstructed porous medium has been employed to represent the morphology. Keywords: ionic system, permselectivity, reconstructed porous media, electroosmotic flow, concentration polarization.
1. Introduction Porous media of various types are involved in practical applications involving ionic electrolytes, e.g., capillary electrochromatography (CEC), nanofiltration, electrodialysis, fuel cells and other membrane applications. One of the important practical problems is to find the relation between the micro-structure of porous media, the distribution of concentrations of ions and the electroosmotic flow through the porous media. Another practical problem is the prediction of the permselectivity of membranes from their known structure, from the distribution of fixed charge in the membrane and from the concentration of the electrolyte. We have applied the methodology of the reconstructed porous medium and employed the solution of the Navier-Stokes, Poisson and mass balance equations to address the above mentioned problems. The previous attempts to describe and model the electroosmotic flow in general porous media followed one of three principal methodologies forced by computational feasibility and/or introduced idealization of the original system: (i)digitally reconstructed porous media with negligible Debye length [1], (ii)networkmodelsofinter-connectedcylindricalpores[2],and(iii)reductionto effective-scale(averaged)1Dmodelsormodelswithsimplegeometry,suchas2D cylindricalmodel[3].
2. Reconstructed porous medium Spatial distribution of phases can be represented in the general case by the so called phase function fi : R3 Æ {0;1} for each phase i. The phase function of the pore phase is defined as [4] ⎧1 if r ∈ pore , f g (r ) = ⎨ (1) ⎩0 otherwise .
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By definition only one phase can be present at any point r ∈ R3. In a discrete form the phase function fg(r) becomes the phase volume function which assigns each finite volume element of space (voxel) a value from the set {0;1}. In a practical implementation the domain on which the phase volume functions are specified is typically a cubic grid of Nx × Ny × Nz voxels, cf. Figure 1. Such a region of real space is called the computational unit cell. The relationship between the unit cell and the porous medium of interest depends on the absolute dimensions of the medium and on the spatial resolution at which the medium is represented (feature dimensions). The unit cell can either contain the entire medium plus some void space surrounding it, or be a sample of a much larger (theoretically infinite) medium. In the latter case the dimensions of the unit cell must be such that the unit cell is statistically representative of the entire medium.
(a)
(b) Figure 1. (a) Porous slot, section of Fig. 1b.; black color is pore and blue color is solid. (b) Reconstructed granular medium. Periodic boundary conditions. The reconstruction of the porous/multi-phase medium is the process starting from the image obtained by the electron microscopy or by other imaging techniques, followed by the evaluation of the suitable morphological descriptors of the image, and concluding by the generation of the spatially 3D porous/multi-phase medium with the same morphological characteristics as those of the original image. In mathematical terms, given the porosity and the suitable morphological characteristics, we want to reconstruct the replica of the porous medium represented by the phase function fg(r) defined on the discrete grid of voxels with coordinates r. The overview of several algorithms of the stochastic or diagenetic reconstruction of the porous media is available in [5]. The reconstructed porous medium is then used as the input for the calculation of effective transport, mechanical and electric properties of the medium [6], or as the input for the modeling of various reaction, transport and transformation processes.
3. Model equations Stationary distributions of velocity and pressure for the incompressible liquid are obtained by the solution of the continuity and Navier-Stokes equations ∇⋅v =0, (1) ρ (v ⋅ ∇v ) = −∇p + η∇ 2 v − q∇φ , where v is the velocity, ρ is the density, p is the pressure, η is the dynamic viscosity, q is the charge density and φ is the electric potential. The term (– q∇φ) is the electric force acting on the unit volume of the fluid. The charge density q is the linear combination of concentrations of ions ci ,
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∑i zi ci ,
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(2)
where F is the Faraday constant and zi is the charge number of the i-th species. Local electroneutrality assumption (q = 0) usually holds in the bulk electrolyte far from charged surfaces. In the system with the Debye length comparable with the characteristic pore size, the distribution of the electric potential is obtained from Poisson equation ∇ 2φ = −q /(ε r ε 0 ) , (3) where εr and ε0 are the relative and absolute permittivity. The steady state local balance of species i in the ionic system without chemical reaction is considered in the form ∇ ⋅ Ji = 0 , with J i = vci − Di ∇ci − F /( RT ) z i Di ci ∇φ , (4) where Ji is the molar flux intensity of the i-th species defined by the Nernst-Planck equation, Di is the diffusivity of the i-th species, R is the gas constant and T is the temperature. The balance (4) is considered for all species, i = 1,…, N . The system of equations (1)–(4) with appropriate boundary conditions for the set of state variables {p, v, φ, q, ci} is solved. The boundary conditions at the surface of the solid phase are n⋅v = 0 , n ⋅ ∇φ = σ /(ε r ε 0 ) , n ⋅ Ji = 0 , i = 1,..., N , (5) where n is the local outer unit normal vector of the solid phase and σ is the surface charge density. The conditions on the boundary of the computational domain depend on the considered case: (i) boundary conditions (5) are applied for the impermeable wall with immobilized zero or non-zero surface charge density, (ii) Dirichlet boundary conditions, or (iii) periodic boundary conditions. For example, Dirichlet boundary conditions on the left and right boundary of the computational domain are left boundary : p = pleft , φ = φleft , ci = ci, left , i = 1,..., N , (6) right boundary : p = pright , φ = φright , ci = ci, right , i = 1,..., N .
When the Debye length is negligible when compared to the characteristic pore size, e.g., in the case of highly concentrated electrolytes, the grid required to solve the model equations (1)–(4) in the vicinity of the solid phase with a non-zero surface charge density σ would have to be very fine. In an alternative approach the very thin electric double layer is excluded from the computational domain, so the electric term (– q∇φ) in eq (1) is not considered and local electroneutrality assumption (q = 0) holds in the electrolyte. The set of state variables reduces to {p, v, φ, ci}, and the system is described by the modified eqs (1’), (2’) and by eqs (4), ∇⋅v =0, (1’) ρ (v ⋅ ∇v ) = −∇p + η∇ 2 v , 0=F
∑i zi ci ,
(2’)
and the boundary conditions at the surface of the solid phase change to n⋅v =0, t ⋅ v = − μ eo (t ⋅ ∇φ ) , n ⋅ Ji = 0 , i = 1,..., N , (5’) where t is the matrix containing vectors spanning the tangential plane of the solid surface. The boundary condition containing the electroosmotic mobility μeo is the von Smoluchowski equation. Model equations were discretized on the rectangular (in 2D) or cubic (in 3D) grid and processed by the finite element method and resulted in the set of linear and nonlinear equations. This set of equations was solved by the Newton method and the system
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of linear equations solved during each iteration was solved by the LU decomposition of the banded matrix in 2D or by the multigrid algorithm in 3D [7].
4. Electroosmotic flow in porous media with negligible Debye length The distribution of the fluid velocity, pressure and electric potential in the spatially 2D porous structure with the assumption of the negligible thickness of the electric double layer was computed. The system was described by the set of equations (1’), (2’) and (4) with boundary conditions (5’) applied on the surface of the solid phase. Periodic boundary conditions were employed on the lower and upper boundary in Figure 2a. Boundary conditions on the left and right boundary are also considered to be periodic, (7) pright − pleft = Δp , φright − φleft = Δφ , ci , right − ci, left = 0 , for i=1,…,N. The obtained solution is visualized on Figure 2 for a spatially 2D domain with a coarse discretization. In the finite element method bilinear base functions are used for state variables {p, φ, ci}, but biquadratic base functions are employed for v. Multigrid algorithm performs well in 3D and computations with reconstructed porous media larger than 100 × 100 × 100 voxels are feasible on personal computers, cf. Figure 2b.
(b)
(a)
Figure 2. Electroosmotic flow in spatially 2D and 3D porous domains (Δp = 0, Δφ > 0). Arrows represent the velocity distribution. (a) The size of the considered domain is 10 × 10 μm.
5. Ionic permselectivity in a narrow capillary and in a porous medium The concentration polarization can develop in the case of the Debye length comparable with the pore diameter. It is demonstrated in Figure 3 for the simple case of migration of ions through the spatially 2D microchannel with a nonuniform distribution of surface charge density σ,
⎧− 5 × 10 -5 C/m 2 for x ∈ 0.67, 1.33 μm
σ =⎨
⎩
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The intensity of the externally applied electric field is E = |–∇φ| = 150 kV/m. Concentration of the uni-univalent electrolyte on the left and on the right boundary is cleft = cright = 0.001 mol/m3. The considered system is described by eqs (1)–(4) and applied boundary conditions are (5) and (6). Co-ions migrate from left to right due to
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the externally applied potential difference, cf. Figure 3b. Because of the electric repulsion between the negatively charged ions and negatively charged walls of the channel the co-ions with concentration c– accumulate at x < 0.67 μm in front of the charged part. The concentration of counter-ions c+ increases in the middle part of the capillary due to the attractive interaction with the wall, cf. Figure 3a. The ratio of molar fluxes (selectivity) of counter-ions to co-ions is J+/J– = 1.69. 3
+ c [mol/m ] 0.0026
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x [m]
0.0
0.0
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(b) Figure 3. Distribution of ionic concentration in the spatially 2D microchannel: a) counter-ions c+; b) co-ions c– .
The concentration distribution in the narrow micro-channel containing charged particles with the surface charge density σ = –2×10-5 C/m2 subjected to the externally applied electric field E = 150 kV/m is displayed in Figure 4. The surface charge distribution on the walls of the micro-channel is ⎧⎪− 2 × 10-5 C/m 2 for x ∈ 0.47, 1.67 μm σ =⎨ 0 C/m 2 for x ∉ 0.47, 1.67 μm ⎪⎩ The concentration of the uni-univalent electrolyte on the left and on the right boundary is cleft = cright = 0.001 mol/m3. Negatively charged co-ions migrate from the left to the right part of the micro-channel. The middle part of the micro-channel containing charged particles is depleted of co-ions and enriched by counter-ions because of electrostatic interactions with charged surfaces. The concentration polarization develops due to the accumulation of negatively charged co-ions at x < 0.47 μm. The ratio of molar fluxes (selectivity) of counter-ions to co-ions is J+/J– = 2.47. y [m]
y [m]
8e-7
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+
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Figure 4. Distribution of ionic concentration in the spatially 2D microchannel filled with charged particles: a) counter-ions c+; b) co-ions c– .
6. Conclusions The principal parameters of the porous media (e.g., membranes) that affect their performance in electric field driven applications are: (i) spatially 3D structure of the porous media, (ii) type and distribution of fixed charged groups in the porous structure, and (iii) type and concentration of the electrolyte. The fixed charge groups in the membrane cause the excess of counter-ions and the exclusion of co-ions in the pores. However, the absolute concentration of the fixed charge itself is not relevant for the permselectivity of the membrane. Most current modeling studies describe the transport processes in porous media represented either as simplified geometrical structures (e.g., cylindrical capillaries), or as effective-scale (i.e., spaceaveraged) pseudo-homogeneous media. We have developed the software allowing to predict the transport and separation properties of porous medium (e.g., perfusion flow and ionic permselectivity) from its micro-structure in applications involving electrodialysis, capillary electrochromatography and nanofiltration [8–11]. Porous materials in electric field applications are either polymeric, anorganic or mixed polymeric/anorganic in their chemical nature. These materials very often exhibit a broad pore size distribution and contain pores in the range of 1 nm to 100 μm. Multiscale modeling of electric field driven processes in porous media with a broad pore size distribution is the next objective of our work. Another objective of our ongoing work is the automatic refinement of the computational grid required to solve problems with the Debye length significantly smaller than the average pore size but still large enough to prevent the use of the von Smoluchowski boundary condition. Acknowledgments. The support from the Czech Grant Agency (project 104/03/H141) and from the Ministry of Education (MSM 6046137306) is acknowledged.
References 1. Mario, S., Coelho, D., Bekri, S., Adler, P.M.: J.C o l.In t.S ci. 223,296-304(2000). 2. Grimes, B.A., Mayers, J.J., Liapis, A.I.: J.C hro m .A. 890,61-72(2000). 3. Vallano, P.T., Remcho, V.T.: A n a l.C hem . 72,4255-4265(2000). 4. Adler P.M.: Porous Media: Geometry and transports. Butterworth-Heinemann, Boston (2001). 5. Kosek J., Stepanek F., Marek M.: Modeling of transport and transformation processes in porous and multiphase bodies, in Advances in Chemical Engineering, Vol. 30 „Multiscale Analysis“, edited by Marin G.B., Elsevier (2005). 6. Torquato S.: Random Heterogeneous Materials: Microstructure and Macroscopic properties. Sprinder/Verlag, New York (2002). 7. Holst M., Saied F.: J. Comput. Chem. 14, 105-113 (1993). 8. Delgado, Á.V.: Interfacial electrokinetics and electrophoresis. Marcel Dekker (2002). 9. Dongquing Li: Electrokinetics in Microfluidics. Elsevier Academic Press (2004). 10. Paces M., Kosek J., Marek M., Tallarek U., Seidel-Morgenstern A.: Electrophoresis 24, 380-389 (2003). 11. Paces M., Kosek J., Kubicek M., Marek M.: Modeling of the perfusion flow in capillary electrochromatography, submitted (2005).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Development of a Multi-compartment Dynamic Model for the Prediction of Particle Size Distribution and Particle Segregation in a Catalytic Olefin Polymerization FBR George Dompazis, Vassilis Kanellopoulos, Costas Kiparissides Department of Chemical Engineering and Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki, PO Box 472, Thessaloniki, Greece 541 24
Abstract In the present study a comprehensive multi-scale, multi-compartment model is developed for the prediction of morphological (i.e., particle size distribution (PSD), particle segregation) and molecular (i.e., molecular weight distribution (MWD)) distributed polymer properties in a catalytic olefin polymerization FBR. The multi-scale description of the FBR utilizes models at four different levels, namely, a kinetic model, a single particle model, a population balance model and a multi-compartment reactormixing model. At the molecular level, a two-site Ziegler-Natta catalytic copolymerization model is employed to describe the copolymerization of ethylene with propylene. To calculate the particle growth and the spatial monomer and temperature profiles in a particle, the random pore polymeric flow model (RPPFM) is utilized. The RPPFM is solved together with a dynamic discretized particle population balance model, to predict the PSD. Moreover, overall dynamic mass and energy balances in the reactor level are derived, in order to calculate the monomer(s) concentration and temperature profiles along the reactor height. The effects of various fluidized bed operating conditions (e.g., fluidization gas velocity, temperature, catalyst feed rate) on the morphological and molecular distributed polymer properties and reactor operability are analyzed. Keywords: Multi-compartment model, Multi-scale reactor model, particle segregation, particle size distribution, polymer distributed properties.
1. Introduction High and medium density polyoelfins are commercially manufactured in gas phase fluidized bed olefin polymerization reactors using high activity transition metal catalysts such as Ziegler-Natta catalysts, Phillips-Chromium oxide catalysts and supported metallocene catalysts. Although polymer particles are assumed to be very well-mixed, particle segregation may occur in large industrial fluidized bed reactors. This means that the polymer particle size distribution at the reactor exit may differ from the PSDs at different locations along the bed height. In a fluidized bed reactor, strong segregation can occur if the bed contains particles of different densities. Density differences and particle size differences are common reasons for particle segregation. Despite its inherent importance, a limited number of papers have been published on the modeling of the particle-size distribution in gas-phase catalytic olefin polymerization processes. Zacca, et al. (1994), developed a population balance model using the catalyst residence time as the main coordinate, to model particle-size developments in multistage olefin polymerization reactors, including vertical and horizontal stirred beds
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and fluidized-bed reactors. Choi, et al. (1994), incorporated an isothermal simplified multigrain particle model, by neglecting the external particle mass and heat transfer resistances, into a steady-state PBE to investigate the effect of catalyst deactivation on the PSD and average molecular properties for both uniform and size distributed catalyst feeds. Yiannoulakis, et al. (2001), extended the model of Choi, et al. (1994), to account for the combined effects of internal mass and heat transfer resistances on the PSD for highly active catalysts. In a recent publication Dompazis, et al. (2005), developed a comprehensive integrated model, accounting for the multi-scale phenomena taking place in a continuous gas-phase ethylene copolymerization FBR to describe the molecular and morphological properties of the particulate polymer. Kim and Choi, (2001) presented a steady state multi-compartment population balance model using the concept of size-dependent absorption/spillage model to investigate the effects of fluidization and reaction conditions on the reactor performance. In what follows, a dynamic multi-scale, multi-compartment model is developed. Extensive numerical simulations are carried out to investigate the effect of critical reactor operating parameters (e.g., fluidization gas velocity, catalyst feed), on the dynamic evolution of the molecular and morphological polymer distributed properties (PSD, MWD etc.), particle segregation, and temperature in the reactor.
2. Kinetic Modeling at the Molecular Level To describe the molecular weight developments over a heterogeneous Ziegler-Natta catalyst, a generalized two-site kinetic model is employed (Hatzantonis et al., 2000). The kinetic mechanism comprises a series of elementary reactions, including site activation, propagation, site deactivation and site transfer reactions.
3. Modeling at the Particle Level To simulate the growth of a single polymer particle, the random pore polymeric flow model (RPPFM) of Kanellopoulos, et al. (2004) was employed. The equations to be solved for the calculation of spatial ethylene and propylene concentrations and temperature profile in a growing polymer particle, as well as the overall particle growth rate, G (D ) , (cm/s) are presented elsewhere (Kanellopoulos, et al., 2004).
4. Calculation of PSD in a Reactor Compartment To calculate the dynamic evolution of PSD in a gas-phase FBR a dynamic population balance model needs to be solved together with the system of differential equations describing the radial monomer(s) concentration and temperature profiles in a single particle (Kanellopoulos, et al., 2004) and the overall mass and energy balances in the reactor level. Accordingly, the bed is divided into N compartments of equal volume (see Figure 1). Each reactor zone is assumed to consist of an emulsion phase compartment and a bubble phase compartment. Polymer particles can be transferred from the emulsion to the bubble phase and vice versa. In all cases, the total mass of solids in each compartment remains constant. The dynamic population balance equation and the overall mass balance in the jth reactor zone can take the following form: Emulsion Phase: ∂n j (D, t ) ∂t
+
[
]=
∂ G (D )n j (D, t ) ∂D
[
]
1 F j −1 n j −1 (D, t ) − F j n j (D, t ) + Ftrwe, j n trwe, j (D, t ) − Ftrew, j n trew, j (D, t ) (1) We, j
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Fi −1 + Ftrwe, j + We, j
∫ G (D)n (D, t )d (ρ πD 6)− F 3
j
j
p
j
− Ftrew, j = 0
(2)
Dmin
Bubble Phase: ∂nw, j (D, t ) ∂t
=
[
]
1 Fre, j +1nw, j +1 (D, t ) − Fre, j nw, j (D, t ) − Ftrwe, j ntrwe, j (D, t ) + Ftrew, j ntrew, j (D, t ) Ww, j
Fre, j +1 − Fre, j + Ftrew, j − Ftrwe, j = 0
(4)
Ue,1 ,Te,1 [M i]e,1
Ub,1 ,T b,1 [M i]b,1
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Bubble Phase
Ftr,1we
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Fp,2 , n p,2(D, t)
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nw,2(D, t)
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Tb,2 Ww,2
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nw,3 (D, t) Fre,j
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Fc , n c(D)
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Fp,N-1 , n p,N-1(D, t)
Zone N
Ftr,Nwe
Fp,N , n p,N(D, t) Product
We,N U0 ,Tin [Mi ]in
N Ftr,New Fre
nw,N(D, t)
Ub,N
Ww,N
Tb,N
N
[Mi ]b,N U0 ,Tin [M i]in
Figure 1. Schematic representation of the multi-compartment model. where n j (D, t ) and nw, j (D, t ) , expressed in (#/g/cm), denote the number diameter density functions of particles in the “ j ” emulsion phase and in the “ j ” wake phase of the bubble phase compartment, respectively. We and Ww is the mass of solids in the emulsion phase and in the wake phase of the bubbles, respectively and Fre is the particle circulation rate (Baeyens and Geldart 1986).
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5. Overall Monomer and Energy Balances Based on the above multi-compartment model (see Figure 1), the mass and energy balance equations for the bubble and emulsion phases can be derived to describe the temporal-spacial variation of monomer concentration and temperature in the bubble and emulsion phases, respectively (Dompazis et al., 2006).
6. Results and Discussion Extensive numerical simulations were carried out by using the proposed model (see Figure 1) to investigate the effects of various reactor operating conditions on the distributed molecular and morphological polymer properties in a catalyzed, gas phase, ethylene propylene copolymerization FBR. 18
U0 = 20 cm/s
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Figure 2. Effect of fluidization gas velocity on the PSD in the emulsion phase. 24
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Figure 3. Effect of fluidization gas velocity on the PSD in the bubble phase. In Figures 2 and 3, the effect of fluidization gas velocity on the particle size distribution in the reactor compartments is shown for a catalyst feed rate equal to 0.1 g/s. In the present multi-compartment model the number of compartments was set equal to three in all model simulations. As can be seen, at low gas velocities, the PSD is shifted to larger sizes from the top to the bottom compartment. As the gas velocity increases, the individual PSDs in each compartment collapse into the same distribution, implying that
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the FBR can be approximated by a single CSTR. The particle size distributions in the wake phase of bubbles are shown in Figure 3. As expected, the amount of small particles in the wake phase of bubbles is substantially larger than that in the emulsion phase. Thus, as a bubble rises in the reactor the PSD of particles in the wake shifts to smaller sizes. Figure 4 depicts the effect of fluidization gas velocity on the temporal spatial variation of temperature in the FBR for a catalyst feed rate equal to 0.1 g/s. Notice that in the first case the fluidization gas velocity is equal to 30 cm/s (segregation mode), while in the second one the corresponding fluidization gas velocity is equal to 70 cm/s (well-mixed operation). As can be seen, at low fluidization gas velocities the reactor temperature increases with respect to its axial position. As the gas fluidization velocity increases, the reactor temperature is almost constant along the bed height. 120,0
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In Figure 5, the dynamic evolution of MWD of polymer particles in the emulsion phase is illustrated along the bed height. As can be seen, the MWD of particles is shifted to larger chain lengths as we move from the bottom to the top of the bed. This can be explained by the fact that the polymerization time of the polymer particles in the top compartment is smaller than the corresponding time of the particles in the bottom compartment. As a result, the molecular weight of the polymer particles in top compartment will be higher due to the higher polymerization rate.
References J. Baeyens and D. Geldart (1986) Solids Mixing. In Gas Fluidization Technology, D. Geldart Ed., Wiley. K.Y. Choi, X. Zhao and S. Tang, (1994), Population balance modelling for a continuous gas phase olefin polymerization reactor. Journal of Applied Polymer Science, 53, 1589-1597. G. Dompazis, V. Kanellopoulos and C. Kiparissides, (2005), A multi-scale modeling approach for the prediction of molecular and morphological properties in multi-site catalyst, olefin polymerization reactors. Macromolecular Materials and Engineering, 290, 525-536. G. Dompazis, V. Kanellopoulos and C. Kiparissides, (2006), Development of a multicompartment model for the dynamic prediction of particle size distribution in a catalytic olefin polymerization FBR. To be Submitted. H. Hatzantonis, A. Yiagopoulos, H. Yiannoulakis and C. Kiparissides, (2000), Recent developments in modeling gas-phase catalyzed olefin polymerization fluidized-bed reactors: The effect of bubble size variation on the reactor’s performance. Chemical Engineering Science, 55, 3237-3259. V. Kanellopoulos, G. Dompazis, B. Gustafsson and C. Kiparissides, (2004), Comprehensive analysis of single-particle growth in heterogeneous olefin polymerization: the random-pore polymeric flow model. Ind. Eng. Chem. Res., 43 (17), 5166-5180. J.Y. Kim and K.Y. Choi, (2001), Modeling of particle segregation phenomena in a gas phase fluidized bed olefin polymerization reactor. Chemical Engineering Science, 56, 4069-4083. H. Yiannoulakis, A. Yiagopoulos and C. Kiparissides, (2001), Recent developments in the particle size distribution modeling of fluidized-bed olefin polymerization reactors. Chemical Engineering Science, 56, 917-925. J.J. Zacca, A.J. Debling and H.W. Ray, (1996), Reactor residence time distribution effects on the multistage polymerization of olefins I. Basic principles and illustrative examples, polypropylene. Chemical Engineering Science, 51 (21), 4859-4886.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Mixing in a T-shaped microreactor: scales and quality of mixing Dieter Bothea*, Carsten Stemichb, Hans-Joachim Warneckeb a
Chair for Mathematics, Center of Computational Engineering Science, RWTH Aachen, Pauwelsstraße 19, 52074 Aachen, Germany b Institute of Chemical Engineering, Department of Chemistry, Faculty of Sciences, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany
The large area-to-volume ratio of microreactors gives prospect of better yield and selectivity than for conventional designs, since diffusive fluxes of mass and heat in micro-devices scale with area, while the rate of changes corresponding to sources and sinks are proportional to volume. Indeed, theoretical considerations of the scaling behaviour (see [1]) support the fact that microreactors allow for faster chemical reactions and provide better thermal control. Moreover, specific applications prove that these advantages of micro-reactors can be realised in order to perform fast exothermic reactions (cf. [2-4]) and to enhance selectivity [5]. For such applications, the mixing of chemical species is of special interest, since it is an essential condition for chemical reactions. To obtain efficient mixing for the short residence times in micro-systems, the contact area between regions of higher and lower species concentration has to be increased significantly. To avoid large pressure drops, secondary flows instead of turbulent flow fields are preferred. In case of a T-shaped micro-mixer, the secondary flow acts mainly in cross directions, i.e. perpendicular to the axial direction, and can be used to mix the two feed streams. To assess and optimise the mixing process, this qualitative picture has to be understood more thoroughly and significant quantitative information has to be added. In particular, the interplay of convective and diffusive transport to bridge the gap between the reactor and the molecular scale, have to be further investigated, since even if no new physico-chemical phenomena occur, new aspects enter the picture in case of micro-systems; cf. [6] and the references given there. The present paper employs CFD-simulations to obtain first steps in this direction. 1. Mixing: a multiscale process Reactive mixing is a multiscale process per se, since educts are usually initially segregated, while mixing on the molecular level is necessary for any chemical reaction to occur. Whatever transport processes are involved on the macro- or meso-scale, the final step to obtain homogeneity on the molecular scale – the so-called micro-mixing – can solely be achieved by molecular diffusion. To avoid adverse effects of imperfect mixing on conversion and selectivity, the total mixing time should be well below the time-scale of chemical reaction. Since typical residence times in micro-channels are significantly smaller than in *
Corresponding author. Tel.: +49-5251-60-3619; Fax: +49-5251-60-3244. Email address: [email protected].
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352 macroscopic systems, the contact area between regions of higher and lower species concentration has to be increased considerably in order to advance diffusive dissipation of concentration gradients. An increase of the contact area can either be reached mechanically or hydrodynamically (cf., e.g. [7, 8]). Here, we only consider hydrodynamically driven mixing of liquids. In continuous processes, mixing in axial direction is often unwanted, since it reduces conversion and strongly influences selectivities. Consequently, mixing should combine complete and instantaneous stirring in cross directions with no relative axial movements (plug flow). While this evidently cannot be achieved completely, secondary flows can be employed to approximate such a mixing behaviour. In case of duct-like geometries this leads to T-shaped reactors in the simplest case (cf. Figure 1) or to zigzag-channels with a T-shaped inlet as a more efficient alternative (cf. [8]). Focusing on TFigure 1. T-shaped micro-mixer. shaped chemical reactors, once the flow type, given in terms of the Reynolds number, is fixed, the optimal dimensions of the mixing channel are to a large extent determined by the needed hydrodynamical residence time τ H . Indeed, simple considerations using in particular τ H = 5τ R and l / d H = 10...100 lead to
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and a typical Reynolds number of about 200 for engulfment flow (explained below), this leads to a hydraulic diameter in the range of 30…100 µm. While scaling considerations give a first hint why microreactors are useful especially for fast chemical reactions [1], they do not answer the question whether cross-directional transport due to secondary flow within a duct of this dimension is fast enough to achieve efficient mixing. In this respect notice first that if the time τ D available for micro-mixing is estimated as τ D = 0.01τ R , in order to avoid limitation by diffusion, then the scale of segregation lS should be reduced to lS = 0.1 D τ R . (2) micro-mixer
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This leads to (3) d H lS = b Re Sc with Sc = ν / D , b = 0.7… 2 for the ratio of the initial (maximum) scale of segregation and those needed for micro-mixing to work fast enough. As an example, Re=200 as above and a typical value of Sc=1000 in aqueous systems yield a ratio in the range 300…1000. In other words, by means of the secondary flow, the initial scale on which the feed streams are segregated has to be decreased by a factor of 300…1000 within a small section of the mixing channel. Depending on the structure and effectiveness of the secondary flow, a gap within the hierarchy of scales may remain that should be bridged e.g. by repeated reorientation or by flow modulation to trigger chaotic advection. Figure 2 illustrates this hierarchy of scales for a T-shaped micro-mixer in case of so-called engulfment secondary flow. 2. Numerical simulation of flow and species transport The numerical simulations presented here are performed for a T-shaped micromixer with rectangular cross sections. The geometry of the micro-reactor consist of two inlet channels, each with a length of 8 mm and a depth of 100 µm. The mixing channel is 16 mm long with the same depth as the inlet channels. The width of the inlet channels are 100 µm each, the width of the mixing channel is 200 µm. For channels of these dimensions and for liquid flow, the transport of mass and momentum is adequately described by the Navier-Stokes equations, which in non-dimensional form read as (4) ∇ ⋅ u = 0 , ∂ t u + u ⋅ ∇u = −∇p ′ + Re −1Δu , complemented by appropriate boundary conditions. The transport of an ideally diluted non-reactive chemical species is governed by the species equation (5) ∂ t c + u ⋅ ∇c = (Re Sc) −1 Δc , where c denotes the dimensionless molar concentration. For a reduction of the computational domain, shortened inlet channels and mixing channel are used (Fig. 1). Preliminary calculations gave a minimum length of 1.5 mm for the mixing channel in order to avoid perturbations of the flow behaviour due to the outflow conditions at this artificial boundary, which only approximate the actual physics. All the results given below refer to a mixing channel length of 1.5 mm and inlet channels with a length of 400 µm. To avoid inlet effects, a fully Figure 3. Flow trajectories for engulfment developed duct flow is used on flow (Re=186) both inlet faces (cf. [9]). At the outlet, pressure is set to a fixed reference value and no-slip boundary conditions are used at fixed walls. The numerical simulations were carried out with the CFD software FLUENT 6.2. The computational domain is meshed by a block structured grid of about 1 million cubic grid cells, which is sufficient to resolve
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354 the characteristic features of the flow. For calculations of species concentration for high Schmidt numbers, the mesh is locally refined to resolve the finest scales. The correspondingly modified grid consists of about 12 million cubic grid cells with a local resolution of up to 0.6 µm. The computations are performed with the parallel version of FLUENT using up to 11 processors (64bit Intel Xeon). The calculations are carried out in the Paderborn Center for Parallel Computing (PC2) on the linux-based cluster system ARMINIUS. 3. Intensity and Scale of Segregation Quantitative measures of the quality of mixing should not only consider the amplitude of the variations present within the concentration distribution, i.e. an intensity of mixing, but also provide a measure for the scale on which gradients persist. For the former, we employ Danckwerts’ intensity of segregation [10] defined by σ2 1 I S = 2 with σ 2 = ( c − c )2 dV , (6) |V | V σ0
∫
where c denotes the mean value of the concentration field c and σ 02 = c ( cmax − c ) with feed concentrations zero and cmax . Based on (6), we employ σ I M = 1 − IS = 1 − , (7) σ0 which has a value of 1 in the homogeneously mixed case and 0 for complete segregation. A number of essentially similar measures which depend on statistical parameters are compared in [11]. These measures are insensitive to the length scales on which segregation occurs. To define a meaningful scale of segregation, information about the structure of the concentration distribution is needed. A well-known quantity to characterise the length scale is the striation thickness s. In case of lamellar structures, s-1 corresponds to the specific contact area. Given any segregated concentration distribution c(x) inside a volume V, the quantity 1 c (8) Φ( V ) = || ∇f || dV with f = cmax |V | V
∫
coincides with the specific contact area inside a volume V; here |V| denotes the volume content of V and || ∇f || is the (Euclidian) length of the gradient of the normalised concentration. Below, we apply an analogous measure to a cross section A instead of a volume V. For a segregated species distribution the quantity Φ then gives the specific contact length inside the cross section. The reciprocal of Φ has the dimension of a length and can be interpreted as an average distance between regions of high and low species concentration. This interpretation is exact in the case of lamellar structures, hence Φ −1 is the striation thickness then. The quantity Φ keeps these meanings for almost segregated concentration fields, while these interpretations lose their significance for close to homogeneous fields. In any case, Φ is the potential for diffusive mixing, since it measures the total driving force for diffusive fluxes within the concentration field.
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The computation of specific contact area or contact line by (8) is also helpful to determine local scales during numerical simulations. Note that if a cross-section contains locally a single tracer filament of width d, then inside an appropriate small part A of that section the filament has length l, i.e. the contact line has length 2l if the contribution of d is negligible. Hence d can be computed as d = 2 f dA || ∇f || dA . (9)
∫
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IM [-]
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4. Flow regimes and mixing Depending on the mean velocity, three different stationary flow regimes can be observed up to a Reynolds number of about 240, where time-periodic flow phenomena set in. At low Reynolds numbers strictly laminar flow behaviour occurs where both inlet streams run parallel through the mixing channel, without formation of vortices. Above a critical Reynolds number a secondary flow in form of a double vortex pair is build due to centrifugal forces. In this regime, symmetry concerning a plane perpendicular to the inlet channels is still maintained. Therefore, mixing across this symmetry plane can solely occur due to diffusion and, hence, the intensity of mixing is approximately zero. With increasing velocity this flow symmetry is destroyed and fluid elements reach the opposite side of the mixing channel as shown in Figure 3. This so-called engulfment flow regime is the most interesting here, since the resulting intertwinement of both fluid streams generates additional contact area and, hence, raises the potential for diffusive mixing. This causes an increase of the intensity of mixing accompanied by a decrease of the scale of segregation. In fact, as can be seen in Figure 4, these quantities show a jump at the critical mean velocity. Above a mean velocity of approx. 1.1 m/s the scale of segregation, computed from the specific contact area, falls to about 50 µm and further decreases to 30 µm. As mentioned above, these values can be interpreted as a mean distance between regions of high and low tracer concentration. This is underlined by Figure 5, which shows the tracer distribution on a cross section inside the mixing zone, 200 µm from the entrance into the mixing channel. This figure also compares numerical and experimental tracer distribution on a certain cross section inside the mean velocity [m s-1] mixing channel and Figure 4. Intensity of mixing (grey) and scale of thus gives a qualitative segregation (black) versus mean velocity. validation. The
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ln (λ conc)
experimental data is provided by the research group of Prof. Dr. N. Räbiger (IUV, University of Bremen); cf. [12]. The bottom part shows the tracer distribution obtained by laser induced fluorescence combined with micro-resolution confocal microscopy (µLIF). The top part displays the result of the corresponding numerical simulation with about 9 million grid cells. Both pictures correspond to a mean velocity of 1.4 m/s and are in good agreement. While the computed concentration profile is symmetric with respect to the centre of the cross section, the Figure 5. Simulated (top) and experimental distribution shows a noticeable experimental (bottom) tracer deviation which might be caused by distribution in the mixing chanasymmetries of the physical channel nel 200 µm behind its entrance geometry. Of course, the size of the finest length scale that occurs depends on the Schmidt number. In the situation of Figure 5, extremely fine structures occur near the center of both 3,5 vortices. This is a y = -0,4554x + 3,4658 3 consequence of the high R2 = 0,9984 2,5 Schmidt number of approx. 2 3600 of the fluorescence 1,5 1 tracer. In the numerical 0,5 simulation these fine 0 structures are to some 0 1 2 3 4 5 6 7 extend smeared by ln (Sc) numerical diffusion. To resolve small scales at Figure 6. Log-log plot of smallest length scale relatively large Schmidt versus Schmidt number (Re=186). numbers of up to 500, locally refined grids of up to 18 million cubic grid cells are used with local cell sizes down to 0.6 µm. To investigate how the size of the smallest structures depends on Sc, we performed numerical simulations for different diffusivities and extracted the smallest size by means of (9). Figure 6 shows the dependence of the smallest scales on Sc at Re=186 and within a cross section at 100 µm from the entrance of the mixing channel. The first four points correspond to Sc=2, 5, 10, 30 and follow a power law with an exponent of about -0.46. The final point (Sc=500) lies above this curve, most probably due to insufficient grid resolution. Analogous evaluation for Re=173 at the same position also yields a power law dependence with exponent -0.47 again, but with a slightly larger constant which corresponds to a coarser convective scale. It therefore appears that the finest structures λ conc scale as λ conc = λ vel Sc , (10) where the reference scale λ vel in particular depends on Re. Note that (10) means that the finest scale is given by the Batchelor scale, although the flow
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field is even stationary. To build such fine structures requires relative motion accompanied by shear. Since the fine structures are dissipated by diffusion, they can only persist if both sub-processes act on the same time-scale, i.e. if λ2vel λ2conc , (11) = ν D which provides a simple explanation for (10). On the other hand, since the tracer distributions are far from being statistically homogeneous, the average length scale is more important than the smallest one. Somewhat surprisingly, in the engulfment flow regime the mean length scale obtained from the specific contact area scales as λ vel , (12) λ conc, av = 1 + a ln (Sc) with certain coefficients a = a (Re) , in contrast to (10). This has to be further investigated.
5. Acknowledgements We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft within the Priority Program SPP1141 “Analysis, Modelling and Calculation of Mixing Processes with and without Chemical Reaction“. We also thank our cooperation partners Prof. Dr. N. Räbiger and Dr. M. Schlüter for providing the experimental data. References [1]
D. Bothe, C. Stemich, H.-J. Warnecke, 2006, Fluid mixing in a T-shaped micro-mixer, Chem. Eng. Sci, 61, 2950-8. [2] M.-A. Schneider, T. Maeder, P. Ryser, F. Stoessel,2004, A microreactor-based system for the study of fast exothermic reactions in liquid phase: characterization of the system, Chem. Eng. J., 101, 241-50. [3] P. Löb, H. Löwe, V. Hessel, 2004, Fluorinations, chlorinations and brominations of organic compounds in micro reactors, J. Fluorine Chem., 125, 1677-94. [4] K. Kawai, T. Ebata, T. Kitazume, 2005, The synthesis of fluorinated materials in microreactors, J. Fluorine Chem., 126, 956-61. [5] J. Yoshida, A. Nagaki, T. Iwasaki, S. Suga, 2005, Enhancement of chemical selectivity by microreactors, Chem. Eng. Technol., 28(3), 259-66. [6] J.M. Ottino, S. Wiggins, 2004, Introduction: mixing in microfluidics, Phil. Trans. R. Soc. Lond. A, 362, 923-35. [7] V. Hessel, H. Löwe, F. Schönfeld, 2005, Micromixers – a review on passive and active mixing principles, Chem. Eng. Sci., 60, 2479-2501. [8] N.-T. Nguyen, Z. Wu, 2004, Micromixers – a review, J. Micromech. Microeng., 15, R1-R16 [9] D. Bothe, C. Stemich, H.-J. Warnecke, 2004, Theoretische und experimentelle Untersuchungen der Mischvorgänge in T-förmigen Mikroreaktoren – Teil 1: Numerische Simulation und Beurteilung des Strömungsmischens, CIT, 76, 10, 1480-4. [10] P.V. Danckwerts, 1952, The definition and measurement of some characteristics of mixtures, Appl. Sci. Res., A3, 279-96. [11] J. Boss, 1986, Evaluation of the homogeneity degree of a mixture, Bulk Solids Handlings, 6, 6, 1207-15. [12] M. Hoffmann, M. Schlüter, N. Räbiger, 2006, Experimental investigation of liquid-liquid mixing in T-shaped micro-mixers using µ-LIF and µ-PIV, Chem. Eng. Sci, 61, 2968-76.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Direct Modeling of Unit Operations on Molecular Level Danijel Babic, Andreas Pfennig Lehrstuhl für Thermische Verfahrenstechnik, RWTH Aachen University, 52056 Aachen, Germany, www.tvt.rwth-aachen.de
Abstract The behavior of a sieve-tray distillation column with three trays at total reflux has been simulated on molecular level until steady state was reached. Sieve trays were created by introducing partly permeable boundaries in the system. The heating at the bottom and the cooling at the top of the column were achieved by influencing the molecular velocities in the corresponding regions of the column. The results show that it is possible to mimic the behavior of distillation columns directly based on molecular simulations. Keywords: molecular simulation, unit operation, distillation.
1. The Goal The behavior of technical equipment is determined by the properties and interactions of the molecules that are involved in the process. Thus it is finally these molecular properties, which determine the technical equipment design in chemical-engineering equipment. It is generally regarded as impossible to bridge the difference in length and time scale between molecular description and technical equipment in a single step. Thus a common approach is to introduce several steps of model hierarchy in between, e.g. describing mass transfer in a single phase and at interfaces as well as describing single drops or bubbles, the behavior of which is then accounted for to describe the entire equipment. Simultaneously it is known that the major effects in unit operations can be described by equilibrium and rate-based approaches, which again have their basis on molecular level. As a result, in depicting the behavior of technical equipment it appears sufficient to describe the path of the components involved through various steps of conditions. These conditions refer to pressure, temperature, concentrations and gradients of these properties as well as interfaces. Information on the behavior of the substances as a function of these conditions can be obtained from molecular simulations with some hundred or at most some thousand molecules, which are easily feasible today. Thus the goal of this work is to develop a simulation tool which is on the one hand based on molecular simulations and which on the other is capable of performing simulations of entire unit operations. Here, as a first step, the feasibility of this approach is shown for a simple example.
2. Advantage of the molecular approach If unit operations are simulated based on molecular behavior a big advantage is that fully predictive calculations will become possible in the future. Based on quantumchemical methods it is already today possible to obtain information which allows to determine thermodynamic data e.g. via the COSMO approach by Klamt (1995) and co-
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workers. Different groups today try to transfer similar methods to the development of models necessary for molecular simulations.
3. Simulation As a first step a demonstrator has been realized based on the molecular-dynamics code MDMIXT of the CCP5 library of the SERC Daresbury Laboratory (CCP5, 2005). This code has been thoroughly redesigned and modified as to allow different compartments and boundaries between them to be accounted for in a simulation. The size of the basic compartments between which the different boundaries can be defined are the elementary cells of the simulation which contain 4 molecules each. The boundaries are defined in two dimensions while in the third dimension the molecules can move freely with periodic boundary conditions. In general all boundaries between the elementary cells are assumed to have no effect unless specific conditions are defined. Thus the molecules are simulated as they move through the system based on solving the equations of motion, where the molecules are depicted as multi-centre Lennard-Jones particles. Typically a time step for integrating the equation of motion of 10-15 s to 3*10-15 s is used, depending on the strength of the interactions. At each time step it is checked whether a molecule has passed a specific boundary between the basic cells. If this is the case the conditions specified for the boundary are applied to the molecule. The major boundaries accounted for are as follows: impenetrable walls The molecules are reflected from the impenetrable walls with respect to their center of mass. partly penetrable walls To mimic the behavior of sieve trays in a column, partly penetrable walls have been defined through which the molecules can only pass in one direction with a specified probability. This probability corresponds to the fraction of the open area of the sieve tray. It is realized by comparing the probability with a random number for each instance a molecule tries to pass the boundary. If the molecule does not pass the boundary it is reflected. heating or cooling boundaries Molecules that pass a heating or cooling boundary have their velocity changed by a certain amount which can be specified. Since molecular velocity directly corresponds to temperature this boundary acts as a heating or cooling boundary. Also feed as well as removal boundaries have been introduced to be able to describe a distillation column with finite reflux ratio. These boundaries modifying the amount of substance in the system are implemented but have not been used in the simulations below, since here total reflux is assumed in the first simulations for simplicity. Different properties of a boundary can be specified simultaneously for each individual boundary present in the system. Thus e.g. an impenetrable wall at the bottom of the column can simultaneously be a heating boundary. This allows a wide variety of behavior and geometries to be depicted. The simulation is based on an NVE ensemble, where the number of particles, the volume and the energy of the system are specified. To adjust the starting temperature the velocities of the molecules are scaled to the desired temperature during initial simulation steps. The potential function is evaluated based on periodic boundary conditions in all three spatial directions. A cut-off radius of half the box length of 0.976 nm has been used.
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Evaluation of the simulation results has been integrated into the program. As a function of column height the profiles of partial and overall density as well as temperature are determined. Additionally the compositions on the trays are evaluated by selecting appropriate compartments above and below the trays, which can be unequivocally attributed to the liquid or the vapor phase respectively. Averaging of the results occurs over a specified number of steps, e.g. 40000 in the simulation presented below. This averaging is realized as pseudo-rolling average. To avoid storing 40000 individual values of the property to be averaged e.g. only the last up to 1000 values are stored individually. For time steps before this only the averages over the 1000 steps are stored, e.g. a total of 40 such averaged values, thus information dating back over roughly 40000 steps is collected. The pseudo-rolling average is then obtained from the last up to 1000 individual values, the 39 averages over 1000 steps before that, and the appropriate fraction of the average over 1000 steps before these 39 values. Thus the information on the current simulation step is accounted for individually, whereas the information on the steps dating back 40000 steps is taken into account only on an averaged basis. The boundaries described above have been used to describe a distillation column with three trays as shown in Fig. 1. A heating region was added at the bottom and a cooling area at the top by influencing the molecular velocities, which represent local temperature. The trays were represented by partly penetrable walls, which mimic the behavior of sieve trays. In this column the molecules are moving from bottom to top induced by the heating at the bottom and the partly penetrable boundaries of the trays. To ensure a molecular motion of the liquid in the opposite direction, gravitation was included on a level where the transport and equilibrium properties of the mixture are not significantly affected. This leads to a counter flow of liquid, which closely mimics the behavior in a technical equipment with cross flow on each tray. Nevertheless gravitation has to be set to a value much higher than on the earth surface, since the potential of gravitation introduced takes relatively small values due to the small height of only 15 nm of the entire simulated system. Here a value of 2.8 × 1013 m/s2 was chosen. Only by choosing such a high value the gravitational potential has any effect as compared to the potential of molecular interaction. In separate simulations the influence of gravitation as well as of the size of the system have been investigated. The corresponding simulations were performed for a system of pure methane, which was set up in a way that a two-phase system corresponding to a vapor-liquid equilibrium resulted. The vapor pressure has been evaluated in these simulations. The results show that the value of gravity as well as the system size in ydirection have a significant influence on the vapor pressure. The values chosen in the simulation shown below have a certain effect on the results which is accepted for this first simulations of the demonstrator, especially since the simulations with pure methane show that already slightly reduced values of gravity lead to strongly improved results. Both of these parameters also have a strong influence on the simulation time required to achieve steady-state results. Since the initial starting configuration chosen representing homogeneous distribution of the molecules of all species throughout the entire system is the most inefficient choice it is expected that improving this starting configuration can significantly speed up the simulation. Thus current activities are directed at choosing and manipulating the composition along the apparatus in a systematic way with the goal to reach steady state significantly faster. To visualize the results of the simulation, output files were generated, which can be used by the ray-tracing tool POV-Ray (2005). With POV-Ray realistic pictures of the molecules in the distillation column can be generated and can be linked together to
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obtain digital movies with appropriate tools. This way it is also possible to generate movies for teaching purposes visualizing the molecular behavior at different conditions also including interfaces. Such movies, which are suitable for visualizing the different states of matter as well as the molecular mobility e.g. in lectures on physical chemistry or thermodynamics, can be supplied by the authors upon request. Additionally an efficient output to screen is generated which can also be switched off and which can be updated e.g. every 1000 time steps. This output is very helpful for setting up the appropriate boundaries, since their effect can directly be seen. Also this output can be stored as individual frames which then can be combined to result in a movie. Thirdly a file format is generated which allows viewing the molecular motion inside a CAVE in 3D-projection. This application has been developed in collaboration with the virtual reality group of the Center for Computing & Communication of RWTH Aachen University. impenetrable wall
cooling boundary partly penetrable wall cell boundary
z y x
heating boundary
Fig. 1: Representation of a distillation column on molecular level
4. Results Based on this configuration a simulation was performed for the column with three trays with a permeability of 60 % as shown in Fig. 1. The size of an elementary cell was 0.976 nm, the size of the system in y-direction was two elementary cells. A mixture of methane + ethane was simulated with as little as 600 molecules in the column. This is roughly the minimum required to obtain results that are only slightly influenced by the finiteness of the system. The starting condition was a uniform distribution of molecules in the entire column. Due to this starting configuration, which can be regarded as worst case, steady state was reached after simulation of 3.5 ns. The simulations were
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performed on a laptop computer with pentium-III processor and 1.13 GHz. The simulation took roughly 9 days. The result after reaching steady state is shown in Fig. 2. The top left diagram depicts the final state of the column together with some numeric output. To the top right the McCabe-Thiele diagram is plotted where the equilibrium line was generated from data taken from Knapp et al. (1982). It can be seen that the simulations generate sieve trays, which approximately correspond to theoretical stages. The equilibrium line is intersected because the concentrations are fluctuating properties in the simulation. The reason for reaching ideal equilibrium behavior with the trays is the small size of the system, where diffusion on each tray is much faster than the convective flow. Since in reality sieve trays have an efficiency considerably less than unity, future work will be directed at increasing the flow rates to generate states on the trays which are farther away from equilibrium. This will simultaneously lead to faster simulations, since higher flow rates directly correspond to shorter times for reaching steady state. In the lower part of Fig. 2 temperature and density profiles are shown along the column. Temperature decreases from bottom to top of the column as expected. Also the partial density of methane increases in the same direction while that of ethane decreases. It can be seen that relatively strong statistical fluctuations occur in the vapor phase while the values for the liquid phase are much more stable on all trays. The reason for this is the higher number of molecules over which the averaging is performed in the liquid due to its higher density.
Fig. 2: Result of a direct molecular simulation of a distillation column at total reflux.
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5. Conclusions Overall the results of this simulation show that a distillation column can be simulated in principle with as few as 1000 molecules. The agreement with e.g. experimental equilibrium information is excellent. This approach will now be extended to allow simulation also of other unit operations, e.g. solvent extraction and to take the rate-determining steps into account more appropriately. By increasing the flow rates the computational effort will be decreased, since steady state can be reached faster. Simultaneously the non-equilibrium character of the processes taking place inside technical equipment can be described more realistically. Additionally the results of the first simulation steps can be used to adjust the composition profile along the column. This will also lead to a much faster approach to steady state.
References CCP5, 2005, http://www.ccp5.ac.uk/, programm library at: http://www.ccp5.ac.uk/librar.shtml, November 14, 2005 A. Klamt, 1995, Conductor-like Screening Model for Real Solvents: A New Approach to the Quantitative Calculation of Solvation Phenomena, J. Phys. Chem. 99, 2224-2235 H. Knapp, R. Döring, L. Oellrich, U. Plöcker, J. M. Prausnitz, 1982, Vapor-Liquid Equilibria for Mixtures of Low Boiling Substances. Dechema Chemistry Data Series, Vol. VI, Dechema, Frankfurt POV-Ray, 2005, http://www.povray.org/, November 14, 2005
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Modelling and Simulation of Fe2O3/Aluminum Thermite Combustion: Experimental Validation Luísa Durães,a Paulo Brito,a,b José Campos,c António Portugala a
CEM Group, Chemical Engineering Dept., Faculty of Sciences and Technology, University of Coimbra, Pólo II, Coimbra 3030–290, Portugal b Chemical Technology Dept., School of Technology and Management, Bragança Polytechnic Inst., Campus de Santa Apolónia, Bragança 5301–857, Portugal c Thermodynamics Group, Mechanical Engineering Dept., Faculty of Sciences and Technology, University of Coimbra, Pólo II, Coimbra 3030–201, Portugal
Abstract A one-dimensional model was built to simulate the non-steady radial combustion propagation on thin circular samples of Fe2O3/Aluminum thermite mixtures. The model considers zero order kinetics and conductive/radiative heat transfer. All the properties of the system are assumed to vary with the temperature and composition during the propagation and phase transitions are also contemplated. These features, not yet considered in the literature, allowed the obtainment of realistic solutions, readily comparable with experimental values measured in an earlier work. The experimental combustion velocities were used to adjust the kinetic constant of the model, in order to give it a good predictive capability. The predicted combustion temperatures and reaction extents were higher than the experimental. This was justified by the heat losses due to the spraying of products away from the combustion system and the incompleteness of the reaction, observed in experimental conditions and not considered in the model. Keywords: Combustion, Fe2O3/Aluminum thermite, Modelling, Finite differences.
1. Introduction The self-propagating high temperature reactions, as the Fe2O3/Aluminum thermite combustion, are hard to follow by experimentation due to fast chemical and physical transformations and the high temperatures achieved. Hence, many studies concerning theoretical prediction of these combustion processes have been published (Moore and Feng, 1995, Makino, 2001) and represent a valuable guideline for experimental work. The Fe2O3/Aluminum thermite reaction has already been simulated in cylindrical geometry with a one-dimensional coordinate system attached to the uniformly propagating combustion wave (Raymond et al., 1998, Shkadinsky et al., 1997, 2000), and with a fixed one-dimensional coordinate system (Brito et al., 2004). However, the availability of experimental results of radial combustion on disk shaped samples (Durães et al., 2006a) has stimulated, in an earlier work, the derivation of a onedimensional model to describe the Fe2O3/Al radial combustion propagation (Brito et al., 2005). The built model congregates several features of simpler models published (Bowen and Derby, 1995, Carrillo-Heian et al., 1999, Graeve et al., 2001, CuadradoLaborde et al., 2003). It considers non-steady propagation with conductive/radiative heat transfer mechanisms and zero order reaction kinetics. Additionally, all the thermophysical properties are assumed to vary with the temperature and composition of the mixture, assumption not yet considered in the literature. Adaptive numerical methods were applied in the resolution, involving a mesh adaptation method based on a
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continuous grid refinement technique (Brito, 1998). The solution profiles are fast moving steep fronts, which are validated in this work by experimental results. In the model specifications, the experimental set-up features are considered, namely the reactant samples geometry, dimensions, compositions and density, confinement materials properties and environment conditions. Five thermite mixtures compositions, one stoichiometric and four over aluminized, were experimented and simulated, and the obtained combustion front propagation velocity, temperature and final products composition are compared.
2. Experimental Procedure Industrial Fe2O3 (1.7 μm, 96 %, Bayer) and aluminum (18.6 μm, 89.3 %, Carob) powders were mixed in stoichiometric and over aluminized ratios (vd. Eq. 1 and Table 1). Chemical and physical characterisation of reactants, and mixtures preparation procedure were presented in previous papers (Durães et al., 2006a, 2006b). Fe 2 O3 + 2 Al → 2 Fe + Al 2 O3
(1)
Table 1. Thermite mixtures composition. Mixture
T100
T112
T127
T142
T159
Fe2O3:Al molar ratio
1:2
1:2.24
1:2.54
1:2.85
1:3.19
Fe2O3:Al mass ratio
1:0.338
1:0.379
1:0.429
1:0.482
1:0.538
1.00
1.12
1.27
1.42
1.59
a
Equivalence ratio a
(O for the oxidation of the existing Al to Al2O3)/(O present in the Fe2O3 of the mixture). thermocouples
50 3
5 10
15 60
5
Stainless Steel circular box PMMA circular lid Thermite mixture Ignition channel
Figure 1. Combustion sample setup (dimensions in mm).
Reactant mixtures were compressed in a stainless steel circular box with an inner PMMA lid (vd. Fig. 1) as described in Durães et al. (2006a). Samples of 1-2 mm thickness were obtained and their porosities varied from 0.30 to 0.50. Samples combustion was initiated in the ignition channel via a nichrome wire instantaneously heated by a capacitor discharge. Radial flame propagation was monitored by digital video-crono-photography. Combustion thermograms were registered at two different radius, using W/Re thermocouples. The obtained radial combustion propagation profiles, rates and temperatures were discussed in Durães et al. (2006a). Finally, the mixtures combustion products were collected and characterized by X-ray diffraction and Mössbauer spectroscopy (Durães et al., 2006b).
3. Model and Numerical Method The built model is based on the following assumptions: i) one-dimensional radial nonsteady propagation; ii) general reaction with mass stoichiometry αAA + αBB → αCC + αDD; iii) limiting reactant A; iv) zero order kinetics; v) conductive/radiative heat
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transfer mechanisms; vi) negligible relative movement between species; vii) sample dimensions and confinement materials as in Fig. 1. Therefore, the energetic and mass partial balances take the form:
ρ M CPM
[
(
∂T 1 ∂ ⎡ ⎛ ∂T ⎞⎤ 4 4 = ⎟⎥ + Q ⋅ r − (U steel/air + U PMMA/air )(T − T0 ) + 2σε M T − T0 ⎢k M ⎜ r ∂t r ∂r ⎣ ⎝ ∂r ⎠⎦
dW A = −α A r dt
)] Z (2) (3)
σ is the Steffan-Boltzman constant, ε the emissivity, WA the mass concentration of A, Z
the sample thickness and subscript M represents the mixture. The reaction enthalpy (–Q) is computed considering the enthalpy variations (CP(T) integral and phase transitions) at constant pressure of reactants and products in the temperature path T0 – T. The reaction kinetics is defined by r = H(T – Treact)K, where H is the Heaviside function and K a nontemperature dependent kinetic constant. The last term in Eq. (2) accounts for heat losses through the sample top and bottom to the surrounding (at T0). The differential problem is completed by the definition of suitable initial and boundary conditions, reformulated in relation to Brito et al. (2005): ⎧0 ≤ r ≤ R 0 ⇒ T = Tigni t=0 ⎨ ⎩ r > R 0 ⇒ T = T0
t > 0; r = 0 ⇒
(4)
∂T =0 ∂r
t > 0; r = R ⇒ k M
(5)
[
(
∂T ′ / air (T − T0 ) + σε M T 4 − T0 4 = − U steel ∂r
)]
(6)
Eq. (4) simulates ignition by a temperature spatial pulse with height Tigni and length R0 at the initial time. On the inner and outer boundaries, a symmetry condition and conductive/radiative heat transfer are considered, respectively. The thermophysical properties vary with the temperature and composition of the mixture. The mixing rules for each property, with i = A,B,C,D and E (air), are: C P M = ∑ ωi C P i ; i
ρM = 1
ωi
∑ρ i
i
⎛ ; k M = ⎜⎜1 ⎝
υi
∑ k + ∑υ k i
i
i
i
i
⎞ ⎟ 2 ; ε M = ∑ υ i ε i (7) ⎟ i ⎠
where ωi and υi are the mass and volumetric fractions of component i. The mixture conductivity is the average value between the conductivity of a serial and a parallel rearrangement of the components on a very narrow film (thickness Δr) centered on each spatial node position. An equivalent conductivity component is introduced in air conductivity estimative, for the radiation on the void spaces of the serial arrangement (k’E = kE + 4σεMT3υEΔr). Phase transitions of the components are considered, over a temperature range (ΔT) of 1 K, each time its transition temperatures are crossed, by means of an equivalent CP: C’Pi = CPi + Li/ΔT (Li – latent heat). The model was solved using non-uniform centered finite difference approximations to estimate spatial derivatives and DDASSL numerical integrator to perform time
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integration. An adaptive strategy based on successive refinement of an initial onedimensional uniform spatial grid was developed and described in Brito (1998). Essentially, the adaptive algorithm converts the original problem in a set of subproblems generated by a subgrid selection procedure, in each time step. This selection is done by comparison of the problem solution obtained over a coarse (level 1) and a fine (level 2) grids. The nodes where an error criterion is not verified are grouped in subgrids over which subproblems are generated and solved. The procedure is repeated until the error criterion is verified on all nodes or a maximum refinement level is reached. In the later case, the procedure is repeated with a smaller time step. To avoid numerical problems that could arise due to the definition of boundary conditions for high refinement level subproblems (Brito et al., 2005), in this work it was chosen to only execute temporal step adaptation, fixing the maximum refinement level as 2. The model solution comprises temperature and composition spatial and temporal profiles. The composition spatial profiles were used to estimate the combustion wave propagation velocity, considering the front position .vs. time. The location of the front was obtained by the 50 % conversion point. For the particular thermite system under study (Eq. 1), the thermophysical properties of the components and its dependence relations with temperature were given in Brito et al. (2005). Out of the temperature ranges available in the literature, properties values were fixed as the last value known. The confinement materials properties were also defined in Brito et al. (2005). The general data for the simulations are presented in Table 2. It was found that the decreasing of the adaptive algorithm tolerance does not affect significantly the results, only leading to greater execution times. Table 2. General data for the simulation. Q T (J/kg) T0 (K) P (Pa) Tigni (K) 0
5322746
298.15
Z (m)
τ (s)
0.0015
a
101325 T’ (K)
0.1
1000
a
2300
Treact (K)
R (m)
R0 (m)
1200
0.025
0.0015
Δr (m) 1×10
υair 0
-5
0.392; 0.384; 0.365; 0.370; 0.350b
DDASSL tol.
Algorithm tol.
Finite diff. points
1st and 2nd level grid nodes
1×10-6
1×10-2
5
41; 81
a
Time and temperature normalization constants; b Experimental mean porosity for T100, T112, T127, T142 and T159, respectively.
4. Discussion of Results and Conclusions As expected, the temperature and composition spatial and temporal profiles are steep propagation waves. They are aproximatelly constant velocity waves with constant maximum temperature, except in the boundary regions. The conversion was complete in all cases. There was a minimum kinetic constant (K) value for self-propagation of the combustion wave: 7000 kg/(m3.s) for T100 and T112, 8000 kg/(m3.s) for T127 and T142, and 9000 kg/(m3.s) for T159. Figure 2 exemplifies the obtained profiles with minimum K. During the radial propagation, the front velocity tended to a constant value (vd. Fig. 3). These steady values are presented in Fig. 4 as function of K. The mean maximum temperature values, for 0.5 ≤ (r/R) ≤ 0.8, are also included. For each mixture, the increasing of K led to higher mean maximum temperatures and wave propagation velocities, as expected. The excess of aluminum in the mixtures improved the heat transfer in the system, by the increase of the thermal conductivity, and enhanced the
Modelling and Simulation of Fe2O3/Aluminum Thermite Combustion 6
0.2
1
5
WA/WA0
T/T'
3 2
0.2 0 0.2
0.4
0.6
0.8
0.05
0.4
0 0
0.1
0.6
1
0 0
0.2
0.4
0.6
0.8
Figure 2. Temperature and composition radial profiles for the T100 combustion with minimum K. Time gap between profiles - 0.3 s. 0.04
T100 T112 T127 T142 T159 T100 Exp.
0 0
20000
40000
60000
T (K)
v (m/s)
0.03
0.01
0.4
0.6
0.8
Figure 3. Typical combustion front velocity .vs. radius. Case of T100 mixture with K=100000 kg/(m3.s).
4800 4600 4400 4200 4000 3800 3600 3400 3200 3000
80000 100000 120000
T100 T112 T127 T142 T159
0
20000
K (kg/(m3.s))
40000 60000 80000 100000 120000 K (kg/(m3.s))
Figure 4. Combustion front velocities and mean maximum temperatures with several K values.
7
7
6 5
6 5
(a)
2 1 0
T/T'
T/T'
combustion propagation velocity. However, as the aluminum excess represents an additional thermal capacitance without contributing to an increase of heat released by the reaction, a higher aluminum excess led to lower mean maximum temperatures. Considering the stoichiometric mixture as a reference, an optimum K value was selected – 90000 kg/(m3.s) – to adjust the predicted combustion front velocity to the experimental value (vd. Fig. 4). Figure 5 presents the temperature profiles for T100 and T159 combustion with the selected K. The other mixtures led to intermediate profiles. Higher values of K led to steeper and more uniform profiles (vd. Figs. 2 and 5). The same was verified for the composition profiles. The calculated combustion front velocities, with selected K and for all mixtures, are given in Fig. 6 and compared with experimental results. A good agreement for T100 and T112 mixtures was reached. For the other mixtures, the predicted values are considerably lower than the experimental. This discrepancy is due to the experimental occurrence of a consecutive exothermic intermetallic reaction, which formed Fe3Al when aluminum was in excess and enhanced the front velocity (Durães et al., 2006a, 2006b). This reaction was not considered in the model. Experimentally, a nearly stoichiometric condition was only achieved between T112 and T127 mixtures, due to the incompleteness of reaction (Durães et al., 2006b). The model always gave complete conversion. The predicted combustion temperatures
4 3
4 3
(b)
2 1 0
0
0.2
0.4
0.6 r/R
0.8
1
1
1
r/R
r/R
0.02
0.2
r/R
0
1
v (m/s)
0.15
0.8
4
369
0
0.2
0.4
0.6
0.8
1
r/R
Figure 5. Temperature profiles obtained in the radial combustion of (a) T100 and (b) T159 mixtures with K=90000 kg/(m3.s). Time gap between profiles - 0.1 s.
Energy loss(J/(m .s))
0.06
v (m/s)
0.05
3
Model Experimental
0.04 0.03 0.02 T100
T112
T127
T142
T159
Figure 6. Experimental and predicted (K=90000 kg/(m3.s)) combustion front velocities.
1.E+10
5000
1.E+08
4000
1.E+06
3000
1.E+04 1.E+02
Conduction/ convection Radiation
1.E+00
Temperature
1.E-02 0
2000
T (K)
L. Durães et al.
370
1000
0 0.005 0.01 0.015 0.02 0.025 r (m)
Figure 7. Temperature and heat losses radial profiles obtained for T100 mixture, with 90000 kg/(m3.s) and t=0.5 s.
were higher than the experimental mean value (≈ 2300 K). This is justified by the heat losses due to the spraying of products away from the combustion system, observed experimentally and not considered in the model. Figure 7 presents the typical heat losses radial profiles calculated for a median time of the propagation. The radiative term is ≈ 100 times higher than the condutive/convective term in the combustion products region, where the temperature is very high. In the reactants region, the magnitude of these terms is comparable and strongly decreases.
References C.R. Bowen and B. Derby, 1995, Finite-difference modelling of self-propagating hightemperature synthesis of materials, Acta Metall. Mater., 43(10), 3903. P. Brito, 1998, Aplicação de métodos numéricos adaptativos na integração de sistemas algébricodiferenciais caracterizados por frentes abruptas, MSc. Thesis, University of Coimbra. P. Brito, L. Durães, J. Campos and A. Portugal, 2004, Aplicação de métodos adaptativos para a simulação de processos de combustão, in: Proc. of CMCE 2004, APMTAC & SEMNI & LNEC, Lisboa, p. 472 & CD-ROM. P. Brito, L. Durães, J. Campos, A. Portugal, 2005, Modelling and simulation of Fe2O3/Al thermite combustion in: Proc. of CHEMPOR 2005, Chem. Eng. Dept., Coimbra, p. 157 & CD-ROM. E.M. Carrillo-Heian, O.A. Graeve, A. Feng, J.A. Faghih and Z.A. Munir, 1999, Modeling studies of the effect of thermal and electrical conductivities and relative density of field-activated selfpropagating combustion synthesis, J. Mater. Res., 14(5), 1949. C. Cuadrado-Laborde, L.C. Damonte and L. Mendoza-Zélis, 2003, Theoretical treatment of a self-sustained, ball milling induced, mechanochemical reaction in the Fe2O3-Al system, Mater. Sci. Eng., A355, 106. L. Durães, J. Campos and A. Portugal, 2006a, Radial combustion propagation in iron(III) oxide/ aluminum thermite mixtures, Propell. Explos. Pyrot., 31(1), 42. L. Durães, B. Costa, R. Santos, A. Correia, J. Campos and A. Portugal, 2006b, Fe2O3/Aluminum thermite reaction intermediate and final products identification, Comb. Flame. Submitted. O. Graeve, E. Carrillo-Heian, A. Feng and Z. Munir, 2001, Modeling of wave configuration during electrically ignited combustion synthesis, J. Mater. Res., 16(1), 93. J.J. Moore and H.J. Feng, 1995, Combustion synthesis of advanced materials: Part II. Classification, applications and modelling, Prog. Mater. Sci., 39, 275. A. Makino, 2001, Fundamental aspects of the heterogeneous flame in the self-propagating hightemperature synthesis (SHS) process, Prog. Energy Comb. Sci., 27, 1. C.S. Raymond, K.G. Shkadinsky and V.A. Volpert, 1998, Gravitational effects on liquid flame thermite systems, Comb. Sci. Technol., 131, 107. K. Shkadinsky, G. Shkadinskaya and B. Matkowski, 2000, Gas-phase influence on quasisteady “liquid flames” in gravitational fields, Comb. Sci. Technol., 157, 87. K. Shkadinsky, G. Shkadinskaya and V. Volpert, 1997, Stability of “liquid flame” combustion waves Chem. Eng. Sci., 52(9), 1415.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Modelling of self-ignition and process upsets in industrial gaseous hydrocarbon oxidation processes Hans J. Pasman,a Michael Fairweather b a b
Delft Uni. of Technology, PO Box 5046, 2600 GA Delft, NL; [email protected] University of Leeds, School of Proc, Env. and Mats. Eng.; [email protected]
Abstract Industrial gas phase oxidation processes of hydrocarbons important for manufacturing intermediates for polymers and oxygenated end-products from various feedstocks have often to be operated sub-optimally as a result of lack of knowledge. Whenever hot hydrocarbons are mixed with oxygen or air problems can arise due to self-ignition. This paper gives an overview of work on relevant mechanisms, mathematical models and their validation being undertaken as part of the EC-supported SAFEKINEX project. Keywords: Hydrocarbon oxidation, explosion models, kinetics, ignition.
1. Introduction Hydrocarbon oxidation reactions can take many pathways, depending on concentration and temperature. With respect to temperature, distinction can be made between low, intermediate and high temperature oxidation mechanisms. The first produces the notorious peroxides, and even di-peroxides, which act as a source of radicals on decomposition at slightly higher temperatures. In a reacting mixture spontaneous explosion can occur by exponential self-acceleration, thermally or by radical branching, i.e. reactions multiplying radical chain carriers. Peroxides are a source of such radicals. Such an acceleration occurrence can trigger an avalanche, and is the cause of so-called cool flames that can even occur in compositions so rich in hydrocarbon that no normal hot flame can propagate. Cool flames produce limited energy and pressure build-up but can, under certain conditions, cause hot flames by multi-stage ignition. Cool flames occur in saturated hydrocarbons after a variable induction period (ignition delay time) at the relatively low temperatures of 250-400 °C. The precise range of compositions in which hot flame explosions can arise in mixtures at elevated temperature (250-300 °C) and pressure (up to 50 bar) is not well known. This is also true for peak explosion pressure and rate of pressure rise, as well as for near-upper explosion limit compositions. In contrast, atmospheric conditions and stoichiometric composition have been widely investigated. There are also open questions in regard to flame acceleration and deflagration to detonation transition at such conditions, particularly for in-vessel events. To avoid run-away or explosion in practice, a margin has to be kept around conditions which entail risk, although higher oxygen concentrations can be advantageous in obtaining higher reaction rates and yield, with less energy consumption, recycle and waste. The core of the problem for more efficient processes and improved operability is the complexity of the interactions between chemistry, heat and mass transfer in the
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flow, and the control of interdependent parameters such as composition, temperature, pressure and residence time. The SAFEKINEX project combines expertise in the areas of hydrocarbon oxidation kinetic modelling and gas explosion safety testing and modelling. The project started in January 2003 for 4 years with a budget of M€3.5 for a consortium of 13 partners distributed over 6 countries and with industry participation. The first two work packages concerned detailed kinetic modelling and explosion testing, although these are now completed. The third and fourth work packages, now in progress, concern explosion modelling and the reduction of detailed kinetic schemes. The latter is a final deliverable that serves as a starting point for incorporation of results in to CFD models. This paper gives an overview of work underway on relevant kinetic models, including models for burning velocity prediction, and mathematical models of explosions, and their validation, being undertaken as part of the SAFEKINEX project. Although a great deal of work of relevance to explosions and their prediction has previously appeared in the literature, the novelty of the work reported here lies in its relevance to industrial gas phase oxidation processes of hydrocarbons, and the chemical kinetic input used as the basis of the work.
2. Modelling kinetics and explosions 2.1. Kinetic models The basis of this work is the computer package EXGAS developed by Battin-Leclerc et al. [1]. This produces a kinetic scheme containing 100’s of reactant species, their thermo-physical and kinetic data, and >1000 reactions for the oxidation of each alkane, alkene, cyclane and aromatic hydrocarbon up to C10 in Chemkin format. Chemkin is a computer package developed by Kee et al. [2] for solving the set of stiff simultaneous ordinary differential equations which constitute the model. It applies to different reactor types, e.g. perfectly stirred, plug flow and burner-type reactors. Similar, more recent, packages are also available, e.g. Chemical Workbench [3] and Cantera [4]. EXGAS contains a primary mechanism generator in which the hydrocarbons and oxygen in the initial mixture react. The primary products are fed to a lumped secondary mechanism generator in which isomers are no longer distinguished and which produces the small species which react to form the end products of combustion in a third step, the so called C0-C1-C2 kinetics. This part of the project was developed by Konnov at Vrije Universiteit Brussels [5]. Validation of EXGAS is mainly against experiments that use an initial homogenous mixture at the desired temperature and pressure. Ignition delay time is then determined under near-adiabatic conditions in a rapid compression machine that produces ignition by compressing the mixture by means of a piston and shock tubes in which a reflected shock wave creates the desired conditions for a sufficiently long duration of time. Laminar burning velocity data obtained from a radiation compensated burner was also used. The kinetic models have been applied to predicting the induction period in (static) autoignition experiments at various temperatures and pressures in glass and steel vessels, the
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extent of pre-ignition reactions in forced ignition gas explosion experiments, the minimum energy required in forced ignition, and most importantly the laminar burning velocities required in explosion modelling. In this regard, an important drawback of Chemkin-type models is the constraint that the reactor space must be homogeneous. This prohibits simulation of problems with temperature distributions, as encountered in practice. Further development of CFD is crucial to resolving such problems. 2.2. Burning velocity Konnov [5] calculates the laminar burning velocity for any composition over the desired range of temperatures and pressures: the basis for further explosion modelling. Next is the sensitivity of the flame to stretching and wrinkling. A curved flame surface can either be reinforced and accelerated, or weakened and decelerated, by the difference in the diffusional properties of the fuel and oxidiser, depending on which is deficient compared to the stoichiometric composition. This is characterised by the Markstein number that can be calculated using full kinetic models for the case of one-dimensional, spherically expanding flames using the finite-difference scheme developed by Maas [6] and refined in the project by Zarzalis and Weiss at the University of Karlsruhe. Through these effects an initially perfectly spherical flame will develop wrinkles and cusps that evolve and become more pronounced as the flame expands further. This leads to self-induced turbulence and flame acceleration. The turbulent flame also has a much larger net burning surface area, which can significantly increase energy release rates. However, for fuel-rich compositions, examined in the project, such flames are nonspherical since they propagate slowly and rise under the influence of buoyancy whilst producing soot. Modelling of such flames is still in its infancy. 2.3. Explosion modelling Modelling of explosions in closed vessels is being carried out by Wolanski and coworkers (Warsaw UT). This model uses conservation laws, the laminar burning velocity, and equilibrium of combustion products to calculate, for a spherically expanding flame, pressure, temperature and flame location with time. The influence of turbulence on the burning velocity, vessel shape (in non-spherical vessels the flame touches the walls at different times), heat loss, venting to atmosphere, and pipeconnected vessels are also being investigated. The descriptive equations, a set of ordinary differential equations, are integrated by means of an explicit method using an appropriate time step, with solution of the non-linear equations describing thermodynamic equilibrium being the most time consuming activity. Significant work remains to be performed, although initial pressure build-up results are encouraging.
3. Model results and comparison with experiments 3.1. Burning velocity and explosion Fig.1 compares data for laminar burning velocity and predictions of the detailed kinetic models, and shows excellent agreement between them.
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Fig. 2 shows an early result derived from the gas explosion model for an explosion in a closed vessel, and compares this with experiment. These predictions were obtained using both the laminar burning velocity and one derived for a flame with a self-induced turbulent burning velocity that develops as the flame expands. The turbulent burning velocity follows from a semi-empirical relation based on the Reynolds number defined with respect to the laminar flame speed and flame radius. Modelling of gas explosions in vessels is seen to be making good progress, although challenging developments to cover aspects such as heat loss, multi-vessel explosions, and near-limit flames remain. 25
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Fig. 1 Adiabatic burning velocities for CH4-O2-N2 flames with different dilution ratios. Crosses: 18% O2 in air; circles: 17%; squares: 16%. Solid lines: modelling based on C0-C1-C2 kinetics and Chemkin Premix package [7].
Fig. 2 Pressure-time history of a stoichiometric methane-air explosion at 1 bar a. Curve which displays peak - experiment. Calculated curve which peaks at ≈ 0.4 s determined using laminar burning velocity. After modification to account for turbulence, final curve, peaking at ≈ 0.3 s, approaches experiment.
3.2. Self-ignition and minimum ignition energy Figs. 3 and 4 demonstrate how self-ignition can be predicted on the basis of the detailed kinetic models described above. The simple heat transfer model possible at this stage, in combination with the kinetics, to account for losses to the ambient during the process is still a severe limitation. As mentioned, it is also possible to calculate spark ignition energy and energy density. Fig. 5 shows results for stoichiometric propane-air mixtures at varying energy deposition times plotted against the deposition radius. For small deposition radii the ignition energy appears largely dependent on the deposition time and tends to a constant value. The minimum ignition energy therefore seems to depend particularly on a certain minimum deposition time. Per unit of volume, the ignition energy (energy density) reaches a minimum at larger radius. Experiments to validate these predictions are under way.
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Fig. 3 Induction time to occurrence of cool flame and self-ignition in 9.5% (rich) n-butane-air mixture at 1 and 10 bar, calculated with detailed kinetic model and measured in 200 ml steel or glass vessel [7]. The best EXGAS derived model shows a reactivity which is slightly too high (same ignition delay time at 35 K lower temperature). The heat transfer coefficient, h, is 1.5 W/(m2K).
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Fig. 4 Model calculation of self-ignition of 78% n-butane and 22% oxygen at 4.1 bara and 500 K in a 20 l steel vessel. If performed at 38.5 K lower temperature the calculation synchronised with the measured consumption of oxygen. Prior to self-ignition the mixture could be ignited and exploded by a fuse wire (h = 2 W/(m2K)).
Induction controlled
Fig. 5 Calculated ignition energies (left) and ignition energy densities (right) of propane-air mixtures versus energy deposition radius rd at different deposition times [8].
4. Discussion and conclusions Fundamental work of relevance to industrial gaseous hydrocarbon oxidation processes has been described, with emphasis on kinetic and explosion modelling. At the present time, these models are still under development, and further validation is required against data to be gathered as part of the SAFEKINEX project. Extensions of the methods described will encompass a wider range of initial, and particularly elevated, conditions, as well as more complex and practically relevant geometries. Work to provide reduced kinetic mechanisms, now under way, will ultimately allow the predictive techniques described to be linked directly with CFD codes capable of predicting the full range of scenarios of interest in practice.
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5. References 1. Buda F., Bounaceur R., Warth V., Glaude P.A., Fournet R., Battin-Leclerc F., Combust. Flame 142 (2005) 170-186. 2. http://www.reactiondesign.com/ 3. http://www.kintech.ru/ 4. http://rayleigh.cds.caltech.edu/~goodwin/cantera/index.html 5. Konnov A.A., Dyakov I.V., Combust. Flame, 136 (2004) 371-376. 6. Maas U., 1988, Doctoral thesis, Ruprechts-Karls University Heidelberg, Germany. 7. Liebner, Ch., Pekalski, A.A., Pasman, H.J., Fire Bridge 2005, Belfast, UK, 9-11 May. 8. Pasman H.J., Bouma R., Zarzalis, N., Weiss, M. CEJEM 2 (2005) 55-69, ISSN 1733-7178. Acknowledgement: Financial support of this work by the European Commission within the Fifth Framework Programme on Energy, Environment and Sustainable Development, contract EVG1CT-2002-00072, Project SAFEKINEX, is gratefully acknowledged, as well as the support by the project work package leaders: Dr. V. Schroeder, BAM; Dr F.Battin-Leclerc, CNRS; and Prof. J.F. Griffiths, University of Leeds, and other project partners. See also www.safekinex.org.
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A simplex search method for experimental optimization with multiple objectives Ernesto Martínez INGAR (CONICET-UTN), Avellaneda 3657, S3002 GJC, Argentina
Abstract Experimental optimization with multiple responses in the face of noise and outliers is a key issue in product/process development and to guarantee product end-use properties. A novel concept of maximimizing the concordance of desirability functions of all responses concerned is presented. The Kendall´s coefficient of concordance W borrowed from nonparametric statistics is used to provide a statistical characterization of optimality in a multiobjective setting. A multi-directional simplex method is proposed in the concordance function framework. Simplex reflection, expansion and contraction operations are based on ranking vertices according to their corresponding values of W. Keywords: Desirability functions, multiobjective optimization, simplex search method
1. Introduction Experimental process optimization whilst accounting simultaneously for several objectives or responses is a challenging problem in product/process development, runto-run optimization, calibration of analytical methods, design of extremum-seeking controllers and automated planning of experiments in chemistry workstations [1-3] However, the standard Nelder-Mead simplex and all of its variants can only handle one performance index which necessarily reqiuires combining several responses into a single objective. In this work, a novel variant of the simplex search method based on maximizing the concordance of the desirability functions for a set of responses is proposed so as to account simultaneously for multiple objectives.
2. Multi-objective optimality and concordance The desirability function approach is one of the most widely used methods in industry for dealing with the optimization of multiple-response problems [2,3]. A desirability function d(yi) assigns real numbers between 0 and 1 to the possible values of each response yi. The value of di(yi) increases as the desirability of the corresponding response increases. There are two types of transformation from yi to di(yi), namely one-
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sided and two-sided transformation. We employ the one-sided transformation when yi is to be maximized or minimized, and the two-sided transformation when yi is to be assigned a target value. 2.1. Case study: reducing NOX-emissions from a boiler In non selective reduction systems (NSRSs), a solution of urea or ammonia is injected into, and mixed with, the flue gases in the hot section of a furnace to reduce harmful emissions of nitrogen oxides (NOx). Typically, a NSRS is operated at a temperature range of 800 to 950 0C. The conversion of NOx to N2 in the presence of NH3 is optimized within this range. For a given temperature, three input variables describing injection conditions are used to minimize NOx content and to maintain NH3 near a 3 ppm target value in the flue gas. The desirability functions are defined as shown in Fig. 1. d1
d2
1
1
0
0 Lower limit
Upper limit
[NOx]
Lower limit
Upper limit
Target
(a)
[NH3]
(b)
Fig. 1. Desirability functions for the NSRS. (a) minimization of NOx-; (b) NH3 target
Table 1 summarizes information regarding input and response variables whilst Table 2 provides a small dataset of inputs and process responses around the reference input point [1.03, 250, 2.5] along with the corresponding values of each response desirability function. Table 1. Input and response variable data Name
Units 3
Type
Range
Ref. Value
Density, x1 Flow, x2 Pressure, x3
kg/m kg/h bar
Input Input Input
1.00 ≤ x1 ≤ 1.06 100 ≤ x 2 ≤ 400 1.0 ≤ x 3 ≤ 4.5
1.030 250 2.5
NOx, y1 NH3, y2
ppm ppm
Response Response
50 ≤ NOx ≤ 150 0 ≤ NH3 ≤ 10
minimization target (3.0)
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Table 2. Data set around the reference point for the NSRS run #
x1
x2
x3
y1
y2
d1
d2
1 2 3 4 5 6 7 8 9
1.031 1.029 1.058 1.020 1.035 1.028 1.012 1.059 1.030
290 245 200 220 260 240 225 210 255
2.2 2.3 4.0 2.9 3.0 2.4 2.0 4.1 2.6
87 88 84 109 90 96 121 83 90
9.6 5.2 8.9 3.3 9.1 5.0 2.0 7.4 6.0
0.63 0.62 0.66 0.41 0.60 0.54 0.29 0.67 0.60
0.057 0.069 0.157 0.957 0.129 0.714 0.667 0.371 0.571
2.2. Correlation and concordance The monotonicity property proposed for one objective in [1] is generalized here using the notion of concordance of desirability functions regarding a given optimum location for the joint response optimization. Def.: Maximal concordance. As the distance to the joint response optimum increases there exist a monotically decreasing trend in the degree of concordance for the corresponding desirability functions. Accordingly, hypothetical optima that are closer to the true optimum should exhibit a greater degree of joint correlation or association than those that are farther away. For experimental optimization, it is required that the chosen measure of concordance be robust against noise and unknown sources of variability present in process data. In this work, resorting to the Kendall´s coefficient of concordance W borrowed from nonparametric statistics [4] is proposed. Suppose we have obtained the set of ranks θ i1 , θ i2 ,…, θ ik ( k ≥ 2 ) for the response desirability functions using the same ordinal scale of increasing distances to an hypothetical optimum location in a data set with n experiments. Let´s denote the sum of ranks given to the ith data point by the k responses as Ri, i=1,2,…,n. The sum of ranks for each desirability function is 1+2+….+n=n.(n+1)/2 , and hence the average rank for each of the n data points is (n+1)/2. If there is no agreement among the models and the model assign ranks to data points almost randomly, each rank for each response would be the average rank (n+1)/2 and the rank sum for each model would be equal to (k.(n+1)/2) because each is the sum of k ranks. The sum of squares of deviations of the actual rank sums around k.(n+1)/2 is denoted by S and defined as n
⎡
S= ∑ ⎢ Ri − i =1⎣
k (n + 1) ⎤ 2 ⎥⎦
2
(1)
On the other hand, if there is perfect agreement among the responses, each of them would have ranks that are all the same and the rank sums would be some permutation of
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the numbers 1k, 2k, 3k, …, nk. The sum of squares of deviations around k.(n+1)/2 in the case of perfect concordance is n ⎡ k (n + 1) ⎤ ∑ ⎢ik − 2 ⎥⎦ i =1⎣
2
(2)
The Kendall´s coefficient of concordance W for the group of k responses is thus defined as the ratio between Eq. (1) and (2) which, after some algebraic manipulations, can be written as W=
12S 2
k n(n 2 − 1)
(3)
The sum of squares of deviations under perfect response agreement is the maximum possible value of S and the therefore the value of W ranges between 0 and 1, with increasing values reflecting an increasing degree of concordance among the k responses. In Appendix D of [4] a table for different values of n and k of the probabilities for the null hypothesis in the case of perfect agreement are provided. Based on the ranks for the distance to a hypothetical optimum and the corresponding ranks for the desirability functions in the NSRS case study the resulting value of the concordance index is W=0.67.
3. Multi-directional statistical simplex method Accounting simultaneously for several responses through the concordance index W requires multi-directional searches of the input space [5]. At any iteration m, where m ≥ 0 , the proposed simplex search method requires n + 1 points (vertices) v o , v1 ,..., v n , which define a non-degenerate simplex in ℜ n . The edges of the current simplex are used to define the search directions based on vertices ranking. For ranking, the value of the concordance index W is calculated for each vertex. Using this information, the algorithm distinguishes the “best” vertex v om in the current simplex as the point exhibiting the highest concordance for the set of responses. The best vertex v om satisfies: W ( v om ) ≥ W ( v im ), i = 1,..., n
(4)
The n edges connecting the best vertex to the remaining n vertices determine a set of linearly independent search directions. 3.1. Simplex reflection, expansion and contraction Given a data set from previous experiments and the concordance values for the vertices in the current simplex, a simplex reflection operation from the best vertex generates n new points for which the k responses will be obtained. The reflection operation is defined along the edges vom v1m and vom v2m . The reflection step is successful if:
A Simplex Search Method for Experimental Optimization
max{W ( v rm ), i = 1,..., n} > W ( v om )
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(5)
i
It is worth noting that checking for a successful reflection using (5) does not necessarily implies doing an actual experiment at each reflected point. However, unless the dataset already has enough information, testing for (5) should be based on actual experimentation at the reflected points. If simplex reflection is successful, a simplex expansion from the best current vertex is considered. The acceptance condition for the expansion step is: max{W ( v em ), i = 1,..., n} > max{W ( v rm ), i = 1,..., n} i
i
(6)
If the expanded simplex is accepted, the next iteration will have the best current vertex and the corresponding expanded points. If only the reflection condition applies, the next simplex will be defined using the reflected vertices from the best vertex. If the current simplex reflection fails to produce a vertex which can exhibit a point with higher concordance than the current best one, a simplex contraction is attemped. The concordance increase condition for the contracted simplex is: max{W ( v cm ), i = 1,..., n} > W ( v om ) i
(7)
3.2. Algorithm The goal of the multi-directional statistical simplex algorithm is to construct a sequence of best vertices {v om } that converges to a maximizer of the concordance index W. To achieve this, the algorithm requires that the values of W for the best vertex be monotically increasing. Thus, when simplex operations fail to satisfy at least one the increase conditions of Eq. (5), (6) or (7)., a simplex restart step is done. The overall logic of the algorithm is given in Fig. 2. The convergence criterion for the algorithm is w0m +1 − w0m ≤ ε
(8)
3.3. Case study (continued) The iteration-to-iteration evolution of the concordance index W is shown in Fig. 3. After 30 experiments, the solution obtained was x1=1,032, x2=284 and x3=3,3. and the corresponding values for the desirability functions are: d1=0.74 and d2=0.82.
4. Concluding remarks The novel concept of maximum concordance for simultaneuous optimización of multiple responses has been presented. The Kendall´s concordance index W has been
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proposed to cope with noise and outliers. A new multi-directional simplex algorithm has been developed in the concordance concept.
References 1. 2. 3. 4. 5.
E. C Martinez, Ind. Eng, Chem. Res. 44, 8796-8805 (2005). G Derringer, Quality Progress, June issue, 51-58 (1994). G Derringer and R. Suich, J. Quality Technology 12, 214-219 (1980). J. D. Gibbons, Nonparametric measures of associations, SAGE Publications (1993). V. J. Torczon, Multi-directional search: a direct search algorithm for parallel machines, PhD Thesis, Rice University, Houston, (1989) Accept expansion Yes
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Fig. 2. Multi-directional statistical simplex algorithm
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Automatic generation of reduced reaction mechanisms for hydrocarbon oxidation with application to autoignition boundary prediction for explosion hazards mitigation R. Portera, M. Fairweathera, J.F. Griffithsb, K.J. Hughesb, A.S. Tomlina a
School of Process, Environment and Materials Engineering and bSchool of Chemistry, University of Leeds, Leeds LS2 9JT, UK
Abstract In this work we present an automatic method for removing species and reactions from comprehensive reaction mechanisms without significant detriment to model performance. Numerical methods are applied to a lean n-butane - air closed vessel system. A method for the automatic construction of closed vessel ambient temperature composition (Ta – φ) ignition diagrams is presented, which is used to evaluate the comprehensive and reduced models. Application of the quasi-steady state approximation to the reduced mechanism has been proven to significantly reduce the number of species with very little loss of output accuracy. Keywords: Combustion, autoignition, lean n-butane-air, QSSA, sensitivity analysis.
1. Introduction Despite our considerable knowledge of the potential hazards associated with the chemical process industries, explosion hazards continue to occur during hydrocarbon processing under partial oxidation conditions. Among the reasons for this is the change of conditions that arise from process intensification, combined with an incomplete knowledge of the oxidation characteristics of the processed materials. The ability to couple chemical kinetics with fluid dynamics and simulate these processes in reactive multi-dimensional flows would be a powerful process engineering tool that would constitute a significant advance in methodologies available to predict such hazards. Detailed combustion kinetic mechanisms contain hundreds of chemical species and thousands of reactions, making them too computationally expensive to be solved in computational fluid dynamics (CFD) codes. By adopting formal mathematical procedures, more compact and computationally efficient kinetic models can be generated by reducing the numbers of species and reactions from the detailed mechanisms. Currently, this involves running full kinetic models with multiple initial conditions in a non CFD-based environment, interpreting the results using local sensitivity methods, identifying and removing redundant species and reactions, and then testing the reduced mechanisms. Many hours can be saved by automating these tasks using programming techniques. In this paper we describe software which can be used to automatically minimise the numbers of chemical species and reactions without loss of important kinetic detail. The codes are based on the use of UNIX shell scripts to completely automate the utilisation of numerical integration and local sensitivity analysis software. Reduced chemical models which can be used in higher dimensional simulations are obtained as output.
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The bench-mark is set by the performance of the full scheme and the criteria for performance of the reduced models are matched to this. As well as being fundamental to the potential hazards, an important basis for validation of the models is the ignition diagram as a function of ambient temperature versus composition or pressure, in which is mapped a wide range of combustion regimes. The construction of the numerically predicted ignition diagram is also a laborious process which is amenable to automatic generation. This new software, encompassing automation in both areas, is applied in the present work to illustrate the accurate reproduction of ignition and cool flame boundaries over a range of operating conditions using significantly reduced kinetic schemes when compared with the full models adopted at the outset.
2. Methodology and Models The comprehensive model to which the methods were applied was derived at CNRSDCPR, Nancy [1] for n-butane oxidation, comprising 125 species in 314 irreversible reactions and 417 reversible reactions. The reversible reactions can be expressed as irreversible pairs equivalent to a total of 1148 irreversible reactions for the full scheme. The resulting system of ordinary differential equations was solved using the SPRINT integration package [2] for a closed vessel system with spatial uniformity assumed. An ambient temperature – composition (Ta – φ) ignition diagram was automatically constructed using developed software which can categorise the various non-isothermal behaviour such as 2-stage autoignition, cool flames, and slow reaction by monitioring temperature and gradient changes in the predicted temperature profiles. The software works by conducting a series of simulations over the selected temperature range of 550 – 750 K at specified intervals of 5 K and at a fixed pressure and composition where exclusively 2-stage ignition occurs. Then a bisection method is employed in which the partial fuel pressure is initially halved (while maintaining the total pressure), and then progressively adjusted in order to locate the boundary between ignition and cool flame or slow reaction behaviour, and similarly for the cases where cool flame behaviour is observed, to locate the cool flame/slow reaction boundary. These calculations proceed until the desired level of accuracy is obtained, in this case to 0.5 torr. Similar software has been developed to compute the pressure – ambient temperature ignition diagram. The resulting Ta – φ ignition diagram was used as the benchmark against which the reduced models were tested. Using the ignition diagram as reference, a number of different operating conditions were selected covering a representative range of the temperature/composition space at which sensitivity analysis and mechanism reduction are to be performed. A shell script was set up to run the integration code at each chosen condition, and manipulate the output data files. Time points from the calculated temperature profiles at the chosen operating conditions were automatically selected on the basis of ΔT and the gradient of each trajectory, as shown in fig. 1. Information related to these conditions and rate data from the mechanism were used to identify necessary species via the investigation of the Jacobian matrix [3] using algorithms incorporated into the SPRINT code originally implemented in the KINALC package [4, 5]. The necessary species include selected important species as defined by the user, and other species for which realistic concentrations are required in order to reproduce the concentrations of important species or important reaction features. The union of identified necessary species was taken at the selected time points and the irreversible consuming and reversible reactions
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of all redundant species removed. The resulting mechanism was then converted to irreversible form for further analysis. Via a similar process, techniques were then used to identify reactions that can be eliminated. Local sensitivity analysis was used to identify redundant reactions by consideration of the rate sensitivity matrix F~ :
~ k j ∂f i F= , f i ∂k j
(1)
where kj is the rate parameter of the jth reaction and fi is the rate of production of species i. The effect of a change of each rate parameter on the rates of production of necessary species is given by a least-squares objective function: 2
⎛ k j ∂f i ⎞ ⎟ . Bj = ∑⎜ ⎜ ⎟ i ⎝ f i ∂k j ⎠
( 2)
A reaction is considered important if it has a Bj value above a user specified threshold. Finally, principal component analysis based on the eigenvalue-eigenvector ~ ~ decomposition of the cross-product matrix FT F , was used to identify redundant reactions. Each eigenvector represents a set of coupled reactions whose relative contributions are shown by the relative size of the eigenvector elements. Thresholds were defined for the significant magnitudes of the eigenvalues and eigenvectors and this provided an automatic way of deciding which reactions can be eliminated [6-8]. Considerable improvement in the performance of the reduced models can be achieved by using subsets of necessary species relevant for each specific time point within the objective function, rather than the combined set of necessary species acquired from the species reduction. This is illustrated in Fig. 2 by comparing reduced mechanisms obtained using Equation 2, with either the full set of species included in the summation i, or time point specific sets as identified by the local Jacobian matrix. A similar result would follow from principal component analysis. 1200
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Fig. 1. Automatically selected time points during simulated 2-stage ignition in the viscinity of the transition from cool flame to ignition. The first time point was automatically selected at 0.003 seconds.
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Fig. 2. Comparison of using all necessary species or a subset at each time point in the objective function. Unbroken line – species reduced, 715 reaction mechanism. Dotted line – subset reduced, 449 reaction mechanism. Dashed line – all necessary species reduced, 449 reaction mechanism.
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3. Application of the Quasi-Steady State Approximation The application of the above sensitivity methods leads to a skeleton mechanism with all redundant species and reactions removed. However, in many cases the level of reduction achieved by such methods is not sufficient for application of the chemical model within complex flow computations. Subsequent reduction may be based on exploiting the time-scales present in the mechanism, with a range of reduction techniques falling into this category including intrinsic low dimensional manifold (ILDM) based methods [9] and methods based on the application of the quasi-steady state approximation (QSSA). QSSA based methods are commonly used in kinetic model reduction by assuming that fast reacting species locally equilibrate with respect to the slower species within the system. The concentration of the QSSA species can then be approximated via the algebraic expression f i q = 0, rather than a differential equation, where the superscript q denotes a QSSA species. In many cases QSSA species can be removed via simple reaction lumping. Alternatively, the concentration of species ci can be expressed in terms of the concentrations of other species in the system and the rate parameters. Such expressions can be solved either analytically or via iterative techniques for sets of highly coupled species. The choice species suitable for application of the QSSA can be determined in a variety of ways including using perturbation methods. The instantaneous QSSA error for a single species, was defined in [10] using a local linear perturbation method as:
Δ c is =
1 fi , c i J ii
(3)
where Jii is the diagonal element of the chemical Jacobian for species i. Although the QSSA errors vary throughout the simulations, peaking during ignition, for many species the errors remain below a certain threshold throughout. Using a tolerance of 1% across all selected time-points for the QSSA error, 31 QSSA species can be automatically identified. Many have fairly simple reaction structures and therefore can be removed via the methods illustrated in the following example.
RH
1
R
2 -2
RO2
3 -3
QOOH
4 -4
6
R' + alkene
O2QOOH 5
OH + product
Fig. 3. Reaction sequence to which the QSSA was applied.
In the reaction sequence shown in Fig. 3, solving the algebraic expressions resulting from the application of the QSSA for the highlighted species can be demonstrated to be equivalent to the lumping of several of the individual reaction steps resulting in the removal of RO2, QOOH and O2QOOH. The central part of the reaction sequence can then be replaced by: 2' R ⎯⎯→ OH + product,
where
k 2' = k 2 (1 − k − 2 (k −2 + k 3 − k 3 k −3 (k −3 + k 4 − k 4 k − 4 (k −4 + k 5 )))).
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Then R can be removed to leave the final reaction sequence: 7 RH ⎯ ⎯→ R ' + alkene 8 RH ⎯ ⎯→ OH + product,
where
⎛ k ⎞ k 7 = k1 ⎜⎜ ' 6 ⎟⎟ ⎝ k 2 + k6 ⎠
⎛ k' k 8 = k1 ⎜⎜ ' 2 ⎝ k2 + k6
and
⎞ ⎟⎟. ⎠
In the simplest approach, k2’ is assumed to be a constant fraction of k2, and set at the fraction calculated in the region of maximum flux through R to OH + product. A rate of production analysis of the full scheme shows this to be a good approximation in this instance, and applying it gives simulated temperature profiles in excellent agreement with those obtained from the original scheme. The ratio of k7 to k8 is not constant, and changes significantly with temperature, favouring k8 at low temperatures and switching over to k7 at high temperatures. Even so, assuming a constant ratio based on that applicable at low temperatures still gives very good agreement in the simulated temperature profiles, with only slight deviation at the later times and higher temperatures where this approximation is no longer valid. A more rigorous approach is to program directly the variation of k2’, k7 and k8 with temperature, although this results in a loss of compatibility of the reduced mechanism with commercial simulation packages such as CHEMKIN. Of the QSSA species identified, 14 were easily removed by applying the method highlighted above resulting in a final mechanism of 58 species and 270 reactions.
4. Model Validation and Application of Sensitivity Analysis Fig. 4 shows the experimental and simulated Ta – φ ignition diagrams for n-butane + air. The qualitative features of the experimental Ta – φ ignition boundary [11], shown in Fig. 4, are captured by the numerical models showing both cool flame and two stage ignition behaviour. The reverse “s” shape of the ignition boundary is displayed by the Slow
750 Reaction
750
700
700
Cool flame
Ta / K
Ta / K
Experiment 650
2-stage ignition
Cool flame
650
2-stage ignition
600
600
Slow Reaction 550 0.0
550
0.5
1.0
1.5
2.0
2.5
% n-C4H10 by volume in air
Fig. 4. Comparison of experimental and full scheme Ta – φ ignition diagrams.
1.6
1.8
2.0
2.2
% n-C4H10 by volume in air
Fig. 5. Comparison of full scheme (solid line), species reduced (dotted line) and QSSA reduced (dashed line) Ta – φ ignition boundaries.
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models and this is an important validation. However, quantitative disagreements with the experiment remain, especially at higher temperatures where the model over-predicts the autoignition temperatures. This may imply a shortcoming in the way that the intermediate molecular products that lead to high-temperature reactions are interpreted. There may also be some discrepancy due to inhomegeneities of temperature in the unstirred vessel [11]. Comparison of the model results shows that both reduced mechanisms illustrated in Fig.5 reproduce the behaviour of the full scheme extremely well. The scheme produced by removal of redundant species from the full scheme produced a mechanism comprising of 72 necessary species and 713 irreversible reactions, generated a Ta – φ ignition diagram that matched that of the full scheme very closely. Further reduction by removal of redundant reactions and applying the QSSA to remove a further 14 species, giving a scheme of 58 necessary species and 270 reactions, also behaved very well, with only minor deviations to the full scheme prediction. It is possible to apply different cut off values in these methods in order to reduce the mechanisms still further but at a cost of a reduced level of agreement with the full scheme. By specifying higher thresholds for the eigenvalues and eigenvectors of principal component analysis, prior to QSSA, it is possible to reduce the numbers of reactions even further. However, the increasing error induced by this reduction was considered to be unsatisfactory since it gave little extra computational saving.
5. Conclusions Software for the automatic construction of ignition diagrams has been developed. Programming techniques have allowed the automatic and systematic reduction of a lean n-butane - air kinetic model, simulated in a closed vessel. Comparisons of the predictions of full and reduced schemes have shown that the numbers of species and reactions have been successfully reduced. Further reductions have been achieved using the quasi-steady state approximation to lump reactions and further reduce species.
Acknowledgement The authors gratefully acknowledge financial support from the EU (EVG1-CT-200200072-SAFEKINEX) and from EPSRC (GR/R42726/01).
References [1] www.ensic.u-nancy.fr/DCPR/Anglais/GCR/softwares.htm [2] M. Berzins, R.M. Furzland, Shell Research Ltd., TNER 85058, 1985. [3] T. Turányi, New J. Chem. 14 (1990) 795-803. [4] www.chem.leeds.ac.uk/Combustion/kinalc.htm [5] T. Turányi, Reliab. Eng. Syst. Safe., 57 (1997) 41-48. [6] S. Vajda, P. Valkó, T. Turányi, Int. J. Chem. Kinet., 17 (1985) 55-81. [7] A.C. Heard, M.J. Pilling, A.S. Tomlin, Atmos. Environ. 32 (1998) 1059-1073. [8] A.S. Tomlin, T. Turányi, M.J. Pilling, in: M.J. Pilling (Ed.), Low Temperature Combustion and Autoignition, Elsevier, Amsterdam, 1997, p. 293. [9] U. Maas, S.B. Pope, Combust. Flame 88 (1992) 239-264. [10] T. Turányi, A.S. Tomlin, M.J. Pilling, J. Phys. Chem. 97 (1993) 163-172. [11] M.R. Chandraratna, J.F. Griffiths, Combust. Flame 99 (1994) 626-634.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Combining HAZOP with Dynamic Process Model Development for Safety Analysis Shimon Eizenberga, Mordechai Shachama, Neima Braunerb a
Dept. Chem. Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel School of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel
b
Abstract A quantitative variation of the HAZOP (Hazard and Operability Analysis) procedure is demonstrated. The process is divided into sections and dynamic models of the separate sections are prepared. Those models are used in the framework of the HAZOP procedure to determine the magnitude of the deviations from normal operation conditions that may lead to serious accidents and to test design modification to improve the safety characteristic of the process. A process involving an exothermic reaction conducted in a semi-batch reactor is used to demonstrate the advantages of the proposed procedure. Keywords: HAZOP; HAZAN; Dynamic simulation; Temperature runaway.
1. Introduction Process hazards analysis is an essential part of the process design activity. In the United States for example OSHA (Occupational Health and Safety Administration) regulations require that major chemical plants perform process hazards analysis on a regular basis when a new process is launched, or a major change occurs in an existing process (Dash and Vakatasubramanian, [1]). HAZOP (Hazard and Operability Analysis) is a widely used procedure for process hazards analysis [1-6]. HAZOP is carried out by a multidisciplinary team of experts in a qualitative manner. The new process is examined systematically, section by section, looking for inadequacies in design, which may lead to serious accidents. A series of guide words (such as "NONE", "MORE OF", "LESS OF" etc.) are used to ensure that all the potential deviations from normal operating conditions are considered. For each deviation the possible causes are listed and the consequences and actions required are considered. Often the action required is a change of the design in order to reduce the probability of a particular deviation, or to reduce the severity of its consequences. In a few cases, where deviation from normal conditions may lead to catastrophic events, HAZOP is often followed by a detailed hazard analysis (HAZAN, [2]), where the probability for the occurrence of such events is evaluated. Recently the addition of dynamic simulation to the HAZOP and HAZAN procedures has been advocated [4-6] as a means to provide quantitative answers regarding the magnitude of the deviations that will lead to severe consequences, the time it takes to reach a "no return" stage of an accident after the deviation has occurred and the action that can be taken in order to prevent the accident. Detailed simulation models of various processes operating under abnormal conditions were carried out, for example by Eizenberg et al. [7] and Shacham et al.[8,9]. The development of a simulation model of a large-scale process operating in abnormal conditions is considered a very demanding, difficult and often even an infeasible task
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[1]. However, in the framework of HAZOP, those difficulties can be alleviated by dividing the process into sections, and modeling each section separately. The aim of this paper is to show the importance of incorporating dynamic simulation in the framework of HAZOP approach. First a model, which represents the process in normal operating conditions, is developed. This model is extended and modified so that it can represent the process behavior adequately when deviations from normal conditions are introduced. The resultant simulation program is used as an integral part of the HAZOP procedure and later it can be used, also, for process safety education and operator training. The proposed procedure is demonstrated in the next section, using a semi-batch reactor in which 2-octanone is produced from 2-octanol (van Woezik and Westerterp [10,11]). In this reactor, small deviations from the appropriate operating conditions may cause sudden reaction of accumulated product 2-octanone, followed by reaction rate and temperature runaway. A dynamic model of the reactor is solved using the Polymath 6.1* numerical software package.
2. Nitric acid oxidation in a semi-batch reactor – an example The nitric acid oxidation of 2-octanol to 2-octanone followed by further oxidation of 2octanone to carboxylic acids was studied by van Woezik and Westerterp [10,11]. The oxidation of 2-octanol is carried out in a two-phase reaction system: an organic liquid phase, which initially contains 2-octanol, in contact with an aqueous nitric acid phase in which the reactions takes place. The reaction can be described with the following equations: r1 A+ B⎯ ⎯→ P + 2B
(1)
r2 P + B ⎯⎯→ X
(2)
where A is 2- octanol, P is 2-octanone , X are the further oxidation products and B is the nitrosonium ion, which also causes an autocatalytic behavior. The reaction is carried out in a semi-batch reactor in which aqueous nitric acid is present right from the start, and the organic component 2-octanol (A) added at a constant feed rate until a desired molar ratio of the reactants has been reached. The 2-octanol reacts to form 2-octanone and carboxylic acid. The heat of reaction is removed by a coolant, which flows through an external jacket. Under normal operating conditions, when the temperature in the reactor does not exceed the limit of approximately 0 °C throughout the reaction, only a very small fraction (about 7.5 %) of the 2-octanone is converted to carboxylic acids. However, if the temperature at any point exceeds approximately 5 °C, runaway conditions develop, which may lead to a maximal temperature of over 200 °C, and conversion of essentially all of the 2-octanone to carboxylic acid. The mathematical model of the reactor and its cooling jacket is shown in Table 1. This model is based on the model presented by van Woezik and Westerterp [10]. The model in Table 1 is presented in a format which enables copying and pasting the column of the *
POLYMATH is copyrighted by M. Shacham, M. B. Cutlip and M. Elly (http://www.polymath-software.com/).
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equations directly into the differential equation solver program of the POLYMATH 6.1 package. Running this model will provide the solution for the reactor in normal operating condition. The model equations are of the form: (output variable) = g (input variables, constants) where g is a function. Table 1 provides also clear documentation of the mathematical model as the "Comment" column includes definition of the output variable of the equation, including the units of this variable. The model equations are presented in an order consistent with the principles of model building [8]. The equations are aggregated around the balance equations. A balance equation is added to the model first. Next the input variables of this equation are specified. Some variables are expressed as constitutive equations (e.g. reaction rates, heat and mass transfer rates), others as thermodynamic property correlations or constants. The addition of new equations is continued as long as there are still input variables that have not been defined as output variables. This structure of the mathematical model makes the model easy to understand and easy to modify for conducting HAZOP studies. In Table 1, the initial and final values of the independent variable (t, time) are defined first. Next the mol balance equations that yield the amount of desired product, 2octanone (in line 3), and the undesired carboxylic acid (in line 5) are entered. The definition of the initial value for the output variable follows the defining equation. Note that the mass balance equations are presented in [10] in dimensionless form. We preferred to use dimensional variables, as it has been shown in [7] that the use of such variables in HAZOP studies is preferable. The equations that specify input variables associated with the mass balance equation are listed in lines 7 through 36 of Table 1. The next balance equation, the energy balance on the reactor content which defines the temperature in the reactor as output variable is entered in line 37, with initial value specified in line 38. The input variables associated with the reactor's energy balance equation are specified in lines 39 through 53. The balance equation on the cooling jacket, which yields the outlet temperature of the cooling media, and the associated input variable specifications are shown in lines 54 through 60 of Table 1. Simulating the operation of the semi-batch reactor using the numerical values shown in Table 1 (which represent normal operating conditions) yields the results, for the key variables, shown in the first column (marked by "100%") of Table 1. The final amount of the desired product: 1-octanol is nP = 3.16 kmol, the final amount of the carboxylic acids is nX = 0.26 kmol and the maximal temperature in the reactor is Tr,max = 1.31 °C. Those results are consistent with the values obtained by van Woezik and Westerterp [10]. After verification of the correctness of the model the HAZOP studies can be carried out. These studies are carried out by dividing the process into various sections and using the guide words: None, More of, Less of etc. to generate a list of potential equipment failures or other deviations from normal operating conditions. Selecting, for example, the cooling jacket section the following list of potential deviations should be considered: 1. No flow in the cooling jacket (None); 2. Reduction of the flow rate in the cooling jacket (Less of); 3. Reduction of the effective heat transfer rate (Less of); 4. Reduction of the effective jacket volume (Less of); and 5. Increase of the inlet temperature of the cooling media (More of). Lest us take, as an example the reduction of the effective heat transfer rate. This rate will, most probably, decrease with time because of scale accumulation in the heat
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transfer surface. In order to carry out the quantitative studies regarding such a reduction the model equations have to be modified. UAcool which is defined in line 48 must be multiplied by the appropriate fraction of the design heat transfer rate value and the simulation has to be carried out using the modified model. The results of such simulations are summarized in Table 2. It can be seen that when the effective heat transfer rate gets below 81% of the design value temperature runaway develops, where Tr,max = 195 °C. Such extreme consequences of a relatively small reduction of the effective heat transfer rate indicate that the safety margin on the heat transfer area is not large enough. The design must be changed (by adding an internal cooling coil, for example) to increase the heat transfer area, if such a change can be justified on an economical basis. After making all the design changes, the model of the process has to be updated and the HAZOP procedure must be repeated using the modified model.
3. Conclusions and Discussion It has been demonstrated using a semi-batch reactor in which an exothermic reaction is carried out [10] that the quantitative HAZOP procedure outlined in this paper can provide more reliable and precise information regarding development of hazardous conditions in chemical processes than the traditional qualitative procedure. It can also provide clear guidelines for process modification to design a process with better safety characteristics. A complete HAZOP analysis was carried out for the nitric acid oxidation example. After completing this analysis and the consequential required process model modifications, the model was exported to MATLAB. A MATLAB GUI interface was constructed, which enables generation of the abnormal conditions that were tested during the HAZOP analysis. The resultant simulation program can be used for process safety education and operators training. Due to space limitations, the details of the complete HAZOP analysis and the training simulation program cannot be provided here.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
S. Dash and V. Venkatasubramanian, AIChE J., 49 (2003) 124. T. A. Kletz. HAZOP & HAZAN, The Inst. of Chemical Engineers, Rugby, U. K., 1999 H. G. Lawley, Chem. Eng. Progr., 70 (1974) 45. R.Wennersten, R, Narfeldt, A. Granfors and S. Sjokvist, Computers chem. Engng, 20(1996), Suppl. A, S665. H. Graf and H. Schmidt-Traub, Computers chem. Engng, 25(2001), 61. G. L. L. Reniers, W. Dullaert, B. J. M Ale and K. Soudan, J. Loss Prev. Process Ind. 18 (2005), 119 S. Eizenberg, M. Shacham and N. Brauner, J. Loss Prev. Process Ind. 17 (2004), 513. M. Shacham, N. Brauner and M. B. Cutlip, Computers chem. Engng, 24(2000) 415. M. Shacham, N. Brauner and M. B. Cutlip, Chem. Eng. Educ., 35 (2001) 268. B. A. A. van Woezik and K. R. Westerterp, Chem. Eng. Process. 41 (2001) 59. B. A. A. van Woezik and K. R. Westerterp, Chem. Eng. Process. 39 (2000) 521.
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Table 1. Mathematical model of the nitric acid oxidation example. No.
Equation
Comment (Output Variable Definition)
1 2 3 4 5 6 7 8 9 10 11 12 13
t(0) = 0.0001 t(f) = 72000 d(Np)/d(t) = (r1 - r2) * Vr0 / (1 - Epsd) Np(0) = 0 d(Nx)/d(t) = r2 * Vr0 / (1 - Epsd) Nx(0) = 0 r1 = k1 * CaOrg * CbAq * (1 - Epsd) r2 = k2 * CpOrg * CbAq * (1 - Epsd) Vr0 = 1.5 Epsd = Vdos1 / (Vdos1 + Vr0) k1 = maA1 * exp(-E1perR / Tr - m1 * H) k2 = mpA2 * exp(-E2perR / Tr - m2 * H) CaOrg = (Theta * NaF - Np - Nx) / (Vdos1 * Theta) CpOrg = Np / (Vdos1 * Theta) CbAq = (Np + Y * NaF) / Vr0 Vdos1 = 0.6 maA1 = 10 ^ 5 mpA2 = 10 ^ 10 E1perR = 11300 E2perR = 12000 m1 = 6.6 m2 = 2.2 H = -.6221 - 3.7214 * wt - 1.5714 * wt ^ 2 Theta = If (t <= tdos) Then (t / tdos) Else (1) NaF = Vdos1 * RhoOctan / MwOctan Y = 0.035 wt = Nn * Mw / (Vr0 * RhoAcid) tdos = 36000 RhoOctan = 820.7 MwOctan = 130.23 Nn = CnAq * Vr0 Mw = 63 RhoAcid = 1500 CnAq = (NnO - Y * NaF - Np - 2 * Nx) / Vr0 NnO = Vr0 * Percent * RhoAcid / Mw Percent = 0.6 d(Tr)/d(t) = (Qr + Qdos + Qcool) / Gamma Tr(0) = 260 Qr = Qnol + Qnone
Starting time Final time (s) Number of moles of 2-octanone (P) from mol bal. Number of moles of 2-octanone (P) at t = t0 Number of moles of carb. acids (X) from mol bal. Number of moles of carboxylic acids (X) at t = t0 Reaction rate of a and b to p[kmol/m3/s] Reaction rate of p and b to x[kmol/m3/s] Initial volume in a reactor [m3] Volume fraction of dispersed phase Specific reaction rate 1 Specific reaction rate 2 Concentr of a in org phase [kmole/m3]
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Concentr. of (P) in org phase [kmol/m3] Concentr. of (B) in aq. phase [kmole/m3] Final volume of the dose [m3] Pre-exponential factor reaction 1 [m3/kmol/s] Pre-exponential factor reaction 2[m3/kmol/s] Activation temperature reaction 1 [K] Activation tempetature reaction 2 [K] Hammett's reaction rate coeff. reaction 1 Hammett's reaction rate coeff. reaction 2 Hammett's acidity function Dimensionless time up to t=tdos Total amount of 2-octanol (a) fed [kmol] Initial concentr. of nitrosonium ion Y=Nb0/NaF Mass concentr. of nitric acid sol [%/100%] dosing time [s], 10h Density of 2-octanol [kg/m3] Molar mass of 2-octanol [kg/kmol] Number of moles of HNO3 [kmol] Molar mass of HNO3 [kg/kmol] Density of pure nitric acid [kg/m3] Concentr. of HNO3 in the aq. phase [kmol/m3] Initial number of mole of HNO3 [kmole] Initial mass concentr of nitr. acid sol. [%] Temp. in reactor (K) from energy balance Temp. in the reactor at t = t0 (K) Sum of the heat of reaction the reactions [W)
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Qdos = Phi * RhoCPdos * (Tdos - Tr) Qcool = UAcool * (Tcool - Tr) Gamma = Gamma0 + RhoCPdos * Phi * t Qnol = r1 * Vr0 * Hnol / (1 - Epsd) Qnone = r2 * Vr0 * Hnone / (1 - Epsd) Phi = Vdos1 / tdos RhoCPdos = 2 * 10 ^ 6 Tdos = 293.15 UAcool = UA0 + (UA1 - UA0) * Theta Gamma0 = 5.4 * 10 ^ 6 Hnol = 160 * 10 ^ 6 Hnone = 520 * 10 ^ 6 UA0 = 1500 UA1 = 2100 d(Tcool)/d(t) = (Fw * (Tcool_IN - Tcool) Qcool / (RhoCoolant * CpCoolant)) / Vj Tcool(0) = 273.15 Fw = 100 / 60 * 10 ^ (-3) Tcool_IN = 260 RhoCoolant = 1000 CpCoolant = 4183 Vj = 1.5
Heat input due to reactant addition [W] Heat removed by the cooling jacket [W] Total heat capacity of the system [J/K] Heat of reaction, 1 [W] Heat of reaction, 2 [W] Volumetric flow rate of the feed [m3/s] Heat capacity of dose [J/m3/K] Temperature of feed dose [K] Cooling surface heat transfer coefficient [W/K] Initial heat capacity of the system [J/K] Specific heat of reaction 1 [J/kmol] Specific heat of reaction 2 [J/kmole] Initial cool. surface heat trans. coeff.[W/K] Final cool. surface heat trans. coeff. [W/K] Cooling water outlet temperature (K) from jacket energy balance Coolant exit temp. at t = t0 (K) Flow rate of coolant [m3/s] Initial coolant temperature [K] The density of coolant [kg/m3] Heat capacity of coolant [J/kg/K] Volume of the jacket [m3]
Table 2. Maximal temperatures and final product amounts for reduced effective heat transfer rates. effective heat transfer rate nP [kmol] nX [kmol] Tcool,max [ °C] Treactor,,max [ °C]
100%
90%
81%
80%
70%
60%
3.16
3.08
0.05
0.14
0.3
0.38
0.26
0.33
3.71
3.3
2.63
2.64
-10.4
-10.2
8.88
6.42
0.3
2.37
1.31
3.9
195
177
147
128
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Validation of a digital packing algorithm for the packing and subsequent fluid flow through packed columns Richard Caulkin, Michael Fairweather, Xiaodong Jia, Richard A. Williams Institute of Particle Science and Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom
Abstract A new packing algorithm, called DigiPac, was validated for packed columns in a paper presented at ESCAPE 15. The columns considered contained mono, binary and ternary mixtures of particles in conventional beds, as well as spherical particles in shell-side beds which contained fixed arrays of tubes in order to represent the more realistic geometries found in industry. Predicted voidages were compared with experimental data, with good agreement observed. The present paper extends this work to simulate both bed structures and, using lattice Boltzmann modelling, the fluid flow and pressure drop through a number of beds for which experimental data are available in the literature. The results demonstrate the usefulness of these combined techniques for the design of unit operations based on packed beds. Keywords: Packed beds, digital packing algorithm, lattice Boltzmann modelling.
1. Introduction Packed beds, due to their high surface area ratio, are used extensively in chemical engineering operations. Common applications include fixed bed catalytic reactors, chromatographic reactors, ion exchange columns and absorption towers. The design and performance prediction of such systems depends on mathematical models that describe the behaviour of fluid flow, heat and mass transfer, and the pressure drop of the fluid through the bed. The models themselves are generally dependant on accurate experimental data describing transport parameters such as effective thermal conductivity coefficients, the wall heat transfer coefficient and effective dispersion coefficients. Much work has been undertaken to quantify the structural properties of packed beds, and the transport processes through them (McGreavy et al, 1986) in order to permit the design of more efficient unit operations. This work has generally resulted in empirical correlations which, despite their usefulness for equipment design, are limited by the range of data used in their formulation. In particular, correlations are generally only available for regular shaped packing materials with a limited particle size range. This paper describes a novel computational approach to predicting the fluid flow through such beds and compares resulting numerical predictions like-for-like with data from the literature. Fluid flow through packed beds is often simulated using conventional finite-volume, computational fluid dynamic (CFD) techniques. Such simulations have, however, been limited to simple packing materials (Taylor et al, 2000) due to the difficulties encountered in representing large numbers of complexshaped objects. The present work takes an alternative approach by using lattice Boltzmann modelling (LBM) which is able to deal with fluid flow through complex
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porous structures more readily than conventional Navier-Stokes equation solvers. A brief summary of the technique used is given below.
2. Digital Packing Algorithm DigiPac is a computer-based tool that uses a novel solution for packing objects using digital means, as opposed to the vector approach used by many other traditional packing models. It is capable of packing particles of any shape into containers of any geometry. The basis of the algorithm is to map the particles, the container and particle movement on to a grid. Then, a single particle is just a coherent collection of pixels (twodimensional) or voxels (three-dimensional) that moves randomly, one grid cell at a time, within the specified boundaries of the container. In two dimensions there are 8 possible directions (4 orthogonal and 4 diagonal) that a particle can move in, all initially with equal probability. In three dimensions there are 26 possible directions (6 orthogonal and 20 diagonal). In order to encourage particles to settle having undergone movement, the upward component of a move is only accepted by the algorithm with a so-called rebound probability. This results in a directional and diffusive particle motion, which aids the particles in effectively exploring every available packing space. Since particles reside and move on a grid one step at a time, collision and overlap detection is a simple matter of determining whether more than one object occupies the same site(s) at any one time. A more comprehensive description of the DigiPac algorithm has been given elsewhere (Jia and Williams, 2001; Caulkin et al, 2005).
3. Lattice Boltzmann Method The Lattice Boltzmann method (Wolf-Gladrow, 2000, and references within) is directly descended from the discretisation and solution of the Boltzmann equation on a cellular automata. In this technique, velocities are allowed to evolve either in a 2D or 3D lattice space, obeying certain collision rules. Solution of the Navier-Stokes equations (Ladd, 1994) is realised by appropriately modelling the collision operator with diffusion limits. LBM has a significant advantage when compared to the earlier approaches to solving the Navier-Stokes equations in that solid boundaries of complex geometry can be easily embedded due to the use of cellular automata. This technique has also been extended to model turbulent flow regimes as well. Since the flow evolution remains independent of the structural complications, effective parallelisation can be achieved, hence making it more appealing for multi-processor interfacing as well. Shortcomings of LBM include (1) it is very demanding on computing resources, particularly physical RAM. For instance, on well-equipped 32-bit PCs, the simulation scale is usually 320x320x320 or smaller; (2) the LBM scheme implemented in our code does not address the issue of a self-consistent coupling between temperature dynamics and heat transfer within the fluid flow; for applications where heat transfer is as important as mass transfer, this is a serious limitation; thermal LBM is an active research area and some significant advances can be expected; (3) the LBM scheme we have implemented uses a uniform body force in place of pressure gradient. This is not a point-wise matching, only a domain averaged one; but it is a necessary sacrifice to avoid the costly solution of the elliptic Poison equation which links a given velocity field to the spatial distribution of pressure (Succi, 2001). Experimental validation of LBM was previously performed by Mantle et al (2001) using MRI velocimetry. As the basis of the LBM and DigiPac techniques are the same, the main advantage of using the lattice Boltzmann method for the current application is
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that it can easily be interfaced with the digitalisation technique. The LBM used in the present work was developed in-house, and conforms to the description given in (WolfGladrow, 2000). The PC used for the simulations was a dual Opteron CPU machine running Linux operating system (64-bit version of SuSE 9.1). Typical simulation time was between 30 minutes and one hour. The code uses multithreads to run on the two processors simultaneously.
4. Results and Discussion Validation of DigiPac was carried out by comparing the simulation results with experimental data in the public domain (Bey and Eigenberger, 1997; McGreavy et al, 1986; Zeiser et al, 2001). In the first set of simulations, three different diameters of spherical particles, dp, were packed into cylindrical tubes of fixed diameter (dt) in order to form three aspect ratios (dt/dp=3.0, 5.0 and 6.0). The resulting mean and radial voidage from these beds are plotted in Figs. 1 and 2. It can be seen that the void fractions compare well with the experimental values. For mean voidage (McGreavy et al, 1986), which is widely used as an overall index of bed structure, deviations were seen in the lower tube-to-particle diameter ratios (<2dt/dp) and can be explained in terms of error due to digitalisation. However, it must be noted that the global voidage furnishes no information about the local properties of a bed. Plug flow is commonly assumed in mono-sized particle beds, i.e. the bed has uniform properties throughout. While this may be the case at the centre of relatively large packed beds, it is certainly not so in the vicinity of the walls. As a result of large void spaces near the wall, flow channels are large in this area, offering less resistance to flow than at the bed centre. An exact replication of the data is never possible because of the random nature of the packing obtained. Typical features of the radial voidage packing structures are that for dt/dp=5.0 the porosity exhibits a minimum (0.3) at the bed centre, while for dt/dp=6.0 a local maximum (0.45) of the void fraction is observed.
Mean voidage
0.7 0.6
Exp (McGreavy, 1968) Sim
0.5 0.4 0.3 1
3
5 7 9 11 13 15 17 Tube-to-particle diameter ratio, dt/dp
19
21
Figure 1. Mean voidage results; DigiPac compared to experimentally derived data for mono-sized spheres of various tube-to-particle diameter ratios In the present work, a Monte-Carlo (MC) simulation of the packing process has been used to obtain the geometrical information for the flow field analysis because simulated structures are easy to produce and correspond well to the random packing of a real particle filled bed. The radial distribution of void fraction for the sphere packed beds with dt/dp=5 and 6 are shown in Fig. 2. For comparison, the curve from the correlation obtained by fitting experimental data is also presented (Bey and Eigenberger, 1997). It can be found that the void fraction distribution of the MC simulated packed bed is in good agreement with correlations derived from experimental investigations. A precise fit can be observed in the region close to the wall, while some deviations exist in the
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central region because the experimental techniques did not allow a fine cut near the centre of the bed. Exp
1
Exp
1
Sim
Sim 0.8 Void Fraction
Void Fraction
0.8 0.6 0.4 0.2
0.6 0.4 0.2
0
0
0
1 2 3 4 Particle diameters from wall, r/dp
5
0
1
2 3 4 5 Particle diameters from wall, r/dp
6
Figure 2. Radial voidage results of DigiPac packed spheres compared to experimentally recorded values for aspect ratios dt/dp=5 (left) and dt/dp=6 (right) Lattice Boltzmann simulations were performed on the packings for three sets of flow conditions, as considered by McGreavy et al (1986) and Bey and Eigenberger (1997). The fluid flow simulations used packed beds with relatively low tube-to-particle diameter ratios since under these conditions any channelling effects become more severe. For the setup of the simulations a suitable sized matrix was used for each bed, with the bed ends being removed prior the LB simulations being undertaken. This was done as the random nature of the packing in these regions is not representative of the packing structure as a whole, and could therefore give false results. In Fig.3, and conforming to experiment, the high velocity predictions match the high radial voidage results of Fig. 2. 4
4 Exp
3 2.5 2 1.5 1
Exp
3.5 Dimensionless Velocity
Dimensionless Velocity
3.5
Sim
0.5
Sim
3 2.5 2 1.5 1 0.5
0
0
0
1 2 3 4 Radial distance from wall (particle diameters)
5
0
1 2 3 4 5 Radial distance from wall (particle diameters)
6
Figure 3. LBM and experimental comparisons of fluid velocity for spheres, dt/dp=5 (left) and dt/dp=6 (right) The radial distributions of axial flow in the above beds are scaled with the velocities from the empty beds. It is seen that the simulated results are generally in good agreement with the values from the literature, with the flow velocities oscillating along the radial direction. According to the LB simulations, the periodic length scale for the velocity oscillation is about the diameter of one sphere. This corresponds precisely with the oscillation of the void fraction distributions of Fig. 2.
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It has been established that close to the entrance of the tube, flow there is at a higher velocity than at the exit, slowly decreasing along the x-axis. This is evidently due to the pressure drop along the length of the tube in the direction of flow. Channeling effects are also seen, particularly in near-wall regions where voidage values are high. In this study, the LB used is a constant pressure model, meaning that body forces increase or decrease as necessary in order to maintain pressure at a constant value. Therefore the pressure gradient must be related to the body force in order to calculate overall pressure drop.
Pressure drop (Pascals)
60 50 40 30 20 Exp; Re=4.08 Sim
10 0 0
0.05
0.1 0.15 Distance from inlet of bed (m)
0.2
0.25
Figure 4. Comparison of measured and predicted overall pressure drop over the length of a packed bed of spheres, dt/dp=3 and equivalent to 0.22m in height Fig. 4 shows the comparative decrease in fluid pressure for a DigiPac simulated spherepacked bed with that of an experimentally derived bed from the literature, with both beds having an aspect ratio (dt/dp) of 3 and equal packing heights. It is seen that at the exit of the predicted bed, the drop in pressure from that at the inlet is a close match to the actual bed. 0.12
0.09
0.06
0.03
0 Figure 5. Cross-sectional images of fluid flow through beds dt/dp=5 (top) and 6 (bottom) at inlet (left), centre (middle) and outlet (right) of bed (scale bar in LBM units).
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Fig. 5 shows a snapshot of the flow at various distances along the beds with dt/dp=5 and dt/dp=6. The areas in white and dark grey show the presence of objects blocking the flow. The bright pixels show high fluid flow velocities at that particular point. It is observed that at the entrance, the velocities are higher than at the exit. Variations in fluid flow velocities exist because of varying packing densities. Channeling or tunneling effects inside the packing can be observed, particularly close to the walls where higher voidage, as seen in Fig. 2, exists due the wall effect.
5. Conclusions The geometries of randomly packed beds have been generated using a Monte Carlo method which has been validated by comparisons with experimental measurements of global and local voidage distribution. Lattice Boltzmann simulations conducted for flow through these packed beds have demonstrated that these results also compare well with experimental data. The results for the local velocity distributions obtained with the LBM show the typical channeling effects close to the walls and subsequent maximum velocity in this region that is observed in experiments. The present numerical approaches also allow a detailed investigation into local velocity distributions inside a packed bed, and thus can be used to improve basic understanding of the underlying transport phenomena in these complex porous structures.
References O. Bey, G. Eigenberger, 1997, Fluid flow through catalyst filled tubes, Chem. Eng. Sci., 52, 1365-1376. R. Caulkin, M. Fairweather, X. Jia, R.A. Williams, 2005, European Symposium on ComputerAided Process Engineering-15, L. Puigjaner, A. Espuna, Eds., Elsevier, Amsterdam, 367-372. X. Jia, R.A. Williams, 2001, A packing algorithm for particles of arbitrary shapes, Powder Tech., 120, 175-186. A.J.C. Ladd, 1994, Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation, Jl. Fluid Mech., 271, 285-309. A.J.C. Ladd, 1994, Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results, Jl. Fluid Mech., 271, 311 – 339. M.D. Mantle, A.J. Sederman, L.F. Gladden, 2001, Single- and two-phase flow in fixed-bed reactors: MRI flow visualisation and lattice-Boltzmann simulations, Chem. Eng. Sci., 56, 523529. C. McGreavy, E.A. Foumeny, K.H. Javed, 1986, Characterisation of transport properties for fixed bed in terms of local bed structure and flow distribution, Chem. Eng. Sci., 41, 787-797.S. S. Ross, A.G. Smith, M.W. Smith, K. Taylor, 2000, CFD modelling of pressure drop and flow distribution in packed bed filters, PHOENICS Journal of CFD & its applications, 13, 399-413. S. Succi, 2001, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford Science Publications, ISBN 0198503989. D.A. Wolf-Gladrow, 2000, Lattice-Gas Cellular Automata and Lattice Boltzmann Models – An Introduction, Springer, Berlin. Th. Zeiser, P. Lammers, E. Klemm, Y.W. Li, J. Bernsdorf, G. Brenner, 2001, CFD-calculation of flow, dispersion and reaction in a catalyst filled tube by the lattice Boltzmann method, Chem. Eng. Sci., 56, 1697-1704.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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A Hybrid Global Optimization Scheme for Process Design and Dynamic Optimization a
Chyi-Tsong Chen , Shih-Tien Penga, Ying-Jyuan Cioub, Cheng-Liang Chenb,* a
Department of Chemical Engineering, Feng Chia University, Taichung 407, Taiwan; e-mail: [email protected] b Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan; e-mail: [email protected]
Abstract A hybrid global optimization algorithm is developed for the solution of process design and dynamic optimization problems. The proposed algorithm consists of a feasible control strategy, information theory and a chaotic searching algorithm. With the feasible control strategy, points that satisfy the constraints and with minimum objective-function values can be located. After determining the potential candidates, a chaotic algorithm is served to generate new points to find the global optima. The information theory is utilized to escape from the local optima. To extend the proposed global scheme to the solution of the dynamic optimization problems, the orthogonal collocation strategy is used for converting the original dynamic problem into a finite-dimensional optimization problem. This effort leads the proposed global optimization scheme directly applicable to the solution of dynamic optimization problems and makes the solution procedure quite easy. The applicability and effectiveness of the proposed global optimization scheme have been tested with some typical optimization problems, and extensive comparisons with the existing simulated annealing algorithm are performed. Keywords: Global optimization; Stochastic approach; Dynamic optimization; Feasible point strategy; Information theory; Chaotic algorithm 1. Introduction Global optimization deals with the mathematical modeling and computation of real-world problems, which has ubiquitous applications in engineering, applied sciences, sciences, and economics. Many practical problems often lead to large nonconvex, and in many cases, have multiple local optima. However, most optimization strategies currently used are local methods, which easily fail and at best, locate only local optima. As a result, the development of global optimization methods have received considerable attention over the last few years. The global optimization techniques can be generally classified into two categories: the deterministic methods (Soland, 1971; Geoffrion, 1972; Hansen, 1980; Tuy et al., 1985; Horst et al., 1992) and the stochastic methods (Kirkpatrick et al., 1983; Holland, 1975; Shannon, 1948). Generally, deterministic methods are usually affected by the initial guesses, and stochastic methods are not efficient to locate the optimum with high precision. In this work, the proposed hybrid algorithm uses the stochastic algorithm to find the solution near the global optimum and in turn the deterministic algorithm is used for locating the global optimum with high precision. The proposed scheme integrates the feasible point strategy, information theory and chaotic algorithm. Based on the feasible point strategy, the chosen points are refined to locate in the constrained region and avoids the difficulty of treating equality constraints in the stochastic algorithm. Only those located in the feasible region are the candidates for further searching. After candidates have been chosen, the information theory is subsequently adopted in the stochastic algorithm for not being trapped by local optima and to culminating in arrival at the global optimum. Furthermore, for generating exploring points in the stage of information theory, a chaotic algorithm is adopted. This can make use of the benefit of chaotic dynamics, which is therefore more efficient than the conventional random procedure used in the stochastic searching. Some illustrated examples are given to demonstrate the applicability and effectiveness of the proposed scheme, and extensive comparisons with existing optimization methods are included as a rigorous base for evaluation.
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402 2. Problem formulation
A majority of optimization problems encountered in chemical engineering can be described by the nonlinear programming problem (NLP) as follows: (1a)
min f ( x) x
subject to
g ( x) ≤ 0
(1b)
h ( x) = 0 (1c) where x ∈ R n is the decision vector, f : R n → R denotes the objective function, and g : R n → R l and n υ h : R → R ( n > υ ) represent, respectively, the inequality and equality constraints. The formulated constraint region is assumed to be nonconvex, due to the presence of multiple local optima, which therefore can make many existing optimization algorithms inadequate to locate the global optimum effectively. Therefore, the purpose of this paper is to present a global optimization scheme for solving the global optimization problem described in Eq. (1).
3. The proposed hybrid global optimization scheme The flow chart of the proposed optimization scheme is depicted in Fig. 1 where the presented algorithm involves two stages. 3.1. The feasible point strategy To avoid the difficulty of tackling the equality constrains in a stochastic algorithm and to determine whether these starting points generated from the parameter space are in the feasible region, a feasible point strategy is applied, in which a constrained infeasibility minimization problem (CIMP) is solved instead of the original optimization problem. The proposed feasible point strategy simultaneously minimizing the performance index ( α ) and decision variables ( x ) is quite different from Elwkeil and Arora (1995) in which only the performance index ( α ) is used for minimizing. This effort makes the proposed algorithm more efficient to find the solutions closer the global solution and thus drive α toward zero as soon as possible at the least iterations. On the basis of the proposed feasible point strategy, we can formulate a new optimization problem as follows: min α
(2a)
α,x
subject to
g (x) ≤ α 1 h ( x) ≤ α 1 −h(x) ≤ α 1 f ( x) ≤ f 0 − ε + α
(2b) (2c) (2d) (2e) where 1 = [1 1, …, 1]T . f 0 and ε denote, respectively, the current value of the objective function and the global optimality tolerance. The CIMP (2) is always feasible with respect to x , and thus the proposed algorithm does not have difficulty with equality constraints. Actually, the problem (2) has infinite number of solutions for x at α = 0 unless f 0 is a global optimum. The concept of the proposed feasible point strategy is depicted in Fig. 2. Therein we adopt a local optimizer (Gill et al., 1998) using successive quadratic programming (SQP) which applies the equivalent of a Newton step to the NLP for finding a local minimum, and then add a new constraint that forces the value of the objective function to be lower than the current local minimum in order for searching a new feasible point. This procedure can be repeated until no feasible points exist. To implement the proposed scheme, a global algorithm is required for finding feasible points and a local algorithm for finding local minima, so a stochastic global algorithm and a deterministic local algorithm are integrated in this step.
A Hybrid Global Optimization Scheme for Process Design and Dynamic Optimization 403 3.2. The information theory (IT) Owing to the presence of multiple local optima of nonconvex optimization problems, a mechanism to jump x i out the local optima is needed for locating the global optimum. This work considers the use of IT searching scheme, which takes into account the data distribution and information of the past reference points for increasing the possibility and efficiency to find the global optimum. In general, the IT consists of three indices including information entropy, information energy and information free energy. The information entropy according to Shannon’s definition (Shannon, 1948) for a variable X , which can randomly take values from a set X , is described as follows: m
(3)
S (x) = ∑ Pi (x i ) ln[ Pi (x i )] i =1
Where ⎛ D2 ⎞ exp⎜⎜ − i 2 ⎟⎟ ⎝ 2σ i ⎠ Pi (xi ) = (4) m ⎛ D2 ⎞ exp⎜⎜ − i 2 ⎟⎟ ∑ i =1 ⎝ 2σ i ⎠ is the probability of the event x i occurring. Di2 = ( x * − x i ) T ( x * − x i ) is a scalar function and σ i2 is the variance of ( x * − x i ) in which x * is the current optimum vector of the decision variable and is the x i = [ x1,i x2, i … xn , i ]T decision vector of the i th set of X . If the variable X can only take a narrow range of values, S (x) is close to 0. If the variable X can take a lot of different values in X each time with a small Pi ( x i ) , S (x) will be a large negative number. As a result, information entropy, S (x) , is a measure of how random a variable is distributed. Its value decreases when the variable is more randomly distributed. In the light of the feasible point strategy, the information energy defined with respect to the satisfied degree for constraints is formulated by U = α −α 0 (5) where α denotes the currently objective function value of the CIMP and α 0 is the value of the minimum α recorded in the optimal search. This index is a measure of how well the objective function performs for the feasible point strategy, but it also associates with the original objective function minimization. After defining the indices of entropy and energy, it is necessary to integrate these two indices due to the presence of inconsistency between the two if considering them separately. To balance them, a composite information index named information free energy is defined (Chen et al., 1998): F = U − TS
(6) where T the annealing temperature, or the weighting factor, is updated by
(7) Tk +1 = Tk exp( −c ⋅ m1 z ) with 0 < exp(−c ⋅ m 1 z ) < 1 . Here, m is the number of data sets, and c and z are user-specified constants. 3.3. The chaotic algorithm for next point generation To improve efficiency of the random procedure in next point generation, the following chaotic algorithm is suggested x k +1 = (1 − β ) x * + β x′k , β < 1 x * (8) where is the current best point, β and x′k are, respectively, given by
(9)
⎛ k −1⎞ ⎟ ⎝ k ⎠
β = 1− ⎜
d
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404
and x′k = x k + ( v − 0.5)(BU − BL ) (10) with integer d to be specified, integer k the iteration index, v ∈ [0, 1] , BL and BU the lower and upper bounds of the variables x . Essentially, the proposed chaotic algorithm is of randomization, ergodicity and regularity, which can search and no repeat throughout the data points by its selfregularity. To avoid locating at the local optima, the generated points are needed to distribute extensively in the initial phase of the search procedure so that the bigger value of β is necessary. On the other hand, while the search procedure is deeply and closer to the global optimum, smaller value of β is preferred.
4. Illustrative examples 4.1. A nonconvex nonlinear program The first example considers an optimization problem of the reactor network design (Manousiouthakis & Sourlas, 1992). The schematic diagram is shown in Fig. 3 and this nonconvex nonlinear programming problem is formulated as follows: min − x 4 x
subject to
x1 − 1 + k1 x1 x5 = 0 x2 − x1 + k 2 x2 x6 = 0 x3 + x1 − 1 + k 3 x3 x5 = 0 x 4 − x3 + x 2 − x1 + k 4 x 4 x6 = 0 x50.5 + x60.5 ≤ 4 0 ≤ x ≤ (1, 1, 1, 1, 16, 16) k 2 = 0.99k1, k3 = 0.0391908 and k 4 = 0.9 k 3. This example constitutes a diffi-
where k1 = 0.09755988 , cult test problem as it possesses two local minima with an objective function value that is very close to that of the global solution. The global optimum is x = (0.771516, 0.516992, 0.204192, 0.388811, 3.035568, 5.097263) with f = −0.388811 , and the local solutions are at x = (0.390, 0.390, 0.375, 0.375, 16, 0) with f = −0.375 and at x = (1, 0.393, 0, 0.388, 0, 16) with f = −0.3881 . Notably, the two local solutions utilize only one of the two reactors; whereas the global solution makes use of both reactors. To explore features and value selection of parameters of the proposed global optimization algorithm, the performance evaluation results are summarized in Table 1. With the nominal parameter setting of N = 5 , ε = 0.1 , and m = 5 , c = 0.1 , z = 10 , d = 4 of information theory, it is found that the least number of iterations for finding the global optimum is obtained. Furthermore, to verify the effectiveness of the proposed algorithm, the performance comparisons with a global optimization strategy of Choi et al. (1999) by using a feasible point strategy with simulated annealing and a deterministic algorithm presented by Gill et al. (1998) are included as a rigorous base for evaluation. As the results shown in Table 2, it is evident that the proposed algorithm has potential to find the global optimum every time and give a smaller value of sum(|constraints|) (precision) than the approaches of Choi et al. (1999) and Gill et al. (1998). 4.2. A dynamic optimization problem (DOP) This is a mixed catalyst problem which exhibiting a singular segment. The problem determines the optimal mixing policy of two catalysts along the length of a tubular reactor. The control variable in this case is mixing ratio of the catalysts and its optimal profile has a singular arc, making this a difficult problem to solve. The problem is taken from Bell and Sargent (2000) and formulated as:
A Hybrid Global Optimization Scheme for Process Design and Dynamic Optimization 405
subject to
max x3 (t f ) u (t )
dx1 = u (10 x2 − x1 ) , x1 (0) = 1 dt dx2 = u ( x1 − 10 x2 ) − (1 − u ) x2 , x2 (0) = 0 dt x3 = 1 − x1 − x2
with 0 ≤ u (t ) ≤ 1 on t ∈ [0, t f ] and t f = 1 . The approximate state profiles, simulation state profiles and optimal control profiles of the proposed scheme with the parameter setting of N = 5 , ε = 0.1 , m = 5 , c = 0.1 , z = 10 and d = 4 , for q = 3 , 5 and 7 are depicted in Fig. 4. The optimal costs calculated by the proposed algorithm are 0.035398 ( q = 3 ), 0.044244 ( q = 5 ) and 0.048987 ( q = 7 ) which is better than the best objective-function value of 0.048080 reported by Bell and Sargent (2000). In Fig. 4, the discrete points are NLP solutions which are used in generating the Lagrange polynomials. These polynomials are used to plot the approximate profiles (dotted lines) which connect all of the discrete points on the graphs. The simulation profiles result from simulating the original model using the proposed optimal control policies. The purpose of comparing the approximate and simulation profiles is to check the validity of the NLP results from which it can be seen that the fit between the approximate and simulation profiles improves as q increases from 3 to 7. However, a larger q did not necessarily result in lower (better) corresponding simulation values of the performance index. Once q increases to some value, the discrepancy between the approximate and the simulation profile is very small which also means that no better corresponding simulation values of the performance index will be obtained. 5. Conclusions A hybrid global optimization scheme for process design and dynamic optimization has been presented. By means of the deterministic local optimizer, optima will be found and a stochastic global algorithm is applied for jumping out of local optima and locating at the global optimum. Under the proposed scheme, the use of feasible point strategy can avoid the difficulty of dealing with equality constraints and chosen points are refined to locate in the feasible region. After candidates have been chosen, the information theory with chaotic algorithm will function as not be trapped by local optima and be able to reach the global optimum efficiently. Furthermore, the proposed scheme with using the collocation method can also be directly extended to tackle with DOPs described by differential and algebraic equations. The practical potential of the proposed scheme has been demonstrated and extensive comparisons with other methods have been performed through illustrated examples. Numerical results reveal that the proposed optimization scheme is superior to the comparable methods and more efficient to find the global optimum within limited test runs. 6. Reference Bell, M. L., & Sargent, R. W. H. (2000). Optimal control of inequality constrained DAE systems. Computers and Chemical Engineering, 24, 2385-2404. Chen, J., Wong, D. S. H., Jang, S. S., & Yang, S. L. (1998). Product and process development using artificial neural-network model and information analysis. AIChE Journal, 44, 876-887. Choi, S. H., Ko, J. W., & Manousiouthakis, V. (1999). A stochastic approach to global optimization of chemical processes. Computers and Chemical Engineering, 23, 1351-1356. Elwakeil, O. A., & Arora, J. S. (1995). Methods for finding feasible points in constrained optimization. AIAA Journal, 33, 1715-1719. Geoffrion, A. M. (1972). Generalized benders decomposition. Journal of Optimization Theory and Applications, 10, 237-260. Gill, P. E., Murray, W., Saunders, M. A., & Wright, M. H. (1998). User’s guide for NPSOL 5.0: A FORTRAN package for nonlinear programming. Technical Report SOL 86-2, Revised July 30, 1998, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, California 94305-4022, 1998.
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Hansen, E. R. (1980). Global optimization using interval analysis: The multi-dimensional case. Numerische Mathematik, 34, 247-270. Holland, J. H. (1975). Adaptations in natural and artificial systems. Ann Arbor, MI: University of Michigan Press. Horst, R., Thoai, N. V., & De Vries, J. (1992). A new simplicial cover technique in constrained global optimization. Journal of Global Optimization, 2, 1-19. Kirkpatrick, S., Gelatt, C. D. Jr., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220, 671-680. Manousiouthakis, M., & Sourlas, D. (1992). A global optimization approach to rationally constrained rational programming. Chemical Engineering Communication, 115, 127-147. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379-423 land, R. M. (1971). An algorithm for separable nonconvex programming problems Ⅱ: nonconvex constraints. Management Science, 17, 759-773. Tieu, D., Cluett, W. R., & Penlidis, A. (1995). A comparison of collocation methods for solving dynamic optimization problems. Computers and Chemical Engineering, 19, 375-381. Tuy, H., Thieu, T. V., & Thai, N. Q. (1985). A conical algorithm for globally minimizing a concave function over a closed convex set. Mathematics of Operations Research, 10, 498-514. Villadsen, J., & Michelsen, L. M. (1978). Solution of Differential Equation Models by Polynomial Approximation. Englewood Cliffs, NJ: Prentice-Hall.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Methodology for decision support among conflicting objectives using process simulators Naveed Ramzana,Werner Witta a
Brandenburgische Technische Universität, Lehrstuhl Anlagen- und Sicherheitstechnik, Burger Chaussee 2, Lehrgebäude 4/5, Cottbus 03044, Germany
Abstract For solving multiobjective decision making problems, which may arise during optimization of operating or design of process plants, a systematic and effective procedure is required. As far as the process or control system has to be modified process simulators like Aspen Plus are widely used. But these simulators are not designed for investigation of other objectives as environment and safety. Due to complex and conflicting nature of multiobjective decision making an integrated optimization tool should be of value. In this paper a systematic methodology and its different stages to deal this problem is presented. The methodology consists of four stages/layers: 1) Problem definition and generation of process alternatives 2) Analysis of alternatives 3) Multiobjective optimization 4) Alternative design evaluation layer. The multiobjective optimization layer is based on goal programming using Aspen Plus and the optimization tool of MatLab. Relevant aspects of the methodology - so far up to now developed - is explained and demonstrated with the help of an industrial separation process. Keywords: Multiobjective decision making, Pareto approach, distillation
Introduction Today chemical process design relies on decisions among conflicting objectives such as economic, environmental aspects and process safety. Increasing and overlapping regulatory demands, pressure to reduce cost of plant construction and operation and at the same time improved performance requirements emphasize to integrate economic, environmental and safety objectives into the chemical process design and modification activities instead of tackling them as end of pipe treatment. Several attempts have been made to integrate environmental objectives in early process design (Palaniappan (2002), Sharratt (1999), Chen Hui (2004)) and in multiobjective optimization procedures (Chakraborty (2002), Alexander (2000)). The commonly used methodologies for detailed characterization of environmental impacts of chemicals, products and processes are e.g. Life Cycle Impact Assessment (Azapagic (1999)), the Waste Reduction Algorithm (Young (1999)), the Methodology for Environment Impact Minimization (Pistikipoulas (1997)) and the Environment Fate and Risk Assessment tool (Chen Hui (2004)). A similar situation we have in the safety area. Several tools ranging from checklists and informal brainstorming techniques to very formal and disciplined methodologies such as Hazard and Operability (HAZOP) studies and Failure Mode and Effect Analysis (FMEA) are in use during process development and early stages of plant design.
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Recently few index-based methods such as EHS-Method, SREST Layer Assessment Method and Total Inherent Safety Index suitable at design phase are presented. However there are very less work come into picture, which involves multicritria decision analysis among process safety and economics. Giuggioli Bussaca (2004) performed multiobjective optimization of industrial safety systems by Genetic Algorithm with respect to availability, economics and worker safety objectives. Kim (2004) proposed a novel decision procedure based on multiobjective optimization to find the investment priority to reduce plant accidents. It is clear that multiple streams of dissimilar information are evolved during the environmental, safety and economic evaluation of the chemical processes. A variety of standard techniques are also available for this purpose. However, the decision making among these conflicting objectives is a complex process. In this article, a systematic framework using process simulators and standard techniques for supporting decision making among conflicting performance objectives is presented.
Systematic methodology The methodology is built around several standard independent techniques. These techniques have been suitably modified/adopted and woven together in an integrated plate form developed in Visual Basic. The methodology presented in figure 1 consists of four layers/stages: 1) Generation of process alternatives and problem definition 2) Analysis of alternatives stage i.e generation of relevant data for comparison of environmental, economic and safety objectives 3) Multiobjective optimization 4) Design evaluation stage i.e decision making from the pareto-surface of noninferior solution. Stage-4 Stage-3
Stage-2
Stage-1 Figure 1. Methodology for decision support among conflicting objectives Stage 1: Generation of process alternatives and problem definition In the first stage following tasks are performed: 1) Definition of the scope of the study 2) Statement of key assumptions and the performance targets such as quality etc. 3)
417
Methodology for Decision Support Among Conflicting Objectives
Identification of the key designs, control, and manipulated variables 4) Definition of the system boundary 5) Identification of constraints 6) Choice of functional unit for all calculations 7) Collection of relevant information about process and chemicals to be handled 8) Generation of different alternatives. For plant documentation a CAD tool like Comos PT is used. Stage 2: Analysis of alternatives This stage is composed of process module, economic module, environmental module, safety module and data manager. In the process module, each alternative is modeled using Aspen Plus and various steady state simulation runs are performed to generate relevant data for use in other modules. An in house economic module for the calculation of total annualized cost is developed in visual basic. The WAR model developed by US Environmental Protection Agency (EPA) is used to calculate environmental effects. Standard safety analysis and risk assessment techniques supported by dynamic simulation are integrated in the safety module. Stage-3: Multiobjective optimization This stage is to set up multiobjective optimization among these conflicting objectives. The aim is to find out the trade-off surface for each alternative. From the trade-off surface, the best compromise design is obtained. The way of strategy to carry out multiobjective optimization using Aspen Plus and MatLab are described in figure 2.
Objective Functions Initial Designs
Multi-Objective Optimizer (Goal Programming)
Nondominated Set Analysis
D A T A M A N A G E R
PEI
Environmental Model
Optimizing Variables TAC
SQP Optimizer
Process Model
Economic Objective Function & Constraints
Aspen Plus TM
(Risk,Cost)
Safety Model
MATLABTM
WAR
OptHazdyn FMEA-PP Event Tree
Excel
Multicriteria Optimal Surfaces (MOSs) Figure 2. Calculation loop for multiobjective optimization (Stage 2,3)
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N. Ramzan and W. Witt
Stage-4: Design evaluation In the final layer/stage, the best compromise designs for each alternative are compared. The Pareto approach is very appropriate for this purpose. The Pareto-approach aims at eliminating alternatives that are clearly dominated, i.e. other feasible alternatives exist, which are better with regard to both or more objective functions.
Case study The stripping unit under study is part of a hydrocarbon recovery plant, which removes hydrocarbons and other solvents from the off-gases of the distillate fraction plants. Water, acetone, methanol, and acetic acid are the main components of the feed stream. The product stream (acetone rich) is separated from the effluent by using live steam injection. The column has a diameter of 0.728 meter and consists of 35 trays. The live steam is entered at stage 35 at temperature 141 ˚C and 3.75 bar pressure. The feed, which is at its bubble point, is entered at stage 16 with a column head pressure of 1.013 bars. The separation targets are: - Distillate: water < 10% - Base: acetone < 2000 ppm, methanol < 1-2% and acidity < 1-3% The aim is to optimize the stripping unit considering general design variables and safety related aspects. Figure 3 shows the original and an alternative design of the stripping unit and different alternatives studied.
General design alternatives (D-I to DVI) Alternative Total ID stages D-I 35 D-II 32 D-III 30 D-IV 35 D-V 32 D-VI 30
Feed stage 16 15 14 16 15 14
With/without sidestream Without Without Without With With With
Safety related alternatives (SS-I to SSVI) Alternative ID SS-I SS-II SS-III SS-IV SS-V SS-VI
Alternative Manual field shut-off within 5 min Manually initiated remote shutdown Automatic SIS Automatic reliable SIS Same as above with effective management procedures High-integrity SIS
* SIS……. Safety instrumented system Figure 3. Process diagram of system along with design and safety alternatives
The full-order tray-by-tray model applying mass and energy balances and equilibrium relations at each tray in the column (MESH Equations) is used for simulation and
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Methodology for Decision Support Among Conflicting Objectives
optimization of each alternative. The process model is configured by RADFRAC model based on well-supported Boston method. NRTL-HOC equations are used as thermodynamic model. The total annualized cost (TAC) and potential environmental impact (PEI) are used as economic and environmental performance objectives. Discussion of the results The optimum solution of each alternative satisfying both economic and environmental objectives is located in the objective space for finding non-dominated solutions or edegeworth pareto optimal design (see figure 4 a). A solution that is not dominated by any other individual solution is said to be non-dominated solution. The non-dominated alternative in the current set of design alternatives is identified. This alternative is considered the best and assigned as rank 1 in the figure 4 a. The alternative D-VI that is equally good as alternative D-IV and D-V in environmental performance objectives but better in economic objectives then all other alternatives so come in rank 1. Then considering this alternative virtually removed from the set of alternatives and next set of non-dominated solutions are identified and assigned rank 2. D-V and D-III come in this rank as shown in figure 4 a. This process continued until all alternatives are ranked.
DV
DII DIII
DVI
300 250 200 150 100
SS-III
1
350
50
SS-II SS-I
DIV
SS-IV
2
4 DI
SS-V
3
SS-VI
400
350 345 340 335 330 325 320 315 310 305 300
Safety System Cos t x 103 [$ / yr]
T A C $ /Y e a r Thousands
The effects of six different safety related design improvements (see figure 3) and typical implementation cost level on the probability of accident and safe shut off are studied. Each alternative is analyzed using event tree analysis. The probability of occurrence of accident scenario x 10-04 per year vs different safety alternatives are shown in figure 4 b. From the results, it is clear that in proceeding from safety alternative SS-I to SS-VI the probability of occurrence accident scenario falls down from 10-01 to 10-05 . The final choice of safety alternative depends on the safety standards to be met and economic advantage.
0 0,1
0
0.2
0.4
0.6
0.8
1
1.2
1
10
100
Accident probabilty x 10
1000 -4
10000
[1/ yr]
PEI/Kg
a) Non-dominated set analysis
b) Comparison of different safety alternatives
Figure 4. Evaluation of different design and safety alternatives
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N. Ramzan and W. Witt
Future Work Increasing social pressure and strict legislations resulted in changing the approach of traditional design practices. An effort is made in this paper to explain the systematic procedure for incorporating multiple conflicting objectives in the design of process plants using current tools with the help of example. The future work is focused to refine this concept and to develop the computer aided tool.
References Azapagic, A.;Clift, R. (1999): The application of life cycle assessment to process optimisation. Comp. & Chem Engg., 23,1509-1526. Alexander, B.;Barton, G.; Petrie, J. & Romagnoli, J. (2000): Process synthesis and optimisation tools for environmental design: methodology and structure. Comp.& Chem. Engg, 24,1195-1200. Chen, H.;Shonnard R.D. (2004): Systematic framework for environmentally conscious chemical process design: early and detailed design stages, Industrial Engg. Chem. Res., 43,535-552. Chakraborty, A.;Linninger, A. (2002): Plant-Wide Waste Management. 1. Synthesis and Multiobjective Design. Industrial Engg. Chem. Res., 41,4591-4604. Giuggioli Busacca,P.;Marseguerra,M.;Zio,E.(2001):Multiobjective optimization by genetic algorithms:application to safety systems,Reliability engineering and system safety,72,59-74 Koller,G.; Weirich, D; et.al (1998):Ecological and economic objective functions for screening in integrated development of fine chemical processes.1. Stream Allocation and case studiesork , Industrial Engg. Chem. Res., 37,3408-3413 Kim. D ; Yeo, Y.; & Moon II. (2004): Multiobjective optimisation for safety related decisions making in chemical processes. Journal of chemical engineering of Japan.37, 2,332-337. Stefanis K.S.; Pistikopoulos, N.E. (1997): Methodology for environment risk assessment of industrial nonroutine releases. Industrial Engg. Chem. Res., 36,3694-3707. Sharrat, P. (1999): Environmental criteria in design. Comp.& Chem. Engg,23,1469-1475. Palaniappan, C. ; Srinvasan, R. ; & Halim, I. (2002): A material-centric methodology for developing inherently safer environmentally benign processes Comp.& Chem. Engg, ,26,757-774.
Young, M.D. ; Cabezas, H. (1999): Designing sustainable processes with simulation: the waste reduction (WAR) algorithm. Comp. & Chem. Engg, 23,1477-1491.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
421
Grey-box stochastic modelling of industrial fedbatch cultivation Jan K. Rasmussena, Henrik Madsenb, Sten B. Jørgensena a
CAPEC, Department of Chemical engineering, Technical University of Denmark, Building 229, DK-2800 Lyngby, Denmark b Informatics and Mathematical Modelling, Technical University of Denmark, Building 321, DK-2800 Lyngby, Denmark
Abstract This paper presents application of a grey-box stochastic modelling framework for developing continuous time models for dynamic systems based on experimental data. The framework is used to develop predictive models suited for control purposes. The main idea behind the framework is to combine stochastic modelling with data to obtain information on parameter values and model (in-) validity. In case the proposed model is falsified the method can point out specific functional deficiencies which facilitate further model development. The developed model can be used for monitoring purposes as well as serve as a basis for advanced multivariable control to reject both intrabatch and interbatch disturbances. The industrial fermentation investigated in this case is production of an amylase enzyme by a filamentous fungus. Keywords: Parameter estimation, grey-box modelling, industrial fermentation.
1. Introduction Fed-batch processes play a very important role in chemical and biochemical industry. Fermentations are widely used in biochemical industry and are most often carried out as fed-batch processes. Present control schemes do not utilise the full potential of the production facilities and may often fail to achieve uniform product quality and optimal productivity. Application of advanced monitoring tools and multivariable control schemes can be a solution to this problem. The introduction of model based control strategies is considered difficult because suitable models are not readily available and require a significant investment in experimental work for their development. First principles engineering models can be used for monitoring and control purposes assuming that they are sufficiently accurate. Parameter estimation in a first principles engineering model can be very time consuming and can cause problems when scaling up from laboratory to industrial fermenters. The latter phenomena can not be investigated in laboratory scale equipment which therefore makes large scale experiments desirable. The approach taken in this paper is to combine first principle engineering models with operational data to produce predictive models suited for control purposes. The method described in this paper is grey-box stochastic modelling which consists of a set of stochastic differential equations (SDEs) describing the dynamics of the system in continuous time and a set of discrete time measurements. An important advantage using this approach compared to using deterministic models is that stochastic models can account for random variations in data and thereby provide a sound basis for hypothesis testing related to model validity based on available data. A framework for greybox stochastic model development has already been developed (Kristensen et al., 2004) and is described in figure 1.
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J.K. Rasmussen et al. Nonparametric modeling
Statistical tests No Yes
First principles engineering models
Model (re)formulation
Model unfalsified?
Experimental data
Parameter estimation
Stochastic state space model
Residual analysis
Figure 1.Grey-box stochastic modelling framework One of the key ideas behind the grey-box stochastic modelling framework is to use all prior information for formulation of an initial first principles engineering model. Unknown parameters of the initial model are then estimated from experimental data and a residual analysis is carried out to evaluate the quality of the resulting model. The next step in the modelling cycle is the model falsification or unfalsification which aims to determine if the model is sufficiently accurate to serve its intended purpose. If the model is unfalsified the model development is completed assuming that data are representative for the intended applications. In case of falsification the modelling cycle must be repeated by reformulating the initial model. In this case statistical tests can be used to provide indications of which parts of the model that are deficient. Nonparametric modelling can be applied to estimate which functional relationships are needed to improve the model. If the developed model is sufficiently accurate it can be used for online monitoring of the process or serve as a software sensor for otherwise unobservable states. Furthermore it can serve as the process model for development and tuning of advanced multivariable controllers. In this paper the modelling framework is considered.
2. Process description The process studied is a cultivation of the filamentous fungi Aspergillus for production of the enzyme amylase. The fermentation is initiated by transferring the contents of a seed tank to the main fermentation tank when a specified transfer criterion has been satisfied. The main fermentation tank contains an initial amount of substrate and the main fermentation process starts immediately after inoculation. The main fermentation is carried out in a batch and fedbatch phase. When the initial substrate in the main fermenter has been consumed by the micro organisms the fedbatch phase is initiated. Feed dosing is started at a low level and increased to its final value within a specified time span. The fedbatch phase continues for the rest of the fermentation duration and the majority of the enzyme is produced in this phase. The fermenters are equipped with sensors for online measurements of different process variables but some values are only available as offline measurements which makes closed loop control more difficult and requires an accurate model for predicting the unobserved variables.
3. Model formulation Initially a very simple first principles model for the fermentation is proposed. Measurements show that only a small amount of enzyme is formed in the batch phase
Grey-Box Stochastic Modelling of Industrial Fed-Batch Cultivation
423
(before the substrate feed begins) and the two phases are therefore modelled separately. Only the batch phase is considered in the model presented in this paper and it is assumed that no enzyme is formed in this phase of the process. If necessary this assumption can be changed in a later model iteration. In order to keep the model simple further assumptions have been made regarding product formation and yields. It is assumed that substrate is converted to only carbon dioxide and biomass and that the yield coefficients for conversion of substrate and uptake of oxygen are constant. The assumed yield coefficients can therefore easily be calculated from information about the initial amount of substrate, the total evolution of carbon dioxide and the total uptake of oxygen. Furthermore the specific growth rate, ȝ, and the oxygen mass transfer coefficient, kLa, are assumed constant. The initial model is given by 3 types of equations; stochastic differential equations, algebraic equations and observation equations. Stochastic differential equations:
x § · dx = ¨ μ x + Fevap ¸ dt + σ 1dw1 V © ¹
(1)
§ 1 · s ds = − ¨ μ x − Fevap ¸ dt + σ 2 dw2 V © Ysx ¹
(2)
(
)
cO § · dcO2 = ¨ − rO2 x + k L a cOsat2 − cO2 + 2 Fevap ¸ dt + σ 3 dw3 V © ¹
(3)
dV = − Fevap dt + σ 4 dw4
(4)
The states considered in this system are: biomass (x), substrate concentration (s), oxygen concentration (cO2) and volume (V). wi are four independent Wiener processes with incremental standard deviations given by ıi. Algebraic equations:
rO2 = Yxo μ
;
rCO2 = Yxc μ
(5)
Here the specific rates of oxygen consumption (rO2) and carbon dioxide evolution (rCO2) are given. These are modelled as being proportional to the specific growth rate, the proportional factors being the yield coefficients. Observation equations:
OUR = rO2 xV + e1 CER = rCO2 xV + e2
e1 ∈ N (0, s12 )
; ;
e2 ∈ N (0, s22 )
(6) (7)
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DOT =
J.K. Rasmussen et al.
cO2 cOsat2
⋅100% + e3
Volume = V + e4
;
e3 ∈ N (0, s32 )
(8)
e4 ∈ N (0, s42 )
;
(9)
For the above modelling 4 variables are used from the experimental data sets. OUR is the Oxygen Utilisation Rate, CER is the Carbon dioxide Evolution rate and DOT is the Dissolved Oxygen Tension (oxygen concentration measured in percent of saturation). The volume is included directly in the model. ei are independent white noise processes taken from a normal distribution with a mean of zero and standard deviation of si.
4. Parameter estimation and results Experimental data are taken from three batches run in pilot plant at Novozymes A/S. All batches have been run under similar conditions and using the same fermentation recipe. Parameters and corresponding standard deviations are estimated using a maximum likelihood method implemented in a computer program called CTSM (Continuous Time Stochastic Modelling). The program solves the stochastic differential equations in continuous time and estimates parameters using discrete time measurements. The initial parameter estimation shows that the uncertainties on parameters ȝ and kLa are very large and additionally the standard deviations of the Wiener processes in equation (1) and (3) are significant. This shows that the assumptions of constant ȝ and kLa do not hold. In order to reveal the time variation of the two phenomena they are introduced as states and two stochastic differential equations are therefore added to the previous four:
d μ = 0 + σ 5 dw5
(10)
dk L a = 0 + σ 6 dw6
(11)
The phenomena are assumed to be constant over time which is clearly not true, but the CTSM approach allows for subsequent estimation of the time wise behaviour of the phenomena and furthermore of their functional dependence of other states. Batch 1
Batch 2
Batch 3
Parameter
Estimate
Std. dev.
Estimate
Std. dev.
Estimate
Std. dev.
x0
5.71E-01
1.30E-01
9.51E-02
8.35E-02
4.19E-01
1.55E-01
cO20
4.16E-04
3.16E-06
4.18E-04
5.61E-07
4.13E-04
8.26E-07
V0
1.17E+03
4.54E+00
1.17E+03
7.97E-01
1.16E+03
4.10E-01
ȝ0
1.62E-01
3.21E-02
1.56E-01
2.97E-02
1.41E-01
3.70E-02
kLa0
9.51E+02
1.06E+02
5.22E+02
6.76E+01
1.04E+03
8.56E+01
Table 1. Estimates from the second estimation using eq. (1-11) with corresponding standard deviations of initial states. Table 1 gives estimates of the initial states in the model as well as uncertainty information (standard deviation). The initial substrate concentration is a known parameter and is therefore not estimated. It is seen that the initial estimates of cO2, V
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Grey-Box Stochastic Modelling of Industrial Fed-Batch Cultivation
and ȝ are similar for the three batches. The initial estimate of biomass (x) varies a lot from batch to batch suggesting that different amounts of biomass have been transferred to the main fermentation tank. The estimate on kLa is similar for batch 1 and 3 but only approximately half the value in batch 2. Batch 1 t-score
Batch 2
P(>|t|)
t-score
Batch 3 P(>|t|)
t-score
P(>|t|)
log ı1 (x)
-31.9147
0
-6.1517
0
-4.7905
0
log ı2 (s)
-0.029
0.9769
-0.1521
0.8792
-0.09
0.9284
log ı3 (cO2)
-5.1117
0
-62.6838
0
-8.2029
0
log ı4 (V)
-0.0581
0.9537
0.0259
0.9794
-0.057
0.9546
log ı5 (ȝ)
-38.6969
0
-35.7205
0
-29.4522
0
log ı6 (kLa)
23.6974
0
6.1096
0
28.4267
0
Table 2. Estimate of significances of state noise terms for model eq. (1-11). The estimates of the standard deviations of the state noise terms are given in table 2. For numerical reasons the logarithmic standard deviations are estimated as this gives a better numerical conditioning of the system. The p(>|t|) value is the fraction of probability of the corresponding t-distribution outside the limits set by the t-score. It can be interpreted as the probability that the parameter is insignificant. If the parameter is insignificant it can be replaced by a fixed value which is smaller than the estimate or even by a zero. If the noise term is insignificant it indicates that the corresponding SDE gives a satisfactory description of the particular state variable. Any model deficiencies will be contained in the noise term. As long as the noise term is significant the corresponding SDE does not match with the experimental observations and should be modified. This is seen for parameters ı1, ı3, ı5 and ı6. This suggests that functional relationships related to x, cO2, ȝ and kLa should be improved. Batch 1
Batch 2
Batch 3
t-score
P(>|t|)
t-score
P(>|t|)
t-score
P(>|t|)
log s1 (OUR)
0.0011
0.9991
0.1948
0.8457
0.001
0.9992
log s2 (CER)
0.0008
0.9993
0.1918
0.8481
0.001
0.9992
log s3 (DOT)
-4.3865
0
-24.651
0
-8.6884
0
log s4 (V)
23.4939
0
-2.7129
0.0072
5.5806
0
Table 3. Estimate of significances of measurement noise terms for model eq. (1-11). The estimates of the standard deviations (logarithmic value) on the measurement noise terms are given in table 3. Again it is seen that similar conclusions can be drawn from all three batches. It is seen that the measurement equations for OUR and CER fit the experimental data very well whereas the equations for DOT and volume should be reexamined. The significant noise in the DOT measurement can be explained by poor calibration of the sensor and the uncertainty on the volume measurement is known to be large.
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Batch 1 Batch 2 Batch 3
0.15
1000
Batch 2 Batch 3
800 kLa (1/h)
Specific growth rate, mu (1/h)
0.2
0.1
600 400 200
0.05 0
10
20 time (h)
30
0 0
10
20 time (h)
Figure 2. Estimates of specific growth rate, ȝ, and oxygen mass transfer coefficient, kLa, as function of time. The specific growth rate (figure 2) is seen to exhibit a very similar behaviour in all three batches. In the beginning the growth rate increases to a maximum value around 7-10h, which can be explained by intracellular formation of proteins due to the new growth conditions in the batch phase. Thereafter a decrease appears and at a certain point (around 15-18h) the growth rate seems to level off for a few hours where after the decrease continues. This phenomenon is not simple to explain but is likely to be due to the cells producing by-products. The evolution of ȝ as shown in the figure is only valid if the yield coefficients indeed can be assumed constant. The kLa shows a very similar behaviour for two of the three batches. For batch 2 it is seen that the value is much lower in the beginning of the process than for the two other batches. This might be due to poor aeration or stirring conditions.
5. Discussion Application of a grey-box stochastic modelling framework used to model a fermentation process has been illustrated in this paper. A stochastic model has been combined with experimental data to obtain information on phenomenological dependencies and uncertainties. Data from three batches run under similar conditions has been used and similar behaviour, in e.g., the specific growth rates is observed. This indicates that this modelling methodology provides a sound basis for development of a model which can capture essential process dynamics. Essential unknown phenomena, e.g. formation of by-products, must be investigated experimentally before the modelling cycle can be continued.
References 1. Kristensen, N. R., H. Madsen and S.B. Jørgensen, 2004, A Method for Systematic Improvement of Stochastic Grey-Box Models, Comp. & Chem. Eng., 28/8, 1431-1449 2. Kristensen, N. R. and H. Madsen Continuos, Time Stochastic Modelling Mathematics Guide, Informatics and Mathematical Modelling, Technical University of Denmark
Acknowledgments The first author thanks the support from Novozymes Bioprocess Academy which made this project possible.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Monitoring and improving LP optimization with uncertain parameters Danielle Zyngier*, Thomas E. Marlin Dept of Chem Eng, McMaster Univ, 1280 Main St. W, Hamilton, ON, L8S4L7, Canada
Abstract Linear Programming (LP) remains the workhorse of applied optimization, having openloop process applications such as production planning, inventory scheduling and closedloop applications such as Real-time optimization and the steady-state determination at each execution of a Model Predictive Controller (MPC). This paper presents new methods for monitoring performance (estimating the degradation due to uncertainty) and for improvement (reducing the uncertainty, when required, through economically optimal experiments) of closed-loop RTO systems. Keywords: RTO, performance monitoring, LP, uncertainty, experimental design
1. Introduction The value of optimization in plant operating planning and scheduling is well recognized. Potentially large benefits are possible when the optimum operation point changes often, i.e., there are significant disturbances in the process or changes in economics (Marlin and Hrymak, 1997; White, 1997). Little research has investigated monitoring, diagnosing and improving the performance of closed-loop economic optimization. In this paper, a systematic procedure is presented that is not limited to a constant active set, provides a measure of economic degradation due to model errors, identifies key parameters for a scenario, and designs plant experiments for improved model accuracy giving better economics performance. This paper investigates a real-time optimization (RTO) system with an LP optimizer with feedback information compensating for disturbances. The method is presented and applied to a case study involving blending gasoline in a petroleum refinery. In this example, five components (Reformate, LSR Naphtha, n-Butane, FCC Gasoline and Alkylate) are blended into the final gasoline product. The quality measurements in this system are the product blend octane and Reid vapour pressure (RVP) that are measured frequently onstream. The uncertainty in the case study lies only in the octane and RVP properties of the feed components to the blend, which are not measured due to the high cost of analyzers. Also, closed-loop simulations include measurement errors typical for industrial systems. A linear blending model was achieved through the use of blending indices (Gary and Handwerk, 1984) and the use of “blend-quality” inequality constraints (Pedersen et al., 1995). The closed-loop RTO solves Eq.(1) at each execution, which is equivalent to industrial blending optimization systems (Pedersen et.al., 1995). In Eq. (1), Fi represents the flow of each feed component, value is the value of product gasoline, costi is the cost of each component, Qi j represents the qualities of each component in the model and the subscript blend refers to the final blended gasoline. The feedback bias terms ε for octane *
Current address: Honeywell Process Solutions, 300 Yorkland Blvd., M2J 1S1, Toronto, Canada.
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D. Zyngier and T.E. Marlin
and RVP are calculated before the optimization problem as the difference between the predicted and measured quality (as in an MPC) and are, therefore, constants in Eq. (1). max ¦ (value − costi )Fi 5
i = Reformate, LSR, n-Butane, FCC Gas, Alkylate j = Octane, RVP
i =1
Fi
5
5
5
i =1
i =1
i =1
(1)
j j,model j + ε j ) ≤ Qblend,max s.t. : Qblend,min ¦ Fi ≤ ¦ Fi (Qi ¦ Fi 5
Demand min ≤ ¦ Fi ≤ Demand max
0 ≤ Fi ≤ Availablei
i =1
2. Five-Step Monitoring and Enhancement Procedure Step 1: Data Rectification: The first step consists of checking the data for consistency with the model and the uncertainty description. This step is needed because the subsequent steps rely on a priori estimates. If these are in error, the user should problem-shoot a major fault in the plant before implementing RTO. This step uses published methods (e.g., Johnston and Kramer, 1995). Step 2: Evaluating CLRTO Performance: The next step is to determine whether model/plant mismatch could lead to a significant profit loss; if a significant loss is possible, we will reduce the mismatch in subsequent steps. The potential loss due to parameter mismatch is the difference (gap) between the "ideal case" (CLRTO with no parameter mismatch) and "actual case" (CLRTO with parameter mismatch). The “worst-case” scenario, leading to the largest profit loss, occurs when the component qualities (within their uncertainty bounds) maximize the profit difference. In Step 2, the maximum profit loss, or the maximum improvement, is evaluated by finding the scenario of component qualities that maximize the profit gap. This approach is formulated as the following bilevel programming problem. max (Pr BC − PrWC ) (2) j , plant j ,ε
Qi
s.t.
max Fi
PrWC = ¦ (value − cost i )Fi 5
max Fi
i =1
¦ Fi (Qi
j,model
i =1 5
¦ Fi (Qi
j,model
Demand
(Q
5
5
j + ε j ) ≥ Qblend,min ¦ Fi
¦ Fi Qi 5
5
j + ε j ) ≤ Qblend,max ¦ Fi
¦ Fi Qi
i =1
i
− Qi
5
5
i =1
i =1
i =1
i =1
0 ≤ Fi ≤ Available i
) Var (Q )(Q −1
5
j ≤ Qblend,max ¦ Fi
Demand min ≤ ¦ Fi ≤ Demand max
max
0 ≤ Fi ≤ Available i j , no min al T
j , plant
5
5
≤ ¦ Fi ≤ Demand
5
j ≥ Qblend,min ¦ Fi i =1
i =1
i =1
min
j , plant
i =1
i =1
i =1
j , plant
5
i =1
s.t .
s.t . 5
PrBC = ¦ (value − cost i )Fi
plant
i
j , plant
− Qi
j , no min al
)≤ χ
2
α , DOF
ε j ¦ Fi = ¦ Fi (Qi j,plant − Qi j,model )
In this problem, the inner optimization problems (PrBC and PrWC) determine the values of the flowrates that yield the maximum profit, while the outer level determines the values of plant component quality parameters that maximize the profit gap. The “worstcase” sub-problem (PrWC) takes feedback into account because final closed-loop steady state profit is considered. Since the “best-case” sub-problem (PrBC) considers a perfect model, feedback is not needed, so a simpler formulation is possible. To facilitate the solution of Eq. (2), the two inner optimization problems are replaced with their Karush-Kuhn-Tucker optimality conditions (Clark and Westerberg, 1990). For the blending case study in this work, the bilevel programming problem
Monitoring and Improving LP Optimization with Uncertain Parameters
429
consisted of 56 variables (due to the inclusion of Lagrange multipliers) and 44 complementarity constraints arising from the optimality conditions of the inequality constraints. In our experience, local optimal solutions were obtained very reliably by using IPOPT-C, an interior point solver tailored to handle complementarity constraints (Raghunathan and Biegler, 2003). Problems converged quickly (< 2.0 CPU seconds) to local solutions for the case studied in this paper. It is worth mentioning that even though this is a non-convex optimization problem, several starting conditions converged to the same optimum point, building confidence in the results. Step 3: Updating Parameters Using CLRTO Data: If the profit gap evaluated in Step 2 is deemed significant (say over $100 per day), we would like to reduce model mismatch by using available measurements to estimate the uncertain parameters. The parameter-updating strategy used here is Bayesian Estimation, which allows for the incorporation of prior estimates of the parameter uncertainty and prevents unnecessarily large plant experiments (Box and Tiao, 1973). The estimated parameter values are obtained by using the following equation (Reilly, 1973).
(
Qt = Var (Q) −1 |t −1 + X TVar ( z ) −1 X
) (Var (Q) −1
−1
|t −1 Qt −1 + X TVar ( z ) −1 z
)
(3)
In Eq.(3), Var(Q)|t-1 is the prior variance-covariance matrix of the parameters, Qt and Qt-1 are the matrices with the new and initial parameter estimates, X is the matrix with input variables and Var(z) is the variance-covariance matrix of the output variables (measurements). In the case study, the output variables (z) considered were blend octane and RVP properties. When using the estimation method in Eq. (3), the parameter uncertainty can be shown to decrease with the experiments according to the following equation (Reilly, 1973).
(
Var (Q) |t = Var (Q ) −1 |t −1 + X TVar ( z ) −1 X
)
−1
(4)
Since the RTO introduces changes to manipulated variable values in response to model error and disturbances, initial closed-loop data from the current batch) is available at no cost and might have information useful in estimating RTO model parameters. Although this is a feedback system without external perturbations, Eq. (3) can be applied because of the lack of disturbances in the component qualities leaving large storage tanks. Step 4: Designing Profit-Based Experiments: If Step 3 does not reduce the potential loss below the threshold, designed experiments can be performed on the plant. The experimental design formulation is presented in the following.
(
min t1.Max Pr Gap − t2 .Prexp Fexp, i
)
(5)
s.t. MaxPrGap = Problem in Eq. (2) Var (Q
plant
(
) |t = Var (Q
plant −1
−1
) |t −1 + X Var ( z ) X exp T exp
j j j j Qmin, exp ¦ Fexp,i ≤ ¦ (Qm ,i Fexp,i + ε ) ≤ Qmax,exp ¦ Fexp,i i
Demand
i
min, exp
)
−1
(5a) (5b) (5c)
i
≤ ¦ F exp, i ≤ Demand i
0 ≤ Fexp,i ≤ Availablei
max, exp
(5d) (5e)
The variables being adjusted by the outermost problem are the component flowrates during the experiment; however, feedback information can be used by including the updated bias information (ε) in the experimental design calculation. This data at new
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operating conditions affect Var(Qplant) through Xexp, thus reducing the uncertainty region, i.e., tightening the upper and lower bounds of the MaxPrGap sub-problem in Eq.(2). The key issue in proposed experimental design strategy is the trade-off between the cost of experimenting and the benefits of an improved model in the closed-loop system, represented by the objective function in Eq.(5). In order to quantify this trade-off in terms of profit, a measure of time is needed. Since the case study in this work is a batch process, t1 the time remaining (days) in the batch after experimentation, while t2 is the time needed (days) to perform the experiment. The formulation in Eq. (5) is a three-level optimization problem, which was solved by using an unconstrained direct search method known as Derivative-Free Optimization (DFO) (Conn et al., 1997). The constraints (5c)-(5f) were replaced by penalty terms in the objective function in Eq. (5). DFO is based on approximating the objective function by a (simpler) surrogate model within a trust region, and then optimizing the surrogate model to obtain an improved point. It has been shown to be globally convergent to a local solution, and to be computationally more efficient than other direct search methods (Wright, 1996). For these reasons, the Matlab implementation of the DFO algorithm developed by Fan (2002) was used. Step 5: Implementing New Values: The new parameter values are inserted in the CLRTO model.
3. Results The five-step procedure was applied to a gasoline-blending system. The process was assumed to have independent initial uncertainties for octane and RVP properties in the 5 components (±0.47 octane numbers or psi). Measurement errors were assumed to be ±0.2 octane numbers and ±0.15 psi for octane and RVP qualities of the final blend, respectively, and 0.5% of the actual rate for flow measurements. These are realistic estimates of the uncertainties encountered in industrial gasoline-blending processes (Szoke and Kennedy, 1984). No gross errors were included in the simulated measurements. The five-step diagnostic procedure was initiated when the blend and real-time optimization began. After 4 RTO runs, the process under closed-loop RTO was achieved steady-state operation with a profit of $8,549/day. The component flowrates can be seen in Fig. (1). 7000 6000
Fi (bbl/day)
5000 4000 3000 2000
Reformate LSR Naphtha n-Butane FCC Gas Alkylate
1000 0 O O O O ate xp1 xp2 xp3 xp4 xp5 xp6 xp7 TO TO TO RT LRT LRT LRT pd R R R E E E E E E E L U C C C C CL CL CL
Figure 1. Component Flowrates during Case Study
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Step 1 – Data Rectification: According to the chi-square test, the data remained within the expected uncertainty. Step 2 – Evaluating CLRTO Performance: Eq. (2) indicated that the “best-case” scenario (PrBC) was $9,810.3/day, while the “worst-case” scenario (PrWC) was $7,604.9/day. This yielded a profit gap of $2,205.4/day, which was deemed too large. Step 3 – Updating Parameters Using CLRTO Data: Data from the first three CLRTO iterations was used to update parameters using the Bayesian Estimation method in Eqs. (3) and (4). After this was done, the largest profit gap in Eq. (2) was equal to $1,785.6/day, which was still considered too large. Step 4 – Designing Profit-Based Experiments: Assuming a blend with 9 hours of operation remaining, and a 15-minute long experiment, the weightings t1 and t2 used were 0.3646 day and 0.0104 day, respectively. After designing seven profit-based experiments (one at a time) according to Eq. (5), the largest profit gap calculated using Eq. (2) was reduced to $82.1/day. Step 5 – Implementing New Values: After applying the improved parameter estimates obtained in Step 4 in the CLRTO model and returning the CLRTO to closedloop operation, the system converged to a new operating point with Reformate, LSR Naphtha and n-Butane being blended to produce gasoline. This new operation yielded a profit of $9,118.1/day. It is important to note that the gasoline blend resulting from the two different operation points (the initial and the final ones in Fig. (1)) both produced gasoline with their maximum RVP and minimum octane, which is expected from an efficient gasoline-blending process (Gary and Handwerk, 1984). The difference between the operating strategies lies in the use of less expensive components in the more profitable operating strategy to achieve the same blend qualities. Assuming a single gasoline-blending batch per day, the benefits obtained from applying the procedure to this case study would be of approximately $208,000/year increased profit over a current state-of-the-art industrial blend optimization system. 3.1. Case with Different Economics (Case 2) Given the same initial RTO model/plant parameter mismatch as in the previous case study (Case 1) but different economic parameters in Cases 1 and 2, the initial operation of the CLRTO yielded the same initial optimal basis as in the preceding case, with LSR Naphtha, n-Butane and Alkylate being blended. The final uncertainties that result from the application of the 5-step procedure to each case after 10 designed experiments can be seen in Table 1. Case 1 Case 2 Octane RVP Final Octane RVP Final uncertaity uncertaity Flowrates uncertaity uncertaity Flowrates (octane) (psi) (bbl/day) (octane) (psi) (bbl/day) Reformate ±0.11 ±0.08 5696 ±0.21 ±0.17 0 LSR Naphtha
±0.37
±0.33
942
±0.40
±0.38
177
n-Butane
±0.46
±0.45
362
±0.46
±0.46
223
FCC Gas
±0.47
±0.46
0
±0.12
±0.10
4500
Alkylate
±0.15
±0.12
0
±0.13
±0.11
2100
Table 1. Parameter Uncertainty after Experimentation
In Case 1, the key decision is between Reformate and Alkylate; in Case 2, the key decision is between FCC gasoline and Alkylate. This case study demonstrates that the
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five-step procedure is able to identify the key model parameters for the specific scenario and to reduce the uncertainty in the key parameters with economically optimal plant experiments. These important features are achieved without guidance from the user.
4. Conclusions A sequential five-step method to monitor and enhance the performance of closedloop RTO systems has been developed in this work. This procedure provides quantitative measure of potential improvement. Since the true plant is not known, the method determines the largest gap between the best- and worst-case profit scenarios due to parameter variation within their uncertainty region. In addition, the method reduces uncertainty to increase profit with the least disruptive experiments possible. In order to reduce the severity and number of plant experiments needed in the five-step procedure, prior knowledge about the parameter distribution is considered when estimating model parameters by using Bayesian estimation. A new cost-effective experimental design strategy was proposed. It considers the trade-off between improving the closed-loop RTO model and the cost of performing the experiment. Besides being performed under the most profitable conditions possible, the experiment also obeys process constraints. This work adds to the current state-of-the-art in monitoring optimization performance. It is not limited to a single active set, can handle open or closed-loop systems, addresses correlated parameter uncertainty, emphasizes the objective value (rather than a measure or information), and provides a new experimental design method. Future extensions include applications for sensor selection to reduce experimentation, at the cost of capital equipment.
References Box,G.E.P., Tiao, G.C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley Publishing Co., Inc., Philippines. Clark, P.A., Westerberg, A.W. (1990). Bilevel Programming for Steady-State Chemical Process Design - i. Fundamentals and Algorithms. Comp. Chem. Engng., 14 (1) 87-97. Conn , A.R., Scheinberg, K., Toint, Ph.L. (1997) Recent Progress in Unconstrained Nonlinear Optimization without Derivatives. Mathematical Programming 79, 397-414. Fan, E. (2002) Global Optimization of Lennard-Jones Atomic Clusters. M.Sc. Thesis, McMaster University, Hamilton, ON, Canada. Gary, J.H., Handwerk, G.E. (1984). Petroleum Refining - Technology and Economics. Marcel Dekker, Inc., New York, NY. Johnston, L.P.M., Kramer, M.A. (1995). Maximum Likelihood Data Rectification: Steady-State Systems. AIChE Journal, 41 (11), 2415-2426. Marlin, T.E., Hrymak, A.N. (1997). Real-Time Operations Optimization of Continuous Processes. Fifth International Conference on Chemical Process Control. AIChE Symp. Series. 316, 156-164. Pedersen,C.C., Mudt,D.R., Bailey,J.K. (1995). Closed Loop Real Time Optimization of a Hydrocracker Complex. NPRA Conference Proceedings, CC-95-121, Nashville, TN. Raghunathan, A.U., Biegler, L.T. (2003) Mathematical Programs with Equilibrium Constraints (MPECs) in Process Engineering. Comput. Chem. Eng., 27, 1381-1392. Reilly, P.M. (1973). Linear Regression from a Bayesian Viewpoint. Lecture Notes. Szoke, R.W., Kennedy, J.P. (1984) Optimizing Control of a Catalytic Reformer with Digital Instrumentation. CC-84-117, 1-10, NPRA Computer Conference. Cherry Hill, NJ. White, D.C. (1997). Online Optimization: What, Where and Estimating ROI. Hydrocarbon Processing, 43-51. Wright, M.H. (1996). Direct Search Methods: Once Scorned, Now Respectable, in D.F. Griffiths and G.A. Watson (eds.). Numerical Analysis 1995, 191-208. Addison Wesley Longman, Harlow, UK.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Assessing the Performance of Batch Reactive Distillations through Conceptual Models José Espinosa INGAR-CONICET, Avellaneda 3657, Santa Fe S3002 GJC, Argentina
Abstract This contribution focuses on the development of a dynamic conceptual model for batch reactive distillations. Assuming a rectifier with an infinite number of stages, pinch analysis is applied to assess the performance of the combined process. Two highly nonideal quaternary mixtures, each one suffering an esterification reaction, are taken as example cases. Keywords: Conceptual Design, Batch Reactive Distillation, Pinch Analysis.
1. Introduction The analysis of batch reactive distillation as an alternative to the conventional configuration of a reactor followed by a batch rectifier needs the development of conceptual models to assess the performance of the combined operation. These models should allow the designer to analyze feasible products and evaluate the influence on conversion of variables like vapor flow rate, total reflux time, reaction plus distillation operation time and reflux ratio policy. This contribution focuses on the development of a dynamic conceptual model for a reboiler-reactor with an attached column above it. The main model assumption is a column with an infinite number of stages. Hence, pinch analysis can be used to predict the performance of the combined operation. For a given instantaneous still composition, the key feature of the method is the estimation of the instantaneous minimum reflux (operation at constant distillate composition) or the instantaneous distillate composition (operation at constant reflux ratio or constant Damköhler number) through pinch analysis. Between the two different methods that have been developed for conventional batch rectification (Espinosa and Salomone, 1999; Espinosa et al., 2005), this contribution explores the usefulness of calculating controlling pinch points from linearization of column profiles at instantaneous still composition (Espinosa and Salomone, 1999) for modeling the combined operation.
2. Conceptual Model for Batch Reactive Distillation and Vapor-Liquid Equilibrium The batch reactive distillation system consists of a reaction still and a rectifier section on top. The process is divided into a start-up period at total reflux and a combined reaction-distillation step, which can be operated at different reflux policies; i.e, constant reflux ratio, constant Damköhler number and constant distillate composition. Only the overall and component material balances around the combined process are necessary to estimate process performance if the following assumptions are made: i) a liquid-phase reaction takes place only in the still, ii) the column is in a pseudo steady state, iii) the vapor and liquid flow rates are constant from stage to stage, iv) pinch points control the geometry of internal profiles. The last assumption implies a column having an infinite
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number of stages and therefore, pinch analysis enables estimation of instantaneous distillate composition (reflux ratio) from a given instantaneous still composition of a column operated at constant reflux ratio (distillate composition) without resorting to satge-by-stage calculations. The key ingredient of the model is illustrated in Figure 1 for the mixture methyl acetatemethanol-water-acetic acid. Figure 1 shows the mass balance line given by the desired distillate composition xD (binary azeotrope methyl acetate-methanol), the vapor feed to the rectifier (vapor yxB* in equilibrium with the instantaneous still composition xB) and the composition xN of the liquid leaving the rectifier lower end. The last composition is calculated as the intersection between the mass balance line and the plane formed by the three controlling pinch points; i.e; xB, xPIII and xPII. Figure 1 also shows the internal profile calculated through simulation in Hysys. The simulated internal profile approximating xB is contained in a plane very close to that estimated by linearization of column profiles at instantaneous still composition (Espinosa and Salomone, 1999) and therefore, good agreement between rigorous and simplified simulation is found. Table 1 shows the numerical results corresponding to Figure 1. In this contribution, the instantaneous Damköhler number is defined as a dimensionless ratio of the rate of production r [kmol/(hm3)] Vs [m3] to the rate of product removal D [kmol/h]. Therefore, from a given Da number the distillate flow rate D is first calculated and then, the instantaneous reflux ratio RDa from a given value of the vapor flow rate is obtained; i.e, RDa = V [kmol/h]/D [kmol/h] -1. Pinch theory is finally employed to estimate the instantaneous distillate composition. All the examples analyzed in this contribution are modelled through the Wilson equation in order to reproduce the highly non-ideal behavior of the vapor-liquid equilibrium. The mixture ethyl acetate-ethanol-water-acetic acid presents three azeotropes, the ternary one being formed by ethyl acetate-ethanol-water and behaving as the unstable node for the system. The system methyl acetate-methanol-water-acetic acid presents two azeotropes with the azeotrope methyl acetate-methanol behaving as the unstable node. MeOH
xPII,Hysys
xPII,est 1.0
still path
xD
0.8
yxB* 0.6
AcH MeAc
0.4
xN xB
0.0 0.2 0.4
xPIII,est W
0.6 0.8 1.0
1.0
0.8
0.6
0.4
0.2
0.2
0.0 0.0
Figure 1. System MeAc-MeOH-W-AcH. Instantaneous minimum reflux and still path at constant distillate composition.
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Table 1. Calculated Rmininst from Hysys and the Conceptual Model Conceptual Model and Hysys; xB = [0.033642, 0.126585, 0.704795, 0.134978] xD
[0.657100, 0.342900,0.000000, 0.000000 ] Conceptual Model: Rmininst=LN/D=1.187
yxB*
[0.318917, 0.287771, 0.363347, 0.029965]
xN
[0.034144, 0.241349, 0.669310, 0.055197]
xD
[0.650698, 0.349302, 0.000000, 0.000000] Hysys : Rmininst=1.5 ; LN/D=1.188
yxB*
[0.318203, 0.288470, 0.363222, 0.030105]
xN
[0.038403, 0.237244, 0.668902, 0.055452]
3. System ethyl acetate-ethanol-water-acetic acid The feed to the still consists of a mixture with composition [0.00 EtAc, 0.45 EtOH, 0.10 W, 0.45 AcH] and molar amount M0= 1 kmol. The vapor flow rate V [kmol/h] is set to 0.5. Taking into account an overall process time of about 12 hours, four different operating policies were simulated with the aid of the conceptual model: i) Operation at total reflux (12 hours); ii) Operation at constant reflux (R= 10); iii) Operation at constant Damköhler number (Da= 0.40); iv) Operation at constant distillate composition (xD= xTAzeo =[0.6317, 0.1111, 0.2572, 0.0000]). The startup time for operating policies at finite values of reflux ratio is set at a value of 2 hours. Therefore, all simulations show the same behavior during the first 2 hours. Table 2 shows the kinetic data for the esterification of ethanol and acetic acid taken from Bogacki et al. (1989). Table 2. Kinetic data for the example cases kf [m3/(kmol s)]
kr [m3/(kmol s)]
Case 1
7.9 E-06
2.7 E-06
Case 2
275
55
Ea [Kcal/mol]
10
The resulting acetic acid percent conversions are [57.87, 70.28, 73.77, 78.23]. All processes with operation policies combining reaction and distillation present an enhancement in the conversion of the heavy limiting reagent with respect to operation at total reflux due to the removal of a distillate product rich in the reaction products ethyl acetate and water. Figure 2(a) clearly indicates that product removal rate achieves its highest mean value when the column is operated at constant distillate composition. Figure 2(b) shows the evolution of reflux ratio through the batch for operation policies different from total reflux. The evolution corresponding to operation at constant distillate composition gives the maximum instantaneous removal rate (minimum instantaneous reflux ratio) of ethyl acetate in a distillate product with minimal feasible amount of reagent ethanol. The maximum in ethyl acetate molar amount at t= 3.6 hours shown in Figure 2(a) for operation at constant reflux tell us that the rate of product removal by distillation is lower than the rate of ethyl acetate production by chemical reaction at the beginning of the combined reaction-separation step. This is due to the fact that the reflux ratio R of 10 is well above the minimum necessary to achieve a product of azeotropic composition. Accordingly, the temperature in the still reaches a pronounced minimum value as shown in Figure 2(c). Only after the product removal by distillation surpasses the production rate by distillation, the reboiler temperature begins to increase as it does
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in conventional distillation. This effect can be seen right after 2 hours for both the process at constant distillate composition and constant Damköhler number. Finally, although operation at Da= 0.4 provides a removal rate greater than the reaction rate during the reaction-distillation step, the resulting reflux ratio policy gives a product of variable composition with a product recovery in the distillate lower than the product amount obtained at constant distillate composition. 50
(a)
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5
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time [h]
Figure 2. System EtAc-EtOH-W-AcH. (a) EtAc molar amount (b) reflux ratio versus time. 368
(c)
Reboiler Temperature [K]
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364 362 360
Da = 0.4
358 356
R = 10
354 352
Total Reflux
350 348
0
2
4
6
8
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time [h]
Figure 2. System EtAc-EtOH-W-AcH. (c) reboiler temperature versus time.
4. System methyl acetate-methanol-water-acetic acid The feed to the still consists of a mixture with composition [0.0000 MeAc, 0.5688 MeOH, 0.025 W, 0.4063 AcH] and molar amount M0= 1 kmol . The vapor flow rate V [kmol/h] is set to 1.25. Taking into account an overall process time of about 1.45 hours, three different operating policies were simulated with the aid of the conceptual model: i) Operation at total reflux (2 hours to approach chemical equilibrium); ii) Operation at constant reflux (R= 1.8); iii) Operation at constant distillate composition (xD= xBAzeo
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=[0.6571, 0.3429, 0.0000, 0.0000]). The start-up time for operating policies at finite values of reflux ratio is set at a value of 0.25 hours. Kinetic data taken from Elgue et al. (2002) are shown in Table 2. The values of both forward and reverse constants correspond to a catalyst amount (H2SO4) of 5 ml. The resulting acetic acid percent conversions are [78.28, 87.48, 91.89]. Figure 1 shows the still path corresponding to the operation at constant distillate composition. During the start-up period, the still path follows the stoichiometric line with the still temperature decreasing as a result of methyl acetate buildup due to the chemical reaction. Once a finite value of reflux ratio is established, the molar fraction of reaction product in the still decreases from its maximum value as shown in Figures 1 and 3(a) and the still temperature evolution follows the typical trend found in conventional distillation. The same behavior for the temperature in the still is predicted in Gfeller (2005) through rigorous simulation of a vessel reactor with a column of five stages above it. Returning to Figure 3(a), the evolution of the compositions in the still resembles that shown in Figure 1(c) elsewhere (Elgue et al., 2002). After the combined reaction-distillation step, two distillative main cuts (not shown) must be drawn off as overhead fractions; i.e, a MeOH-W cut with traces of the remaining methyl acetate at the end of the previous step and a water rich cut. At the end of the process an acetic acid rich fraction remains in the reaction vessel. Both the MeOH-W cut and the residuum in the still are recycled to the next batch.
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extr. rate at V = 0.5 0.10 0.05
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(b)
0.30
0.00 0.25
r [kmol/h] at V = 0.5 0.50
0.75
1.00
1.25
1.50
time [h]
Figure 3. System MeAc-MeOH-W-AcH. (a) Still composition (b) MeAc extraction & reaction rate versus time.
In order to show the influence of vapor flow rate on conversion several simulation runs of the conceptual model were performed at a value of reflux ratio R= 1.8. The analysis of results indicates that there is an optimum value of the vapor flow rate V for which the conversion of the limiting reagent achieves a maximum value. Figure 3(b) shows the reaction rate and removal rate at two different values of the vapor flow rate. From Figure 3(b) it is clear that the rate of product removal at the low value of the vapor flow rate remains below the rate of production by chemical reaction for more than 30 minutes (or 45 minutes when both start-up and combined operation steps are taken into account). On the other hand, at V = 1.25 kmol/h the rate of methyl acetate removal surpasses the reaction rate from the very beginning of the combined operation step. As a
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result, a higher conversion is achieved in the last case. However, a further increment in V results in a lower conversion due to losses of the reagent methanol by distillation. In other words, the constant reflux ratio R= 1.8 is not high enough to maintain a product of nearly azeotropic composition during most of the time of the reaction-distillation step. An optimum in conversion was also found by simulating processes with different startup times. In all the cases, the reflux ratio was set to 1.8 and the overall operation time to 1.45 hours. Whilst a very short start-up period gives rise to losses of reagent methanol due to the reflux ratio is below the minimum one to achieve a distillate stream with high product content, a very high start-up time gives rise to a decoupling between reaction and distillation. Both extreme cases will lead to a decrease in conversion with respect to the optimum operation. The maximum in the considered example was a start-up time around 0.10 hours. This behavior was also experienced through rigorous simulation of the system in Elgue et al. (2002).
5. Conclusions and Future Work From the results obtained with the aid of the conceptual model based on pinch theory, it seems that the operation policy at constant distillate composition is the most appropriate since the limiting reagent conversion is maximized by eliminating the volatile product from the reaction mixture at the maximum feasible rate (under the constraint of a given value of the vapor flow rate V) with minimum loss of other volatile reagents. The instantaneous minimum reflux ratio, as calculated from pinch theory, depends on the instantaneous still composition and the selected distillate composition and it does not necessarily monotously increase its value as operation proceeds as shown in Figure 2(b). A similar behavior is found for temperature in the still. Operation at constant reflux, on the other hand, is very easy to implement in practice although expected conversion values are lower than those that can be obtained operating the system at constant distillate composition. Therefore, the operation control of a column operating at a variable reflux policy is a challenging problem and it will be the subject of future research work.
References Bogacki, M. B., K. Alejski and J. Szymanowski, 1989, The Fast Method of the Solution of a Reacting Distillation Problem, Comput. Chem. Eng., 13 (9), 1081-1085. Elgue, S., L. Prat, M. Cabassud, J. M. Lann and J. Cézerac, 2002, Optimsation of a Methyl Acetate Production Process by Reactive Batch Distillation, in ESCAPE 12, J. Grievnik and J. van Schijndel, Eds., Elsevier, 475-480. Espinosa, J. and Salomone, E., 1999, Minimum reflux for batch distillations of ideal and nonideal mixtures at constant reflux, Ind. Eng. Chem. Res., 38 (7), 2732-2746. Espinosa, J., S. Brüggemann and W. Marquardt, 2005, Application of the Rectification Body Method to Batch Rectification, in ESCAPE-15, L. Puigjaner and A. Espuña, Eds., Elsevier, 757-762. Gfeller, J., 2005, Internal report at ÉCOLE POLYTECHNIQUE FEDÉRALE DE LAUSANNE. Hysys User Manual, 1999, Hyprotech Ltd. : Calgary, Canada.
Acknowledgements The author would like to acknowledge the financial support provided by ANPCyT and CONICET of Argentina.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
An Integrated Stochastic Method for Global Optimization of Continuous Functions Mekapati Srinivasa and G. P. Rangaiaha a
Department of Chemical & Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576
Abstract Stochastic methods have attracted growing interest in the recent past as they require less computational effort to provide the global optima. Some of the well known methods are Genetic Algorithm (GA), Differential Evolution (DE) and Tabu search (TS). Each of these methods has a unique feature of escaping from the local minima and/or improved computational efficiency. Though each of these methods has its own advantage(s), they may be trapped in the local minima at times because of the highly non-linear nature of the objective function. In this work, an integrated stochastic method (ISM) is proposed by identifying and then integrating the strong features of DE and TS. A local optimization technique is used at the end to improve the accuracy of the final solution and computational efficiency of the algorithm. The performance of ISM is tested on many benchmark problems and challenging phase equilibrium calculations. The former contain a few to hundreds of local minima whereas the latter has comparable minima. The results show that the performance of ISM is better compared to DE and TS. Keywords: Differential evolution; Taboo search; Integrated stochastic method; Benchmark problems; Phase equilibrium calculations
1. Introduction Global optimization methods and their applications are attracting greater attention and interest due to the non-convex nature of the objective functions and the need to find the global optimum. In general, these methods can be classified into two categories: deterministic and stochastic (Pardalos et al., 2000). The former methods guarantee the global optimum under certain conditions whereas the latter do not. However, stochastic methods do not require such restrictive conditions, find global optimum with good success rate and are also computationally more efficient than the latter. Among the many, DE (Storn and Price, 1997) and TS (Chelouah and Siarry, 2000) are some of the promising methods reported in the literature. DE is a population based direct search method especially for non-linear and non-differentiable continuous functions. It mimics biological evolution by performing mutation, crossover and selection steps as in GA to escape from the local minima. The main advantages of DE are its capability to escape from the local minima with a few parameters and fast convergence to the global optimum compared to GA (Karaboga and Cetinkaya, 2004). TS, developed by Glover (1989) for combinatorial optimization, has been used for continuous optimization (Teh and Rangaiah, 2003). It is a meta heuristic algorithm that guides the heuristics to escape from the local minima. The main feature of TS is it avoids re-visits to the same place during the search thus providing good computational efficiency. Both DE and TS have their own merits and limitations. In this study, an attempt has been made to identify and
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combine the strong features of DE and TS, to develop a new method (ISM) with good reliability as in DE along with good computational efficiency like in TS. To the best of authors' knowledge, this is the first attempt to develop an integrated method combining the strengths of selected stochastic methods. ISM is tested thoroughly and systematically on benchmark and phase equilibrium problems.
2. Description of ISM ISM is developed by combining the reliable escaping mechanism of DE (i.e., mutation, crossover and selection steps) with the concept of TS (i.e., avoiding the re-visits during the search using tabu list). This is because our extensive experience showed that DE is more reliable compared to TS whereas the latter is computationally efficient than the former. The authors have chosen DE instead of GA because the former has only a few parameters and computationally efficient compared to GA (Karaboga and Cetinkaya, 2004). The proposed algorithm works better compared to DE and TS alone because by implementing TS concept in DE, i.e., ISM avoids the revisits to the same place during the search providing good computational efficiency while preserving the good reliability of DE. For example, the number of function evaluations taken by ISM and DE for Easom function to locate the global minimum up to a six decimal accuracy is 1855 and 2135 respectively. ISM (Figure 1) starts by choosing the optimal values for the parameters: population size (NP), amplification factor (A), crossover constant (CR), tabu list size (tls), tabu radius (tr) and maximum number of generations (Mgen). The algorithm initially generates a population of size NP using uniformly distributed random numbers to cover the entire feasible region. A boundary violation check is performed to see if any infeasible solution is generated; the infeasible points are replaced by generating new individuals.
Start
Set the parameters to optimum values
Generate the population and evaluate the objective function at each individual
Send the evaluated points to the tabu list
Set generation = 1 Generation = generation + 1 Mutation Crossover Perform tabu check and reject the points near to those in tabu list
Selection and updating of tabu list
Has stopping criterion satisfied? No Yes
Local optimization from the best point
Stop and print the result
Figure 1: Flow chart of ISM
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The objective function is evaluated at each individual, and the best point is captured. The evaluated points are then sent to the tabu list, which will be used to ensure that the algorithm does not search again close to these points. The three main steps: mutation, crossover and selection are performed on the population. The mutant individual is generated for each randomly chosen target individual (Xi, G) in the population by i = 1, 2, 3, …, NP (1) Vi, G+1 = Xr1, G + A (Xr2, G – Xr3, G); where Xr1, G, Xr2, G and Xr3, G are the three random individuals chosen in the population of the current generation G, to produce the mutant individual for the next generation, Vi, G+1. The random numbers r1, r2 and r3 should be different from the running index, i and hence NP should be 4 to do mutation. A has a value between 0 and 2, controls the amplification of the differential variation between two random individuals. In the crossover step, a trial individual is produced by copying some elements of the mutant individual to the target individual with a probability equal to crossover constant (CR). The trial individual is then compared with the points in tabu list and is rejected if it is nearby to those in tabu list for the next step. The new individual then competes with Xi ,G for a place in the next generation; generally, a greedy criterion such as objective function value is used to select the best point for further generations. The tabu list is updated and the process of mutation, crossover and selection is repeated until a stopping criterion such as Mgen is satisfied. The best point found over all the generations is further refined using a local optimization technique, namely, quasi-Newton method.
3. Implementation and evaluation of ISM The code for ISM is developed in FORTRAN and the performance of the method is compared to that of DE, GA and TS. The FORTRAN codes for TS and GA are taken from Teh and Rangaiah (2003), and DE code is taken from the website http://www.icsi.berkeley.edu/~storn/code.html. A local optimization technique is used at the end of each of these methods to improve the computational efficiency. For the quasi-Newton method, an inbuilt IMSL subroutine is used. The methods are evaluated based on the reliability and computational efficiency in locating the global optimum. The reliability is measured in terms of success rate (SR) i.e., the number of times the algorithm successfully located the global optimum out of 100 trials. A run is said to be successful only if the global optimum is obtained with a fractional error of 10-6 in the objective function value. The computational efficiency is measured in terms of number of function evaluations (NFE) required to locate the global optimum. The gradient in the local optimization is calculated numerically, and the NFE includes both the function calls for evaluating the objective and function calls for the gradient.
4. Application to benchmark problems To asses the performance of ISM, it has been applied to several benchmark problems involving 2 to 20 variables and a few to hundreds of local minima. A brief description of these functions and their global minima are given in Teh and Rangaiah (2003). The stopping criterion used is the maximum number of generations, and not convergence to the global minimum. We have used the former because, in reality, global minimum of application problems is unknown a priori. The parameters of DE, GA, TS and ISM are tuned using six out of twelve test functions. The optimal parameters thus obtained are used for the remaining functions too. The tuning is performed to achieve good reliability and computational efficiency, and is performed by varying one parameter at a time while the rest are fixed at their nominal/recent optimum values. The nominal
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values of parameters in all the methods are chosen based on the literature and preliminary experience with some of the functions. 4.1. Results and discussion The results for solving benchmark problems by TS, DE, GA and ISM are given in Table 1. Each problem is solved 100 times, each time by generating a random initial estimate. The results are compared in terms of SR and NFE, which is the average over 100 trials. SR is 100% unless otherwise stated. It is clear from Table 1 that the reliability of ISM is equal to that of DE, GA and is better than that of TS. This is perhaps due to the different escaping mechanisms from local minima corresponding to each method. To escape from local minimum, ISM performs mutation and crossover over a set of individuals as in DE and GA whereas TS makes use of best point obtained in the current iteration to generate neighbors for the next generation, even though it is worse than the best points obtained in the previous iterations. The reliability of TS is less for Easom function because of the flat objective function. As the function is flat, all the neighbors generated in TS will have the same function value trapping the search in that region. On the contrary, GA, DE and ISM located the global minimum region with the help of mutation and crossover. The reliability of all the methods is high for Shubert function though it has 760 local minima. This may be because locating one of the several global minima (around 18) is sufficient to achieve the best function value. The reliability of TS for Rosenbrock functions is less because of the associated narrow global minimum region.
#
Functions GP ES SH ROS2 Z2 H3 ROS5 Z5 ROS10 Z10 ROS20 ZAK20
Table 1: SR and NFE for different test functions GA DE TS ISM NFE NFE SR NFE SR NFE 20013 3224 100 918 100 976 20007 3224 90 1040 100 2401 20007 3246 100 1033 100 2252 20038 3247 99 2021 100 1799 20003 3223 100 1009 100 420 20009 4824 100 987 100 812 20197 8194c 76 5275 99 4463 20077 8026 100 2629 100 1110 21977a 16661d 74 17051 97 6303 20246 16031 100 8491 100 2477 25378b 34294e 82 44869 99 12341 21170 32041 100 19157 100 5715
Note 1: SR is 94, 89, 99, 98 and 96 at a, b, c, d and e respectively. Note 2: # GP – Goldstein and Price; ES – Easom; SH – Shubert; H3 – Hartmann 3 variable; ROS2, ROS5, ROS10 and ROS20 – Rosenbrock 2, 5, 10 and 20 variables; Z2, Z5, Z10, Z20 – Zakharov 2, 5, 10 and 20 variables.
NFE of DE and GA is more than that of ISM by a factor of 1.34 (ES) to 7.67 (Z2) and 2.05 (ROS20) to 20.5 (GP) respectively. NFE of ISM is less compared to TS for high dimension problems (a factor of 0.36 (ROS10) to 0.27 (ROS20)), even though it is more for low dimension problems (a factor of 1.06 (GP) to 2.3 (ES)). Overall, the computational efficiency of ISM is better compared to DE, GA and is comparable to that of TS. Both DE and GA are highly reliable but the former seems to be computationally efficient than the latter. Though TS has low NFE for low dimensional problems, its reliability is less compared to that of ISM and DE. By implementing TS concept in DE i.e., in ISM, we are able to maintain reliability as in DE and at the same
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time with low NFE. The NFE is increasing with number of variables (ROS2 to ROS20) with all the methods due to increase in the size of solution space which makes all the algorithms to generate more number of points to locate potential (global) regions.
5. Application to phase equilibrium problems Phase equilibrium calculations play a significant role in the design, simulation and optimization of chemical process. Development of robust and efficient method for phase equilibrium has long been a challenge and still it is. The objective is to calculate the number of moles of each phase and its composition at equilibrium given the pressure, temperature, components and feed composition. Basically, methods for multiphase equilibrium calculations can be categorized into two types: equation-solving approach and Gibbs free energy minimization. The modern trend is the treatment of phase equilibrium problems by direct minimization of Gibbs free energy. The objective function in this approach is a highly non-linear and non-convex requiring reliable and efficient global optimization. Several researchers have applied different global optimization methods both deterministic (Burgos-Solórzano et al., 2004) and stochastic (Rangaiah, 2001) using this approach. A review of the works on free energy minimization, and problem formulation can be found in Teh and Rangaiah (2003). The examples considered in this study include vapor-liquid equilibrium (VLE), liquid-liquid equilibrium (LLE) and vapor-liquid-liquid equilibrium (VLLE) problems involving multiple components and popular thermodynamic models. More information about the examples and local and global minima can be found in Teh and Rangaiah (2003). Parameters of DE, GA, TS and ISM are tuned in a similar way as for the benchmark problems. Three out of 10 examples are chosen for tuning, and the optimal parameters thus obtained are used to solve the remaining functions. 5.1. Results and discussion Table 2: SR and NFE of different methods for solving phase equilibrium problems TS ISM Example Problem GA DE number Type SR NFE NFE 1 (2) VLE (2) 20017a 7607 99 1348 5882 2 (6a) VLE (3) 20084 11440 96 1618 5660 3 (6b) VLE (3) 20086 11445 96 1639 5755 4 (9) VLE (9) ----------5 (11a) LLE (2) 20017b 7628 98 1432 6534 6 (11b) LLE (2) 20024 7619 100 1359 5255 7 (12) LLE (2) 20026 7624 100 1367 5352 8 (13) LLE (3) 20025c 11436 94 1575 9838 9 (17) VLLE (6) 20238d 15351 68 5648 12965 10 (18) VLLE (6) 20262 15355 81 5486 12723 Note: SR is 99, 97, 73 and 75 at a, b, c and d respectively. Bracketed number in the 1st column represents example number in Teh and Rangaiah (2003) and that in the 2nd column refers to the dimension of the problem.
Each example is solved 100 times, each time starting from a different randomly chosen point in the feasible region, and the performance results of the methods are summarized in Table 2. SR is 100% unless otherwise stated. The reliability of ISM and DE is 100% for all examples except for example 4, and is slightly less for GA especially for examples 8 and 9. The reliability of TS is comparable to that of ISM and DE for VLE
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and LLE problems except for example 8. This is due to the presence of comparable minima (i.e., function values at the trivial and global minimum are -0.35430 and 0.360353 respectively) in that example. As the minima are comparable, better regions where the function value is less than the local minimum become narrower and narrower and the algorithm fails to explore good points causing low SR. On the other hand, ISM and DE were able to escape from the local minimum with their mutation and crossover mechanisms. For example 4, all the methods failed because of comparable minima (function values at local and global minima are -161.5364 and -161.5416 respectively) and more number of variables (9). For VLLE, the reliability of TS is less compared to ISM and DE because of the comparable minima (stated in Teh and Rangaiah, 2003) in these problems. The reliability and computational efficiency of DE is better compared to GA. Computational efficiency of ISM is better compared to both DE and GA but is less than that of TS. NFE of GA and DE is more than that of ISM by a factor of 1.6 to 3.8 and 1.2 to 2.0 respectively. NFE of ISM is more than that of TS by a factor of 2.3 (example 9) to 4.6 (example 5). Although TS has good computational efficiency, its reliability is less for VLLE problems. On the other hand, ISM has good reliability like DE and is also more efficient than DE. Overall, the performance of ISM is better than that of GA and DE, and is comparable to that of TS. Comparison of CPU times with stochastic (i.e., GA and TS) and deterministic methods for some of these examples can be found in Teh and Rangaiah, 2003.
6. Conclusions and Future work A new method, namely, ISM is developed by effectively integrating the strong features of DE and TS. The method is first tested for a set of benchmark problems involving 2 to 20 variables and a few to hundreds of local minima, and then for challenging phase equilibrium calculations involving several components and multiple phases. ISM successfully located the global minimum for all the problems, and the performance of ISM is better than that of GA, DE and TS. Our future work includes the implementation of simulated annealing concept in ISM to improve its performance further and its application to various problems such as phase stability analysis and parameter estimation in models.
References F. Glover, 1989, Tabu search: part 1, ORSA Journal on Computing, 1, pp. 190-206. G. I. Burgos-Solorzano, J. F. Brennecke, M. A. Stadtherr, 2004, Validated computing approach for high pressure chemical and multiphase equilibrium, Fluid Phase Equilibria, 219, pp. 245255. G. P. Rangaiah, 2001, Evaluation of genetic algorithms and simulated annealing for phase equilibrum and stability problems, Fluid Phase Equilibria, 187-188, pp. 83-109. N. Karaboga and B. Cetinkaya, 2004, Performance comparison of genetic and differential evolution algorithms for digital FIR filter design, Advances In Information Systems: Third International Conference, LNCS, 3261, pp. 482-488. P. M. Pardalos, H. E. Romeijin and H. Tuy, 2000, Recent developments and trends in global optimization, Journal of Computational and Applied Mathematics, 124, pp. 209-228. R. Storn and K. Price, 1997, A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11, pp. 341-359. R. Chelouah and P. Siarry, 2000, Tabu search applied to global optimization, European Journal of Operational Research, 123, pp. 256-270. Y. S. Teh and G. P. Rangaiah, 2003, Tabu search for global optimization of continuous functions with application to phase equilibrium calculations, Computers and Chemical Engineering, 27, pp. 1665-1679.
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The ProMoT/Diana Simulation Environment M. Krasnyk,a* K. Bondareva,a O. Milokhov,a K. Teplinskiy,b M. Ginkel,a A Kienleac a
Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany b Computer Science Faculty, National Technical University of Donetsk, Artem Str. 58, 83000 Donetsk, Ukraine c Otto-von-Guericke-University Magdeburg, P.O. Box 4120, 39106 Magdeburg, Germany
Abstract We introduce the object-oriented modeling tool ProMoT and the simulation environment Diana suitable for numerical analysis of problems in chemical engineering and systems biology. The key aspects of this environment are flexible structured models, an efficient modular numerical kernel, and the use of the scripting language Python as a powerful command line interface. The implementation is based on CAPE-OPEN interfaces to allow a modular software design and easy extensions of the system. The contribution discusses the design and implementation rationale of the simulation environment. Keywords: object-oriented modeling tool, dynamic simulation, numerical analysis.
1. Motivation With the increasing capabilities of modern computers, it appears possible in academia and industry to describe and analyse more complex problems with higher resolutions. To make this possible a good and ”easy to use” modeling tool is required, that allows to create sophisticated models and analyse them with robust and efficient simulation code. Usually engineers have the choice between well known commercial simulation environments, like VisSim, Simulink, DyMoLa, gPROMS, etc. These software packages are well established with considerable libraries in different engineering fields and have own communities. There are also efforts to develop standards for the exchange of models like Modelica or the Systems Biology Markup Language. We want to contribute our modeling and simulation environment ProMoT/Diana (the names stand for ”Process Modeling Tool” and ”Dynamic sImulation And Nonlinear Analysis” tool). The special characteristics of ProMoT and Diana are, that they build on open source, freely available numerical code, provide sophisticated modeling and simulation capabilities for large models found in process engineering and systems biology, allow to be connected with other tools by common interfaces, allow to be easily extended for new requirements arising in academic research and will be freely available for academic groups. The following sections will discuss the general design of ProMoT and some aspects of the software design of Diana along with a short application example.
2. Modeling with ProMoT 2.1. Model setup ProMoT is a general modeling tool for the object-oriented development of dynamic equation-based models. Its main fields of application are process engineering and systems biology. ProMoT does not contain a simulation system itself but it has been originally *
Corresponding author. Tel.: +49 391 6110 377, mailto:[email protected]
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developed as modeling front-end for the simulation environment DIVA (Mohl et al., 1997). The model description used inside the tool is generic and in essence boils down to a tree of objects containing a symbolic system of ordinary differential and algebraic equations which describe continuous behaviour. The models can also contain descriptions of discrete behaviour that interact with the continuous equation set. The discrete part is described using the formalism of a deterministic, boolean valued Petri net. This formalism has been chosen for several reasons: i) places and transitions can be easily connected to the continuous model as boolean variables or discrete events respectively; ii) Petri networks are easy to modularise, allowing for encapsulated descriptions of local discrete processes; iii) local discrete parts can act asynchronously but can also be connected to describe the synchronisation of events. These characteristics render Petri networks as a good tool to describe discrete model changes based on physical phenomena as well as programmed controllers for process engineering models (Mangold et al., 2005). The model description inside ProMoT is quite general and is not limited to work with one specific simulation framework. Therefore the models can be exported for different simulation environments like Matlab, Diva and last but not least for our new simulation environment Diana. Models in ProMoT can be setup using the modeling language MDL or by aggregation and connection of existing modules in a graphical editor. For more information about this topic see (Ginkel et al., 2003; Waschler et al., 2005). The structuring follows the multi-level framework of network-theory, which is more extensively described in (Mangold et al., 2005; Gilles, 1998). Based on this framework, several modeling libraries have been implemented for different application fields like e.g. reaction and separation systems with vapour-liquid equilibrium (Waschler et al., 2005), membrane reactors, fuel cells and models in systems biology (Ginkel et al., 2003). When the user has finished the work on the model, the modeling tool allows performing different static tests of the dependency structure of the equation set to detect modeling errors and helps to debug the model. Furthermore the equation structure allows optimizing the equation set for efficient simulation; this is further elaborated in (Mangold et al., 2005). The derived equation structure is also exported to most of the simulation environments as equation pattern to allow efficient calculations using numeric algorithms and sparse matrix techniques. 2.2. Code generation design Code generation starts with an instantiated and optimised model. Since there are different output formats with own characteristics, the code generation is not implemented in methods of the model but in separate classes that interact with the model in a protocol similar to the strategy design pattern (Gamma et al., 1995). In this way it becomes easier to maintain multiple output formats without continuously changing all model classes. In the case of code-generation for Diana, the original structure of the model is not exported, because the simulator only supports equation sets with fixed structure and structured computation would be less efficient for large scale models. The generated code implements a subclass of the generic abstract model class of Diana. The generic class realizes most functionalities and interfaces for memory allocation, variable access, computation of approximate Jacobian matrices by finite differences and generic methods to evaluate the state and transitions of the Petri network. The specific model class must realize the specific initialisation of the model, the calculation of intermediate variables and residuals. It can also implement some optional functions. If the model has a discrete part it will implement a specifically structured Petri network and
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the respective trigger functions. If theree ar specific requirements on accuracy, e.g. if sensitivity analysis or continuations are desired, the model can implement methods for analytic calculation of Jacobians. Analytic Jacobians are derived using the symbolic differentiation algorithm implemented in the computer algebra system Maxima. We have implemented some special functions to generate optimized code for Jacobian matrices calculation of iterative equations and for derivatives of higher order.
3. Numerical analysis with Diana 3.1. General design rationale The generic interfaces for solvers and models in Diana are based on the specifications developed by the CAPE-OPEN community ke (Jaret al., 1999). These interfaces are designed to be used across an inter-process communication interface and they allow exchanging specific models and numeric algorithms without difficult changes of the sources. We use these interfaces wrapped into the Python scripting language as a command line user interface. The creation of model and solver instances is in general performed in a factory design pattern, where the class for the object is identified by name. This abstraction is useful, because it will allow to get the instances in different ways: i) as a direct python instance ii) as an instance created from a dynamically loaded C++ shared library or iii) as an remote instance published by an external CAPE enabled simulation package. Currently we work with shared libraries. 3.2. Dynamic simulation Dynamic simulation in the Diana environment is performed by a set of numerical differential algebraic solvers. All solvers implement the ICapeNumericDAESolver interface. This allows to use the solver instance as a ”black-box” integrator in Python scripts. The setup of the solver can be changed by a set of parameters: e.g. the relative error tolerance, the type of the internal linear algebra solver, etc. This parameter concept is derived from CAPE-OPEN; parameters have a name and a value, contain also a specification with default value and a documentation text, which makes them selfexplanatory. In the same manner adjustments to models take place. Solvers accept different reporting interfaces to process results of simulation directly. This allows to generate simulation logs or plots of simulation results. In Diana different integration codes have been used, like BDF or Runge-Kutta, this allows to solve different problem types, like systems of stiff ordinary or differential algebraic equations. 3.3. Parameter continuation A continuation superclass has been developed for the Diana nonlinear analysis suite that allows to perform a numerical continuation of the solution for a specified subtask. The superclass realizes a predictor-corrector method with tangent or chord predictor and local or pseudo-arclength parameterisation of the corrector. A Newton method with line search algorithm has been used as the corrector method. Continuation subclasses define the specific problem description by specifying residual vector and Jacobi matrix. Parameter continuation is used for the calculation of steady state solution branches and critical boundaries in the parameter space. During continuation the user has the possibility to determine the stability of the steady state. Stability analysis is applied to linearized model near steady state and based on the generalised eigenvalue problem. Such kind of analysis allows to detect events where real eigenvalue or pairs of the complex conjugate eigenvalues cross the imaginary axis.
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Two additional continuation subclasses are used to find solution curves with conditions specified by an augmented system. The first augmented system is used for tracking Hopf points with pure imaginary eigenvalues. The second one is used for singularities with zero real eigenvalue. If the model has been generated with support for the calculation of higher order derivatives, it is possible to add supplementary conditional equations to find more complex singularities of the steady state continuation curve. Conditions for limit points, isola points, pitchfork bifurcation points, and winged cusp points have been implemented. More information about methods and applications of the singularity analysis can be found in (Krasnyk et al., 2005). 3.4. Optimisation Numerical optimisation is used to solve a variety of problems in process engineering and systems biology. There are currently running projects in our working team that involve model identification and system reengineering, experimental design and optimal control of engineering processes. These problems or optimisation tasks mostly use some model of a physical or chemical process, but involve additional specific calculations of objective functions and constraints that can not be carried out directly within the simulation of the model. Our main goal is to provide a kind of glue software with common interfaces that allows to integrate different problem specifications with the different algorithms. The user should be required to program as little as possible, but must be able to customise the calculations for his specific optimisation task. In order to integrate algorithms which can solve different optimisation tasks we propose a software architecture, where the optimisation task is abstracted to a common interface separate from the algorithm. This interface allows to access optimisation variables, to compute the objective function and optional inequality and equality constraints. Optionally also sensitivity matrices can be calculated by the task, but this is only required, if gradient based algorithms should be applied. Different algorithms can be applied to this abstracted interface, and the implementation effort integrate to a new algorithm is reduced to a minimum since all model and problem specific calculations are in the responsibility of the optimisation task. There are different preimplemented concrete optimisation tasks, which implement the abovementioned interface. On the one hand there are fully preimplemented tasks, e.g., parameter fitting, where simulated time courses of a model are adjusted to measurements from different experiments. In this case the user only configures the optimisation task by supplying the model, the measured values and specific parameter sets. On the other hand there is the possibility to implement completely custom objective functions and constraints for an optimisation with Python functions. In this way the optimisation is highly flexible and the user is able to adjust the calculations very easily. Currently there are a deterministic multiple shooting algorithm and a genetic algorithm implemented (Teplinskiy et al., 2005).
4. Application example The application is considered the dynamic simulation of a counter-current fixed-bed reactor with self-sustained oscillations. The model is represented by a system of discretized parabolic partial differential and algebraic equations. For details with a full model description and discussions of the results see (Mangold et al., 2000). The ProMoT compiler mdl2diana generates C++ source files for the model and uses an external compiler to produce a shared library with the model code. The following Python script realizes a simple dynamic simulation of the reactor model:
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import diana, sys main=diana.GetDianaMain(sys.argv); mmanger=main.GetModelManager(); model=mmanger.CreateModel(diana.CAPE_CONTINUOUS, "GzfbrIdwt.so"); model.Initialize(); eso=model.GetActiveESO(); epar=eso.GetParameters(); evar=eso.GetStateVariables(); ri=main.CreateReportingInterface("basic"); ri.SetComponentName("data"); print "increase inlet temperatures of both tubes"; epar["t1_ein"].SetValue(600.0); epar["t2_ein"].SetValue(600.0); print "adjust inlet concentrations"; epar["xa1_ein"].SetValue(0.003); epar["xa2_ein"].SetValue(0.003); # create solver sfactory=main.GetSolverFactory(); solver=sfactory.CreateSolver(diana.CAPE_DAE, model, "ida.so"); solver.Initialize(); solpar=solver.GetParameters(); solver.SetReportingInterface(ri); ri.Add(solpar["T"]); for i in xrange(epar["n_z"].GetValue()): ri.Add(evar.ItemByName("t1[%d]"%(i+1))); solpar["Tout"].SetValue(100); solpar["Tend"].SetValue(1000) solver.Solve(); print "reduce inlet temperature in order to get oscillations" epar["t1_ein"].SetValue(300.0); epar["t2_ein"].SetValue(300.0); solpar["Tend"].SetValue(6000); solver.Solve(); ri.WriteData("oscil.dat");
The evolution of the temperature profile in the reactor is shown in figure 1.
Figure 1. Results of dynamic simulation of the counter-current fixed bed reactor
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5. Conclusion and future plans ProMoT/Diana is an open-source and extensible modeling and simulation environment for process engineering and systems biology. It allows for object-oriented modeling, model exchange and provides an extensible architect ure for numeric model analysis. By the use of Python it allows flexible extensions for users that are not experienced programmers. In the future it is planned to extend the set of available numerical algorithms. Also a network layer will be developed based on CAPE-OPEN that allows to connect Diana with other simulation packages via a CORBA middleware.
References E. Gamma, R. Helm, R. Johnson, and John Vlissides. Design Patterns: Elements Reusable ObjectOriented Software. Addison Wesley, 1995. E.D. Gilles. Network theory for chemical processes. Chem. Engng. Technology, 21:121–132, 1998. M. Ginkel, A. Kremling, T. Nutsch, R. Rehner, and E.D. Gilles. Modular modeling of cellular systems with ProMoT/DIVA. Bioinformatics, 19:1169–1176, 2003. M. Jarke, J. Koeller, W. Marquardt, L. von Wedel, and B. Braunschweig. CAPE-OPEN: Experiences from a standardization effort in chemical industries. Technical report, Lehrstuhl fr Prozesstechnik, RWTH Aachen, 1999. M. Krasnyk, M. Ginkel, M. Mangold, and A. Kienle. Numerical analysis of higherorder singularities in complex process models in ProMoT. In Puigjaner L. and A. Espuña, editors, European Symposium on Computer Aided Process Engineer ing - ESCAPE-15, pp. 223–228. Elsevier, 2005. M. Mangold, F. Klose,nda E.D. Gilles. Dynamic behavior of a counter-current fixedbed reactor with sustained oscillations. In S. Pierucci, editor, European Symposium on Computer Aided Process Engineering - ESCAPE-10, pp. 205–210. Elsevier, 2000. M. Mangold, O. Angeles-Palacios, M. Ginkel, R. Waschler, A. Kienle, and E.D. Gilles. Computer aided modeling of chemical and biological systems - methods, tools, and applications. Industrial & Engineering Chemistry Research, 44(8):2579–2591, 2005. K. D. Mohl, A. Spieker, R. Köhler, E. D. Gilles, and M. Zeitz. DIVA - A simulation environment for chemical engineering applications. In ICCS Collect. Vol. Sci. Pap., pp. 8–15. Donetsk State Techn. University, Ukraine, 1997. K. Teplinskiy, V. Trubarov, and V. Svjatnyj. Optimization problems in the technological-oriented parallel simulation environment. pp. 582–587, Erlangen, 2005. SCS Publishing. R. Waschler, O. Angeles-Palacios, M. Ginkel, and A. Kienle. Object-oriented modeling of large–scale chemical engineering processes with ProMoT. Mathematical and Computer Modeling of Dynamical Systems, 2005. in press.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Strategy and framework for solving signal-based MIDO problems R.H. Nyströma, I. Harjunkoskia, R. Frankeb a b
ABB Corporate Research, Wallstadter Str. 59, D–68526 Ladenburg, Germany ABB AG, PTSP-T29, Kallstadter Str. 1, D–68309 Mannheim, Germany
Abstract A strategy for solving mixed-integer dynamic optimization (MIDO) problems is presented. Special emphasis is on problems, where auxiliary time-dependent variables enter the model as inputs. These auxiliary variables are based on the binary variables that are used to formulate the problem through a linear transformation. The solution of such MIDO problems using a dynamic optimizer is also discussed. Keywords: Mixed-integer optimization, dynamic optimization, scheduling, production optimization.
1. Introduction Mixed-integer dynamic optimization (MIDO) problems arise when binary decision variables enter a dynamic optimization (DO) problem. Examples are control structure selection in combination with DO (Bansal et al., 2002), production sequencing in combination with DO (Chatzidoukas et al., 2003, Nyström et al., 2005), or model predictive control (MPC) using discrete variables (Gallestey et al., 2003). An MIDO formulation can be used for solving production optimization problems including sequencing and trajectory optimization within both transition (off-spec) stages and production (on-spec) stages (Nyström et al., 2005). The problem is relevant e.g. within the polymer industry and used as basis for an example problem in the present study. An MIDO problem can be solved by standard methods for solving mixed-integer nonlinear problems (MINLP), typically Outer Approximation (OA) or Benders Decomposition (BD), see the overview on MINLP methods by Grossmann (2002). Due to the numerical and algorithmic complexity of these methods, the solvers available for solving MIDO problems are scarce. One example is the optimization tool DyOS (Brendel et al., 2003). This study presents a solution strategy of MIDO problems for the publicly available dynamic optimization tool HQP (Franke & Arnold, 1997). Matlab is used for interfacing with HQP, and common MILP solvers are used for solving the master problem. The model is a compiled Matlab S-Function. A simple benchmark problem derived from Nyström et al. (2005) is used as illustrative example. The solution approach is signal-based, i.e., the binary decision variables enter the model using model inputs. Alternatively, auxiliary variables can be used, so that the binary variables are present only in the master problem. This reduces the effort for modelling and provides for more flexibility. In the example discussed, the auxiliary variables entering the model are the sequence-dependent bounds used in stage path constraints and stage endpoint constraints. This gives the advantage that the problem can be
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rescaled to any degree (i.e., for any number of products to be produced), without having to increase the number of inputs to the model or having to modify it in any other way. For keeping track of which binary or auxiliary variable enters which optimization stage, an input index mapping approach is used. The linearizations or Lagrange multipliers of the DO problem are obtained by an approach inspired by Bansal et al. (2003). Thus, the binary variables or auxiliary variables are “activated” for optimization in a last iteration. At the same time, they are being fixed by additional equality constraints. Thus, linearizations or Lagrange multiplicators w.r.t. these variables can be obtained, although they are treated as constants during the main DO effort.
2. OA and BD prerequisites Outer Approximation (OA) and Benders Decomposition (BD) are standard methods for solving MINLPs and can readily be used for solving MIDO problems, see e.g. Bansal et al. (2002) and Oldenburg et al. (2003). In both methods, the MIDO problem is split up into a primal problem, which is a DO (the reduced MIDO problem with fixed binaries), and into a master problem, which is an MILP. The master and primal problems are solved consecutively. During the course of the iteration, the lower bound (LB, from the master problem) approaches the upper bound (UB, provided by the primal problem), and the iterations are typically interrupted when the LB is close enough to the UB. Integer cuts in the master problem are commonly used to ensure that the same set of binaries is only tested once. The methods OA and BD differ in how the master problems are constructed from the primal problem solutions. In OA, the master problem is formed using the linearizations of the objective function and of the constraints, essentially T
min UB u
§ dJ · s.t. J k + ¨ ¸ (u − u k ) ≤ UB , k = 1...K © du ¹ k
(1)
T
§ dc · c k + ¨ ¸ (u − u k ) ≥ 0 , k = 1...K © du ¹ k
(2)
where K is the total number of iterations up to the present point, Jk is the value of the primal problem solution in the kth iteration, uk are the decision variables of the MIDO problem (both binaries and continuous) giving the solution of the kth primal problem, ck is the vector of inequality constraints in the kth primal problem, and UB is an auxiliary variable. The master problems become quite large, since they contain entries from all constraints and costs in all previous iterations. By the extended penalty approach, OA can be expanded by slack variables, which make the approximate solution of nonconvex problems possible. See Grossmann (2002) for more details. The master problem in BD is simpler than in OA. Here we have min UB u
T ª § dJ · T º § dJ · s.t. J k + ¨ ¸ (u − u k ) + z kT «c k ¨ ¸ (u − u k )» ≤ UB , k = 1...K © du ¹ k »¼ ¬« © du ¹ k
(3)
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where zk is the resulting vector of Lagrange multipliers for the constraint vector ck in the kth iteration in the primal problem. BD uses much smaller master problems; only one constraint is added per iteration. Additionally, only the binary variables are typically used in the master problem in BD. Bansal et al. (2003) give a practical method for numerically solving the BD and OA problems. Specifically, when a master problem solution has been obtained, an additional optimization can be made, whereby the binary variables are not constants, but decision variables, which have been “frozen” using additional equality constraints. This facilitates the extraction of the Lagrange multipliers and gradients needed in the master problems. The Lagrange multipliers of the additional equality constraints are shown to be equivalent to the Lagrange multipliers of the overall problem. The approach is adopted in this study.
3. Using the dynamic optimization solver HQP for primal problem solution The solver HQP (Franke & Arnold, 1997) is a multi-stage dynamic optimization solver. It implements a sparse Sequential Quadratic Programming (SQP) algorithm for the solution of large-scale nonlinear optimization problems. The sparse structure arising from the treatment of dynamic optimization problems in discrete time is exploited. An Interior Point method is employed to efficiently solve quadratic sub-problems with a high number of inequality constraints. HQP offers useful features for the treatment of primal problems in BD and OA. Its S-Function interface supports passive and active inputs (i.e., inputs that are not or that are used as decision variables, respectively), decimation of decision variables (i.e., the decision variables can be set to be constant for a particular number of optimization intervals), multiple samples within optimization intervals, and fixing inputs using equality constraints. Also, the Lagrange multipliers and gradients can be obtained. The ability to switch between active and passive inputs, as well as being able to fix active inputs using additional equality constraints is highly useful for adopting the approach by Bansal et al. (2003). The model used by HQP is a dynamically linked library (DLL) with the S-Function interface. Such a model can be obtained by several tools, for example Dymola. In order to calculate gradients with respect to inputs to the model, an input index map is formed together with the input profiles. The input index map has the same dimensions as the input matrix, and contains information on the input variable appearing at different time samples. For instance, if decimation is used to keep an input constant for 10 samples at a time, then this should also show up in the input index map. Thus, the 10 gradients w.r.t. to 10 sample times can be mapped to the same variable. Extracting the information needed by OA and BD requires some steps. After the primal problem has been solved, 1. Quadratic programming (QP) matrices are dumped from HQP. 2. Equality constraints corresponding to the fixation of the binary/auxiliary inputs are located in the in the QP matrices (rows in the A and b matrices of the QP problem). 3. Adjoint variables (state variables) are eliminated, because the gradients with respect to the inputs are needed, but partial derivatives with respect to states and inputs are available. Theoretically, it is possible to include the adjoint variables in the master problem, but this gives huge MILPs and requires treatment of equality constraints.
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4. Gradients of the objective function and the inequality constraints are obtained as a result of the elimination. 5. Gradients are recalculated to correspond to inputs instead of input differences. 6. Gradients of the original variables are calculated by using the input index map and summing all gradients pertaining to a variable. 7. Lagrange multiplicators of the equality constraints found are obtained from HQP. 8. Lagrange multiplicators are recalculated to correspond to inputs instead of input differences. 9. Lagrange multiplicators of the original variables are calculated by using the input index map and summing all Lagrange multiplicators pertaining to a variable. 10. Objective function and solution inputs at the optimum are obtained from HQP.
4. Using auxiliary inputs instead of binary variables The solver HQP uses models, which are compiled binary files (S-Function DLLs). The model contains a fixed amount of inputs, outputs, states and parameters. Since it needs to be recompiled each time its structure is changed, it is desirable to formulate an optimization problem such that the model can be used as such with varying problem setup. A class of problems in which this approach is feasible are multi-stage problems, e.g. with the structure presented in (Nyström et al., 2005), where a production optimization problem is formulated. The aim is to produce a number of different products. Each product grade is specified by bounds on quality variable outputs. The task of the master problem is to determine the sequence in which the products are to be produced. The primal problem optimizes the transition and production stages. Thus, the primal problem is described by a range of stage-dependent output constraints that are functions of the binary variables. These constraints are most conveniently handled by letting auxiliary inputs enter the model. The auxiliary inputs contain stage-dependent bounds, which are used to impose the output constraints. The binary variables used in the master problem need not enter the primal problem. Thus, the number of inputs to the model does not grow with a growing number of products. Further, the model does not need to contain all information required to form the master problem. For example, if time-slot binary variables were directly fed to the model, 81 binaries would be needed for 9 products, distributed so that 9 binaries simultaneusly enter the model. Inside the model, the constraints would be calculated from the binaries, thus the model needs to contain product-related information specific to the overall production optimization problem, but not relevant to the dynamic optimization problem in question. Replacing this approach by one where auxiliary variables enter the model instead is highly useful and should be so for most problem classes, where a changing problem size leads to a varying number of binary variables. Figure 1 summarizes the concept of using auxiliary variables instead of binaries in the primal problem. Master problem
Binary variables
Linear transformation
Auxiliary Primal problem inputs
Objective function, solution variables, gradients, Lagrange multipliers
Figure 1. Using auxiliary inputs instead of binaries in primal problem.
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It is straightforward to show that the OA and BD master problems can be formulated using gradients with respect to the auxiliary variables plus the linear transformation instead of using gradients w.r.t. the binary variables. For instance, let a denote auxiliary variables, which are obtained from the binary variables b with the transformation a = Ab+B. Then the expression showing up in BD and OA can be manipulated according to T
T
T
T
T
§ dx · § dx · § dx · § da · § dx · ¨ ¸ (a − a k ) = ¨ ¸ A(b − bk ) = ¨ ¸ ¨ ¸ (b − bk ) = ¨ ¸ (b − bk ) © da ¹ k © da ¹ k © da ¹ k © db ¹ k © db ¹ k
(4)
The way to generate the transformation matrices depends on the problem. In the production optimization problem described above, an example is given by bound U qbound = U qspec (auxiliary variable) is the upper bound of the ,s , p ξ p , s , where U q , s
¦ p
quality signal q in stage s, ξ p, s is a binary time-slot variable with decision to produce/not produce product p in stage s, and U qspec , p is a product specification constant giving the upper bound of quality signal q for product p.
5. Example A simple version of the production optimization problem in Nyström et al. (2005) was constructed, using a minimal linear process model. The number of products to be produced is 6, there are two manipulated inputs and two quality variable outputs. The test problem has the favourable property that fixing the time scaling of the stages reduces it to linear. The full problem is highly non-convex, and the augmented penalty approach must be used to get any kind of result with OA. For solving the master MILP problems, ILOG CPLEX and/or LP_SOLVE are used. Figure 2 shows the iterations using OA/AP. Due to the non-convexity of the problem and the augmented penalty approach used, the LB and UB do not mean lower and upper bounds in the strict sense, as in the case of a convex problem. Also, the LB reaching the UB is not useful/sufficient as a termination criterion. In the present case, a good point to terminate the algorithm is instead after the 5th iteration, where the UB ceases to decrease, hereafter the LB violently increases. After this point, no significant improvement of the UB is obtained within reasonable time. The found solution 2.14 is reasonably close to the global optimum 2.13. A detailed study on solving this type of problems using DyOS (Brendel et al., 2003) is presently ongoing (Prata et al., 2005).
6. Conclusion A strategy and framework for solving MIDO problems with the publicly available dynamic optimization solver HQP has been discussed. The emphasis is on problems where the binary variables enter the dynamic model as time-dependent signals. Instead of binary variables it may often be favourable to use auxiliary variables, which are linear functions of the binaries. This facilitates the modeling, and typically decreases the size of the dynamic model.
7. Acknowledgement The European Union is gratefully acknowledged for the support of the postdoc work of R. Nyström under the Marie Curie Fellowship, contract no. HPMI-CT-2001-00138.
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2.8 2.7 2.6
UB LB master solution primal solution
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2.5 2.4 2.3 2.2 2.1 2 1.9 1.8 1
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Figure 2. Sample result of solving a highly non-convex MIDO problem with OA.
References Bansal, V., Perkins, J.D., Pistikopoulos, E.N. (2002). A case study in simultaneous design and control using rigorous, mixed-integer dynamic optimization models. Industrial and Engineering Chemistry Research, 41, 760–778. Bansal, V., Sakizlis, V., Ross, R., Perkins, J.D., Pistikopoulos, E.N. (2003). New algorithms for mixed-integer dynamic optimization. Computers and Chemical Engineering, 27(5), 647–668. Brendel, M., Oldenburg, J., Schlegel, M., Stockmann, K. (2003). DyOS User's Guide Version 2.1. LFPT, Prof. Dr.-Ing. Wolfgang Marquardt, RWTH-Aachen University. Chatzidoukas, C., Kiparissides, C., Perkins, J.D., Pistikopoulos, E.N. (2003). Optimal grade transition campaign scheduling in a gas-phase polyolefin FBR using mixed integer dynamic optimization, in: Chen, B., Westerberg, A.W. (Eds.), Process Systems Engineering 2003, Elsevier, pp. 744–747. Franke, R., Arnold, E. (1997). Applying new numerical algorithms to the solution of discrete-time optimal control problems, in: Warwick, K., Karny, M. (Eds.), Computer-Intensive Methods in Control and Signal Processing: The Curse of Dimensionality, Birkhäuser Verlag, Basel, 105– 118, see also http://hqp.sourceforge.net/ for download of HQP. Gallestey, E., Stothert, A., Castagnoli, D., Ferrari-Trecate, G., Morari, M. (2003). Using model predictive control and hybrid systems for optimal scheduling of industrial processes. Automatisierungstechnik, 51(6), 285–293. Grossmann, I.E. (2002). Review of nonlinear mixed-integer and disjunctive programming techniques. Optimization and Engineering, 3, 227–252. Nyström, R.H., Franke, R., Harjunkoski, I., Kroll, A. (2005). Production campaign planning including grade transition sequencing and dynamic optimization. Computers and Chemical Engineering, 29, 2163–2179. Oldenburg, J., Marquardt, W., Heinz, D., Leineweber, D.B. (2003). Mixed-logic dynamic optimization applied to batch distillation process design. AIChE journal, 11, 2900–2917. Prata, A., Oldenburg, J., Marquardt, W., Nyström, R., Kroll, A. (2005). Integrated scheduling and dynamic optimization of grade transitions for a continuous polymerization reactor. In preparation.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
“Smart Models” - a framework for adaptive multiscale modelling Eric S. Fragaa1, Gary Willsb, Michael Fairweatherc, Tony Perrisd a
Department of Chemical Engineering, University College London Advanced Knowledge Technologies, Southampton University c School of Process, Environmental & Materials Engineering, University of Leeds d Consultant, Woking, UK b
Abstract The business environment facing the ECPI (European Chemical & Process Industry) is changing at an ever-increasing rate, bringing with it new challenges to process engineers which the current generation of CAPE tools are illequipped to help them address. This paper puts forward a view of some of the challenges and offers some thoughts on a potential way forward. Keywords: multi-scale modelling, smart models, adaptive models
1. The Business Challenge The ECPI is now confronted with a step-change in its global business environment. For example, analyses by CEFIC SusChem [1], Chemicals Vision 2020 [2] and EUREKA Project 2311: CAPE-21 [3] have identified a number of major commercial, technological and societal challenges in the global marketplace. SusChem concludes that failure to address these challenges effectively will result in a dramatic decline in the ECPI's competitiveness and profitability, with a damaging impact on the EU's economy.
2. New CAPE Challenges in a Changing Business World Process modelling and simulation have long been established as the primary weapon in the process engineer's armoury but there are a number of key areas where today's capabilities struggle to address the new business challenges: • complex materials & mixtures, multi-phase systems, particulates; • models of sustainable processes, including “natural” raw materials and biological species; • more detailed unit operations models (micro- and bulk-mixing, surface interactions, turbulence, etc); • models in manufacturing: models are typically not transferred from engineering to manufacturing; • extended enterprise models typically contain only a very approximate model of the plant itself ; • virtual organisations (VO): specialisation and collaborative working between companies, often on a world-wide basis; • adaptive models and links to measurement systems; and, 1
Author for correspondence: [email protected]
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• performance: radically-improved performance will deliver significant benefits in both design & operations. There seems to be an infernal triangle of trade-offs between model performance, quality/detail and scope. For a given technology, this triangle seems to have a constant area - we cannot “break” it in any meaningful way without a radical change in technology. Charpentier and McKenna [4] also recognised the need for an integrated multidisciplinary and multi-scale approach to meet the needs of the CAPE industry today. They recommend simultaneous research in four directions: 1. total multi-scale control of the process to increase selectivity and productivity; 2. process intensification by design of novel equipment based on scientific principles and new production operating methods; 3. the synthesis of structured products, combining several functions and properties required by the customer, with special emphasis on complex fluids and solids technology; and, 4. the implementation of multi-scale and multidisciplinary computational modelling and simulation for real-life situations. It is the latter item that we will concentrate on in this paper.
3. Modelling & Simulation Technologies 3.1. The multi-scale challenge Many new business opportunities/high-value-added areas are inherently multi-scale they involve complex micro/nano-scale materials (including bio-materials and structured/heterogeneous “natural” materials). These business activities also operate at multiple timescales, from reaction kinetics to supply and distribution chains. Behaviour at each scale directly or indirectly affects that at all other scales and so it is necessary to be able to model simultaneously at all relevant scales, to integrate (or “aggregate”) the modelling at the different scales and thus to predict “inter-scale interactions”. For example, there is an increasing need to be able to model micro/nanoscale interactions at interfaces between phases. Catalysis, bio-activity, particle formation & growth, etc., depend on micro-mixing and interfacial effects at phase boundaries. Hitherto, models have typically either ignored this continuum of scales in both time and space or have tackled them separately (i.e. there have been several models operating at different scales, with human intervention to interpolate and rationalise between them). Today’s and tomorrow’s pressures are such that this approach is no longer viable and that we now need a fundamental re-think, aimed at devising a new cohesive framework for multi-scale modelling: a single effective capability providing a flexible, consistent and systematic approach to modelling and simulation across all the scales – “from molecules to the marketplace” – and embracing the currently disparate technologies, such as MD (molecular dynamics), CFD, CAPE and business models. 3.2. A multi-scale model? It is important to recognise that the individual scales (or levels of detail of representation) represent “slices” of a pseudo-continuum – from molecules to business chains – and that, for any given problem, a different combination of scales may be
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required. Flexibility must therefore be maintained for users to select the most appropriate approach and to modify it as their needs evolve. For instance, depending on the purpose at hand, the following might be required: • A reactor model may incorporate a broad variety of phenomena and sub-processes: molecules and their reactions; bio-materials and mixtures, organisms; multicomponent, multi-phase fluid mixtures and their properties; particles, pellets and beds; mixing and fluid flow; phase aggregation and separation; internal equipment items (e.g. stirrers, heater/coolers); and so on. Basic information on, for example, properties, kinetics, thermodynamics, etc, may be predicted from first-principles MD or bioscience. • Such a model might then be incorporated into a whole-flowsheet model (for example, to examine the potential impact of recycling, awkward separation issues or flexibility, dynamics and control). • Finally, the whole may be embedded into a business model of the supply and distribution chains, for example, to optimise their dynamic capabilities to match the dynamic requirements of the marketplace and to assess the potential impact of both technical and commercial uncertainties. Such “modelling across the scales” will enable a variety of problems to be addressed: • Micro/surface mixing/flows will determine how catalysts behave and thus which reactions occur and how fast they occur. This will enable the prediction of, for example, hotspots and the impact of processing history on product micro-structure and properties. • Optimised integration of kinetics, hydrodynamics and thermodynamics, for example, in the development of novel process equipment (i.e. multi-function, intensification, miniaturisation). • Integrated development of products, processes and business chains for the global optimisation of the extended enterprise, as an integrated whole from molecules to marketplace. • Improved processing of “structured” materials, such as wastes and renewables. • “Indirect” measurement and control, including soft sensors. 3.3. Can this be delivered using today s technologies? With today’s technologies, such multi-scale models might be attempted, for example, by embedding MD into kinetics/thermo/properties (K/T/P) calculations, K/T/P into CFD, CFD into CAPE and CAPE into a business modelling program (!) . A typical approach hitherto has been to select two modelling systems at two different scales and then to develop special-purpose code (typically both problem- & system-specific) to enable them to exchange information. However, a number of serious problems arise with such an approach, even on such a limited basis: • Such an approach can only be attempted by highly-skilled modelling experts, not by the practising engineer. • The number and nature of the “scales” would effectively be pre-defined and may be entirely inappropriate for the problem at hand (and the appropriate scales may well change as study of the problem evolves). • Serious problems would remain regarding the adequacy of the model and its “fitness for purpose”. For example, CFD has serious limitations in such areas as multi-phase flows, complex rheology and/or surface-scales [6] and MD currently struggles with simulations involving complex molecules.
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• Such a deeply layered integration of pre-existing, self-contained systems will be large, difficult to solve and compute-intensive, both to develop and to use, and, more importantly, success is by no means guaranteed. G.D. Ingram et al. [5] have shown that micro-scale and macro-scale models cannot be combined with complete freedom and that those that can be combined are limited and restricted. They do provide a preliminary compatibility table of frameworks. For all practical purposes, an effective multi-scale modelling capability cannot be delivered by combining today’s systems. A fundamentally new approach is required, one which is designed from the outset to encompass the full range of length and time scales and to operate effectively within the context of a virtual organisation.
4. “Smart Models” - a Flexible Framework for Adaptive Multi-scale Modelling & Simulation Performance/quality conflicts are not new! In some contexts, methods have been developed to alleviate the problem. For example, variable step-length integrators and adaptive mesh techniques and now standard in most numerical codes for solving differential equations. The key features of these techniques are typically that they are auto/adaptive, and therefore embody metrics to define the “quality” of the model and/or the solution, and that the resulting models are implicitly “mixed granularity” in that the step lengths or meshes can be on different scales in different parts of the model. The success of these methods then raises the question: Can we apply such adaptive/granularity concepts to model complexity itself, rather than just to the solution methods? We are therefore developing the concept of “smart models”, a novel hierarchical modelling architecture which incorporates “intelligence” to selectively and to adaptively manipulate the models and incorporate detail and complexity only in those areas of the model which are critical to providing an adequate solution and remove or approximate such detail and complexity where it is not. The concept thus reflects how an engineer might tackle such a problem by hand: start out with some very simple models/assumptions of the individual units, get some idea of flowsheet conditions and then start digging a bit deeper where it seemed justified. This process would then be lo oped until the user felt satisfied that the quality/accuracy of the results was suitable for the purpose at hand. The engineer thus adapts his/her approach on the basis of perceived “fitness for purpose” and uses a “mixed granularity” model, adapting and modifying the choice of granularity (i.e. of specific scales from the overall continuum) as the solution evolves. Thus, we can imagine a modelling and simulation capability within which the following hold: • A model consists of a hierarchy of “layers” of increasing detail, complexity and sophistication, potentially spanning the entire pseudo-continuum of length and time scales, from molecules to business chains. • Each layer contains a model definition of some kind and that model has a number of parameters. • Each layer/model accepts parameters from “below” and calculates the parameters required by the layer “above”. • “Intelligence” (using e-Science capabilities, such as ontologies, languages, agents, etc) could be incorporated at any appropriate point to define and manipulate the models, parameters and solution methods, assess the quality/fitness-for-purpose of
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the model and then decide how “deep” to go under any particular circumstances to satisfy the needs of the user Such an architecture has a number of real advantages: • It is flexible and extensible and is capable of transparent, scaleable and effective operation on different kinds of computer architecture (from single processors, through parallel architectures, clusters and to the Grid), as the scale and scope of the model and the computing power requirements evolve.2 • It provides a flexible, rational, consistent and transparent basis for mixed-granularity models. • It is straightforward to incorporate “foreign modules”, “legacy code”, adaptive models, etc., and to integrate with laboratory or plant systems. • Models can be distributed via the Grid and thus suitable for use in a virtual organisation. • The performance and robustness of the simplified models could be predicted and “guaranteed”, for example, for safety-critical applications • Fitness for purpose can be indicated to the user. Note that although this paper is limited to discussions on models, knowledge based working environments will also be required to support the complex and crossdisciplinary model-building activities, to manage the resulting models and their use across the lifecycle and to support the decision-making processes which will be based on their results.
5. Challenge to the CAPE Research Community: Practical First Steps The present generation of tools has its origins more than 20 years ago. The new challenges are very much more complex than problems addressed to date and delivery of a comprehensive new CAPE-ability and its adoption into widespread industrial practice is a large and multi-disciplinary activity which will inevitably take a number of years (and the resources required are beyond the resources of even the largest centres acting individually). The challenge to the CAPE research community is, therefore, to so organise itself to undertake such initiatives in an efficient and effective manner and to avoid the duplication and fragmentation of its efforts. As has been remarked, th e longest journey begins with the first step , the objective of this paper is to stimulate discussion and debate (and, where necessary, provoke arguments!) which will lead us to some “practical first steps : what is/are the best way(s) to deliver an effective solution; how can we maximise the synergistic opportunities whilst not hindering a thorough exploration of the alternatives?
References 1. http://www.suschem.org 2. http://www.chemicalvision2020.org 3. http://www.eureka.be
2
Note that there have been some attempts to implement process simulators (today's/yesterday's generation) on high performance computers (usually SIMD). These have not been very successful (a) because SIMD is probably the wrong choice and (b) they involved the implementation of a (serial) code without any meaningful redevelopment/restructuring. What we are proposing is a new generation, designed for such architectures (particularly CSP/MIMD)
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4. J. C. Charpentier & T. F. McKenna (2004). Managing complex systems: some trends for the future of chemical and process engineering. Chemical Engineering Science 59:16171640. 5. G. D. Ingram, I. T. Cameron & K. M. Hangos (2004), Classification and analysis of integrating frameworks in multiscale modelling. Chemical Engineering Science 59:21712187. 6. CAPE-21 Definition Phase Report, see http://CAPE-Alliance.ucl.org.uk
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Process design using ionic liquids: Physical property modeling Adolfo E. Ayalaa, Luke D. Simonia, Youdong Lina, Joan F. Brenneckea and Mark A. Stadtherra a
Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN 46556 USA
Abstract Ionic liquids are a relatively new class of materials with properties that make them attractive for a wide variety of engineering applications. For design purposes, it is useful to have a relatively simple model (i.e., excess Gibbs energy model or equationof-state model) capable of describing the physical properties and equilibrium behavior of ILs and IL solutions. We consider here the performance of two selected models, NRTL applied to the modeling of liquid-liquid equilibrium and an electrolyte equationof-state applied to the modeling of aqueous mean ionic activity coefficients. In each case we focus on issues in parameter estimation, and use an approach based on interval mathematics to solve the parameter estimation problem globally. Sample results are presented and suggest that the models considered here may be useful for correlation of data, but may not be well suited for prediction. Keywords: Ionic liquids, Parameter estimation, Phase equilibrium, Physical properties
1. Introduction Room-temperature ionic liquids (ILs) are a relatively new class of materials that have attracted significant interest in the context of environmentally-conscious process design. These materials are salts but are liquids at room temperature. ILs have no measurable vapor pressure (i.e., they do not evaporate) and thus, from a safety and environmental viewpoint, have several potential advantages relative to the traditional volatile organic compounds (VOCs) used as solvents for reactions and separations, including elimination of hazards due to inhalation, explosion and air pollution. ILs also have many other interesting properties, including a wide liquidus range, that may make them attractive for a wide variety of engineering applications [1]. Thus, for engineering design purposes it is useful to have a relatively simple model (i.e., excess Gibbs energy model or equation-of-state model) capable of describing the physical properties and equilibrium behavior of ILs and IL solutions. As a basis for such models there is available today an increasing amount of physical property and phase equilibrium data [e.g., 2], as well as results from molecular simulation studies [e.g., 3]. The overall goal of this project is to evaluate the performance of a variety of models for computing the physical properties and phase behavior of ILs. Since the degree of dissociation in IL solutions is unclear, and there are molecular simulation results [4] that suggest that it may be small even in some dilute solutions, both electrolyte and nonelectrolyte models should be considered. In this abbreviated paper, we will present some initial results for two selected models. The first is the standard NRTL excess Gibbs energy model, without electrolyte extension, which will be applied to liquidliquid equilibrium problems involving ILs. The second is the equation-of-state (EOS) model given by Myers et al. [5], which is an extension of the Peng-Robinson EOS to
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electrolyte systems, and which will be applied to aqueous IL solutions. It should be noted that, beyond standard EOS models, excess Gibbs energy models, and their electrolyte extensions, approaches such as QSPR [e.g., 6] and COSMO-RS [e.g., 7] also have potential in this context.
2. Methodology 2.1. Parameter Estimation The models to be studied involve parameters that must be estimated from experimental data. Estimation of parameters in these models requires either the solution of a nonconvex global optimization problem, or the solution of a nonlinear equation system. Failure to find the globally optimal parameters for a thermodynamic model, and using locally optimal parameters instead, can have significant consequences in subsequent calculations, as demonstrated by Gau et al. [8] and Ulas et al. [9]. The use of locally optimal parameters can lead to rejection of a model that may perform satisfactorily when using globally optimal parameters. For example, Gau et al. [8] showed that using the globally optimal parameters improved the predictive capability of a model. In a problem involving the prediction of homogeneous azeotropes using the Wilson equation, they showed that using the locally optimal parameters given by Gmehling et al. [10] resulted in incorrect predictions of the number of azeotropes, but when using the globally optimal parameters the correct number of azeotropes was predicted. Therefore, in evaluating a model, it is important in doing parameter estimation that the method used will guarantee finding the globally optimal parameters. The interval-Newton approach provides such a methodology. 2.2. Interval-Newton Approach For general background on interval mathematics, including interval-Newton methods, there are several good sources [e.g., 11]. The interval-Newton approach provides a method for computing all the solutions of a system of nonlinear equations, and doing so with mathematical and computational certainty. It can be applied directly to a parameter estimation problem formulated as an equation solving problem, or, for a problem formulated as an optimization problem, it can be applied to the equivalent system of equations (local optimality conditions). An important feature of this approach is that, unlike standard methods for nonlinear equation solving and/or optimization that require a point initialization, the interval-Newton approach requires only an initial interval, and this interval can be chosen to be sufficiently large to enclose all possible results of interest. Intervals are searched for solutions using a powerful root inclusion test based on interval mathematics. This test can determine with mathematical certainty if an interval contains no solution or if it contains a unique solution. If neither of these results can be proven, then typically the interval is bisected and the root inclusion test applied to each subinterval. On completion, an intervalNewton/generalized bisection (IN/GB) algorithm will have determined narrow enclosures of all the solutions to the equation system of interest. In an unconstrained optimization problem, these solutions represent the stationary points, including all local minima, so the global minimum can be readily determined. Alternatively, IN/GB can be applied in connection with a branch-and-bound scheme, which will lead directly to the global minimum without finding any of the other stationary points. In recent years, we have applied this technique to many problems in the modeling of phase behavior [e.g., 8,12].
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3. Results 3.1. Modeling IL-Alcohol-Water Systems Using NRTL One potential application of ILs is in the separation of fermentation broths. Thus, modeling the liquid-liquid equilibrium of IL-alcohol-water systems is of interest. The ternary system IL-octanol-water is also of particular interest since octanol-water partition coefficients are widely used as a measure of the potential ecological impact of a compound. The applicability of an excess Gibbs energy model (NRTL) to such systems is considered here. NRTL parameters for the underlying binary systems were determined using the procedure described below, and then these were used to predict the ternary behavior for comparison to experimental data. When a binary system at constant temperature and pressure is modeled using a twoparameter excess Gibbs energy model, the parameters can be determined directly by using the equal activity condition for each component. This provides two equations: § xD · ln J 1D ln J 1E ln ¨ 1E ¸ © x1 ¹
0
(1) D
§ 1 x1 · ln J 2D ln J 2E ln ¨ E ¸ © 1 x1 ¹
0,
where JiS is the activity coefficient and xiS is the mole fraction of component i in phase S. The activity coefficients are functions of composition, involving two unknown parameters. Thus, by substituting the experimental values of xiS into Eq. (1), there remain only two unknowns, in this case the NRTL binary interaction parameters 'g12 and 'g21 (we will consider the NRTL nonrandomness factor to be fixed at D = 0.2). For the NRTL model, it is well known [13] that there may be multiple solutions to this nonlinear equation system. Thus, it is important that we be able to determine all solutions. This is done using the IN/GB approach discussed above, which is guaranteed to find all solutions within a specified initial parameter interval, here chosen to be very large, namely [5u106, 5u106] J/mol for each parameter. Since Eq. (1) is a necessary but not sufficient condition for phase equilibrium, solutions must be tested to determine if they correspond to thermodynamically stable phase equilibrium. This can be done by using IN/GB to perform tangent plane analysis, as described by Tessier et al. [14], which guarantees that the phase stability problem is solved correctly. If multiple parameter solutions remain after testing for stability, then physical considerations must be used to choose the most appropriate values, since each solution represents an exact fit to the experimental data. For example, parameter values leading to the prediction of multiple miscibility gaps in the binary system can be eliminated, since this behavior is not observed experimentally. Or one can consider whether or not the sign and/or magnitude of the parameters are physically reasonable. Using IN/GB in this framework, the NRTL parameters for alcohol-water, IL-water and IL-alcohol binaries were determined for a number of cases. Some sample results are shown in Tables 1 and 2 for the case in which the IL is 1-butyl-3methylimidazolium hexafluorophosphate ([bmim][PF6]) and the alcohol is octanol. For the octanol-water binary, four parameter solutions to Eq. (1) were found. Solutions 1 and 2 predict stable states, but solution 2 shows two immiscibility gaps, so it is discarded. Solutions 3 and 4 predict unstable phases so they are discarded. Parameter solution 1 corresponds to the literature value [13] for this binary. For the [bmim][PF6]water binary, two solutions to Eq. (1) are found and both predict stable states.
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However, solution 2 has a 'g12 value that is extremely large and can be considered physically unreasonable; therefore, it is discarded. The same procedure was used to determine the NRTL parameters from binary data at other temperatures of interest, and for other ILs and alcohols. A linear dependence of the parameters on temperature is observed. In general, NRTL can be used quite successfully to correlate the available binary LLE data involving ILs. Table 1 NRTL parameters (J/mol) for n-octanol/water at T = 313K (Data from [13]) Solution No. 1 2 3 4
'g12 99.52 58859.91 14293.37 12783.56
'g21 22303.92 22356.01 120783.35 120877.45
Stable? TRUE TRUE FALSE FALSE
Table 2 NRTL parameters (J/mol) for [bmim][PF6]/water at T = 298K (Data from [15]) Solution No. 1 2
'g12 -556.43 -209974
'g21 16569.13 21658.84
Stable? TRUE TRUE
Using parameter sets found from binary data as described above, the NRTL model was applied to predict IL/alcohol/water ternary diagrams. For the systems considered, the model predicts a three-phase region in IL/alcohol/water systems for alcohols with alkyl chains longer than ethanol. The three-phase region becomes smaller as temperature is increased and eventually disappears at a ternary upper critical solution temperature. In general, however, compared to experimental data [16] for such systems, NRTL does not show the right qualitative behavior, since a three-phase region has not been observed. From these results, it is reasonable to presume that, even though NRTL can successfully correlate binary data, it is not well suited to predict ternary phase diagrams from binary data. Therefore, we are investigating the use of the electrolyte NRTL model [17] as a tool for modeling these systems. 3.2. Modeling Aqueous Solutions with an Electrolyte Equation of State In this section, we present some sample results for parameter estimation with the Myers et al. [5] EOS, applied to the modeling of mean ionic activity coefficients in aqueous solutions. This is a three parameter model. Two of the parameters are the van der Waals attraction parameter, a, and excluded volume parameter, b. The third parameter is V , which is the hydrated radius of ions in solution. Myers et al. [5] have used experimental mean ionic activity coefficient data to estimate these parameters for a very large number of electrolytes in aqueous solution, and found that in general it correlates these data better than the (two-parameter) electrolyte NRTL model. However, the a values found span several orders of magnitude, and in general it is not clear that the parameter values reported are consistent with their physical interpretations. Previously [18], using local optimization methods, we have shown that the Myers et al. [5] EOS correlates activity coefficient data well for aqueous solutions of tetra-alkyl ammonium and choline salts, compounds whose structure is similar to that of some ILs. Again, the fit is somewhat better than that obtained [19] with electrolyte
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NRTL, though again the parameter values span a wide range and have no clear physical meaning. This work also demonstrated that the least squares function arising in the parameter estimation for this EOS could have several local minima, and that thus global optimization methods really should be used. Most existing data for mean ionic activity coefficients of aqueous electrolyte solutions was measured some time ago and does not include any volumetric data. However, more recent activity coefficient data, such as that currently being obtained for ILs, may also be complemented with density data as a function of composition. These two types of data can be combined to obtain data in the form of mean ionic activity r as a function of solute density Ui = xiU = xi/v, where v indicates the coefficient J exp,i molar volume of the solution and xi the mole fraction of the solute. We show here some sample results for the model parameters a, b and V when estimated from data in this form using the IN/GB methodology to globally minimize a simple least squares function. The problems used were designed to test the sensitivity of the optimal parameter values to small changes in the volumetric data used. The activity coefficient data used, along with five slightly different sets of volumetric data, are shown in Table 3. Thus, there are five sets of activity coefficient vs. solute density data for which parameter estimation can be done. These data sets should be regarded as hypothetical test problems. Since this parameter estimation problem is an unconstrained optimization problem, the IN/GB approach can be applied to solve the first-order optimality condition for stationary points. It is assumed that the initial parameter interval is selected to be sufficiently large that it contains the global optimum in its interior. Since all stationary points in the initial parameter interval will be found, this also guarantees that the global minimum will be found. Table 4 shows the results of the parameter estimation for the data given in Table 3. It is observed that a wide range of V values are obtained, varying by multiple orders of magnitude. Apparently, at least for these test problems, the optimal value of the V parameter in the EOS is quite sensitive to small changes in the volumetric data used. This parameter represents the radius (given here in Å) of hydrated ions in solution and so the values found do not all appear to be physically meaningful. Our experience with these simple test problems, together with our previous experience [18], and the original results of Myers et al. [5], all suggest that optimal parameter values may vary over a wide range and often appear not to have physical significance. This is suggestive of a model that may correlate well, but not be well suited for prediction. We are currently investigating electrolyte extensions of other equation-of-state models for application to ILs and IL solutions.
4. Concluding Remarks We have considered here the performance of two selected models, NRTL applied to the modeling of liquid-liquid equilibrium and an electrolyte equation-of-state applied to the modeling of aqueous mean ionic activity coefficients. An interval-Newton approach was used to ensure that parameter estimation problems were solved globally. The sample results presented suggest that the models considered here are useful for correlation of data involving ILs, but may not be well suited for prediction.
Acknowledgments This work was supported in part by the State of Indiana 21st Century Research and Technology Fund, the U. S. Department of Energy, and the U. S. Department of Education Graduate Assistance in Areas of National Needs (GAANN) Program.
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Table 3 Data for EOS test problems: Molar volume and mean ionic activity coefficients versus solute molality m. Molar Volume (cm3mol-1) case
m = 0.1
A B C D E
m = 0.2
m = 0.3
m = 0.4
m = 0.5
18.241 18.318 18.394 18.163 18.161 18.160 18.299 18.433 18.567 18.273 18.382 18.489 18.352 18.539 18.724 Mean Ionic Activity Coefficients 0.7980 0.7520 0.7370
18.469 18.158 18.699 18.596 18.909
18.544 18.157 18.830 18.703 19.092
0.7280
0.7280
Table 4 Parameter estimation results for test problems using data in Table 3. case
a (Pa·m6mol-1)
b (cm3mol-1)
A B C D E
0.435 0.763 4.757 3.080 3.689
15.134 16.194 17.634 18.187 18.008
V (Å) 1244.8 7126.2 52.1 234.3 114.1
References [1] J. F. Brennecke and E. J. Maginn, AIChE J., 47(2001), 2384. [2] S. V. N. K. Aki, B. R. Mellein, E. M. Sauer and J. F. Brennecke, J. Phys. Chem. B, 108(2004), 20355. [3] J. K. Shah and E. J. Maginn, J. Phys. Chem. B, 109(2005), 10395. [4] T. I. Morrow and E. J. Maginn, personal communication, 2005. [5] J. A. Myers, S. I. Sandler and R. H. Wood, Ind. Eng. Chem. Res., 41(2002), 3282. [6] D. M. Eike, J. F. Brennecke and E. J. Maginn, Green Chem., 5(2003), 323. [7] O. Spuhl and W. Arlt, . Eng. Chem. Res., 43(2004), 852. [8] C.-Y. Gau and M. A. Stadtherr, Fluid Phase Equilib., 168(2000), 1. [9] S. Ulas, U. M. Diwekar and M. A. Stadtherr, Comput. Chem. Eng., 29(2005), 1805. [10] J. Gmehling, U. Onken and W. Arlt, Vapor-liquid Equilibrium Data Collection, DECHEMA, Frankfurt/Main, Germany, 1977-1990. [11] E. Hansen and G. W. Walster, Global Optimization Using Interval Analysis, Marcel Dekkar, New York, NY, 2004. [12] G. Xu, W. D. Haynes and M. A. Stadtherr, Fluid Phase Equilib., 235(2005), 152. [13] J. M. Sørensen, W. Arlt, Liquid-Liquid Equilibrium Data Collection, DECHEMA, Frankfurt/Main, Germany, 1979-1980. [14] S. R. Tessier, J. F. Brennecke and M. A. Stadtherr, Chem. Eng. Sci. 55(2000), 1785 [15] J. L. Anthony, E. J. Maginn and J. F. Brennecke, J. Phys. Chem. B, 105 (2001), 10942. [16] V. Najdanovic-Visak, J. M. S. S. Esperanca, L. P. N. Rebelo, M. N. da Ponte, H. J. R. Guedes, K. R. Seddon, H. C. det Sousa and J. Szydlowski, Phys. Chem. B, 107 (2003), 12797. [17] C.-C Chen, H. I. Britt, J. F. Boston and L. B. Evans, AICHE J., 28 (1982), 588. [18] A. Ayala and M. A. Stadtherr, AIChE Annual Meeting, Paper 167aa, 2004. [19] L. S. Belvèze, J. F. Brennecke and M. A. Stadtherr, Ind. Eng. Chem. Res., 43(2004), 815.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Study of Non-linear dynamics in Reactive Distillation for TAME synthesis using Equilibrium and Non-equilibrium models Amit M. Katariya,a Ravindra S. Kamath,a Sanjay M. Mahajani,a Kannan M. Moudgalyaa a
Department of Chemical Engng., IIT Bombay, Powai, Mumbai-400076, India
Abstract The work emphasizes on non-linear dynamic effects in dynamic simulation with both Equilibrium (EQ) and non-equilibrium models. Both rigorous EQ and NEQ models result in high-index DAE systems. The variables responsible for the higher index in each case are identified and accordingly, model simplifications are made without compromising on essential dynamic terms. For certain conditions, dynamic simulation with EQ model leads to limit cycles. However, dynamic simulation with more realistic and rigorous NEQ model with identical operating conditions as the EQ model does not reveal any oscillatory behavior. Keywords: Reactive distillation, Non-equilibrium model, TAME synthesis, dynamic simulation, non-linear dynamics.
1. Introduction TAME (tert-amyl methyl ether) produced by etherification of iso-amylenes with methanol is a top contender for the replacement of MTBE as a gasoline additive. The reaction characteristics and process conditions make TAME synthesis a potential candidate for Reactive Distillation (RD). Reactive Distillation is state-of-the-art multi-functional reactor concept that integrates reaction and separation in a single process unit. In several commercial processes, the implementation of RD technology has resulted in significant capital and/or energy savings. Two different types of models are available in the literature for reactive distillation: the equilibrium stage (EQ) model and the non-equilibrium (NEQ) model (also called as rate-based model). The EQ model assumes that the vapor and liquid leaving a stage are in equilibrium. The NEQ model, on the other hand, assumes that the vapor-liquid equilibrium is established only at the interface between the bulk liquid and vapor phases and employs a transport-based approach to predict the flux of mass and energy across the interface. The EQ model is mathematically much simpler and computationally less intensive. On the other hand, NEQ is more consistent with real world operations. A few case studies of TAME synthesis using RD has been appeared in the literature (Subbawala and Fair, 1999; Mohl et. al., 1999; Baur and Krishna, 2002; Baur et. al., 2003; Peng et. al., 2003). Most of them except Peng et. al. (2003) are restricted to steady state analysis. In this work, we present a comparative study of study state and dynamic simulation using both EQ and NEQ models and the non-linear dynamic effects associated with them.
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470 Column pressure = 4.5 bar Isopentane + MeOH Reflux ratio = 1.5
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Catalyst loading = 29518 eq[H+] Pre−reacted feed 1196.1 kmol/h 325K Stage location = 29 Methanol 0.13042 2M1B 0.00798 2M2B 0.07018 TAME 0.13126 isopentane 0.66016
Stripping section 10−theoretical stages
Packing height = 5.58m Number of segments = 98
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Fig. 1. The conceptual column configuration used for EQ simulation along with the hardware design derived from it.
2. Simulation results for EQ model 2.1. Model description and column configuration The EQ model is represented using the well known MESH equations. There are three reversible reactions in this system i.e. two reactions for TAME synthesis from the two isomers of isoamylene and isomerization of isoamylene. The reaction is assumed to be pseudo-homogenous. The kinetic model of Faisel et al. (2000) is used to calculate the reaction rates. The UNIQUAC model is used to calculate the liquid-phase activity coefficients with interaction parameters obtained from HYSYS process simulator. The model results in a system of differential-algebraic equations (DAE) and is solved using the DIVA simulation environment (Mangold et. al., 2000). The conceptual column configuration for the study is shown in Figure 1. The steady state simulation results have been validated for the design and operating parameters of Subbawala and Fair (1999) and Baur and Krishna (2002). The existence of oscillations was also confirmed through the bifurcation diagram showing the presence of Hopf bifurcation (Katariya et. al., 2006). 2.2. Dynamic simulation Apart from steady state simulations, dynamics simulations also need to be investigated for this configuration as the feed composition of the C5 cut (source of isoamylene) is prone to variation depending on the upstream conditions. It is to be noted that a rigorous dynamic model using the EQ concept results in a higher-index DAE system unless correlations for pressure drop and liquid holdup as function of liquid and vapor flow rates are incorporated as algebraic equations in the model. The model has been simplified by neglecting the dynamic changes in the total molar and energy holdup. Using this simplified dynamic EQ model, the column dynamics has been studied for disturbance (in form of step changes) in the isoamylene concentration in the feed. The
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analysis was performed for two different values of Damkohler number (Da = 1.0 and Da = 2.5) and the corresponding behaviors for open loop column simulation are shown in Figure 2a and 2b respectively. For Da = 1.0, the response is nearly first order and the system reaches the corresponding steady state. However, for Da = 2.5, a state of sustained oscillations is observed. The amplitude of the oscillations increases with an increase in the size of the step change. To the best of our knowledge, sustained oscillations have not been reported for TAME synthesis.
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3. Simulation results for NEQ model 3.1. Model Description and Hardware Specification To examine the effect of column hardware and heat and mass transfer resistances on the observed non-linear behavior in the EQ model, a rigorous NEQ model has been developed. We refer to Powers et al. (1988) for detailed model implementation and computational aspects. In case of the NEQ model, the specification of hardware design information (column diameter, tray or packing type and geometry etc.) is mandatory. A packed column has been selected for the NEQ simulations. Each continuous section of the packed column is vertically divided into a number of segments. The packing selected for the reactive and non-reactive (rectifying and stripping) sections are KATAPAK-S and Sulzer-BX respectively. The preliminary column design (diameter and section heights) for the NEQ model is derived from the steady state results of the EQ model and is shown in Figure 1. 3.2. Influence of number of segments The effect of the number of segments in the packed sections of the RD column on the steady state results using the NEQ model is shown in Figure 3. When the number of segments in a particular section is chosen to be same as the number of corresponding theoretical stages in the EQ model, a significant difference in the composition profile predicted by the two models is seen but only in the stripping section. When the number of segments is very large, there is essentially no back-mixing in the column. Figure 3 shows that back-mixing in packed columns strongly affects the composition profile. In real columns, back-mixing and other non-ideal conditions cannot be eliminated and hence an appropriate number of segments should be used. However, this number cannot
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be determined apriori. For steady state simulations, the number of segments was finally restricted to 193 at which the composition was not changing significantly.
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Fig. 3. Comparison of the steady state Fig. 4. Effect of height of the stripping section composition of isoamylene and methanol for the on the isoamylene conversion. EQ model and NEQ model with various number of total segments.
3.3. Influence of height of the stripping section Apart from a different bottom composition, the steady state result using the NEQ model showed a much lower conversion (67.56%) as compared to 84.57% of the EQ model. Since the primary objective was to compare the non-linear dynamic effects of EQ and NEQ models, getting similar conversions and end composition specifications is essential. It was observed that the height of the stripping section influences the column performance significantly. Hence, this height was varied in order to obtain a performance similar to that of EQ model. An optimum stripping height (refer to Figure 4) was realized. A height of 1.4m was selected for the new design since it not only gives a conversion close to optimum but also the conversion and end composition are very similar to that given by the EQ model. The steady state composition and temperature profiles of the NEQ model using this new design is plotted along with that of the EQ model in Figure 5. Even though the top and bottom compositions and temperature profiles are similar, certain sections of the stripping zone show different compositions. 0
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For the NEQ model, the liquid and vapor flow rates in the packed sections are not responsible for the higher index as algebraic equations for these variables in form of pressure drop and holdup correlations are incorporated in the model. However, it is the liquid and vapor flows associated with the condenser and the reboiler that can pose the high-index problems if such equations (e.g. controller equations) are not explicitly considered in the model. Assuming that these equations are not available, to eliminate the index problem, some of the differential equations need to be converted to algebraic equations by neglecting the dynamics. It can be proved through a detailed index analysis that at least the following differential equations need to be converted to algebraic equations:1) Energy balance for the condenser (because condenser load does not appear in any other algebraic equation). 2) Energy balance for the reboiler (to account for reboiler duty, bottom flow rate or vapor flow from the reboiler depending upon the bottom specification). 3) Total material balance for the reboiler and condenser (to account for either vapor or liquid flow) Apart from this rigorous model, we refer to ‘constant holdups’ dynamic NEQ model as the one in which differential equations for all the total material and energy balance are converted to algebraic equations, as was done in the case of the EQ model. To confirm the authenticity of the sustained oscillations observed in the dynamic EQ model, the same analysis was repeated with the rigorous dynamic NEQ model using the new hardware design. As seen in Figure 6, no oscillations have been seen and the system always reaches the corresponding steady state. Thus, the oscillatory behavior that existed in the EQ model disappears in the NEQ model for the desired conversion and purity. However, it must be noted that the parameter space wherein the non-linear dynamic effects are observed in EQ model, is likely to shift in the case of NEQ model simulations. Such a possibility can be ascertained only through continuation (coupled with stability analysis) with respect to all possible parameters and their combinations.
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3.5. Comparison of dynamics of EQ and NEQ models Starting from a steady state with same operating conditions, the dynamics of simplified EQ model, rigorous NEQ model and the ‘constant holdups’ NEQ model for a 2% step increase in the pre-reacted feed flow rate have been studied. As seen from Figure 7, both the NEQ models show a slightly different dynamics but reach the same steady state while the EQ model reaches a different steady state as expected. Both the NEQ models show almost similar computation times since the total number of equations (differential and algebraic) is the same. However, the rigorous NEQ model was much more difficult to converge for larger step changes because of stiffness issues. The convergence properties of the ‘constant holdups’ NEQ model were very similar to that of the EQ model with almost no problems up to ± 5% step changes in operating parameters. The computation time for the NEQ model was observed to be almost 15 times that of the EQ model
4. Conclusion The results of steady state and dynamic simulations using both EQ and NEQ models are compared for the synthesis of TAME. The criteria to avoid high index problems in dynamic simulation are identified. Synthesis of TAME in RD may be associated with non-linear dynamic effects like limit cycles, which are confirmed by dynamic simulation using an EQ model. However, NEQ model simulations in the same parameter space do not reveal such phenomena. The detailed studies using bifurcation diagrams coupled with stability analysis are necessary to identify their presence and investigate their causes.
References H. Subawalla, J. R. Fair, 1999, Ind. Eng. Chem. Res., 38, 3696-3709. K. D. Mohl, A. Kienle, E.D. Gilles, P. Rapmound, K. Sundmacher, U. Hoffmann, 1999, Chem. Eng. Sci., 54, 1029-1043. R. Baur, R. Krishna, 2002, Chem. Eng. Processing. 41, 445-462. R. Baur, R. Krishna, R. Taylor, 2003, Chem. Eng. Processing. 42, 211-221. J. Peng, T.F. Edgar, R.B. Eldridge, 2003, Chem. Eng. Sci., 58, 2671-2680. H. Faisal, C. E. Syed, R. Datta, 2000, J. of chem. Eng. Data, 45, 319-323. M.A. Mangold, E.D. Kienle, K. D. Gilles, 2000, Chem. Eng. Sci., 55, 441-454. A. M. Katariya, S. Mahajani, K. Moudgalya, 2005, Submitted to Ind. Eng. Chem. Res. M.F. Powers, D. J. Vickery, A. Arehole, R. Taylor, 1988, Comput. and Chem. Eng., 12, 1229-1241.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
An agent-oriented architecture for modeling and optimization of naphtha pyrolysis process Xiaodan Gao, Bingzhen Chen, Xiaorong He Department of Chemical Engineering Tsinghua University,Beijing,100084,China
Abstract An agent–oriented architecture for modeling and multi-objective optimization of naphtha pyrolysis process is proposed in this paper. The system contains five main agents: interface, parameter data, reaction kinetic analysis, simulation and optimization. The proposed utility-risk negotiation method of the agents enhances the effectiveness of implementing the multi-objective optimization. A case study on the multi-objective optimization of an industrial naphtha pyrolysis furnace has demonstrated the effectiveness of the proposed agent-oriented architecture and the negotiation method. Keywords: naphtha pyrolysis, multi-agent, multi-objective, negotiation 1. Introduction Naphtha pyrolysis is one of the most important processes in chemical industry. A simulation system of naphtha pyrolysis furnace has been developed[1], and the optimization of periodic operation for the naphtha pyrolysis furnace has been studied by using heuristic optimization approaches[2]. The process of optimization frequently needs to run the simulation module for taking back useful information. As the existing object-oriented simulation and optimization system is not efficient for the interaction between the simulation system and the optimization one in the real-time optimization, so a multi-agent based architecture has been proposed for the following three reasons. First, in the real-time optimization, the simulation module accepts the up-to-date operation condition of the reaction furnace and modifies the standard database embedded in the simulation module. All of the changes occurred in the simulation package needs to pass to the optimization agent as soon as possible . Second, the modification of the standard database will not explicitly influence the simulation results in some case. Therefore, it needn’t being sent to the optimization module for saving time. So the two tasks could be asynchronous and parallel and operated independently. Last, the existing optimization method didn’t consider about the multi-objective optimization problem. In the naphtha pyrolysis process, there are many kinds of product. One manager may hope ethylene’s yield be more important but another one may think more of propylene’s yield. And one of the most efficient methods to settle the problem is negotiation, which is very popular in multi-agent system.. As the single-objective optimization problem has already consumed a lot of computing time for the large-scale problem, the multi-objective optimization should be calculated on a set of distributed computers. In one word, for integrating the techniques, such as multi-object negotiation, distributed calculation and communication, the agent-based architecture would be very useful. In this paper, an agent-oriented architecture for modeling and optimization of naphtha pyrolysis is developed, and an utility-risk negotiation method is proposed. In a case study, a multi-objective problem is
475
476 resolved to demonstrate the effectiveness of the agent oriented optimization system.
2. MUTI-AGENT ARCHITECHTURE CAS(Complex Adaptive System) theory has provided a new concept, the adaptive agent, which views the behavior of the complex system as a result of the coordinative operation between the adaptive agent and the environment. The agent is a self-contained, concurrently executing software process, which is able to communicate with other agents via message passing and can learn to be adapted to the environment through artificial intelligence method， especially reasoning strategy. Some agent architectures have been proved useful in the development of chemical process design for its predominant capacity. Eo,Chang, Shin, and Yoon[3] proposed an agent-based framework for the diagnosis of chemical processes. Agents take care of a set of process units and communicate observations with each other. They use a knowledge base pertaining to the units to take decisions. Nirupam Julka, Rajagopalan Srinivasan and I.Karimi[4] proposed a decision support system based on an agent-based supply chain management framework. John D.Siirola, Steinar Hauan and Arthur W.Westerberg[5] studied the multi-objective optimization using distributed agents and proved that the agents can interact through the direct and indirect sharing of information. In this work, individual agents interact asynchronously through passing message among threads. The structure is consisted of several agents: interface, parameter data, reaction kinetic analysis, simulation and optimization. Each of them has an explicit aim and can send or accept information conveniently and some of them have the adaptive ability of searching better strategy of solving problem, which will be described below. 2.1 Interface agent The interface agent is responsible to accept the modifications from the environment including operators or the DCS. The agent always keeps active to the outside world. If any change occurs, for instance, the kind of the naphtha changes, the flow rate of the feed changes, the agent can memorize the data and send them to other agents that need to be updated. The information about naphtha characteristics and operation condition should be sent to parameter data agents, and sets of analysis data of products will be sent to the reaction kinetic analysis agent. The interface agent also show the simulation or optimization results to the operators, and it could also provide the profile of product yields, pressure and temperature along the reactor tube or time. 2.2 Parameter data agent The parameter data agent accepts the values of naphtha characteristics, operation condition and furnace structure continuously from the interface agent to update the parameters and provide them to the reactor model agent. 2.3 Reaction kinetic analysis agent The kernel of the reactor model is the reaction description. And as known, the naphtha reaction is very complicated. In this paper, Kumar’s molecule reaction model[6] has been used to predict the main yields of the product. But in this model the selectivities of the first-order reaction always changes along with the change of the composition of naphtha. So, before using this model, it is necessary to get the selectivities of the first-order reaction of
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477
naphtha pyrolysis through estimation or experiments. The reaction kinetic analysis agent calculates the selectivites of the first-order reaction based on the sets of analysis data of products taken from interface agent, and put them into its standard database. Then, while the reactor mathematical model agent runs, the reaction kinetic analysis agent accepts the feed characteristics information, then figures out the selectivities of the first-order reaction by using a fuzzy matching method and affords them to the reactor model agent. 2.4 Reactor model agent After receiving messages from both the reaction kinetic analysis agent and the parameter data one, the reactor model agent is activated to run the simulation program and provides the product yields, temperature and pressure profiles along the reactor tubes. The reactor model agent is usually used for two tasks. One is used by the reaction kinetic analysis agent to estimate the selectivities of the first-order reaction. The other one is served as a subsystem of the optimization agent. As independent agents, reactor model agent and optimization
agent will operate to do their own task, at the same time will exchange information effectively with each other. The latter agent accepts the calculated results from the former one and puts them into the optimization program to optimize the objective function. 2.5 Optimization agent The optimization agent is built on a framework for solving multi-objective problems with non-convex objective functions. Each objective is a sub-agent itself and is supported by an advanced simulated annealing algorithm for conducting optimization[2]. The utility-risk negotiation mechanism of agents makes them easily communicate and obtains the final optimization result through the process of competition and compromise. The utility-risk negotiation algorithm can be described as follows: The sub-agents are: Agi , i = 1, 2… , n , where n is the number of objectives (1) The sub-agents optimize themselves independently, and propose the optimal solutions as their initialized suggestions: δ1 , δ 2 …δ n (2) The sub-agents calculate their own utility function with the suggestions: utility Agi ( δ1 ), utility Agi ( δ 2 ),…utility Agi ( δ n ), i = 1, 2… , n
if for one Agi , utility Agi ( δ i ) ≥ utility Agi ( δ j ), ∀j ∈1, 2,… n, j ≠ i
Agi is satisfied and quit the negotiation, if not, turn to step 4. Here the utility function can be defined differently for different problems. (3) The sub-agents calculate their conflict risk so that to decide which sub-agent should yield to others. The conflict risk demonstrates the difference between the solutions suggested by the agent itself and the ones by others. The less the risk value, the agent will be more satisfied to accept other suggestion and yield. The conflict risk can be defined as:
riskAgi =
utilityAgi (δ i ) − ∑ W j utilityAgi (δ j ) j
utilityAgi (δ i )
n
,
∑W j =1
j
= 1, j ≠ i
Where W j demonstrates the importance degree of Ag j that Agi considers.
X. Gao et al.
478 (4) The yielding agent will find another solution worse than the one suggested before. The step that it yields should be large enough to make others yield in the next period of negotiation, and small enough to protect its own utility. Turns to step 3. The agent-based architecture is shown in figure 1
figure1. The agent-based architecture for simulation and optimization of naphtha pyrolysis 3. A CASE STUDY In this case study on multi-objective optimization of naphtha pyrolysis, an agent-based system has been established using multi-thread method to implement parallel calculation of multi-task on one computer. And the interaction among the agents can be implemented by passing messages among threads. The characteristics of the naphtha and the operation conditions are listed in table1, and the furnace is GK-IV furnace. Table1.Feed characteristics and operation conditions IBP(℃) 10%(℃) 31
49
Density(g/) 0.6931
50%(℃)
90%(℃)
89
138
Feed rate(t/h) Water/naphtha 14
FBP(℃)
NP(%)
166
34.8
Outlet
Inlet
pressure(MpaG) temperature(℃)
0.6
0.05
633.75
IP(%)
N(%)
33.8
24.5
O A（%） （%） 0
Highest Lowest COT
COT
824
838
The developed agent-based system is very efficient to modify its parameters along with the changes by operators, and the reaction kinetic agent and reactor mathematical agent can accept this modification rapidly. The multi-objective negotiation method has been used. Two important objectives are studied here.
− Objective A(ObjA): Z ={ z , zmin z j zMK } 1 2
MK
∑Y j =1
e
j
6.9
Modeling and Optimization of Naphtha Pyrolysis Process Objective B(ObjB):
min
Z ={ z1 , z2
zj
479
MK
zMK }
− ∑ (Ye j + Ypj ) j =1
Where MK is the number of pseudo states, z is a vector of operation variables (COT), Ye j is the ethylene yield of the jth state, Ypj is the propylene yield of the jth state. The utility function is defined as follows[7]:
utilityA =
ObjA(δ ) − min A max A − min A
utilityB =
ObjB(δ ) − min B max B − min B
Where δ is the solution proposed by agent A or agent B. MaxA and MinA is the maximum and the minimum value of objective A in the feasible region of agent A, and so as the agent B. First, there is only one pseudo state is studied for the simplification of the optimization problem. Therefore, here only one operation variable is optimized, and the two objectives have the same degree of importance. So the risk function is the same.
riskA =
utilityA(δ A ) − utilityA(δ B ) utilityA(δ A )
riskB =
utilityB(δ B ) − utilityB(δ A ) utilityB(δ B )
Table 2 shows the process of the negotiation. Table2. The negotiation of sub-Agent A and subagentAgent B with the same importance degree Negotiation UtilityA(A) UtilityA(B) UtilityB(A) UtilityB(B) Times
Risk(A)
Risk(B)
Who yields next
1
2.63E-06
5.55E-06
1.000001
0.999997
0.999994
B
2
1
7.82E-02
5.55E-06
0.953497
0.921764
0.999994
A
3
0.935753
7.82E-02
9.22E-02
0.953497
0.916392
0.903342
B
4
0.935753
0.154994
9.22E-02
0.902951
0.834364
0.897931
A
5
0.870727
0.154994
0.181191
0.902951
0.821995
0.799334
B
6
0.870727
0.230563
0.181191
0.848173
0.735207
0.786375
A
7
0.804919
0.230563
0.26699
0.848173
0.713558
0.685217
B
8
0.804919
0.304972
0.26699
0.789275
0.621115
0.661727
A
9
0.738325
0.304972
0.349464
0.789275
0.586941
0.557234
B
10
0.738325
0.37825
0.349464
0.726368
0.487692
0.518889
A
11
0.670938
0.37825
0.428521
0.726368
0.436237
0.41005
B
12
0.670938
0.533761
0.428521
0.576033
0.204456
0.256083
A
13
0.602752
0.533761
0.504072
0.576033
0.114461
0.124926
A
14
0.533761
0.533761
0.576033
0.576033
0
0
A
utility Ag B ( δ A ) ≥ utility Ag B ( δ B )
838
utilityA(δ A ) − utilityA(δ B ) utilityA(δ A )
The result is shown in table 3.
riskB = 0.4
utilityB(δ B ) − utilityB(δ A ) utilityB(δ B )
824
Keep! 825 837 Keep! Keep! 826 836 Keep! Keep! 827 835 Keep! Keep! 828 834 Keep! Keep! 829 833 Keep! Keep! 831 832 Keep! 831 Keep! 829.5 829.5
Second, the importance degree of the objectives is changed, A is more important, so the risk function is changed:
riskA = 0.6
δB
(COT) (COT)
1
Succeed!
δA
X. Gao et al.
480 Table3. The negotiation of Agent A and Agent B with different importance degree Negotiation UtilityA(A) UtilityA(B) UtilityB(A) UtilityB(B) Times
Risk(A)
Risk(B)
Who yields
δA
δB
(COT)
(COT)
1
1
2.63E-06
5.55E-06
1.000001
0.599998
0.399998
B
838
824
2
1
7.82E-02
5.55E-06
0.953497
0.553058
0.399998
B
Keep!
825
3
1
0.154994
5.55E-06
0.902951
0.507004
0.399998
B
Keep!
826
4
1
0.230563
5.55E-06
0.848173
0.461662
0.399997
B
Keep!
827
5
1
0.304972
5.55E-06
0.789275
0.417017
0.399997
B
Keep!
828
6
1
0.37825
5.55E-06
0.726368
0.37305
0.399997
A
Keep!
829
7
0.935753
0.37825
9.22E-02
0.726368
0.357468
0.349247
B
837
Keep!
8
0.935753
0.450427
9.22E-02
0.659562
0.311189
0.344107
A
Keep!
830
9
0.870727
0.450427
0.181191
0.659562
0.28962
0.290114
A
836
Keep!
10
0.804919
0.450427
0.26699
0.659562
0.264244
0.23808
B
835
Keep!
11
0.804919
0.533761
0.26699
0.576033
0.202126
0.214601
A
Keep!
831
12
0.738325
0.533761
0.349464
0.576033
0.166239
0.157331
B
834
Keep!
13
0.738325
0.602752
0.349464
0.504072
0.110173
0.122687
A
Keep!
832
14
0.670938
0.602752
0.428521
0.504072
6.10E-02
6.00E-02
B
833
Keep!
15
833 0.670938 0.670938 0.428521 0.428521 0 0 Succeed! It can be observed that when the preference changes, the optimized results also move to satisfy the agent A more obviously.
4. CONDLUSION The proposed multi-agent architecture for the simulation and muti-objective optimization of naphtha pyrolysis can be efficient enough to satisfy the real-time optimization. The reactive kinetic agent can improve the accuracy of the simulation results based on updating the selectivites of the first-order reaction timely. The proposed utility-risk negotiation method of multi-objective optimization can give a result that satisfactorily reflects the preference of the decision-maker. A case study of the optimization of an industrial naphtha pyrolysis furnace has been implemented, which demonstrates the effectiveness of the proposed agent-oriented optimization system. REFERENCES 1. Gao Xiaodan, Chen Bingzhen, He Xiaorong,2005, Estimation of the selectivities of the first-order reaction in Naphtha Pyrolysis process based on Fuzzy matching method, 2005 AIChE Spring National Meeting, Conference Proceedings, 47c, p 1007-1011
2. Gao Xiaodan, Chen Bingzhen, He Xiaorong, 2005, Study on the Optimization of Periodic Operation for the Naphtha Pyrolysis Furnace, PSE ASIA 2005 Conference Proceedings, P027, 240-244 3. Eo, S. Y., Chang, T. S., Shin, D. & Yoon, E. S Cooperative, 2000, Problem solving in diagnostic agents for chemical processes, Computers and Chemical Engineering, 24 (2-7), 729. 4. Nirupam Julka, Rajagopalan Srinivasan, I.Karimi, 2002, Agent-based supply chain management-1: framework, Computers and Chemical Engineering, 26, 1755-1769.
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Modeling and Optimization of Naphtha Pyrolysis Process
481
5.
Pramod Kumar, Deepak Kunzru, Modeling of Naphtha Pyrolysis, 1985, Ind Eng. Chem
6.
John D.Siirola, Steinar Hauan, Arthur W. Westerberg, 2003, Computing Pareto Fronts Using Distributed Agents, Process Systems Engineering 2003 Proceedings, 334-339. Michael Wooldridge, An introduction to multiagent systems, Chichester, England : J. Wiley, c2002.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
483
On Model Portability Heinz A Preisig a
a∗ †
, Tore Haug-Warberga and Bjørn Tore Løvfalla
Department of Chemical Engineering, Trondheim Norway
With the software of various commercial providers becoming mature, portability of models and associated data comes higher and higher on the agenda. CAPE-Open has accomplished a compromise getting various players on the market to communicate and agree on generating and accepting wrappers for their process models and also thermo data, both key issues in the chemical engineering’s software world. Improving a step beyond wrappers requires a more basic approach, which allows taking advantage of the model structure. The design method and the used representation of the Modeller project have already proven a great degree of portability as models can be mapped into all major solver environments including Matlab, gProms, and other DAE solvers. We attempt to line out the motivation, the present and the future of this approach. Keywords: Computer-aided, modelling, process systems engineering 1. Synopsis Modelling is a core activity of any science or engineering-based profession. It is the basic means for describing the behaviour of processes, systems, activities or what ever term is being used for parts of natural processes and human-invented abstract artefacts. Having a description provides insight into the behaviour and enables various design activities that make it possible for humans to shape their world to their desire: We design and run technical systems and modify, cultivate, utilize natural systems. Thus modelling, defined on this level, is an omnipresent activity. 2. Issue Portability With modelling activities being everywhere and being so central, people’s creativity has been instrumental in generating a wide and rich variety of modelling tools and instruments. However, while disciplines do grow rather independently, there is the glue of a common science base and the incentive of a common global objective that inspires people to seek cross-overs. Portability is all about cross-over. There are a number of issues of which one should be aware of when talking about portability, some of which we shall briefly mention: Problem solvers: Solvers such as gProms, Diva, Ascent, Modelica, Matlab toolboxes etc. are more or less tailored to solve a specific class of problems. The specialisation is usually driven by increasing the efficiency of the solver but has usually the effect of narrowing the class of problems that can be solved. Problem and solvers must thus match and consequently models are written for the solver, which results in a poor portability of the model. Computing environments: Since a lot of work is driven by the availability of technology, it is no surprise that it is often the technology that dictates the solution method and its implementation in contrast to what it should be, namely that the methodology dictates the choice of technology. Data bases: Today, with the fairly world-wide accessibility of machines on the network, the concept of data servers has become a common way of providing access to data. This centralised solution has the obvious advantage of maintaining control over the data and its models in terms of security and ∗ [email protected] † In
parts supported by the Norwegian Research Council through the Gas Research Institute
484
H.A. Preisig et al.
maintenance, but has also the obvious disadvantage of having to handle large and ever increasing number of users and access requests. 3. The CAPE Community’s Reaction CAPE-Open went the only possible path in that it tried to find a compromise matching the players in contrast to proposing a complete restructuring of the tools. As a result CAPE-Open was not a revolution, but did achieve to overcome some of the commercial protectionism. There are probably two major achievements to be recognised: 1) a generic wrapper for process models, 2) a generic wrapper for process-relevant data, particularly thermo data. These wrappers have not only been established, but what is most important, they have also been accepted by the provider industry and are used increasingly. As mentioned, such a result could only be achieved because a compromise could be found. It had however also the effect that the involved products have not really changed. Thus the limitations (memory limitations in the 60ties and 70ties on execution and data, in the 80ties mostly on execution and speed and in the 90ties speed and complexity) that were introduced over the life-time of the product are prevailing. 4. One Step Back - Two Forwards The only way to improve things further is to exploit the structure of the models and the associated data. There have been a number of efforts in this field. In Germany there are two efforts to report, namely at RWTH Aachen [12,18,7] and the the Gilles group at the Max-Planck-Institut f¨ ur Dynamik komplexer technischer Systeme in Magdeburg, Germany [6,17]. In the US it is the Carnegie Mellong Design Centre that had the Ascent effort [5,14], Stephanopulous at MIT [15,16] and Linninger [11]. Another effort worth mentioning is a collaboration of Hungary and Australia [8]. The effort of this group, called the Modeller project is currently in the fourth generation. The first three versions were by T Y Lee [10], A Mehrabani [13]and M R Westerweele [19]. They constructed a tool, which aids in synthesizing and maintaining models. The tool uses a canonical description built on the foundation of science, which is looked at through the glasses of system theory casting the modelling framework into an even higher level of abstraction. The results are reported in a sequence of papers and the mentioned PhD theses. Key observations are: • The conserved extensive quantities form a minimal state-space (minimal as defined in the framework of system theory. • The dynamics are described by the set of conservation laws of all primitive systems. • The plant’s structure can be mapped into a network of primitive systems connected by the exchange of extensive quantities. The given directions in this graph are the reference co-ordinates for the respective flows. • What is described by the extensive quantities can be transposed (reactions, phase changes etc.). • Couplings between primitive systems are given by the continuity of the flow of extensive quantities and the continuity of the force field driving the respective flow. • Any quantity introduced as part of the description of the flows and transposition must consecutively be the result of a series of mappings from the minimal state space. Constructing models on this formulation yields models in the basic form, at least in terms of macroscopic field theory. This enables the mapping into any form as for example implemented in the various commercial solvers as Westerweele demonstrated in his realization of the Modeller tool generating code for Matlab solvers, two local solvers and Modelica, gProms to be written at this point in time. The product is well on its way to commercialisation. Having realised this step forward, which certainly goes beyond the framework as laid out by CAPEOpen, we have come a step forward towards centralising synthesis and maintenance of process models, the next hurdle is then to be taken, namely the integration of (process) data. 5. The Model The model consists of a set of equations and definitions. It may be split into three section, the first two, the Core and the Properties, are describing the behaviour of the plant, the Controller capture any information processing units that extract information about the state of the plant and possibly steer the plant by manipulating the flows of extensive quantities exchanged between primitive parts of the system.
On Model Portability
485 Core:
Notation: x :: vector of conserved quantities and their transitional forms y :: vector of secondary states p :: vector of properties Θ :: vector of parameters
Properties: pf := i pr := i pt := i
pf pr pt
i
i
i
y, pf , Θfi , i+1 ∀i y, pr , Θri , i+1 ∀i y, pt , Θti , i+1
∀i
x˙
:=
x ˆ
:=
x ˜
:=
y
:=
Fx ˆ + Rx ˜, x ˆ y, pf , pu , 0 x ˜ y, pr , 0 y x, y, pt , 0
(1) (2) (3) (4)
Controller: (5) (6)
x˙ c pu
:= c(xc , y, ys , Θc ) ,
:= pu (xc , ys , y, Θc ) .
(8) (9)
(7)
The core is established by defining first the physical topology of the process, which is a network of primitive capacities and flows of extensive quantities between these capacities. This information can be represented as a directed graph, with the direction being the reference coordinates giving meaning to the direction of the flows. This graph represents the whole of the system. The level of details is often referred to as the granularity of the model. Since this graph can become very large, one needs to arrange it in a hierarchical representation, a subject which is discussed at other places. The nodes in the graph represent the capacities, in our case control volumes that can accumulate conserved quantities, latter forming the fundamental state space x. Given the capacities and given the type of flows, the dynamics of the process can be described in the form of the conservation principles (Equation 1). The topology maps into the matrix F and the transposition (reaction, phase changes etc) mapping into the matrix R. These ˜, which in turn must be equations define two new variables, namely the flows x ˆ and the transposition x specified in Definition (2) and (3). The transfer laws and the kinetics are selected from a data base, which has been built consistently, a subject we will have to discuss elsewhere. These two definitions introduce secondary state variables y, which are a rather colourful agglomeration of intensive quantities such as driving forces (temperature pressure and chemical potential), but also others such as concentrations and densities etc, and extensive quantities such as geometrical quantities volume, areas etc. The second section is part of the model, but has a somewhat different flavour, for which reason it has been separated here. The core model defines a set of sets of ”properties”, often also called parameters, which characterise the transfer, kinetics or transformation. For the purpose of clarity and for reasons that will become more explicit later on, we chose to not use the term ”parameter” here, but rather reserve it strictly for variables that are typically instantiated as having been fitted to models using experimental data, thus are the result of a identification. These properties are attached to the respective transfer, transposition, transformation or themselves, latter making the definition recursive as shown in Definition (5) - (7). The third section captures the control units, representing possibly dynamic information processing units, which, by the way, also would connect to the discrete-event dynamic control components adding the feature of describing overflows and other event-driven physical mechanisms but also connect to the high-level control units such as warning and alarm units, sequential control installations and not at least to the planning and scheduling layers on the top. Instantiation: These three packages of equations are of purely algebraic. No numerical information is included. In order to formulate a feasible mathematical problem this set must be augmented with another set of definitions, which assigns numerical values or series of values to some of the above/defined variables including the parameters. For instance in order to define a simulation problem, all initial conditions and parameters must be defined. This instantiation operation can be completely split off from the model editing tool and can be accommodated in a separate software component. It should be noted that this instantiation also includes the thermodynamic data.
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Additional information: In addition it is meaningful to augment the model with additional information that one may have. This could be derivatives, which help in solving sets of nonlinear algebraic equations, or optimisation etc. It could also be information about a set of save starting points, set of free variables, etc. It may also be convenient to carry along information about the hierarchical structuring of the plant model and information such as names and tags, etc. Latter can be handy in constructing graphical user interfaces for programs using the model or results derived from it. Besides the very last-mentioned components, which can readily be captured in structured arrays, lists, dictionaries or the like, dependent on the implementation language, the discussed representation contains thus mainly algebraic information consisting of equations, definitions, functions and variables. Their representation shall be discussed next. 6. Language Representation Besides the stiffness problem associated with the dynamic equations, the variable transformations and the properties are known to absorb a very significant part of the computing time. The thermodynamic equations in particular are known to be implicit and notoriously difficult to solve for the roots. Since the roots are computed at every instant in time the integrator requires during the integration process, it is certainly of interest to seek methods, which take advantage of the structure of the equations. For nonlinear systems, root solvers are faster if they get derivative information, typically first or second order. Symbolic derivatives are not only useful for root solvers, but can also be useful for tailored integrators, parameter estimation algorithms, which most often are optimisations, and sensitivity analysis, linearization and quadratic to higher-order approximations to mention the main ones. Again, we want to take advantage of the structures present in the equations (definitions). We find three forms very frequently in the above set:
f (ξ, ϑ)
:=
g(ξ, ϑ)
:=
λi (ξ, ϑ)
λij (ϑ)ξi ξj
{i,j}
h(ξ, ϑ)
{i}
:=
λ(ϑ)ξi ξj
:= ξ T Λ(ϑ)ξ
{i},{j}
Many of the model equations, in particular the flows and the transpositions are often of the form f , the first in the above set. These are for example heat transfer and kinetic expressions with a few variables. Many of the transformations (Definition 4) and the properties (Definitions 5-7) are multicomponent thermodynamic expressions of the form g and h, where the ξ terms represent usually molecular interactions. The λ, λi and λij are “lambda” functions (of modest complexity) which separate the free variables ξ to the extent possible from the ϑ, which most frequently are a combination of secondary states (often pressure and/or temperature) and parameters. Relatively little can be done with model components that belong in the class of f -functions, but for the thermodynamic models g and h taking advantage of the structure yields potentially huge benefit particularly when the number of ξ-variables increases beyond 4-6. Model h has the most structure. It is typical for descriptions such as one uses for an activity coefficient model or the attractive and repulsive terms in an equation of state. Model g has more structure than f but less than h since all the λij functions may be different. It requires therefore a list implementation which is less efficient (especially during differentiation) than the linear algebraic formulation of g. As part of the ongoing research on the implementation of operations on thermodynamic objects, we created our own, small, meta-language mainly because of the need for the development of an efficient differentiation algorithm, we need a language that enables efficient manipulation of the expressions. The key to the definition and consecutive realisation was to recognise that the complexity of representing λ
equations can be captured in a generic form A {B1 , B2 . . . } with A being a container, {B1 , B2 . . . } a list of containers, an operator modified by the lambda function λ. This representation is able to capture any of the representations making up the model. Such ”objects” exist probably the earliest in APL, where the A and Bi are called the nouns, the operator the verb and the modifier λ the adverb. Yorick [9] is another language that has a similar traversing mechanism and also Matlab can be seen
On Model Portability
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to have them in the form of structured cells, though the latter is not quite obvious to recognise. The operator is a mapping/traversing recipe, whilst the λ-function is a primitive operator tree with simple scalar operators ( +, −, log, etc.). This main structure can thus capture unitary, binary, list and vector and matrix operations. The implementation is motivated by modern languages such as Ruby [4], Python [3], O’Haskell [1] and OCaml [2], to mention some of the main developments, all of which combine the object-oriented programming world with the functional one. The representation has proven to be very useful when differentiating expressions of the above structure, in particular the structures g and h. The differentiation operation is largely dispatched down to the λ-function, which consist of standard primitives, thus is relatively easy to implement. The traversing|mapping operation is a little more diffuse, but it is currently suggested that the set of op erators { , } (dot product and summation) form an Abalian group and are thus closed. This subject is currently being researched in more detail. The core of the representation we call a Pandora-box, which implements the modified traversing operation of object A on the list {Bi }. The language, which is put on top of an existing language used to represent the basics of what is used in the representation of the λ-function is very small and would have to be passed on with the so-encoded model. It has only four different constructs: 1) One to construct new structures, called containers. 2) One iterator, which iterates through the containers. 3) One that reduces the dimensionality of the containers, and finally 4) one which cycles the elements in the containers. As we indicated earlier, this representation deals only with the problem of representing the model as an algebraic object and does not include any numbers. The instantiation of variables, mostly parameters is done separately. This re-enforces our definition of parameters as quantities that are represented explicitly as numerical information, which either is given or is the object of our computations. 7. Towards Self-Contained Models In the previous section we described the contents of a model in terms of an algebraic structure that may be supplemented with additional information, most of which is also of algebraic nature, such as the derivatives but also trees (linked lists, dictionaries ...) and textual information. Truly self-contained models are essentially not possible, because the language representing the model is then also part thereof. This leads to the recursive problem of language definitions. Thus a language definition must be passed on. This definition should be minimal and easy to realise in any implementation language. The model itself should have maximum documentation value, meaning, that besides the language all information can be re-constructed from the portable model. This implies that the model must bootstrap using the minimal language implementation. The model must thus satisfy some conditions on its representation. For example context freeness would be a nice property to have. The model should also contain ALL model components; this includes the thermo data and other data that may be required for the characterisation of the model. In this context one can also make a case that the fundamental ”rules” to construct the behaviour or the ”physics” | ”mechanism” of the process must be part of the model because it defines the context of the model explicitly and allows checking on this as it is used in another environment. At this point in time it is felt that the Pandora meta-language is very suitable for a realisation of such a representation and tools that use it consecutively. 8. Conclusions Portability of process descriptions can only be improved if one exploits the structure of the models and puts it on a more basic foundation. Wrapping is only a fix and hides the existing problems with the representation of the different parts. The abstraction developed as part of this project facilitates tighter control over the construction and maintenance of the model, as its definition is on the level of choosing basic model structures and mechanisms such as convective flow, diffusion-driven flow in contrast to defining equations describing such mechanisms. It also includes all time-scale assumptions in the form of the basic topology chosen to represent the system as well as later assumptions (fast flows, fast reactions, small capacities). Further, the Pandora-box approach exploits the structure of thermodynamic functions and thus simplifies the symbolic computation of derivatives hopefully easing the solution of
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the notoriously implicit algebraic equations. The resulting model is not specialised towards any specific problem but can be used for different tasks including design, parameter identification, controller design etc. The generic description also makes it suitable for mapping it into any type of programmed framework. The result is not a wrapper, but a process of generating tailored process models. If one also has the feature to choose between different solvers, it is very much thinkable that tailored solvers can be constructed efficiently and full advantage can be taken from the structure in the solver process. It is envisioned that this will have the effect of refocusing attention on the solvers and make them to exploit the structure and characteristics of the problem. Also, which is probably more interesting, they should check on the made assumptions, such as validity ranges and model switching or the like. Highly portable models have a number of components: • Topology of the plant model, namely the special decomposition being applied. • Hierarchical decomposition of the topology. • Details on nature of the extensive quantities being conserved and where they are present in the system as well as their transposition. • Details on kinetics of the transposition. • Details on the transport of extensive quantities. • Complete set of state variable transformations connecting the secondary variables with the fundamental extensive quantities, being the ones that are conserved. • Augmented with the material properties associated with the transport systems, the transposition kinetics and the nature of the capacities. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
15. 16. 17. 18. 19.
Haskell - home page. http://www.haskell.org/. Ocaml - home page. http://caml.inria.fr/. Python - home page. http://www.python.org/. Ruby - home. http://www.ruby-lang.org/en/. Westerberg A and Piela P C. Equation-based process modelling. Technical report, Carnegie Mellon Univeristy, 1994. E D Gilles. Netzwerktheorie verfahrenstechnischer prozesse. Chem Ingr Tech, 69(8):1053–1065, 1997. J¨ org Hackenberg. Computer support for theory-based modelling of process systems. PhD thesis, RWTH Aachen, Germany, 2005. K M Hangos and I T Cameron. A formal representation of assumptions in process modelling. Comp & Chem Eng, 25:237–255, 2001. Lawrence Livermore National Lab. Yorick – unofficial home page. http://www.maumae.net/yorick/doc. T Y Lee. The Development of an Object-Oriented Environment for the Modelling of Physical, Chemical and Biological Systems. PhD thesis, Texas A & M University, College Station, TX, USA, 1991. A A Linninger. Metamodelling – a novel approach for phenomena-oriented model generation. In Foundations of Computer-Aided Process Design. Malone M, Trainham J, and Carnahan B, 2000. W Marquardt. Rechnergestuetzte erstellung verfahrenstechnischer prozessmodelle. Chemical Engineering Technology, 64:25–40, 1992. A Z Mehrabani. Computer aided modelling of physical-cyhemical-biological systems. PhD thesis, University of New South Wales, Sydney, Australia, 1995. P C Piela, T G Epperly, K M Westerberg, and A W Westerberg. Ascend: An object-oriented computer environment for modelling and analysis: The modelling language. Comp & Chem Eng, 1(1):53–72, 1991. Henning G Stephanopoulos G and Leone H. Model.la. a modelling language for process engineering–i the formal framework. Comp & Chem Eng, 14(8):813–846, 1990. Henning G Stephanopoulos G and Leone H. Model.la. a modelling language for process engineering–ii. multifaceted modelling of process systems. Comp & Chem Eng, 14(8):847–869, 1990. Ginkel M Tr˜ nkle f, Zeitz M and Gilles E D. Promot: A modelling tool for chemical processes. Mathematical and Computer Modelling of Dynamical Engineering, 6(3):283–307, 2000. Wedel von L. Uniting heterogeneity: an environment for heterogeneous mdoel management in chemical process engineering. PhD thesis, RWTH Aachen, 2003. M R Westerweele. Five Steps for Building Consistent Dynamic Process Models and Their Implementation in the Computer Tool MODELLER. PhD thesis, TU Eindhoven, Eindhoven, The Netherlands, ISBN 90-386-2964-8 2003.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Utility systems operational planning optimization based on pipeline network simulation Luo X.L.a, Hua B.a, Zhang B.J.a, Lu M.L.b aThe Key Lab of Enhanced Heat Transfer and Energy Conservation, Ministry of Education of China, South China University of Technology, Guangzhou, 510640, China bAspen Technology Inc. 10 Canal Park, Cambridge, MA 02141, USA
Abstract In this paper, an analysis on the negative impact of steam parameters change on utility system and process operations is conducted, which is not reported previously. In order to avoid the impact, a method that integrates utility system operational planning optimization with pipeline network simulation has been developed. A multiperiod mixed integer non-linear programming (MINLP) model is developed and a decomposed iteration algorithm is applied to solve this model, which is demonstrated in an industrial case study. Keywords: Utility system; Optimization; Pipeline network; Simulation; MINLP
1. INTRODUCTION Utility system is an important part of the process industry and several efforts on the design and operation of utility system have been reported. Papoulias and Grossmann (1983) described mixed integer linear programming (MILP) model for the optimization of the structure and parameter of utility system under fixed demand. Hui and Natori (1996) presented a mixed-integer formulation of multiperiod design and operational planning for utility system and discussed the industrial application of this problem. Oliveira Francisco (2004) extended the multiperiod synthesis and operational planning model by including the global emissions of atmospheric pollutants issues coming from the fuels burning. However, most of these efforts assume fixed steam and power demand from processes and emphasize on the utility production and distribution, and none of them consider the utility parameters change within the pipeline network. In fact, the steam parameters usually fluctuate greatly when steam passes through pipeline network. Some of the steam branches reaching the process users are degraded greatly and fail to meet the requirement and even impact the process operation. Therefore steam sources parameters or flowrates must be increased to overcome the above problem. Inevitably, the operation of the whole utility system deviated from the initial conditions and the optimal schedule without involving pipeline network simulation was not the optimal solution. This paper presents a method that integrates utility system operational planning optimization with pipeline network simulation and a multiperiod mixed integer non-linear programming (MINLP) model is developed. The MINLP model involves a large number of continuous, discrete decisions variables, non-linear constraints and is difficult to solve. A decomposed iteration approach is applied to solve this problem and the optimal solution can be reached within a reasonable time. A case study shows that using the proposed method, significant savings can be obtained by optimizing the utility system operation at no capital investment as well as steady and safety process operation can be ensured.
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2. PROBLEM FORMULATION The utility system operational planning problem is modeled with a complex MINLP model that is an extension of the multiperiod models described by Iyer and Grossmann (1997). The objective function includes the fixed and variable costs of equipments, costs of purchased power and steam for each period as well as the changeover costs between periods. Following are the objective function and the constraint equations of the proposed utility system multiperiod operational planning optimization problem. Objective function MinCost = ∑ ∑ CEF n × Ynt + ∑ ∑ ( ∑ FFnit × C fi + t
n
t
n
i
（1）
CWF nt × C cw + C n × ZO nt ) + ∑ C w × WFt + ∑ ∑ C sr × SFrt t
t
r
The mass balance of equipment n
∑F
− ∑ Fn ,out ,t = 0
n ,in ,t
in
n ∈ N,t ∈T
out
（2）
The energy balance of equipment n
∑F
h
n ,in ,t n ,in ,t
in
− ∑ Fn ,out ,t hn ,out ,t − Wnt − S nrt hnrt = 0 out
n ∈ N,t ∈T
（3）
The equipment capacity bounds
Ω
L F n , in
Ω
L F n , out
≤ F n , in , t ≤ Ω
≤ F n , out ,t ≤ Ω
U F n , in
U F n , out
n ∈ N,t ∈T
（4）
n ∈ N,t ∈T
（5）
The power and steam demand constraints WF t + ∑ W nt ≥ DW t t ∈T
（6）
n
∑S
SF rt +
nrt
≥ DS
rt
n
The equipment changeover constraints
ZO
≥ Y nt − Y n , t − 1
nt
r ∈ R, t ∈ T
（7）
n ∈ N,t ∈T
（8）
i ∈ I ,t ∈T
（9）
The fuel supply bounds constraints
∑ FF
nit
≤ Ω UFi , t
n
Steam enthalpy is the function of pressure and temperature
h = f ( P, T )
(10)
Pressure drop in pipe j in period t 2 ⎛ ε 0 .25 ⋅ L j m jt 1 ⎜ 0 .11 j Δ p jt = 0 .811 + ∑ζ 4 ⎜ 1 .25 ρ ρ jt , aver ⋅ d j ⎝ dj jt , aver k ∈ K j Temperature drop in pipe j in period t π ⋅ (d j + 2δ 0 j + 2δ j ) ⋅ (T jt,aver − Ta )L j ΔT jt = d j + 2δ 0 j + 2δ j d j + 2δ 0 j + 2δ j 1 ln + 2λ j ⋅ m jt ⋅ C p, jt d j + 2δ 0 j α jt ⋅ m jt ⋅ C p, jt
jk
⎞ ⎟ ⎟ ⎠
j ∈ J ,t ∈T
（11）
j ∈ J ,t ∈T
（12）
Steam flowrate of pipe j that pass through node d in period t
∑m j
djt
=0
d ∈ D, t ∈ T
Steam sink parameter bounds constraints
（13）
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PsL ≤ Pst ≤ PsU
s ∈ S ,t ∈T
(14)
TsL ≤ Tst ≤ TsU
s ∈ S ,t ∈T
(15)
The improved drive turbine model involves steam parameters change (Liu Jinping, 2002)
W = a + b ⋅ mT ,act − mT ,act (ΔP1 ⋅ K ΔP1 + ΔT1 ⋅ K ΔT 1 )
(16)
The calculated steam heat flowrate considered parameters change
m st , act = f ( Pst , T st , m st , dem )
s ∈ S ,t ∈T
(17)
3. MODEL SOLVING STRATEGY It usually fails to guarantee robust convergence and global optimality if the MINLP model presented above is applied directly. Fortunately, it is possible to overcome this difficulty by decomposing the problem into two sub-problems and solve them by a three-step iteration algorithm developed in this paper. One sub-problem is utility system operational planning optimization (USOPO) that involves the MILP model as illustrated from Eqs.(1) to Eqs.(9). The other sub-problem is pipeline network simulation (PNS) that involves non-linear model as illustrated from Eqs.(10) to Eqs.(13). To solve this model, the first step of the proposed algorithm solves the USOPO problem (MILP model) under fixed utility demand and steam parameters. This is followed by the second step of rigorous PNS (non-linear model) and the steam parameters are calculated for every sink. In the third step, the steam sink parameters are checked and the steam source parameters or flowrates are adjusted according to Eqs. (16) and (17). The linear optimization of the first step is repeated, followed again by the pipeline network simulation of the second step. It iterates until the problem converges. This is illustrated in Figure 1 where k denotes the iteration times. The global optimality of the MILP model and the rigorous of pipeline network simulation guarantee the global optimality of the decomposition algorithm. Besides, the algorithm is characterized by rapid convergence of no more than five iterations to reach reasonably small error levels. In this paper, the USOPO problem model is developed using the General Algebraic Modeling System-GAMS and solved using Cplex while the PNS problem is solved using PIPEPHASE 8.1 package. B egin
M IL P O p tim izatio n P ipe n etw ork sim m ulatio n
Ps L ≤ Pst ≤ Ps U
TsL≤ Tst ≤ TsU
A dju st so urce flow rate m so u ce
N
Y K +1 st
− PstK ≤ ε
K +1 st
− TstK ≤ ε
P N
A d ju st so u rce p aram eters P sorce T source
T
E nd
Y
Fig.1 The solving procedure of utility system optimization
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492
4. CASE STUDY Figure 2 shows the flowsheet of the utility system and the pipeline network of a Chinese oil refinery. Table 1 gives the data of steam source parameters and flowrates for the operation of the utility plant in a year horizon consisting of three periods of equal length. In table 1, the symbol “/” denotes that the data is under calculating. The estimated demand of steam flowrate of each sink in each period is illustrated as mest in Table 3. The utility system component model (Luo XiangLong, 2006) is applied in this case. ARGG B3 L01
B2
B1
L13 L12
J0 1 L04 L05 V1
J02 L10
M TBE
P08
L16
P02
J0 4
L09
L17
P03
H PH C
G as Sep
L18
BT2
BT1
L08 P04
L14
J0 3
L06
L07
L15
P01
L03
L02
L19
J0 5
L20 L23
L21
J06 L22
CT1
U tility P la n t
L11
P09 C o n d e n se r
P07 A c ry lic F ib re s P la n t
P06
P05 L u b e P la n t
P o ly m e r P la n t
Fig.2 The flowsheet of utility system and pipe network of an oil refinery
Using the proposed method and algorithm, the optimal results are obtained in the third iteration. The minimum annual cost is $9,286,328 and more than $800,000 is saved compare with the planning schedule made by the refinery. The optimal utility operational planning results are illustrated in Table 2 where mori and mopti denote the optimal flowrates schedule before and after pipeline network simulation respectively. The simulated steam sink parameters and flowrates in each period are illustrated as Pact, Tact, mact in Table 3. Table 2 also shows that the utility optimal planning schedule changes greatly even though the steam demand does not increase very much. And the steam sources parameters need not to change in this case but the steam sinks parameters do change a lot as shown in Table 3. Table 1 The steam sources parameters and flowrates Psource
Tsource
msource (t/h)
(MPa)
(℃)
1
2
3
MP
3.43
430
∕
∕
∕
P01
Mp
1.02
285
73
45
40
P01
LP
3.39
425
50
60
50
P02
LP
1.00
290
43
45
46
Source
Steam
B1, B2, B3
Table 2 The utility system optimization result comparison
Utility Systems Operational Planning Optimization
Period
493
mori (t/h)
mopti (t/h)
B1
B2
B3
T1
T2
T3
V1
B1
B2
B3
T1
T2
T3
V1
1
60
0
130
85
85
27.4
8
60
0
130
85.0
85
21.9
5.5
2
0
0
130
82
0
0
28
0
30
130
52.6
85
20.0
0
3
60
0
0
41
0
22.0
0
60
0
0
40.0
0
11.5
0
Table 3 The estimated and simulated steam parameters and flowrates demand by steam sinks mest (t/h)
Pact (MPa)
Tact (℃)
mact (t/h)
Sink
Steam
1
2
3
1
2
3
1
2
3
1
2
3
P02
MP
85
80
69
3.34
3.34
3.35
418
418
416
87.5
82.4
71.0
P03
LP
50
40
15
0.90
0.95
0.99
280
286
286
50.5
40.1
15.1
P04
LP
50
45
25
0.93
0.95
0.98
277
276
264
50.5
45.5
25.5
P05
LP
55
50
30
0.78
0.86
0.96
276
274
265
55.5
50.6
30.6
P06
LP
40
30
10
0.80
0.88
0.97
276
272
251
40.3
30.4
10.3
P07
LP
50
40
25
0.72
0.84
0.95
266
261
247
50.5
40.9
25.9
5. CONCLUSION This paper presents a new method to overcome the negative impact of steam parameters change in the pipeline network. A decomposed iteration algorithm is presented to solve the complex multiperiod mixed integer non-linear programming problem. A case study shows that using the proposed method the optimal operational planning schedule can be obtained in a reasonable time as well as steady and safety process operation can be ensured. Compared with the previous efforts, the method presented in this paper brings the optimization much closer to the real engineering operation, and the scheduling is more applicable. Besides, if power demand is considered, more significant benefits can be expected by applying the proposed method.
ACKNOWLEDGEMENTS The authors wish to acknowledge the financial supports from the Major State Basic Research Development Program (G2000026307).
NOMENCLATURE Sets I/i N/n D/d
set of fuel J/j set of pipe set of equipment S/s set of steam sink set of pipeline network node Kj/k
T/t set of period R/r set of steam main level set of pipe local resistance
Parameters a, b Cfi Cw CEFn
model coefficients unit price of fuel i unit price of purchased power maintain cost of equipment n
Ccw Cn Csr dj
unit price of cold water equipment startup cost unit price of steam of level r diameter of pipe j
X.L. Luo et al.
494 DWt Ta δj DSrt KΔP1 KΔT1 mst,dem ΩFn,inL ΩFn,outL ΩFn,inU ΩFn,outU ΩFi,tU PsL, PsU TsL, TsU λj ζjk
power demand in period t Lj length of pipe j surrounding temperature δ0j thickness of pipe j insulation thickness of pipe j εj roughness of pipe j demand of steam of level r in period t specific decreased power production for the inlet pressure drop specific decreased power production for the inlet temperature drop steam flowrate demand of sink s before steam parameters change in period t lower bound of inlet steam rate of equipment n lower bound of outlet steam rate of equipment n upper bound of inlet steam rate of equipment n upper bound of outlet steam rate of equipment n upper bound flowrate of fuel i in period t lower and upper bound of steam temperature of sink s lower and upper bound of steam temperature of sink s heat conductivity of insulation of pipe j local resistance coefficient of pipe j
Binary variable Ynt ZOnt
1 if equipment n operate in period t; 0 otherwise 1 if equipment n incurs startup cost in period t; 0 otherwise
Variable Cp,jt mjt Fn,in,t Fn,out,t CWFnt FFnit hn,in,t hn,out,t hnrt mT,act Pst SFrt Snrt Tst Tjt,aver Wnt αjt ρjt,aver ΔP1 ΔT1
steam specific in pipe j in period t W power production of turbine steam flowrate of pipe j in period t WFt purchased power in period t stream flowrate enter equipment n in period t stream flowrate exit equipment n in period t cold water flowrate of unit n in period t flowrate of fuel i that equipment n consumes in period t specific enthalpy of stream that enter equipment n in period t specific enthalpy of stream that exit equipment n in period t steam enthalpy of level r produced in equipment n steam flowrate demand of turbine after steam parameters change steam pressure of sink s in period t purchased steam flowrate of level r in period t steam production rate of level r in equipment n in period t steam temperature of sink s in period t average steam temperature of pipe j in period t power production of equipment n in period t heat exchange coefficient between insulation of pipe j and surrounding average steam density in pipe j in period t pressure drop of steam enter turbine temperature drop of steam enter turbine
REFERENCE Papoulias, S. A, Grossmann, I. E, 1983, A structural optimisation approach in process synthesis—I: utility systems. Comput. Chem. Engng, 19: 481-488 Chi-Wai Hui,Yukikazu Natori, 1996, An industrial application using mixed-integer programming technique: a multiperiod utility system model. Comput. Chem. Engng, 20(Supplement 2): 1577-1582.
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A. P. Oliveira Francisco, H. A. Matos, 2004, Multiperiod synthesis and operational planning of utilitysystems with environmental concerns, Comput. Chem. Engng, 2004, 28: 745-753. Iyer, R.. and Grossmann, I. E., 1997, Optimal multiperiod operational planning for utility systems. Computers and Chemical Engineering, 21(8), 787–800. Liu Jinping, 2002, Study on the Theory and Application of Hierarchical modeling of Steam Power System in Process Industry, Ph.D thesis, SCUT ,China. LUO XiangLong, HUA Ben and ZHANG BingJian, 2006, Optimal multiperiod operational planning for steam power system of petrochemical industry, Computers and applied chemistry, 23 (1): 41-45.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Particle swarm for the dynamic optimization of biochemical processes Jianming Zhanga, Lei Xiea,b,*, Shuqing Wanga a
National Key Laboratory of Industrial Control Technology, Institute of Advanced Process Control, Zhejiang University, Hangzhou 310027, P. R. China b Department of process dynamics and operation, Berlin University of Technology, Sekr. KWT 9, Berlin 10623, Germany
Abstract This paper presents a new approach, DYN-PSO, for the dynamic optimization based on the principles of particle swarm optimization (PSO). DYN-PSO enables the direct call of simulation tool and facilitates the dynamic optimization task for biochemical engineers. The presented approach has the potential to obtain the global optimal solution and is implemented as a toolbox in MATLAB. The proposed DYN-PSO is applied to optimize the inducer and substrate feed profiles of a fed-batch bioreactor in order to maximize the production amount of chloramphenicol acetyltransferase. Keywords: Dynamic optimization; Particle swarm optimization (PSO); Biochemical processes; DYN-PSO
1. Introduction The aim of dynamic optimization is to optimize a performance index, such as production quantity, unit cost, operation/grade transition time etc, by manipulating timevarying process variables. For biochemical processes, start-up, shut-down and product changeover are frequent operations and all these events need dynamic optimization to determine the optimal polices within the dynamic operation period. The model-based approaches have been developed for solving dynamic optimization problems described with differential algebraic equations (DAE) in recent decade (Biegler and Grossmann, 2004). The core of such approaches is reformulating the DAE as a set of algebraic equations with orthogonal collocation method etc. and employing successive quadratic programming to solve the corresponding large-scale nonlinear optimization problem. Although these approaches are computationally effective, the difficulties of the mathematical treatment, such as discretization and sensitivity computation, prohibit the chemical engineers from using mode-based approaches readily in industrial practice. The stochastic approaches, however, provide a possibility of optimization easily with modeling and simulation techniques at hand. As direct search methods, the stochastic approaches are carried out through successive simulations. In recent studies (Faber et al., 2005; Li et al., 2000; Rajesh et al., 2001), simulated annealing and ant colony methods have been reported for solving the dynamic optimization problems. Due to the intractable nature of the dynamic optimization and its importance in biochemical engineering, it desirable to explore other avenues for developing good stochastic approaches for this problem. In this paper, the capability of particle swarm optimization(PSO) for dynamic optimization is explored. *
Corresponding author. Tel: 086-571-87951125, Fax:086-571-87951445 Email address: [email protected]
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As an evolutionary computation technique and general global optimization tool, PSO was first proposed by Kennedy and Eberhart (1995). PSO has many advantages over other stochastic techniques such as it can be easily implemented and has great capability of escaping local optimal solution (Parsopoulos and Vaahatis, 2002). To the authors best knowledge, the use of PSO for biochemical process dynamic optimization has not been reported in the open literatures. In this study, a PSO based DYNamic optimization algorithm DYN-PSO and an easy-to-use toolbox is developed in the MATLAB environment and SIMULINK is utilized as the DAE solver. The rest of the paper is structured as follows. In Section 2, the general formulation of dynamic optimization is presented in brief. The traditional PSO algorithms is introduced in Section 3. Section 4 presents the DYN-PSO algorithm and its implementation issues in detail. The performance of the proposed optimization scheme is demonstrated on a typical fed-batch bioreactor operation optimization in Section 5. The optimal inducer and glucose feed profiles are obtained using the DYN-PSO approach to maximize the production amount of chloramphenicol acetyltransferase(CAT). Finally, some conclusions are discussed in Section 6.
2. Dynamic Optimization Formulation The formulation of dynamic optimization problems under study is given as follows: min ϕ (z (t f ), u(t f )) u ( t ), t f
s.t. dz (t ) = F (z (t ), u(t )) dt H (z (t ), u(t )) ≤ 0
(1)
u L ≤ u(t ) ≤ uU z (0) = z 0 , t0 ≤ t ≤ t f
where z (t ) ∈ ℜnz is the vector of differential state profiles with initial condition z0, u(t ) ∈ ℜnu represents the profiles of control variables, [t0, tf] defines the finite time horizon and the scalar objective function value φ is minimized at the final time tf, F, H define the DAE system, inequality constraints of state variable path, respectively. uL, uU represent the bound constraints on u(t) profiles. A more detailed and general formulation of dynamic optimization can be found in Biegler and Grossmann (2004).
3. Conventional Particle Swarm Optimization PSO is an algorithm first introduced by Kennedy and Eberhart (1995). As an evolutionary algorithm, PSO imitates the social behavior of organisms such as birds flocking and fish schooling. Each solution candidate of the optimization problem(called particle), flies in the problem search space looking for the optimal position according to its own experience as well as to the experience of its neighborhood. The performance of each particle is evaluated using a predefined fitness function, which capturing the characteristics of the optimization problem. The velocity and position are updated according to the following equations at the jth iteration: ⎧Δxij +1 = w ⋅ Δxij + c1R1j (pij − xij ) + c2 R 2j (p gj − xij ) ⎪⎪ j +1 j j +1 (2) ⎨xi = xi + Δxi ⎪1 ≤ i ≤ P,1 ≤ j ≤ IMAX ⎪⎩
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where Δxij +1 ∈ ℜn , called the velocity for particle i, represents the position change by this swarm from its current position in the jth iteration, xij ∈ ℜn is the particle position, pij ∈ ℜn is the best previous position of particle i, p gj ∈ ℜn is the best position that all
the particles have reached, c1, c2 are the positive acceleration coefficient, w is so called inertia weight, R1i , R 2j are diagonal matrices whose diagonal elements are uniformly distributed random numbers between [0, 1], P is the population size and IMAX is the maximum iteration number.
4. PSO for DYNamic Optimization (DYN-PSO) Particle Swarm Optimization, in its original form, can only be applied to the unconstrained and static problems. In this section, a novel PSO based dynamic optimization algorithm DYN-PSO is presented. The key aspects of DYN-PSO algorithms include control profile parameterization, bound and state variables path constraints treatment, DAE solver integration, PSO parameters setting etc. All these issues and a general DYN-PSO framework are given below. 4.1. Control profiles parameterization The aim of DYN-PSO is to find the profiles of control variables u(t) that optimize the objective function φ. In this study, the piecewise linear representation is adopted to approximate the optimal profiles of u(t) (Li et al., 2000; Rajesh et al., 2001). The time horizon [t0, tf] and control profiles for each component of u(t)=[u1(t), u2(t), …, unu(t)]T are discretized over Nt different points: t j = [0, t1, j , t2, j ,..., tm , j ..., t Nt − 2, j ,1]T × t f u j = [u0, j , u1, j , u2, j ,..., ui , j ..., u Nt − 2, j , u Nt −1, j ]T 0 ≤ tm , j ≤ tn , j ≤ 1, ∀1 ≤ m ≤ n ≤ N t − 2
(3)
1 ≤ j ≤ nu , 0 ≤ i ≤ N t − 1, So the control profiles are represented by Nt×nu discretized control values, (Nt-2)×nu time points and the final time tf, which can be handled by PSO algorithm as the optimization variables: x = [t f , tm , j , ui , j ]T ∈ ℜ(2 Nt − 2) nu +1 (4)
With the piecewise assumption, any u(t) can be evaluated by interpolating the control values on its adjacent discretized time points and thus can be used by the DAE solver, e.g. SIMULINK, to evaluate the objective value in Equation (1). 4.2. Bound and state path constraints treatment Bounds constraints on the control variable, u L ≤ u(t ) ≤ uU and discretized time points 0 ≤ tm , j ≤ tn , j ≤ 1 are easy to implement by utilizing a modified scheme of particle position updating in PSO algorithm, which is described in Subsection 4.4 and 4.5. With respect to the state variables path constraints H (z (t ), u(t )) ≤ 0 , the penalty terms are added to the objective function to provide the information on constraint violations. In current study, the penalty terms are in the form of: tf
Pe(u(t ), t f ) = ∫ b(t ) H (z (t),u(t))dt t0
The penalty constant b(t) takes the following form: ⎧ B, H (z (t ), u(t )) > 0 b(t ) = ⎨ ⎩0, H (z (t ), u(t )) ≤ 0
(5)
(6)
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where B is a positive constant, e.g. of value of 500. The penalty term Pe(tf) decreases to zero if and only if no constraints are violated over [t0, tf]. Adding the penalty term to the objective function and utilizing the x definition given in Equation (4) lead to the following DYN-PSO problem formulation: min ϕ (z (x), x) + Pe(x) x
s.t. z (x) = DAE _ Solver ( F , x)
(7)
x L ≤ x ≤ xU z (0) = z 0 where the state variables profiles z(x) is obtained by calling available DAE simulation packages.
4.3. PSO with restart There are several ways of determining when the PSO algorithm should stop. The most common adopted criterion is reaching a maximum number of iterations IMAX, as utilized in Section 3. However, it is pointless for PSO to proceed if the algorithm no longer possesses any capability of improvement. In this study, an improved PSO algorithm with restart is presented, i.e. a new particle population will be generated when current one has no potential to explore better solutions. Such potential is measured with the following criteria which indicates whether all the particles are clustered around the same spot: max( xi − x j ) < ε ,1 ≤ i ≤ j ≤ P (8) i, j
where xi − x j
Σ
Σ
= (xi − x j )T Σ(xi − x j ) is the norm of a vector, Σ is a positive
weighting matrix, e.g. Σ = diag −1 (xU − x L ) . ε is the predefined tolerance, e.g. 10-3. With the restart scheme, the exploring capability of PSO algorithm is further improved and it is more possible to find global solution in limited iterations. 4.4. PSO parameters setting There are some rules of thumb to set the PSO parameters. Population size (P) This parameter is crucial for PSO algorithm. A small population does not create enough interaction for the emergent behavior to PSO to occur. However, the population size is not problem specific, a value ranging from 20 to 50 can guarantee the exploring capability of PSO, especially for unconstrained optimization problems (Parsopoulos and Varhatis,2002). To our experience, for DYN-PSO problems (7), which includes the state variables path constraints, 60 to 100 population size are usually enough. Inertia coefficient (w) This parameter controls the influence of previous velocity on the current one. Roughly speaking, large w facilitates global exploration, whilst low w facilitates local exploration. According to the suggestion of Parsopoulos and Varhatis(2002), a initial value around 1.2 and gradual decline towards 0 can be taken as a good choice of w. Acceleration coefficient (c1, c2) Proper tuning of these parameters can improve the probability of finding global optimum and the speed of convergence. The default value is c1= c2 = 2 (Kennedy, 1998). Velocity limit( Δx L , ΔxU ) To guarantee the particles only fly in the feasible space defined by the constraints , x L ≤ x ≤ xU , the current velocity for ith particle at jth iteration is limited to the range defined as below:
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Δxij +1 ∈ [Δx L , ΔxU ] = [x L − xij , xU − xij ] If current velocity exceeds the above limits, it is simply reset to the nearer bound.
(9)
4.5. DYN-PSO framework In this study, the DYN-PSO algorithm is implemented in MATLAB and MATLAB/SIMLINK is chosen as the DAE solver. A toolbox integrating DYN-PSO and SIMULINK is also developed and can be obtained from the authors. The general framework of the DYN-PSO is illustrated in Figure 1. MATLAB / SIMULINK
Initialize the particle position, velocity, etc.
Particle fitness evaluation (objective function value in equation (7))
Update the particle position (according to equation (2))
Particles clustered around the same spot?
No
Maximum iteration number reached?
Yes
Restart
Yes
Optimal solution
No
DYN-PSO Main Loop
Figure 1. General Framework of DYN-PSO algorithm
It should be noted that after the particle positions are initialized and updated, the discretized time points tm,j are sorted for each 1 ≤ j ≤ nu to satisfy the ascending constraints on tm,j defined in (3).
5. Application Study The fed-batch bioreactor considered in this study is utilized to produce the cloned chloramphenicol acetyltransferase (CAT) with Escherichia coli JM105. The aim of the dynamic optimization is to find a optimal inducer and substrate feed profiles that maximize the production rate. A dynamic mathematical model has been developed for this process by Chae et al.(2000) and validated with experimental data. This model includes 5 state variables, i.e. biomass concentration (X), substrate concentration (S), product production (P), inducer concentration (I) and the fermentation volume (V). The control variables are the glucose feed rate(substrate: Fs) and arabinose (inducer: Fi). The process model is described in Equation (10). Due to space limit, we refer to Chan et al.(2000) and Mahadevan and Doyle III (2003) for details about the parameter values and initial conditions. dX X dS Fs μX S = μ X − ( Fs + FI ) = SF − − ( Fs + FI ) dt V dt V YX / S V dPf
Pf dI FI I S I − qI X − ( Fs + FI ) = π X − k3 Pf − ( Fs + FI ) = dt V dt V V dV = Fs + FI dt And the dynamic optimization function is formulated as follows: min − [ Pf (t f )V f (t f )]
(10)
Fs ( t ), FI ( t )
s.t.
Equation(10),
t f = 30
(11)
0 ≤ FI , Fs ≤ 0.5 L ⋅ h −1 and V ≤ 4 L ∀t ∈ [0, t f ]
The results of DYN-PSO algorithms (with Nt=8, B=100, P=80, IMAX=400 and the algorithm restarts at the 245th iteration) are illustrated in Figures 2 and 3. Figure 2
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shows the optimal feed profiles for the inducer and substrate. Figure 3 illustrates the amount of production formed with respect to time. The final optimal value for the production amount is 0.99525. In contrast, the results obtained by Mahadevan and Doyle III (2003) is 0.91. There is an improvement about 10%.
Figure 2. Optimal feed profiles
Figure 3. Amount of production formed
6. Conclusions A PSO based dynamic optimization algorithm DYN-PSO is presented in this paper and an easy-to-use toolbox based on MATALB has been developed. This algorithm facilitates the dynamic optimization task for the chemical and biochemical engineers with the simulation techniques at hand. Implementation details of DYN-PSO is reported in this paper. Applications in the operation optimization of fed-batch bioreactor demonstrate the efficiency of proposed method.
Acknowledgments This work is partially supported by National Natural Science Foundation of China with grant number #60421002 and #70471052.
References L.T. Bielgar, I.E. Grossmann, 2004, Retrospective in optimization, Computers and Chemical Engineering, 28, 1169-1192. H.J. Chae, M.P. Delisa, H.J. Cha, W.A. Weigand, G. Rao, W.E. Bentley, 2000, Framework for online optimization of recombinant protein expression in high-cell density escherichia coli cultures using GFP-fusion monitoring, Biotech. Bioeng., 69, 275-285. R. Faber, T. Jockenhövel, G. Tsatsaronis, 2005, Dynamic optimization with simulated annealing, Computers and Chemical Engineering, 29, 273-290 J. Kennedy, R. Eberhart, 1995, Particle swarm optimization, In Proc. IEEE Int. Conf. Neural Networks, Perth, 1942-1948. P. Li, K. Löwe, H.A. Arellano-Garcia, G. Wozny, 2000, Integration of simulated annealing to a simulation tool for dynamic optimization of chemical processes. Chemical Engineering and Processing, 39, 357-363. R. Mahadevan, F.J. Doyle III, 2003, On-line optimization of recombinant product in a fed-batch bioreactor, Biotechnology Progress, 19, 639-646. K. E. Parsonpoulos, M.N. Varhatis, 2002, Recent approaches to global optimization problems through particle swarm optimization, Natural Computing, 1, 235-306. J. Rajesh, K. Gupta, H.S. Kusumakar, V.K. Jayaraman, B.D. Kulkarni, 2001, Dynamic optimization of chemical processes using ant colony framwork, Computers and Chemstry, 25, 583-595.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A-priori Identification of Critical Points for the Design and Synthesis of Flexible Process Schemes Zorka Novak Pintarič, Zdravko Kravanja University of Maribor, Faculty of Chemistry and Chemical Engineering, Smetanova 17, SI-2000 Maribor, Slovenia
Abstract An approach for the a-priori identification of vertex and nonvertex critical points needed for the design and synthesis of flexible chemical plants is presented. The critical values of uncertain parameters are identified by maximizing design variables one-by-one, whilst simultaneously optimizing the economic objective function. In this procedure, the uncertain parameters are transformed into the continuous variables between prespecified lower and upper bounds. This task can be accomplished in two alternative ways: firstly, with an explicit derivation of Karush-Kuhn-Tucker (KKT) optimality conditions and secondly, with a two-level formulation. The values of the uncertain parameters obtained are then merged into the final set of critical points. These are then used for the discretization of infinite uncertain problem, in order to obtain flexible design at the optimum objective function. Keywords: uncertain parameters, flexibility, design, critical points
1. Introduction The problem of designing flexible process schemes involves the selection of design variables so that the control variables can be adjusted for every realization of uncertain parameters, in such a way that feasible operation is achieved at the optimum objective function. The most common approach to this task is two-stage programming (recourse) formulation where design variables are selected at the design stage, while optimizing the expected objective function. At the operating stage, the control variables are determined in order to achieve feasible operation [1, 2]. The solution to this problem is usually concerned with the discretization of the original model in order to firstly, evaluate the expected objective function and secondly, to assure design flexibility. As discretization increases the dimensionality of the problem significantly, an approach has been proposed [3] where the expected value of the objective function was approximated at the single central basic point (CBP) while flexibility was assured at the selected vertex points. The latter were identified through the sequential examination of vertices which was time consuming for nontrivial design problems. Another drawback is that in some cases only vertex points may not guarantee feasibility. In order to circumvent both problems, we propose an approach where flexibility is assured by applying minimum set of (non)vertex critical points. The main emphasis of this paper is thus on an a-priori systematic identification of critical points without enumeration of all vertices. For the sake of simplicity the objective function is evaluated at the nominal point. Hallemane and Grossmann [4] defined the critical points as a combinations of those uncertain parameters that limit their variations. In this work, the term critical point relates to those values of uncertain parameters that determine the optimal oversizing of process units for the prespecified range of uncertainty. Critical points are identified by
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maximization of the design variables, while simultaneously optimizing the economic objective function. This idea was realized firstly by means of the heuristic approximate one-level procedure [5]. In this contribution two alternative and more rigorous methods are proposed: the KKT, and two-level formulations. Both methods are able to recognize critical points in-advance without exhaustive enumeration of numerous points. Two examples will be presented to illustrate the proposed approach.
2. Identification of critical points The mathematical model of the design problem under uncertainty is, in general, an infinite problem:
min Z = C ( z,d,θ ) z,d
s.t. g ( z,d,θ ) ≤ 0 d = gd ( z,θ )
(P1)
z,d ∈ R, θ L ≤ θ ≤ θ U In the model (P1), θ is a vector of uncertain parameters that can take any value between the lower and upper bounds, θL and θU. z and d are the vectors of the control variables and the continuous design variables (sizes of process units), respectively. Z is a scalar objective variable, C the economic objective function, g a vector of inequality constraints and gd represents the design specifications. For the sake of simplicity, the state variables are eliminated from the problem and expressed as the implicit functions of z, d and θ. The problem (P1) is usually solved through the discretization of infinite uncertain space over the set of points. These have to be selected so as to assure desired design flexibility at the optimum objective function. The main idea of this work is to identify in advance the points with the largest values of design variables (critical points) for the specified uncertain domain. This is done by maximizing design variables one by one, while allowing uncertain parameters to obtain any value between the specified bounds. As many uncertain points exist with the same maximum value of particular design variable, the point with the optimum objective function should be selected so that the optimality of the final solution is not compromised. 2.1. The KKT formulation In order to identify the critical points, the uncertain parameters in the infinite problem (P1) are transformed into the continuous variables between the lower and upper bounds. If the KKT optimality conditions of the problem (P1) are then written with respect to the control variables, z, and the design variables, d, the problem (P2)i is obtained, in which the uncertain parameters (now variables) are the only degrees of freedom. In (P2)i, IC represents the index set of inequality constraints, gs, μs are the corresponding Kuhn-Tucker multipliers and λi are the Lagrange multipliers of design constraints, gd,i. The model (P2)i is then solved for the maximization of each di, i∈DV, yielding, as a result, the values of uncertain parameters at which the maximum di is obtained at the minimum cost function, C.
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max di θ
∂ ( g d,i − di ) ∂g ∂C + ∑ μ s s + ∑ λi =0 ∂z s∈IC ∂z i∈DV ∂z
∂ ( g d,i − di ) ∂g ∂C + ∑ μ s s + ∑ λi =0 ∂d s∈IC ∂d i∈DV ∂d g s ( z,d,θ ) ≤ 0
μ s g s ( z,d,θ ) = 0
(P2)i
∀i ∈ DV
s ∈ IC , i ∈ DV
g d,i ( z,θ ) − di = 0
μ s ≥ 0, z , d , λi , μ s , θ ∈ R and θ L ≤ θ ≤ θ U 2.2. The two-level formulation Applying KKT conditions explicitly to large and complex problems is impractical. However, the KKT derivation can be avoided by using the following iterative two-level problem for identification of critical points: Lower (control-design) level at iteration k for design variable di, i∈DV:
min Z ( z , d , θ k -1 ) = C ( z , d , θ k -1 ) z,d
g ( z , d , θ k -1 ) ≤ 0 d = gd ( z , θ k -1 )
(LLP)ik
z,d ∈ R Upper (uncertainty) level at iteration k for design variable di, i∈DV:
max di ( z k ,θ ) = g d,i ( z k ,θ ) θ
(ULP)ik
θ ∈ R and θ L ≤ θ ≤ θ U At the lower level, the uncertain parameters are fixed to the values obtained at the upper level of previous iteration, θ k-1, while the control and design variables are adjusted so as to optimize the objective variable, Z. The optimum control variables obtained at this level are then fixed at the upper level, zk, where the design variable under study, di, is maximized with uncertain parameters relaxed into the continuous variables. The uncertain point obtained is then used at the lower level of the next iteration. The value of the design variable obtained at the upper level represents the upper bound, which is compared with the one obtained at the lower level. The algorithm continues until the values of the design variable being studied and the uncertain parameters at both levels, equalize.
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3. Optimization for flexible design The points obtained by solving the KKT or two-level formulations for all design variables are finally merged into the set of critical points (CP) based on the influence of uncertain parameters on the design variables as described in [5]. Systematic setcovering formulation [6] can also be applied. The set of critical points is used in the flexible design problem (P3) to assure flexibility, while the expected objective function in this work is approximated at the nominal point, θN:
min Z N = C N ( x N , z N , d , θ N )
xN , zN ,d
s.t. h( x N , z N , d , θ N ) = 0
h( xc , zc , d , θc ) = 0
g( x N , z N , d , θ N ) ≤ 0
g ( xc , zc , d , θc ) ≤ 0
d ≥ gd ( x N , z N , θ N )
d ≥ gd ( xc , zc , θc )
c ∈ CP
(P3)
x N ,z N , xc , zc , d ∈ R, θ L ≤ θc ≤ θ U 4. Example 1 - vertex critical points This example considers a heat exchanger network composed of four hot and three cold streams (Fig. 1). The problem comprises 10 uncertain parameters with the nominal values and bounds as given in Table 1. The mathematical model includes the equations for heat balances, constraints on temperature differences and design correlations for the heat transfer areas of eight exchanger units. The total cost is minimized, composed of operating and annualized investment costs. The two-level formulation was applied on this example, however, both formulations will be illustrated in detail in an extended paper. 4.1. Identification of critical points The results of the two-level formulation are given in Table 2 where superscripts L and U denote the lower and upper values, respectively, while 0 indicates that uncertain parameter has no effect on the design variable.
Figure 1. Heat exchanger network of Example 1
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Table 1: Uncertain parameters of Example 1
θL θN θU
T1 K 490 500 510
T5 K 410 430 450
T11 K 328 340 352
CF2 kW/K 15 18 21
CF7 kW/K 18 20 22
U1 kW/m2K 0.33 0.41 0.49
T9 K 288 300 312
CF4 kW/K 15 17 19
U6 kW/m2K 0.41 0.48 0.55
T9 K 0 312U 0 0 0 288L 0 0
CF4 kW/K 0 0 19U 19U 19U 0 15L 0
U6 kW/m2K 0 0 0 0 0 0.41L 0 0
TCWIN K 295 300 305
Table 2: The results of two-level formulation max A1 A2 A3 A4 A5 A6 A7 A8
T1 K 510U 0 0 0 0 0 0 490L
T5 K 0 450U 0 0 0 410L 0 0
T11 CF2 K kW/K 0 0 0 0 352U 21U 352U 0 0 0 0 0 328L 15L 0 0
CF7 kW/K 18L 0 0 0 0 0 0 22U
U1 kW/m2K 0.33L 0 0 0 0 0 0 0
TCWIN K 0 0 0 0 305U 0 0 0
From the above results, the set of the two critical points was identified by applying the set covering formulation, θcC=(T1, T5, T11, CF2, CF7, U1, T9, CF4, U6, TCWIN): θ1C=(510, 410, 352, 21, 18, 0.33, 288, 19, 0.41, 305), and θ2C =(490, 450, 328, 15, 22, 0.33, 312, 15, 0.41, 305). 4.2. Optimization for flexible design Finally, the discretized problem was solved simultaneously at the nominal point and two critical points giving the cost of 933 119 $/yr and A=(304, 186, 91, 28, 55, 40, 6, 16) m2. It should be noted that exactly the same result would be obtained by solving the problem at the nominal point and all 210=1024 vertices. This indicates that our approach reduces the number of points by several orders of magnitude, without compromising the flexibility and optimality of the result. Namely, the flexibility index of the solution, determined through 1024 vertex points, is equal to or greater than 1 which indicates that the design obtained has exactly the desired flexibility for feasible operation over defined intervals of uncertain parameters. The size of the discretized model when applying only two critical points together with the nominal point is 134 variables and 97 constraints. When applying the entire set of vertices the size is considerably larger, around 42 000 variables and 33 000 constraints. In the first case, the solution consumes less than 1 s of CPU time on a PC with a 1.6 GHz Intel Pentium 4 processor and 0.512 MB RAM, however, in the second case the CPU time is around 90 s. The convergence characteristics of the proposed two-level algorithm are also favourable. Only two or three main iterations were needed for all design variables in order to equalize their values, as well as the values of uncertain parameters obtained at the two levels.
5. Example 2 - nonvertex critical points In this example, a new uncertain parameter θF, was added to ten uncertain parameters
from previous example, with the lower, nominal and upper values being [0.7, 1, 1.3], respectively. It is assumed that the heat transfer coefficient U8 varies with θF in the following way: U 8 = −35 ⋅ θ F3 + 105.8 ⋅ θ F2 − 103.6 ⋅ θ F + 33.4 .This constraint then imposes the minimum U8 and thus, the maximum A8, at the nonvertex critical point of θF=0.838.
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5.1. Optimal design with vertex points By applying the whole set of 211=2048 vertices and the nominal point simultaneously, the cost obtained amounts to 930 410 $/yr. The areas of the exchangers are A=(304, 186, 91, 28, 55, 40, 6, 14) m2, however, the flexibility index of this design determined at the vertices and at the inner points is smaller than 1. This implies that the flexibility of the design determined by means of vertex points is insufficient. 5.2. Optimal design with the reduced set of critical points The two-level formulations were applied for 8 design variables. Two critical points were identified, θcC=(T1, T5, T11, CF2, CF7, U1, T9, CF4, U6, TCWIN, θF): θ1C=(510, 410, 352, 21, 18, 0.33, 288, 19, 0.41, 305, 0.838), and θ2C =(490, 450, 328, 15, 22, 0.33, 312, 15, 0.41, 305, 0.838). Simultaneous optimization at the nominal point and two critical points yields the cost of 949 710 $/yr. The areas of the exchangers 1-7 are equal as in 5.1. while A8 amounts to 31 m2 (compared with only 14 m2 above). The flexibility index of this design determined at vertices and at the inner points is equal to or greater than 1. This then indicates that minimal value of U8 was recognized properly in this case and that exactly desired flexibility of the design is assured over the specified uncertain space when applying the identified critical points. However, as shown in 5.1., ignorance of inner critical point leads to a nonflexible design with an underestimated cost function.
6. Conclusions The proposed KKT and two-level formulations are able to recognize vertex and nonvertex critical points and avoid explicit enumeration of numerous points. Both formulations identify the critical points by relaxation of uncertain parameters into the continuous variables and, maximization of design variables while simultaneously optimizing the objective function. These algorithms are executed only for each design variable generating the critical points, whose number is less than or equal to the number of design variables. Additional advantage of the two-level procedure is that it avoids explicit derivation of the KKT conditions. This makes it suitable for handling more complex and, possibly, larger problems. Optimal design problem is formulated and solved simultaneously at the nominal point and at the critical points to establish a compromise between the economic objective and the desired flexibility. Although the exactness of the two-level approach has not yet been proved theoretically the results of the examples indicate that optimal and feasible designs are obtained with minimum set of identified critical points.
References 1. 2. 3. 4. 5.
E.N. Pistikopoulos and M.G. Ierapetritou, Comput. Chem. Eng. 19 (1995) 1089. G.M. Ostrovsky, Y.M. Volin and M.M. Senyavin, Comput. Chem. Eng. 21 (1997) 317. Z. Novak Pintarič and Z. Kravanja, Comput. Chem. Eng. 28 (2004) 1105. K.P. Halemane and I.E. Grossmann, AIChE J. 29 (198) 425. Z. Novak Pintarič and Z. Kravanja, Proceedings on the European Symposium on Computer Aided Process Engineering-15. Elsevier, 20A (2005) 91. 6. I.E. Grossmann and R.W.H. Sargent, AIChE J. 24 (1978) 1021.
Acknowledgements The authors are grateful for the financial support of the Slovenian Ministry of Higher Education, Science and Technology (Program P2-0032 and Project J2-6637).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Using water cascade analysis to synthesize water use network in batch process Shaoguang Wang, Shiqing Zheng, Xia Yang, Yugang Li Research Center for Computer and Chemical Engineering, Qingdao University of Science and Technology, No. 53 Zhengzhou Road, Qingdao 266042, China
Abstract A new method based on water cascade analysis (WCA) is presented to synthesize water use in batch process in this paper. This method includes two key steps: determine the target and design the water network. The fresh water demands for overall process or certain time intervals with and without storage tank is identified. Also the determination of overall pinch point, the storage capacity and the storage tank numbers is discussed. A two dimensional water network diagram, the time-purity level diagram, is introduced to help target and design the water network. We use two cases to illustrate the method, the results show it can achieve the optimal batch process water network efficiently. Keyword: Water use network, Batch process, WCA, Water minimization
1. Introduction Water resource and pollution have become a serious problem in the modern world. The pressure of the water shortage, the drive toward the environment sustainability and increasing water treatment cost encourage more process plants to find new ways to reduce the fresh water consumption and wastewater generation. Batch process plays an important role in process industry and accounts for about 4050% of chemical industry, most of batch process industries consume huge amounts of water and generate almost the same amounts of wastewater. In batch process, the operation condition and resource consumption vary with time, and the stream of mass or energy is discontinuous. All these cause the batch process very complex. Over the two decades, the development of systematic techniques for water reduction, reuse, and recycling within a process plant has seen extensive progress for continuous process, but the research for batch is much less. Grau, Graells (1996) developed a mathematical technique for waste minimization with emphasis on waste generated during changeover. A similar technique was developed by Almat´o, Sanmart´, Espu˜na (1997). This technique utilizes storage tanks to override the time constraint in the exploration of reuse and recycle opportunities. Maria Almató, Antonio Espuña (1999) adopted the mathematical programming to optimize the network and get the assignment coefficient between tank and operation. Yao and Yuan (2000) developed a discrete time mathematical model for waste minimization in batch processes by optimizing production campaigns. The methodologies mentioned above can be broadly classified into mathematical and graphical techniques. The advantage of mathematical technique lies in its ability to solve multi-contaminants and multi-objectives problems. But it can’t guarantee the optimality of the solution for its non-convexity. The graphical techniques such as watersurplus and composite curve need trial-error procedure to find the pinch point and water target containing complicated iterative steps.
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The numerical table technique discussed in this paper is a improved WCA technique which can determine the pinch location and water target in maximum water recovery (MWR) network. It is also a kind of WPA(Water pinch analysis).
2. Problems description and assumptions Given a set of water using operations running in batch mode, design the water reuse network so that the fresh water consumption and wastewater generation can be minimized. The assumptions are listed as below: 1. Maximum outlet concentration. Only in this way, can the fresh water demand and waste water be minimized. 2. Mass load transfer should be fulfilled because it is the main task of the process. 3. Optimal scheduling. Each operation has its own fixed starting and end time. 4. The limiting flowrate requirement should be satisfied. The targeting of minimum fresh water and water network design with and without storage tank will be discussed below. How to determine the number and capacity of the tank will also be explored because of its relation with utility investment and cost. In batch process, the time dimension should not be neglected. Thokozani Majozi(2005) illustrates the difference of using reused water between batch and continuous process.
3. Water cascade analysis (WCA) WCA will be introduced to establish the minimum fresh water target. In each time interval, the fresh water demand will be identified. Sum that of all the time intervals up and then the overall water demand will be determined. Then apply this method to the batch process with and without storage tank. To achieve the objective, one has to establish the net water flow rate as well as the water surplus and deficit at different water purity levels within the process under study. The interval water balance table(IWBT) has been introduced for this purpose. In the next section, we demonstrate how the WCA technique serves as a good utility targeting tool for a batch MWR network through a literature case study. 3.1. Case study 1 Case study 1 is taken from Dominic Chwan(2005), the time interval is listed in table 1. The first step in the WCA is to set up the IWBT to determine the net water source or demand at each purity level. The second is to establish the time-dependent WCD (Water Cascade Diagram) and WCT (Water Cascade Table）. Table 1 Time interval for case 1 Number of
Time interval
Duration of time
Water Demands,
Water Sources,
interval, k
(h)
intervals, (h)
WSD,i (ton)
WS,i (ton)
1
0.0-1.0
1.0
WD,1=20
WS,1=20
2
1.0-3.0
2.0
WD,2=200, WD,4=20
WS,2=200, WS,4=20
3
3.0-3.5
0.5
WD,2=50, WD,3=20
WS,2=50, WS,3=20
4
3.5-5.0
1.5
WD,3=60
WS,3=60
3.1.1. The determination of overall pinch point The processes to determine the overall pinch are shown in Fig 1. The value in the rectangle is the amount of water demand or resource at each purity level. C is the contaminant concentration (in ppm). P is the purity deduced from the contaminant
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concentration. “-” represents net water demand, “+” means net water resource. MFW，k is the fresh water demand in time interval k. So if the overall pinch exists, the total amount of fresh water demand is 185 ton. The pinch purity level is at P=0.999900 where cumulative pure water surplus is 0, achieved through the same procedure as Fig 1.
Fig.1 Water cascade diagram and pure water cascade for WCA technique (whole period)
3.1.2. Without storage tank In water using process, time constraint is impossible to break. So water from some process units can not be reused or only partially be reused even if its purity can meet the requirement of other process inlets. For these processes, the overall pinch doesn’t exist but the local point of time interval does. For conventional WPA, the processes above the pinch dose not discharge the effluent and the process below the pinch does not use fresh water. But in terms of each time interval in batch process, this is not always the case. For the process above the pinch, the water from its outlet will be discharged as effluent if it can’t be reused during the relevant period. For the process below the pinch, the water demand will be satisfied with fresh water only if it can’t be met by the reused water during certain period. So for batch process without storage tank, there exists the pinch for time interval, namely local pinch. The target for each time interval can be determined initially according to the WCA procedure. Completely above the local point in some time intervals, the water demand can only be satisfied by fresh water, so the initial water target can not be reduced. But for the time intervals that some water streams coexist, it is possible to reuse one stream outlet to be another stream inlet directly, so the initial fresh water demand can be reduced. In this case, the local pinches for the three preceding time intervals are all at P=0.9999. Only for the fourth time interval, the pinch is at P=0.9992. We introduce the two-dimensional time-purity level water network diagram, in which time is set to be X-dimensional, purity is set to Y-dimensional. Through the analysis of the maximum outlet concentration, mass load transfer and limiting flowrate requirement, the complexity of the initial water network can be reduced. The two-dimensional time-purity level water network diagram for case 1 is shown in Fig. 2. During 0-1.0h, there is only WD,1, the fresh water demand is 20 ton, it can’t be reused because no other water demand in the same time interval, so has to be discharged. During 1.0-3.0h, there are WD,2 and WD,4, the fresh water demand is 100 ton, WD,4 lies below the pinch and its inlet purity requirement is lower than the pinch purity, so it can reuse WD,2, the reused water amount is 11.42 ton, which can be calculated according to the mass load transfer requirement. During 3.0-3.5h, WD,2 is completely
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above the local pinch and needs 25 tons of fresh water. WD,3 goes through the local pinch, so it can be satisfied by fresh water and reused water (10 ton of fresh water and 20 ton water from WD,2). During 3.5-5.0h, WD,3 is completely above the local pinch, it must be satisfied by fresh water, 47.5ton. So the overall fresh water demand is 202.5 ton, leading to further reduction by 46.5% compared with Manan ZA, Foo DCY, Tan YL (2004). Then the water network is designed accordingly, as shown in Fig 3.
Fig 2 Time-Purity level water network diagram without storage system
Fig 3 Time-water network diagram for batch water-using process without storage tank
From the procedures discussed above, it can be seen that the targeting and network design for batch process without storage tank are different from conventional WPA due to time constraint. The obvious difference is that the targeting and network design for these processes should take place simultaneously. In addition to that, some water from the processes above its local pinch has to be discharged because of time constraint. This is also an important difference from the WPA for continuous process. It can be easily seen from above that the approach of Foo and Manan neglects the features of batch process. For the flowrate of batch process unit it is actually unnecessary to keep it even through the whole process under the preconditon of the limiting flowrate being guaranteed.The water flow for process 2 is fixed, but to maximize the reuse water, some water can be transfer from the previous time interval to later one. So the approach presented in this paper which combine WCA with diagram proves better. 3.1.3. With storage tank With storage tank existing, the water generated in the previous time intervals can be stored and reused in later processes. So the water consumption can be lowered. Some constraints can be broken. So the targeting of the minimum fresh water and design of water network can proceed just like the WPA for continuous process. From the above we can see that only the local pinch of 3.5-5.0h time interval is not consistent with the overall pinch. When MFW=56.25(initial result), the pure water surplus is 0.002625 at the overall pinch purity level, which corresponds to fresh water surplus 26.25 ton, so the fresh water demand of this interval is 56.25-26.25=30. The fresh water demand for each time interval with storage is shown in Fig 4. 20 ton
D1/S1 20 ton 125 ton
73.58 ton
D2/S2 20 ton
185ton
40 ton
ST
20 ton 185 ton
D3/S3 11.42 ton 11.42 ton
D4/S4
0
Fig 4 Time-Purity level two-dimensional water network
1.0
2.0
Time(h)3.0
4.0
5.0
Fig 5 Time water network
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water-using process with storage tank
To design the water network with storage, the following problems must be solved: (1) Determine the number of the storage tanks It can be easily seen from Fig 4 that WD,1 and WD,2 are above overall pinch point and the water from their outlet can be reused, and WD,3 and WD,4 reused water from WD,2, so there needs one tank to store water from WD,1. (2) Determine water demand for each time interval Targeting process for time interval 0.1-1.0h,1.0-3.0h and 3.0-3.5 are the same with the section 3.1.2. During 3.5-5.0h, WD,3 can be satisfied by stored water in tank. So the total reused water can be calculated based on the mass load transfer requirement, namely 40 ton. (3) Determine the amount of stored water The storage of the spent water in the 3rd time interval can be reused in the 4th time interval. The local pinch of the 4th time interval rises and is gradually consistent with the overall pinch. So the bottleneck constraint of the time to process is broken and the initial fresh water demand is reduced. And the water network is designed, as shown in Fig 5. The time-purity level diagram is easier to be used in designing the water network than the approach of Foo and Manan because it can illustrate the relationship among purity, water-using process and time, and can underline the importance of pinch point of the water network simumanteously. 3.2. Case study 2 Case 2(Wang YP, Smith R.1995) was introduced to demonstrate the versatility of WCA. 3.2.1. Without storage tank The fresh water demand for each time interval: 0-0.5h: 40 ton, the local pinch is at P=0.9998; 0.5-1.0h: 43.75 ton, the local pinch is at P=0.9996; 1.0-1.5h: 37.5 ton, the local pinch is at P=0.9996. The water using processes are all above their respective local pinch. So the water demands have to be satisfied by fresh water. The fresh water target is 121.25 ton. The water network designed is shown in Fig 6. 3.2.2. With storage tank The final fresh water target is 102.5 ton. P=0.9998 is the overall pinch purity. The local pinch should be subject to overall pinch. In 0.5-1.0h time interval, the local pinch purity is lower than the overall pinch purity. At overall pinch purity, the cumulative pure water surplus is 0.00125, which corresponds to fresh water surplus 0.00125/0.0002=6.25. So the fresh water demand for this time interval is 43.756.25=37.5. Similarly, the fresh water demand is 25 ton for 1-1.5h time interval . The water design result is the same as the literature.
68.75 ton
D1/S1 81.25 ton
121.75 ton
40 ton
40 ton
D2/S2
121.75 ton 12.5 ton
12.5 ton
0
D3/S3 0.5
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Fig 7 Batch network design for case study 2 without storage system
4. Conclusion A new method to determine the maximum water recovery network using WCA technique is discussed in this paper. The two-dimensional time-purity level diagram is introduced to help WCA synthesize water network in batch process with and without storage tank. The results of case1 and 2 indicate the effectiveness of this technique. This approach can avoid the difficulty in getting the optimal solution by mathematical programming and tediousness of graphical techniques. It is a promising approach because it is simple and convenient. And we conclude: (i)If the local pinch purity of a time interval is higher than or equal to the overall pinch purity, the water in this time interval can be stored to be reused. If the local pinch purity of a time interval is lower than the overall pinch purity, it means pure water surplus. It can utilize some water resources with higher pinch purity which occurs prior to it. (ii) During the process of targeting the MWR network, the overall pinch purity is much more important than local pinch. The overall pinch divides the total network into two main thermodynamic regions. The region above overall pinch is the most constrained part and is the bottleneck of the maximum water recovery network. (iii)There only exists local pinch for the water using process without storage tank. If the water using process is completely above the pinch, its water demand can be met only by fresh water. If it is completely below the pinch, its water demand can be satisfied by reused water. If it is partially above the pinch and partially below, its water demand can be satisfied by fresh and reused water. (iv) Compared to other methodologies, water storage is relatively simple and not only increases the water reuse opportunities, but also weakens the dependency on water availability.
Reference Dominic Chwan Yee Foo, Zainuddin Abdul Manan and Yin Ling Tan. Synthesis of maximum water recovery network for batch process systems. Journal of Cleaner Production, Volume 13, Issue 15, December 2005, Pages 1381-1394 Grau, R., Graells, M., Corominas, J., Espun˜a, A., & Puigjaner, L.(1996). Global strategy for energy and waste analysis in scheduling and planning of multiproduct batch chemical processes. Computers and Chemical Engineering, 20, 853–868. Kim J-K, Smith R. Automated design of discontinuous water systems. Transactions of the Institute of Chemical Engineers, Part B 2004;82(B3):238e48. Manan ZA, Foo DCY, Tan YL. Targeting the minimum water flow rate using water cascade analysis technique. AIChE Journal 2004;50(12):3169-83. Maria Almató, Eduard Sanmartí, Antonio Espuña, Rationalizing the water use in the batch process industry, Computers chem. Engng., 1997, Vol. 21, Suppl., S971-S976 Maria Almató, Antonio Espuña, Luis Puigjaner. Optinization of water ude in batch process Industries. Comp. Chem. Engng. [J], 1999, 23:1427-1437 Thokozani Majozi. Wastewater minimisation using central reusable water storage in batch plants .Computers & Chemical Engineering, Volume 29, Issue 7, 15 June 2005, Pages 16311646 Yao, Z. L., & Yuan, X. G. (2000). An approach to optimal design of batch processes with waste minimization. Computers and Chemical Engineering, 24, 1437–1444. Wang YP, Smith R. Time pinch analysis .Transaction of the Institute of Chemical Engineers, Part A 1995;73:905-14
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Multiobjective optimization of multipurpose batch plants using superequipment class concept Andrej Mosat, Laurent Cavin, Ulrich Fischer, Konrad Hungerbühler Safety & Environmental Technology Group, Institute of Chemical and Bioengineering ETH Zürich, ETH-Hoenggerberg, 8093 Zurich, Switzerland
Abstract We present a novel approach for solving different design problems related to single products in multipurpose batch plants: the selection of one production line out of several available, additional investment into an existing line or plant, and grass-root design of a new plant. Multiple objectives are considered in these design problems. Pareto-optimal solutions are generated by means of a Tabu Search algorithm. In the novel approach the concept of superequipment has been defined as an abstract model, which is capable of performing any physico-chemical batch operation. Each superequipment is transformed into a real equipment unit, for example a reactor, during or after the optimization in order to evaluate performance parameters of a design. This novel concept uses an implicit definition of a superstructure and essentially optimizes on the transfers between different equipment units. On the basis of two case studies we demonstrate that the application of the superequipment concept offers a number of advantages for the investigated design problems. The comparison with optimization results obtained with a conventional Tabu Search algorithm revealed that the superequipment method identifies the Pareto-optimal solutions in significantly reduced computation time. Keywords: Tabu Search, multiobjective optimization, batch process, superequipment
1. Introduction Chemical companies currently face different problems related to the global market changes. In speciality chemicals and pharmaceuticals production customers require smaller amounts of product orders, faster delivery times and on demand production. Some of the tasks which are closely related to industrial problems of batch processing can be solved by help of optimization techniques. Usage of such methods allows for faster screening of production capacities, plant line selection for given product, decision making, planning and assessment related to the plant and process. Relatively few articles have been presented that assess the optimal design for a single batch process. In the optimization of multipurpose batch plant designs (MPBP), Wellons and Reklaitis (1989, 1991) proposed a MINLP method for scheduling and optimization. Povoa and Macchietto (1993) described the possibilities of using combinatorial optimization for solving design related problems in multipurpose batch plants. Cavin et al. (2004, 2005) presented a Tabu Search algorithm for identifying the optimal design of a single process in a multipurpose batch plant. We present a novel approach for solving different design problems related to single products in multipurpose batch plants using the concept of superequipment as an abstract model utilizing a virtual unit, which is capable of performing any physicochemical batch operation. Each superequipment is transformed into a real equipment unit, during or after the optimization in order to evaluate performance parameters of a design. This novel concept will be demonstrated on two case studies.
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2. Multipurpose Batch Plant Optimization using Superequipment Class 2.1. Problem Definition The new approach is used for solving multipurpose batch plant design problems in which is given a chemical recipe (i.e. a series of chemical/physical tasks, the capacity requirements for each task of recipe per unit of final product, the base duration of each task at the input scale, recipe constraints), plant and equipment data (i.e. equipment description including detailed specifications such as e.g. operating T-P ranges), economic data (i.e. detailed cost composition on campaign basis and investment costs where applicable), design heuristics (i.e. which equipment class is capable of performing which recipe operation classes) and one or more objective functions. The objective is to determine a set of Pareto-optimal and structurally diverse layouts for the process, i.e. allocation of recipe tasks to equipment units, structure and order of the final recipe (e.g. in parallel or in series use of units) 2.2. Superequipment If we think about the retrofit problem of additional investment into an existing plant line, we see that the combinations of equipment on buy list and their characteristics (i.e. unit size, lining material, options, TP range) require exponential solving time. During a standard Tabu Search (TS) optimization it is necessary to generate a number of combinations in form of designs, where the allocation of each new unit should be varied in the design in order to have a good chance of finding the global optimum. Superequipment concept has been developed as alternative approach. It simplifies the combinatorial problem, because one superequipment unit substitutes any unit from the buy list. Superequipment is not a real unit. It stands for a model of a unit, where each piece of superequipment can be transformed into a real apparatus in the final design. In extension of a standard TS approach (see Cavin et al., 2005) the superequipment class S is defined in the way that any operation from the operation classes present in the operation-to-equipment assignment matrix A can be conducted in the superequipment: S := ∪ Ai .EqClass
(1)
i
We define a superequipment unit as an unit belonging to the equipment list E: Superequipment _ unit := ( E .EqClassID = S )
Figure 1: Transformation of a superquipment unit into a real equipment unit.
(2)
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Each piece of superequipment used in a design must be transformed into an existing real unit at some point during the optimization. The superequipment transformation example in Figure 1 shows three operations, where the second operation is conducted in the superequipment (original design). After transformation two possible solutions exist - a reactor (middle design) or an extractor (bottom design). This flexibility in the class type, size, lining material is maintained through the optimization process up to the final results list, so that the decision maker can see all possible proposals. 2.3. Optimization Algorithm Formulation The method aims at finding the optimal assignment of recipe blocks into given equipment units. The transfers between the units are also determined and define the moves of Tabu Search. As an input, a base case layout and initial batch size, cycle time, operation durations, temperatures, pressures and other data are required. During each iteration one design is altered and all neighbors originating from that particular design are evaluated for their parameters. The design parameters, such as task durations, volume and time requirements for each block are adjusted and scaled accordingly. The algorithm has been designed for handling multiple objectives, where prioritized optimization objectives can be selected from a list comprising e.g. production rate, number of equipment units in design, batch size, productivity per total nominal volume of significant equipment units used in a design, net present value (NPV) of a project with or without investment and payback period. More details on the algorithm and its mathematical implementation can be found in Cavin et al. (2005).
3. Results of Case Studies and Discussion The novel concept using superequipment class will be applied to two problems, i.e. investigating possible investment into an existing line and the selection of one production line from a number of available facilities. 3.1. Investment Scenario for an Existing Plant and a Given Recipe The Vitamin C case study is based on the Reichstein synthesis and was selected for its simplicity to demonstrate the basic principle of superequipment concept. We define a small batch plant (Plant C4) and examine investment possibilities in order to maximize the NPV of a project with investment. There is only space for two additional unit installations within the production building. A comparison of superequipment concept and conventional TS optimization (Cavin et al., 2004, 2005) is presented. The recipe comprises 8 blocks each consisting of several unit operations. The base plant C4 consists of 5 reactors (1 x 10 m3, 4 x 6.3 m3) and one 1.2 m3 centrifuge. The plants
Figure 2. Investment case study: optimal designs for base plant C4 (left) and plant C4S considering investment as identified with superequipment approach (right).
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Table 1: Investment case study using superequipment approach: subset of Pareto-optimal designs. Design Investment rank [#] 1 react+centr. 35 centrifuge 49 react.+cryst.
Equip. Investmen Productivi Nr. of size t ty units [m3] [kUSD] [kg/h] [pcs.] 10; 1.2 820 27.8 8 1.2 340 23.1 7 16; 6.3 800 21.3 8
Payback time [year] 6.6 12.5 7.2
NPV
Campaign time [kUSD] [years] 1300 4.1 1130 4.9 1180 5.4
C4r, C4ce and C4rce used for conventional optimization have one 10 m3 reactor, one 1.2 m3 centrifuge, or both units in addition to the base plant, respectively. The plant C4S consists of the base plant plus two superequipment units that can be transformed into reactors or centrifuges. For the base plant (Plant C4) the optimal production rate is 20.2 kg/h when implementing a design with 6 equipment units (see Figure 2 left). In examining possible investment options two superequipment units are added to this plant, resulting in designs with zero, one or two superequipments in addition to the units from plant C4. A selection of feasible investment options is listed in Table 1, where the designs are Pareto-optimal in the listed objective functions. The best design according to productivity and NPV utilizes additionally one 10 m3 reactor and one 1.2 m3 centrifuge (see Figure 2 right). Objective function values are: productivity 27.8 kg/h, batch size 0.24 t/batch, and payback time of 6.6 years. Cycle time is reduced from 730 min to 525 min by better utilizing the equipment and parallel assignment of the two centrifuges. For the standard optimization the mean CPU time required for finding the apparent global optimum, i.e. the best value found in a specified number of runs, was 3510 s. The superequipment method requires on average 2430 s; here the problem is less constrained and requires less iterations to reach the apparent global optimum. Furthermore the superequipment approach has the significant advantage that only the maximum number of additional units has to be specified for the investment scenario while an explicit definition of a larger number of equipment is required in the conventional approach. 3.2. Plant Selection for given Recipe This case study shows production of a fine chemical used in the pharmaceutical and photo industry. The synthesis is based on a reactant known under commercial name Quinaldine. The recipe and basic process simulation of 4-(2-quinolinylmethoxy)phenol, referred to as product H, have been presented by Petrides et al. (2002). The aim is to show how in a single run diverse plant lines can be optimized by means of superequipment method and can be compared with each other. The recipe comprises 12 blocks each consisting of several unit operations. Three production lines are available: C10 (8 reactors of different size, 4 centrifuges, 4 filters, 0 crystallization units), Q2 (11 / 2 / 3 / 2), and C11 (21 / 8 / 9 / 5). The superequipment plant consists of 24 superequipments to be transformed into equipment from each line. Table 2 provides a comparison of parameters of designs mapped completely to the three plant lines. Design #3 is the overall best in terms of productivity. This is also the best design found for plant C11. The large number of 24 units used for production implies increased costs for this design (NPV=23.0 mio. USD) and thus not reaching the optimum for the NPV. This design is actually performing all recipe blocks in parallel and thus creating overhead in cleaning and labour demand. The productivity per total nominal volume of significant equipment is low (0.71 kg/(h m3)).
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Table 2: Plant selection case study: subset of designs as obtained with superequipment approach sorted according to productivity. Plant line IDs 1, 2 and 3 refer to lines C10, Q2, and C11, respectively. Bold numbers indicate optimal productivity in each line. Design ID (#)
Plant line ID (#)
Prod. rate [kg/h]
Batch size [t]
Nr. equip. [pcs.]
NPV [mio. USD]
3 7 1 4 1 2 4 1
3 3 1 1 3 2 2 2
135.1 133.0 130.9 130.9 90.3 74.0 68.8 58.4
1.27 1.24 1.60 1.60 0.97 0.69 0.63 0.63
24 21 13 14 13 13 14 13
23.0 22.8 23.3 23.2 22.1 21.0 20.6 19.3
The best productivity in the plant C10 can be achieved by design #1 being only 3% less effective than design #3, but it utilizes only 13 units, reducing the costs and reaching the optimum in NPV equal to 23.3 mio. USD for the whole campaign. Plant Q2 is dedicated for small productions and has no 10 m3 reactor, therefore the best achievable productivity is 74.0 kg/h with design #2. All of the resulting Net Present Values are rather similar, because of the high fraction of raw material and solvent costs. Thus the primary criteria for decision might be for example productivity, number of units used or batch size. If the campaign time is the main issue, design #3 should be implemented in plant C11. On the contrary, if the free capacity of a big plant (plant C11) was required for upcoming campaigns, design #1 in plant C10 could be a good compromise between performance, NPV and number of units used while keeping the large plant free. Note that the same design can have multiple instances in the results table. Since the equipment units differ in each plant, the performance indicators also differ. Figure 3 shows the relationships of productivity vs. number of used equipment units for the three plants. A subset of designs with best productivity for given number of units and given plant is displayed. A minimal number of 9 units is needed to process the 12 recipe blocks. The maximal number is 24 (design #3). The design #3 can be matched only with plant C11 without additional investment. The mapped designs show a trend of increasing productivity with increasing number of equipment. The best designs with 15 equipment units (Figure 3) do not include two centrifuges in parallel mode as in the best designs with 14 units. This increases the cycle time. If a design with 15 units and two parallel centrifuges is mapped into plant C10 or plant C11, an additional reactor (as compared to the best design with 14 units), which is placed in series with another reactor, becomes the volume bottleneck and no bigger unit is available. In the plant Q2, the apparent global optimum, i.e. the best value found in a specified number of runs, with 74.0 kg/h and 13 units has not been found with the conventional TS method (200 iterations without finding an apparent global optimum and 40 restarts requiring ca. 230 minutes). This can be explained by too constrained searching space and neighborhood, where the only possibility of diversification in current implementation of TS is in the restarts. However, superequipment method is less constrained due to the ''chameleon'' property of each unit and the diversification process is ensured naturally by moving through the solution space almost without limits.
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Production Rate [kg/h]
120 100
#1 80 60
#2
40 superplant plant C10 superplant plant Q2 superplant plant C11
20 0 8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Number of units used []
Figure 3: Plant selection case study: productivity vs. number of equipment units after matching superequipment designs to plants C10, Q2 and C11. Designs with maximal productivity for a given number of units are displayed. Designs #1-3 refer to Table 2.
In total 26009 s were needed to optimize the three plants with the standard TS while the superequipment plant optimization with 24 units and three lines required 11340 s. Again the superequipment concept offered practical advantages and reduced computation time.
4. Conclusions In the novel approach the concept of superequipment has been defined as an abstract model, which is capable of performing any physico-chemical operation. Each superequipment is transformed into a real equipment unit, for example a reactor, during or after the optimization in order to evaluate performance parameters of a design. Corresponding transformation heuristics need to be defined for each problem type. For different fields of application the superequipment concept means a considerable saving in computation time and effort because the optimization problems are reduced in size, repetitive optimization runs are avoided and the search space is less constrained.
References Cavin, L., A. Mosat, U. Fischer and K. Hungerbühler, 2005. Batch process optimization in a multipurpose plant using Tabu Search with a design-space diversification. Comput. Chem. Eng. 29: 1770-1786. Cavin, L., U. Fischer, F. Glover, and K. Hungerbühler, 2004. Multiobjective process design in multipurpose batch plants using a Tabu Search optimization algorithm. Comput. Chem. Eng. 28: 458-479. Petrides, D.P., A. Koulouris, and P.T. Lagonikos, 2002. The role of process simulation in pharmaceutical process development and product commercialization. Pharm. Eng. 22: 1-8. Povoa, A.B.P., and S. Macchietto, 1993. Optimal design of multipurpose batch plants. 1. Problem formulation. Comput. Chem. Eng. 17: S33-S38. Wellons, M.C, and G.V. Reklaitis, 1991. Scheduling of multipurpose batch chemical plants. 1. Formation of single-product campaigns. Ind. Eng. Chem. Res. 30: 671-688. Wellons, M.C., and G.V. Reklaitis, 1989. Optimal schedule generation for a single-product production line. I. Problem formulation. Comput. Chem. Eng. 13: 201-212.
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Integrated design of energy-saving chemical process systems: strategy, methods and implementation Gennady Ostrovskya, Yury Volina, Dmitry Dvoretskyb, Stanislav Dvoretskyb a b
Karpov Institute of Physical Chemistry, ul. Vorontsovo pole, 10, Moscow, 105064, Russia Tambov State Technical University, ul. Sovetskaya, 106, Tambov, 392000, Russia
Abstract New statements of the problem of integrated design of energy-saving multiproduct chemical process systems (MCPS) under uncertainty have been formulated. Approaches to the development of modified (fast) algorithms of calculating the flexibility function of energy-saving MCPS and solving two- stage stochastic optimization problems have been defined for the following practically relevant cases: 1) the values of uncertain parameters ξ at MCPS running stage can be determined precisely at any moment of time; 2) vector ξ consists of two subvectors ξ1 and ξ2, of which vector ξ1 can be identified at MCPS running stage and the domain of uncertainty of vector ξ2 is the same as at the design stage. Modified algorithms of solution of the above-mentioned problems are proposed implementing branch and bound procedure. An example of integrated design of energy-saving MCPS for azo dyes production is given. Keywords: integrated design, uncertain parameters, energy-saving multiproduct chemical process systems, modified (fast) algorithms, flexible automated production of azo dyes
1. Introduction When carrying out integrated design of energy-saving multiproduct chemical process systems (MCPS) the following basic problems are solved iteratively [1, 2]: 1) optimal product range Ω ∗ is determined; 2) alternative variants of MCPS unit setting and its automatic control system (ACS) class and structure, which provide for the fulfillment of hard and/or soft (probable) flexibility requirements, are chosen; and 3) through solving one- and/or two-stage stochastic optimization problems, design d , controlling z and tuning parameters of MCPS-ACS complex are defined. Methodology of integrated design under uncertainty [3] allows determining optimal coefficients of MCPS-ACS complex age margin. They provide for the fulfillment of
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design constraints g on product quality, process safety, environment saving and technical and economic parameters regardless of the uncertainty of initial physical, chemical, technological and economic data. It should be pointed out that in this case average indexes of MCPS energy- and resource-saving reach optimal values and are on a par with the world standards in the field. The paper analyses possible statements of optimization problem under uncertainty of initial data and reformulates them with regard to peculiarities of energy-saving MCPS; certain approaches to their solution will also be discussed.
2. New statements of energy-saving MCPS integrated design problem and modified algorithms of their solution. Let us obtain the flexibility condition and state the problems of stochastic optimization under uncertainty for practically relevant cases.
Problem A. At the running stage all parameters can be determined precisely at any moment of time (either by direct measuring or as a result of solving the reverse problem on the basis of information obtained through measuring) and controlling variables can be used to fulfill the design constraints. For this case the flexibility condition can be written in the form of F ( 1 ) ( Ω ∗ , d ) = max min max g j ( Ω ∗ , d , z ,ξ ) ≤ 0 ξ ∈Ξ z j∈ J
and optimization problem is stated as following:
( {
}
(1)
)
f1∗( Ω ∗ ) = min M ξ f ∗( Ω ∗ , d , z ,ξ ) | F ( 1 )( Ω ∗ , d ) ≤ 0 ,
(2)
f ∗( Ω ∗ , d ,ξ ) = min( f ( Ω ∗ , d , z ,ξ ) | g j ( Ω ∗ , d , z ,ξ ) ≤ 0 , j = 1,..., m ) ,
(3)
d
z
where M ξ { f ( • )} is expectation value of the goal function f ( • ) .
Problem B. Vector of uncertain parameters ξ consists of two subvectors ξ 1 and ξ 2 . At a certain moment of time during running the values of ξ 1 ∈ Ξ 1 are known, and ξ 2 can assume any value from the domain Ξ 2 . For this case flexibility condition is formulated as F ( 2 ) ( Ω ∗ ,d ) = max min max max g j ( Ω ∗ ,d , z ,ξ ) ≤ 0 . ξ 1 ∈Ξ 1 z ξ 2 ∈Ξ 2 j∈J
(4)
Let us consider the representation of optimization criterion. For a certain moment of time the problem of stochastic optimization is stated as following: ⎧⎪ ⎫⎪ f ∗( Ω ∗ ,d ,ξ 1 ) = min M ξ 2 ⎨ f ( Ω ∗ ,d , z ,ξ ) | max g j ( Ω ∗ ,d , z ,ξ 1 ,ξ 2 ) ≤ 0 , j = 1,...,m⎬ . z ⎪⎩ ⎪⎭ ξ 2 ∈Ξ 2
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The expectation value of ξ from the value f ∗( Ω ∗ , d ,ξ 1 ) should be considered as the optimization criterion of the problem as a whole. As a result we shall have the following problem statement: 1
{
}
f 2∗ ( Ω ∗ ) = min⎛⎜ M ξ 1 f ∗ ( Ω ∗ ,d ,ξ 1 ) | F ( 2 ) ( Ω ∗ ,d ) ≤ 0 ⎞⎟ . ⎠ d ⎝
(5)
Using the discrete approximation of the expression (5) with the help of quadrature formula we shall get f 2∗ ( Ω ∗ ) ≅ min
∑
γ il f ( Ω ∗ ,d , z i ,ξ 1i ,ξ 2l ) d ,z i i∈I ,l∈I 1 2
(6)
under the conditions max g j ( Ω ∗ ,d , z i ,ξ 1i ,ξ 2 ) ≤ 0 , j = 1,...,m , i ∈ I 1 ; ξ ∈Ξ 2
(7)
F ( 2 )( Ω ∗ , d ) ≤ 0 ,
(8)
2
where γ il = γ i ⋅ν l ; ν l ,γ i are weighting factors and I1 , I 2 are sets of approximation point indexes in domains Ξ 1 and Ξ 2 respectively. If the function of distribution density is unknown, weighting factors and sets of approximation point indexes are assigned by user on the basis of intuition and process knowledge. Let us consider the calculation of flexibility function and the solution of optimization problem in more detail. Let F ( 2 ) ( Ω ∗ , d ) be represented as F ( 2 ) ( Ω ∗ ,d ) = max h( Ω ∗ ,d ,ξ 1 ) , ξ 1 ∈Ξ 1
where h( Ω ∗ , d ,ξ 1 ) = min max max g j ( d , z ,ξ 1 ,ξ 2 ) . z ξ 2 ∈Ξ 2 j∈J
Consequently the calculation of flexibility function F ( 2 )( Ω ∗ , d ) becomes the maximization of function h( Ω ∗ , d ,ξ 1 ) on the domain Ξ 1 . We shall use the branch and bound method for maximization, according to which the function h( Ω ∗ , d ,ξ 1 ) maximum is found by division of domain Ξ 1 into subdomains. To implement the branch and bound method an algorithm of calculation of the upperbound estimation of function F ( 2 )( Ω ∗ , d ) must be developed. Let us obtain this algorithm. We shall change the order of the operations when calculating F ( 2 )( Ω ∗ , d ) and (2) designate the resulting expression as F mod ( Ω ∗ , d ) . Then we shall obtain
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524 (2) Fmod ( Ω ∗ , d ) = min max max g j ( d , z ,ξ ) , z j∈J ξ ∈Ξ
(9)
where Ξ = Ξ 1 × Ξ 2 . Thus, the expression for the upperbound estimation of the flexibility function F ( 2 )( Ω ∗ , d ) is identical to the expression for the upperbound estimation of the flexibility function F ( 1 )( Ω ∗ , d ) in case A [4]. The source also describes the implementation of branch and bound method for calculating the flexibility function F ( 1 )( Ω ∗ , d ) . The procedure for calculating function F ( 2 )( Ω ∗ , d ) will be similar to that. The difference will be in that in the case of problem A the entire domain Ξ is searched, and in the case of problem B only domain Ξ 1 . Let us now consider the problem
{
}
fiU = min M ξ f ∗( Ω ∗ , d ,θ ) , d
(10)
(i ) Fmod ( Ω ∗ ,d ) ≤ 0 ,
where i = 1 or i = 2 ; θ = ξ , if i = 1 ; θ = ξ 1 , if i = 2 . It can be easily proved [5] that f i∗ ≤ fiU . Therefore the solution of the problem (10) provides the upperbound estimation of the optimal value of optimization criterion for problems (2) or (5). (i ) Similar to the function Fmod ( Ω ∗ , d ) which has been introduced for the entire domain (i ) ∗ Ξ or Ξ 1 , we may introduce the function Fmod, s ( Ω , d ) for any subdomain Ξ s ⊆ Ξ 1 1 or Ξ s ⊆ Ξ . Let Ξ or Ξ be divided into subdomains Ξ s : ⎧ Ξ , i = 1; Ξ 1 ∪ Ξ 2 ∪ ... ∪ Ξ N = ⎨ 1 ⎩Ξ , i = 2.
We shall consider the problem
{
}
fiU ,N = min M ξ f ∗( Ω ∗ , d ,θ ,
(11)
(i ) (i ) ∗ ∗ Fmod, 1( Ω , d ) ≤ 0 ,..., Fmod,N ( Ω , d ) ≤ 0 .
(12)
d
In [5] it has been proved that for i = 1 the following inequalities are true: f1∗ ≤ f1U ,N ≤ f1U .
This means that the division of Ξ into subdomains increases the upperbound estimation. The same is true for problem B.
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The modified method of solving optimization problem (2), (5) makes full use of this feature. At each iteration the upperbound estimation fiU ,N , i = 1,2 is calculated on the basis of problem (11), (12) solution and certain subdomains Ξ s are divided. However, the following rule applies: at a given iteration only those subdomains Ξ s for which the constraints (12) are active are submitted to division.
Modified algorithm.
Let us designate the subdomains into which the domain Ξ is divided at iteration ν as Ξ s(ν ) ( s = 1, N (ν ) ) . Step 1. Let ν = 0 . Select the initial division of domain Ξ into subdomains Ξ s( 0 ) ( s = 1, N ( 0 ) ) and the initial value of vector d ( 0 ) of the vector d . (ν ) Step 2. Solve the problem (11), (12). Let f and d (ν ) be optimal values of the goal function and vector d of design parameters. S (ν ) of Determine the set active constraints numbers ∗ ( ν ) ( ν ) Fmod,s ( Ω , d ) = 0, s ∈ S . The following correlations become obvious:
Step 3.
Fmod,s ( Ω ∗ , d (ν ) ) > Fmod,i ( Ω ∗ , d (ν ) ), ∀s ∈ S (ν ) , i ≠ s
. Step 4. If set S is empty, then the solution of problem (11), (12) has been found, otherwise go to step 5. Step 5. Check the fulfillment of condition r( Ξ s(ν ) ) ≤ ε 1 ∀s ∈ S (ν ) , where r( Ξ s ) is the size of subdomain Ξ s and ε 1 is a preset small number. If this condition is fulfilled, the iterative procedure may be ended. Otherwise go to step 6. Step 6. Divide each subdomain Ξ s(ν ) ( s ∈ S (ν ) ) into two subdomains Ξ s(ν + 1 ) ,Ξ s(ν + 1 ) ( s ∈ S (ν ) ) . (ν )
1
2
Step 7. Let ν := ν + 1 and go to step 2. As Ξ s(ν + 1 ) ⊂ Ξ s(ν ) , Ξ s(ν + 1 ) ⊂ Ξ s(ν ) , then 1 2 (ν ) (ν +1 ) (ν ) (ν +1 ) ( ν ) ( ν + 1 ) F ≥F , F ≥F and f ≥ f . mod,s
mod,s1
mod,s
mod,s2
The above-stated algorithm can determine the local minimum of problem (11), (12). The method of branch and bound is used as the basis of this algorithm. At each iteration that subdomain Ξ s is divided for which the upperbound estimation of the value F is maximum. The condition r( Ξ s(ν ) ) ≤ ε 1 ∀s ∈ S (ν ) guarantees that the iterative procedure will end only if domains Ξ s(ν ) ( s ∈ S (ν ) ) are small enough. In fact, the search may be ended with the fulfillment of condition f (ν ) − f (ν +1 ) ≤ ε 2 , where ε 2 is a relatively small number.
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3. Integrated design of energy-saving MCPS for azo dyes production. When carrying out integrated design of energy-saving MCPS-ACS complexes, twostage problems of stochastic optimization have been solved with the use of branch and bound procedure. Progressive continuous methods of thin organic synthesis process realization (diazotization, azocombination, nitration, etc.) have been applied to azo dyes MCPS unit setting, as well as highly productive small-scale turbulent tube reactors of diffuser-contractor type and swirl inert flow dryers have been implemented. We have considered closed and open loop systems as alternative classes of automatic control systems; they are designed for the stabilization of MCPS technological regimes, adaptive static optimization, dynamic optimization, programmed control and optimal control of MCPS transient regimes. Synthesis of energy-saving control of nonlinear chemical processes within the MCPS has been realized with the help of the method of analytical designing of optimal regulators. Through imitation research the optimal variant of energy-saving MCPS-ACS complex has been selected. Flexible automated production of azo dyes designed in accordance with the methodology of integrated design has the following technical and economic indexes: capacity range is 100-5000 tons per year of dry dye; azo dyes output is 98-99%, which is 2-3% higher than the existing productions; energy costs are cut by 10-15% in comparison with the existing productions; capacity for metal is decreased by 20% average; machine utilization is increased by 30%; lead-time is reduced 2-3 times; and demand for maintenance staff is decreased by 30% (due to high level of production automation).
4. Conclusion. Computer software for the integrated design of energy-saving computer-aided MCPS has been developed with the help of modified (fast) algorithms of two-stage stochastic optimization problem solution as recommended in the work. These MCPS are of critical importance for the development of new and reequipment of existing energy-consuming multiproduct productions of organic semiproducts and dyes, varnishes and paints, chemicals and polimeric materials additives, films and photomaterials, fuels and lubricants, chemical fertilizers, etc.
Reference. [1] D.Dvoretsky, S.Dvoretsky and V.Kalinin, European Symposium on Computer Aided Process Engineering (ESCAPE’14): Proceedings (2004) 397-402. [2] D.Dvoretsky, S.Dvoretsky and V.Kalinin, 7th World Congress of Chemical Engineering, Glasgow, Scotland: Congress Manuscripts on CD-ROM (2005). [3] L.T.Biegler, I.E.Grossman, A.W.Westerberg, Systematic methods of chemical process design, Prentice Hall, Upper Saddle River NJ, 1997. [4] G.Ostrovsky, Y.Volin, M.Senyavin, T.Berezhinsky, Theoretical Foundations of Chemical Technology, Vol. 28, No. 7 (1994) 54-61 (in Russian). [5] G.Ostrovsky, Y.Volin, M.Senyavin, Comp. Chem. Eng., Vol. 21, No. 3 (1997) 317-325.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Generic hybrid models of solvent-based reactive systems combined with membrane separation system Piotr T. Mitkowski, Gunnar Jonsson, Rafiqul Gani CAPEC, Department of Chemical Engineering, Technical University of Denmark, DK 2800 Lyngby, Denmark.
Abstract Multi-step reactions are commonly found in pharmaceutical and biochemical processes where reactions progress in organic solvents or in aqueous-organic solutions. Usually desired products have to be separated from residual reactants and/or undesired products. Moreover, products may be heat-sensitive which renders the conventional thermal separation processes infeasible. In order to make the process economically feasible, one alternative is to increase the product yield by combining the reactor with a membrane separation unit or with better solvents, or both. Through model-based computer-aided techniques, it is possible to select better solvents and identify membrane-based separation operations which when combined with the reactor would increase process productivity. A systematic modelling framework for investigation of hybrid reactors-separator operations is presented and its application is highlighted through a case study. Keywords: Hybrid process modelling, membrane-based separation, solvent selection
1. Introduction In pharmaceutical, fine chemicals and biochemical manufacturing reactions are most often carried out in a batch or semi batch reactor followed by multiple separation and cleaning steps. Irrespective of whether these reactions are equilibrium or kinetically controlled, on-site removal of products usually enhance the yield and lead to reduced reaction times. Sometimes, the removal of products also reduces the undesired side reactions. In the cases where solvents are used, it can either be recycled or substituted with another more appropriate solvent. For all these reasons, it is beneficial to couple the reactor with a separation unit. The products of the above mentioned reactions are usually heat sensitive, so in order to avoid thermal degradation the separation technique should operate at temperatures lower than the degradation temperature of the compounds. One option could be membrane-based separation processes where the separation proceeds because of the selectivity imparted by the membrane, based on either the difference in size or the chemical potential of the molecules. This could be a good choice in the cases [1] where the reactor effluent contains desired products having molecular weights (Mw) in the range of 300-1000, smaller by-products (Mw between 50-150) and much larger enzymes/catalyst. Also, membrane separation techniques enjoy advantages such as low operational costs, high selectivity, modular design and lower environmental impact. Membrane separation techniques like pervaporation and nanofiltration have been extensively studied [1-3]. Pervaporation has been used in the production of MIBK
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(methylisobutylketone) [4] and MTBE (methyl tert-butyl ether) [5]. Nanofiltration is emerging as an option in separation of molecules with Mw ranging from 500 – 2000 from dilute solutions. Now the membranes which are resistant to degradation by organic solvent are also commercially available. These membranes are fairly reasonable option when the separation is based on size. Coupling of reactor and separation unit is called hybrid process since the two processes influence the performance of each other and the optimisation of the design must take into account this interdependency. Lipnizki et al. [6] highlighted two types (R1 and R2) of hybrid processes consisting of reactor and membrane-based separation based on the type of molecule to be separated. These hybrid processes are presented in Fig. 1a and 1b where the separation unit is physically set apart. However, it is also possible to integrate the membrane separation process with the reactor unit which is usually referred as the membrane reactor (see Fig. 1c). In type R1, the separation process removes the product from recycle loop around the reactor. Type R2 is an example of integration where by-product is removed from hybrid system. The objective of this work is to present a model-based methodology for design/analyse of hybrid process systems.
Fig. 1 Hybrid process layouts, a) type R1, b) type R2, c) internal membrane unit
2. Model-based design methodology of hybrid systems Design of hybrid process system consisting of reactor and membrane-based separation units is usually carried out through trial-and-error approaches involving experiments. Even through they are acceptable in terms of reliability, they are time consuming and expensive while the solution is ad-hoc by nature. Based on a modelbased framework for systematic analysis, it is possible to design hybrid process systems to find improved process design alternatives in terms of process output parameters such as reaction yield, selectivity, processing time and environmentally friendly solvents. A model-based framework for systematic investigation of hybrid process systems is presented in Fig. 2, where the workflow for every step is indicated by the grey-boxes, while the needed models and data are indicated through the white-boxes. Based on the knowledge of reactant properties like size of molecules, temperature of degradation, partial pressure etc, and reaction kinetics, conditions of reaction are defined (step 1). The process output depends on process parameters such as product purity, reaction yield and process time. The objective of the step 2 is to specify these process parameters in order to determine the values of process variables such as temperature, permeability, membrane area etc. which will give the desired process output. In the next step (step 3), influence of solvent on reaction as well as on the process design is considered. A short list of chemicals which could be the potential solvents is generated based on the method of solvent selection given by Gani et. al. [7] and their performance evaluated in the hybrid process. This method includes use of computer-aided molecular design tool
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ICAS-ProCAMD [7]. The properties of solvent which play the biggest role in specific reaction are reactivity of solvent, miscibility with products, polarity, melting and boiling point, vapour pressure, selectivity and EHS properties. Influence of solvent on membrane-based separation method also needs to be considered due to membrane stability and fluxes. Step 4 combines all collected knowledge with membrane separation models to identify the feasible membrane-based separation techniques. The membrane parameters like diffusivity, solubility etc. used in membrane model should represent the available membranes. In the last step, hybrid process configuration has to be chosen and operational limits defined in terms of process yield, reaction time and membrane area. If these constraints are satisfied, a feasible design is obtained; otherwise, decisions from earlier steps will need to be reviewed. This methodology consists of an efficient and systematic generate and test approach, which is able to save time and resources by avoiding duplication of work and efficient decomposition of the problem into integrated sub-problems (as highlighted in Fig. 2). Reaction kinetics Reactants properties
Step 1: Reaction data collection Step 2: Define/determine process demands? Step 3: Choose appropriate solvent Step 4: Find feasible separation method
Solvent database
Membrane Separation Model
Membrane database
Reactor Model Step 5: State Process Conditions
Hybrid Process Model Feasible Design
Fig. 2 Methodology of design/analyze hybrid process system 2.1. Generic model for the hybrid reactor-membrane process The model-based framework needs a generic hybrid process model from which problem specific models can be generated. This generic hybrid process model contains process and property sub-models for both reactor and separation units. These equations are derived from mass, energy and momentum balances, which form a DAE system of equations. The differential equations are the states of the system at discrete time points and algebraic equations are the constitutive and control equations. The generic form of different types of model equations used in the hybrid process model is given as: ⎡ ∂n ⎤ (1) ⎢⎣ ∂t ⎥⎦ = [ Flow in ] − [ Flow out ] + [ Recycle] + [ Reaction] Recycle ≡ 0 = g M ( J i , Am ,υr ) (2)
Reaction ≡ 0 = g r ( K i , K eq , TR ,V , r , t )
Flow in ≡ 0 = g F (υin , Ci ,in , Tin )
Flow out ≡ 0 = g F (υout , Ci ,out , Tout )
(3) (4) (5)
Where Am – membrane area, Ci – concentration, Ji – component flux through the membrane, Keq – equilibrium constant, Ki – Michaelis-Menten constant, r – reaction
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rate, t – reaction time, T – temperature, V – reactor volume, υ – volumetric flow, subscripts: i – components, in – inlet, out – outlet, R – reactor, r – retentate. In addition to the above, constitutive models such as properties models, equilibrium relations, etc. are needed for the constitutive variables. For hybrid process design, the objective is to combine Flow out and Recycle terms (Eq.1) into a single term representing the effluent from the hybrid system, which depends on membrane related variables (A, Ji or υr), directly into reactor model. Advantage of such a reformulated model is simplicity to investigate the performance of the hybrid system. Moreover, this operation reduces number of variables and the degrees of freedom. Using this generic model and the specific details of any problem, the specific hybrid reactor-membrane process can be generated and tested.
3. Case study: Enzymatic esterification Application of the model-based framework is illustrated through an enzymatic esterification reaction. Data used in this study is published by Egger et al. [8] and other data is being generated through in-house experiments. Step 1: Reaction data collection Enzymatic esterification reaction can be represented schematically as: ⎯⎯ → ABE ←⎯ ⎯⎯ →C +W + E A + B + E ←⎯ ⎯ ⎯ Where: A – lysophosphatidylcholine, B – oleic acid, C – phosphatydylcholine, E – enzyme phospholipaze-A2, W – water. Although, this kind of reaction has been studied at temperatures equal to a higher then 50°C, all data used in this work has been obtained in ambient conditions. Egger et. al. [8] reported equilibrium yields in various water activity conditions and substrate concentrations, which has been correlated and verified here to generate the kinetic model. All reactants except water are heat sensitive. Molecules A and C have Mw between 500 – 700 while Mw of B is 282. Step 2: Process demands Reaction, which is kinetically controlled, has a low product yield. The objective is to increase the process productivity by removing the water. Moreover, reaction requires an inert organic solvent. Step 3: Solvent selection Based on information obtained from literature [8] toluene was chosen as the solvent. Other likely solvents generated with ICAS-ProCAMD include ethylacetate, isopropylacetate, hexane and many more (note that only toluene has been consider in this study). Step 4: Separation method selection Pervaporation (PV) is chosen as the membrane-based separation technique because of possibility of introducing hydrophilic membranes that would allow only water to permeate. Step 5: Process conditions and feasible design The proposed hybrid process system is of type R2 (Fig.1b). This set-up is investigated under assumptions that: reactor is well mixed, reaction occurs only in the reactor
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volume, in the liquid phase, reaction medium density is constant, water flux in PV is constant and fluxes for all other components present in the system are neglected. From the generic hybrid model (Eq. 1), the problem specific hybrid process model is generated (Eq. 6). dN i = − J i A + Vν i r ρenz (6) i = A, B, C ,W dt Where ρm,i is molar density of component i, ρe enzyme density and ν stechiometric coefficient. Note that in the above, only mass conservation is used because no heat effect was reported. The accumulation in the membrane process is neglected because change of state variables along the length and time (steady state) are assumed constant. Reaction kinetics is described by reversible Michaelis-Menten kinetics: C ACB ⎛⎜ CC CW 1 ⎞⎟ 1− K mA K mB ⎜⎝ C ACB K eq ⎟⎠ r = rmax ⎛ C C ⎞⎛ C C ⎜⎜1 + A + C ⎟⎟⎜⎜1 + B + W ⎝ K mA K mC ⎠⎝ K mB K mW
(7)
⎞ ⎟⎟ ⎠
All other constitutive variables are assumed constant. Process yield is defined as ratio of moles of desired product (phosphatydicholine) to initial of limiting reactant (lysophospchatydycholine), (Yield = NC/NA0). The hybrid model is solved in the ICASMoT [9] modelling environment, which is a computer aided modelling tool with option of model translation analysis and solution. With the generated problem specific hybrid process model, three scenarios have been investigated in terms of process yield and superiority of the hybrid process over batch reaction. Performance of hybrid system is strongly dependent on the membrane area (Am) and component fluxes (Ji). For reactor coupled with pervaporation unit (RCPV), two cases with different values of factor JW.Am (JW – water flux) have been studied. Process yield is improved from 8% to 16.5% by removing water from the system using a reasonable design for a PV-unit. Values for the different design variables for the three scenarios are given in Table 1 while the yield-time behaviour is shown in Fig. 3. 0,18 0,16
RCPV2 0,14
Yield = N C /N A0
0,12
RCPV1
0,1 0,08 0,06 0,04 0,02 0 0
5
10
15
20
25
30
35
40
Time [h]
Fig. 3 Comparison of hybrid process systems with batch in terms of process yield A membrane which should be able to match the design values of water flux is a cross-linked polyvinyl alcohol membrane. Set-up RCPV2 is recommended for further experimental studies together with experimental verification of membrane performance.
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532 Table 1 Process parameters and process yields V0 [dm3] JW *Am [mmol/h]
Batch 1 -
RCPV 1 1 0.005
RCPV 2 1 0.01
CA0 [mmol/dm3]
10
10
10
CE0 [mmol/dm3]
400
400
400
CW0 [mmol/dm3]
39.5
39.5
39.5
KA =KB = KC = KW [mmol/dm3]
4.9
4.9
4.9
1.04E-04 40 7.8
1.04E-04 40 11.2
1.04E-04 40 16.5
rmax[mmol.mg-1.h-1] t [h] Yield [%]
4. Conclusions A model-based framework for systematic investigation of hybrid systems consisting of well mixed reactors and membrane separation units has been presented along with the application to a relevant case study. The work-flow and the corresponding data-flow for the methods and tools needed by the model-based framework have been developed. Problem specific hybrid process models are generated and used for specific reaction systems under investigation. From this work, it is clear that hybrid processes could show their advantages where difficulties exist to incorporate other separation methods. Reactor combined with membrane separation unit gives significant increase in process yield by overcoming limitations of kinetically controlled reactions and also by reducing the process time. Experimental trials needed to verify the hybrid process is reserved for the final step, thereby saving time and resources.
Acknowledgment Author is please to acknowledge to the PRISM the Marie Curie Research Training Network, European Community’s Sixth Framework Program.
References [1] J.A. Whu, B.C. Baltzis, K.K. Sirkar, Modelling of nanofiltration – assisted organic synthesis, Journal of Membrane Science, 163, (1999), 319-331. [2] F. C. Ferreira, S.Han, A.Boam, S.Zhang, A.G.Livingstone, Membrane aromatic recovery system (MARS): lab bench to industrail pilot scale, Desalination, 148, (2002), 267-273. [3] J.T.Scarpello, D.Nair, L.M. Freitas dos Santos, L.S.White, A.G. Livingstone, The seperation of homogeneous organometalic catalysts using solvent resistant nanofiltration, Journal of Membrane Science, 203, (2002), 71-85. [4] C. Staudt-Bickel, R.N.Lichtenthaler, Integration of pervaporation of the removal of water in production process of methylisobutylketone, Journal of Membrane Science, 111, (1996), 135141. [5] M. Matouq, T. Tagawa, S. Gotp, Combined process for production of methyl tert-butyl ether from tert-butyl alcohol and methanol, Jurnal of Chemical Engineering of Japan, 27, (1994), 302-306. [6] F. Lipnizki, R.W. Field, P-K. Ten, Pervaporation-based hybrid process: a review of process design, applications and economics, Journal of Membrane Science, 155, (1999), 183-210. [7] R. Gani, C. Jim’ene-Gonz’alez, D.J.C. Constable, Method for selection of solvents for promotion of organic reactions, Computers and Chemical Engineering, 29, (2005), 1661-1676. [8] D. Egger, E. Wehtje, P. Adlercreutz, Characterization and optimisation of phospholipase A2 catalyzed synthesis of phosphatidylcholine, Biochimica et Biophysica Acta,1343,(1997),76-84 [9] M.Sales-Cruz, R. Gani, 2003, Computer-Aided Chemical Engineering, vol. 16: Dynamic Model Development, Eds. S.P. Asprey and S. Macchietto, Elsevier, Amsterdam.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
On the Numerical Calibration of Discrete Element Models for the Simulation of Bulk Solids Torsten Gröger a, André Katterfeld b a b
ITASCA Consultants GmbH, Leithestraße 111, 45886 Gelsenkirchen, Germany IFSL, OvG-University of Magdeburg, PF4120, 39106 Magdeburg, Germany
Abstract Due to the rapid increase of the computational power direct particle simulations, such as simulations on the basis of the Discrete Element Method (DEM), become increasingly popular in the field of bulk solids handling and processing. In order to obtain realistic simulations these projects require an accurate characterisation of the bulk solid properties in the Discrete Element Model. Therefore, the so called calibration of bulks solids deserves particular attention. The purpose of the numerical calibration is the adjustment of microscopic parameters, such as particle stiffness and inter-particle friction, in order to fit the macroscopic numerical behaviour, e.g. stress-strainbehaviour, measured on real experiments. The paper discusses the influence and effects of the microscopic parameters and explains the need for the development of new calibration methods. Keywords: Materials Handling, Process Engineering, Discrete Element Simulation, Particle Flow Code, Calibration
1. Introduction In the past years the interest of materials handling industries and materials processing industries in Discrete Element Simulations has risen noticeable. The main reason for this is the enormous increase in the computational power available on the PC market. Today, ITASCA performs large scale simulations with more than 300,000 particles by means of the Particle Flow Code (PFC3d) for regular consulting jobs. Transition chutes as shown in Fig. 1 are representative examples for this.
Fig 1.: The depicted conveyors and transition chutes are examples for large scale simulations for material handling industries.
Discrete Element Simulations can be considered as numerical experiments, which enable the contact-less measurement of microscopic quantities. These data cannot only be used to visualize the simulated process in a very illustrative manner but also to
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compute macroscopic quantities, such as stresses and mean velocities, which are of particular interest for the design and the optimisation of equipment. In contrast to continuum mechanical methods the Discrete Element Method enables modelling of both fast flowing and resting zones of particulate materials with the same constitutive equations. This requires that all important microscopic quantities can be determined and mathematically modelled. However, experiences from consulting projects at ITASCA as well as research projects running at the IFSL, University of Magdeburg show that a lot of open questions exist regarding the calibration of the microscopic models. Issues arising from that are not restricted to mere numerical questions but rather concern the fundamental understanding of the characterization of flow properties by microscopically parameters.
2. The Principle of the Discrete Element Method The Discrete Element Method was developed by Cundall (1979) and a lot of detailed descriptions have been published ever since. Therefore only a brief survey will be given here. For algebraic modelling, the particles of bulk solids need to be represented by well defined geometrical objects. For performance reasons, spheres or sphere conglomerates are preferred. The particles themselves are assumed to be rigid however they are allowed to overlap. These overlaps are regarded as contact deformation from which an elastic contact force arises. Dependent on the applied contact model (Fig. 2) other types of contact forces can contribute to the total contact force. Accumulating all contact forces on a particle delivers the resulting force and moment for this particle. With the mass and the momentum of inertia the Newtonian equation can be integrated for a very short time step. This places a particle onto its new position and hence a new contact detection has to be performed as existing contacts may have vanished or new contacts may have formed. The described cycle needs to be executed in a loop until the desired process time is reached.
Fig. 2: Example of a contact model for spherical particles. Spring – elastic force-displacementlaw, dashpot – viscous damping law, frictional element – Coulomb friction, meniscus – liquid bridge (attractive force)
3. Contact properties 3.1. Elastic contact properties In the simplest case the elastic contact deformations can be modelled by a linear spring law. However, for spherical particles a Hertzian law is more appropriate. Only in very rare cases where the real particles exhibit a spherical shape Young’s modulus and Poisson’s ratio of the solid material can be used directly. If more complex particles are modelled by spheres this simplification needs to be compensated by a calibration of the contact law. For geo-mechanical applications the particle stiffness is adjusted by means of numerical triaxial tests with the goal to fit a measured macroscopic stress-straincurve. With models of very coarse geo-materials numerically stable simulations can be achieved with the realistic stiffness and the realistic masses. For quasi-static processes it
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is often applicable to up-scale volumes and/or masses in order to achieve numerical stability. Unfortunately, the majority of processes from the field of materials handling and process engineering exhibits both fast flow regimes and comparatively small particles, which do not allow a mass or volume scaling. In order to obtain numerical stable time steps that enable a reasonable computing time (less than a month for most consulting jobs), though, the particle stiffness needs to be reduced. For instance, large scale simulations on high-end PC’s require the stiffness of minerals to be decreased by a factor of 100 or higher. Therefore, it is currently not possible to calibrate the particle stiffness for the majority of applications from process engineering and materials handling. It is recommended to choose the particle stiffness as high as the overall computational time allows it. 3.2. Damping Very often the size of the simulated particles is large enough that global damping effects of the surrounding medium can be neglected. For fine particles or surrounding fluids an appropriate damping law can be applied if needed. However, it is essential for most cases of handling and processing of bulk solids to consider the contact damping. Usually, contact damping is modelled in dependency on the relative velocity of the contact partners and occasionally dependent on the contact deformation. Except for nearly spherical particles that enable the measurement of the rebound height of a dropped particle no experiments are known that could be used for a calibration procedure. Practically relatively high contact damping coefficients are required. It is noted that higher damping forces can be achieved for a larger contact stiffness. 3.3. Friction In process engineering and materials handling the macroscopic friction angle of bulk solids is of particular importance. Besides cohesion, friction determines the flow properties of a particulate material significantly. Simultaneously, it is one of the most complex parameters since macroscopic friction is the result off particle friction and rolling friction on the microscopic level as well as the particle shape, the standard deviation of the particle size distribution, the packing structure and the packing density. In general, shear tests are performed numerically and experimentally in order to compare the inclination of the yield loci, which is a measure of the macroscopic friction. Fig. 3 shows examples of simulated yield loci. It is evident that the particle shape has a considerable influence on the macroscopic friction angle
Fig. 3: Yield loci obtained from simulated shear tests. The inclination is a measure for the macroscopic friction. Cohesion (intersection with the ordinate) was caused by liquid bridges.
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Unfortunately, the depicted particles composed of a number of spheres demonstrate two disadvantages. Firstly, with an increasing number of primary spheres the computational effort increases, too, and secondly in sections the particles can roll without any resistance. Therefore, it can be of advantage to introduce a rolling resistance (moment) that arises from an offset of the contact force from the centre of mass as depicted in Fig. 4.
Fig. 4: Examples for an offset of the contact force from the centre of mass.
There are a number of factors that can be responsible for the force offset, such as the deformation due to rolling (Fig. 4 left), the particle shape (Fig. 4 middle) and asperities on the surface of the particles (Fig. 4 right), as well. These effects can all be covered with the coefficient of rolling friction, which is multiplied with the particle radius to obtain the amount of the offset (lever of the force). Fig. 5 shows the influence of the particle friction coefficient and the rolling friction coefficient on the macroscopic friction of a particulate system that is subjected to direct shearing in a Jenike shear cell. Obviously, the same macroscopic friction can be obtained from different combinations of rolling friction and particle friction (e.g. along the lines between two hatched areas). Since it is desirable to find the pair of coefficients that is valid for all flow conditions, regardless if it is slow shearing or fast flowing material, a single type of experiment seems to be insufficient for the determination of the two unknowns. Therefore ITASCA and the Institute of Materials Handling (IFSL) investigate further methods of measuring the macroscopic friction. Currently, the angle of repose formed in a rotating drum as well as formed by a vertical cylinder is investigated (Fig. 6).
34.0 33.0 32.0 31.0 30.0 29.0 macroscopic 28.0 friction [°] 27.0 26.0 25.0 24.0 23.0 22.0
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Fig. 5: Simulated macroscopic friction angle [°] dependent on the particle (Coulomb) friction µ[-] and rolling friction µr[-] for spheres d=2.3 to 2.6 mm in a shear tester.
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Although no results can be presented, yet, it is reasonable to assume that in the process of forming the angel of repose the coefficients of friction and rolling friction have a differently weighted influence compared to shearing a consolidated system. This will lead to diagrams of the same type shown in Fig. 5. However different gradients are expected to be apparent. Hence, overlaying two of these diagrams should deliver an intersection at the desired macroscopic friction coefficient that delivers the pair of frictional coefficients that is representative for the majority of flow conditions. The described procedure is numerically expensive and further research is needed to find short cuts for the calibration process.
Fig. 6: Different experimental methods for the investigation of the angle of repose and their numerical representation.
4. Cohesion Macroscopic cohesion may arise from a number of microscopic causes, such as Vander-Waals-Forces and liquid bridges. The attractive forces on the microscopic level are comparatively well investigated and several mathematical models exist, which can be embedded in the contact model. Apart from sintering processes attractive forces become relevant for particle sizes smaller than 1mm (Fig. 7). 102
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Fig. 7: Influence of microscopic forces on the macroscopic tensile strength in dependency on the particle size.
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Since smaller particle sizes are usually associated with a higher number of particles, large scale DEM-simulations are often restricted to relatively coarse particle systems. Therefore, only the relatively large forces arising from liquid bridges are currently of particular interest for the simulation of industrial applications. Fig. 8 shows two yield loci used for the calibration of cohesion of a wet particulate system consisting of glass spheres with a mean diameter of d=684µm. The calibration process was straight forward for this particular system as the surface tension could be taken from a table and the volume of the bridge could be calculated from the water content (Gröger et. al. 2003). For bulk solids used in industrial applications this will not be possible in most cases. However, the procedure of calibration by means of direct shear tests is comparatively simple if the surface tension is known. The yield locus can then be shifted along the ordinate by varying of the volume of the liquid bridge.
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Comparison of yield loci obtained from shear experiments and simulated shear tests on wet particle systems (d=684µm).
5. Summary Several microscopic parameters used for the direct simulation of particulate systems, such as powders and bulk solids have been discussed and their influence of the flow behaviour was explained. Currently, not all parameters can be calibrated to represent the properties of particulate systems realistically. In case of the elastic properties this is caused by the limitations of the available computational power. In other cases, such as contact damping and friction, fundamental experimental methods for the determination of these properties are still to be developed. The methods from geo-mechanics and soil mechanics are not sufficient to calibrate the more complex flow behaviour of materials from the fields of materials handling and process engineering.
References P.A. Cundall, 1979, Cundall, P.A.; Strack, O. D. L.: A discrete numerical model for granular assemblies. Geotechnique 29, No. 1, 47-65 T. Gröger, U. Tüzün, D. Heyes, 2003, Modelling and Measuring of Cohesion in Wet Granular Materials, Powder Tech, 133, 203-215
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A Heat Transfer model of a scraped surface heat exchanger for Ice Cream Peter M.M. Bongers Unilever Food and Health Research Institute, O. van Noortlaan 120, 3133 AT Vlaardingen, The Netherlands
Abstract A mathematical model of an ice cream freezer was developed by considering the freezer barrel as a series of well mixed stages and employing heat and mass transfer equations. The model was solved using a commercial simulation package to give predictions of product temperature, mechanical dissipation and heat transfer rate. These predictions were found to agree closely with experimental measurements. The process model has the potential to predict local temperature and shear conditions within an ice cream freezer and therefore represents an important first step towards systematic freezer design and performance optimisation and scale-up based on product quality considerations. Keywords: dynamic modelling, heat exchangers, ice cream, validation.
1. Introduction Freezing of ice cream is performed in a scraped surface heat exchanger, where rotating scraper blades continually remove frozen product from the cooled surface and thus maintain a high heat transfer rate. It is within the freezer barrel that much of the product structuring occurs. These product structuring mechanisms include ice crystallisation, aeration and fat de-emulsification. The quality of the final product depends to a large degree on how these structuring processes have been carried out. In order to optimise the freezing process for a given product formulation or to maintain a desired product quality on scale-up, it is necessary to know the local conditions inside the heat exchanger and how these change with operating conditions. Since direct measurement of temperature and shear conditions in a freezer barrel is difficult to achieve, a mathematical modelling approach has been applied in this work to predict these quantities.
2. Why a model The development of processes within ice cream manufacturing has been progressed by a trial-and-error approach. The processes are allowed to evolve over time. Using this way of working, a huge number of experiments need to be conducted. In addition, before a ‘final’ embodiment of a process, a large number of equipment modifications have to be done and tested. The advantage of the trial-and-error way of working is that little fundamental understanding is needed. Disadvantages are that it takes a lot of resources (people and capital), limited understanding is build and disseminated, while all ice cream manufacturers can do the same. The solution is to design the process. In the design phase of a process, all available knowledge has to be harvested and compiled in to a ‘model’. Such a model will be the
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documentation of available knowledge in an exploitable format; i.e. this model can be used to: • Identify the knowledge gaps in the product-process interactions and enable focus on these bottlenecks. • Scale-up in one single step from bench-scale equipment to factory scale-equipment, hence enabling a significant reduction in time-to-market • Fault diagnosis by comparing the actual working of the process with the desired performance. • Performance improvement.
3. Mathematical model of freezer The principle of describing a process mathematically is to use the most simple model that fulfils the purpose. The predictions of the model can be viewed as a chain of phenomena, in which a rough description of all phenomena provides better predictions than a detailed description of only one phenomenon. Within the phenomena that can be described, the following are considered within the scope of the model: Material properties: Compressible fluid having a non-Newtonian rheology Energy sources: Heat transfer, scraping friction, crystallisation and viscous dissipation Product formulation specifics: thermal conductivity, ice phase curve, specific heat The mathematical model was developed by considering the freezer as a series of stages (continuous stirred tank reactor). Heat transfer into coolant Evaporation Temperature
Rotational speed
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Mechanical dissipation Figure 1 Model approach
Mass, energy, impulse balances were formulated for each of the stages in the model. 3.1. Mass balance:
mgas dm = win − wout . Using a ratio r = between the air phase and the dt mliquid + mgas liquid/solids phase, the mass balance can be written for each of the two components:
dm gas dt
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= rin win − rout wout = (1 − rin )win − (1 − rout )wout
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3.2. Energy balance:
d (mh ) = (win hin − wout hout ) + Qviscous + Qscraping + Qcrystalisation + Qrefrigeration dt In which the energy generating terms mechanical (viscous dissipation and scraping), cooling and crystallisation are taken into account. The changes in ice phase volume determine the crystallisation energy. Dissipation of mechanical energy due to shaft rotation is significant in ice cream freezers, accounting for as much as 50% of the heat removed by the refrigerant [2]. This dissipation was assumed to come from two main sources: viscous dissipation, due to fluid flow around the dasher, and scraping friction between the blades and the barrel wall. Viscous dissipation was calculated using laminar mixing theory [3]: Q viscous = τγVelement . The viscosity of ice cream was described using a Power Law equation
∂v τ rz =− K z ∂r
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constants are determined experimentally. The consistency has been modelled using an Arrhenius equation of Temperature and air phase volume. Scraping friction was estimated using an empirically-derived equation based on the thickness of the frozen layer at the wall 5 3
Q scraping = c1s (T - Tcoolant ) N 1.5 r L section N blades , in which c1 and c2 are determined c2
experimentally and Nr is the rotational speed. Heat transfer to the coolant (through the barrel wall) was calculated by assuming that the product-side wall heat transfer coefficient (htc) was the limiting resistance and therefore the coolant-side htc could be ignored. The product-side htc was estimated using an empirical correlation [1] based on penetration theory:
α = 2 ρ C p N blades N r Dλ 3.3. Impulse balance The flow as a function of the pressure drop elements is described by [4]: 1
ρπD n ⎛ Δp D ⎞ n ⎛ Dcore ⎞ ⎜ ⎟ ⎜1 − w= ⎟ D ⎠ 8(2n + 1) ⎜⎝ 4 KLsec tion ⎟⎠ ⎝ 3
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In which Dcore is the diameter of the core. The pressure in each of the elements is calculated using the mass of the gas in the element and treating the air fraction as an ideal gas. The model was implemented in c++ and solved using MATLAB-SIMULINK simulation package [5] to give predictions of product temperature, mechanical dissipation and heat transfer rate.
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4. Experimental verification of model Validation of model predictions was performed by processing a dairy fat ice cream in a fully instrumented pilot-scale freezer (60-240 kg/hr capacity)*, see Figure 2.
Torque P T
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Measurements of the product temperature within the freezer barrel were obtained using thermocouples mounted on the dasher. Overall heat transfer coefficients were obtained from experimental data by employing a heat balance over the barrel. The experimental data is compared to the model predictions in Figure 3.
heat transfer coefficient [W/degCm2]
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Mechanical dissipation (rotor torque) due to viscous dissipation and scraping friction was determined from shaft torque measurements. The experimental data is compared to the model predictions in Figure 4. *
The experiments have been executed by collegues in our R&D laboratory in Colworth, UK.
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Model predictions of mechanical dissipation and overall heat transfer coefficient compared well with data measured experimentally.
5. Conclusions and future work The mathematical model developed in this work is capable of predicting both individual rates of heat transfer and energy dissipation and product temperature changes. It therefore has potential to predict local temperature and shear conditions within an ice cream freezer, since shear is closely linked to dissipation. This type of information will enable process optimisation and scale-up to be based on criteria, which are important to product structuring and therefore quality.
6. Nomenclature Cp D h K L r m n Nnlades Nr s t T Q V v
specific heat diameter enthalpy consistency length gas/liquid ratio mass power law constant number of blade rows rotational speed ice crystals content time temperature total amount of heat flow volume velocity
J/(kgoC) m J/kg [-] m [-] kg [-] [-] 1/s [-] s o C J/s m3 m/s
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References 1. Trommelen, A.M. (1967) Heat transfer in a scraped surface heat exchanger. Trans. Inst. Chem. Engrs. 45, T176-T178. 2. Russell, A.B., Cheney, P.E., Wantling, S. Influence of freezing conditions on ice crystallisation in ice cream. J. Food Eng. no. 39, pp. 179-191, 1999. 3. Godfrey, J.C. (1985) Mixing of high-viscosity fluids. Chapter 11 in: Mixing in the Process Industries (Harnby, N., Edwards, M.F., Nienow, A.W., eds.), Butterworth and Co. 4. Fredrickson, A.G., R.B. Bird (1958). Non-Newtonian flow in annuli, Industrial and Engineering Chemistry, vol.50, no.3, p.347-352. 5. SIMULINK (1992) - A Program for Simulating Dynamic Systems, The Mathworks Inc., Natick, MA, USA.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Computer-aided forecast of catalytic activity in an hydrotreating industrial process using artificial neural network, fuzzy logic and statistics tools F. Jiménez1, V. Kafarov1, M. Nuñez2 1 2
Industrial University of Santander, Tel. +57 +76344746, Bucaramanga, Colombia Colombian Institute of Petroleum, Piedecuesta, Colombia
Abstract Complex hierarchic organization of heterogeneous catalytic systems makes impossible to design a theory to quantitatively predict catalytic activity based on strict laws of heterogeneous catalysis. For this reason here we present a methodology based on computer-aided and mathematical tools to solve the problem of choice of catalysts for a hydrotreating industrial process. Twenty-four hydrotreatment catalysts with different textural and physicochemical properties were aged in an industrial reactor during the length of the run. First, an evaluation of the uncertainty of the industrial experimentally collected data is made based on the Hotelling T2 statistic tool. Next, the application of artificial neural networks (ANN) to determine the influence of each selected physicochemical variable on the catalytic activity (capacity of metals removal: contents of nickel, vanadium and molybdenum) is proposed. Then, a methodology based on the application of several kinds of multiple regression (MLR), including Brandon’s method (which provides a mathematical model using priority of selected variables) is applied. Similarly, we also propose the application of ANN to correlate process variables as well as the utilization of fuzzy logic tools to obtain qualitative models. In this manner it was possible to compare the results obtained from different mathematical methods and to make decisions about the optimum chemical compositions and texture of the industrial catalyst. Keywords: Industrial reactor, Statistic, Fuzzy Logic, Neural Network, Hydrotreating.
1. Introduction The strategy followed in the present work for the computer aided forecast of the catalytic activity from textural and physicochemical properties of the catalysts is schematically shown in Figure 1. Twenty-four fresh hydrotreatment catalysts with different textural and physico-chemical properties were used during the course of the investigation. Six properties were selected by experts and measured to the fresh catalysts: porosity (%), superficial area (m2/g), pore diameter (A), equivalent size (mm), initial content of molybdenum (wt.%) and nickel (wt.%). The catalyst samples were properly placed inside the hydrotreatment industrial reactor, and submitted to the real conditions of process during almost eight months corresponding to the length of the run. At the end of the run, the aged catalysts were recovered and their activity (measured as the capacity to retain metals) was repetitively determined. This information constitutes the preliminary experimental database. To evaluate the r reliability of this database, the classification and selection of this obtained information is necessary, and for this the application of Hotelling's T2 statistic is proposed. Once the data was filtered, a final experimental database (activity, and textural and physico-chemical properties of the
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catalysts) with high reliability was available. The next step to follow deals with the training of neural networks to obtain the sequential importance of the physico-chemical variables on the catalyst activity (capacity of metals removal). This order can be established according to the classification of the significance of each variable during the process (rank of influence on metals removal). The computer-aided forecast for the activity from the selected design parameters of the catalysts (porosity, area, etc.) is based on three different mathematical methods: •Traditional statistic: several types of MLR, and Brandon Method •Artificial neural networks, specially useful due to its property to establish highly non-linear relations in complex processes such as hydrotreating; and finally •Fuzzy logic, which takes into account data that cannot be expressed in quantitative form but with descriptive verbal characters. Using this methodology is possible to take decisions about the optimum textural and physicochemical properties in these hydrotreating catalysts. Figure 1. General Scheme of Investigation CATALYST FRESH SAMPLES PHYSIC-CHEMICAL PROPERTIES INDUSTRIAL EXPERIMENT ACTIVITY MEASURES
Superficial Area Pore Diameter Porosity Equivalent Size Initial %Nickel Initial %Molybdenum
PRELIMINARY MULTITUDE OF EXPERIMENTAL DATABASE FABRICATION OF CATALYST WITH SELECTED PROPERTIES
STATISTIC FILTER T2-HOTELLING FINAL MULTITUDE OF EXPERIMENTAL DATABASE
COMPUTER AIDED FORECAST OF CATALYTIC ACTIVITY
OPTIMAL PROPERTIES OF FRESH CATALYSTS
Artificial Neural Network Parameters influence level on activity Black Box Model Statistics Methods: Multiple Regression Brandon Method
Fuzzy Logic Method
2. Physico-chemical Properties Six physicochemical properties for 24 chosen catalysts were measured: porosity (%), superficial area (m2/g), pore diameter (A), equivalent size (mm), initial content of
Computer-Aided Forecast of Catalytic Activity in an Hydrotreating Industrial Process 547 molybdenum (wt.%) and nickel (wt.%) (Table 1). These data were obtained by different characterization methods: mercury porosimetry (Autopore II 9220), scanning electronic microscopy (Cambridge Stereoscan 240), and nitrogen sorption (Micromeritis ASAP 2000 C).
3. Industrial Experiment Based on a methodology developed at The Colombian Institute of Petroleum (ICP) together with the Refinery of Barrancabermeja (CIB) for the aging of catalysts in metal baskets inside the industrial reactor (Nuñez et al, 2000), samples of the selected catalysts are introduced during load and recovered during unload of the industrial reactor. Since the conditions throughout the reactor vary, a method based on the use of a reference catalyst was introduced. All the catalyst samples were aged inside two fixed-bed industrial catalytic reactors with a catalyst load of approximately 40 tonnes each (in all cases the samples were placed in central axis of the reactor). The length of the run was 241 days and the process was operated at a LHSV of 1.0 h-1, a pressure of 1500 psia and a maximum temperature of 400 °C.
4. Activity Measurements The recovered baskets were identified and through appropriate methods the testing catalysts were separated from the reference catalysts. For the two catalysts atomic absorption (Perkin Elmer 5-100) was used in a repetitive way to determine the contents of nickel, vanadium and molybdenum. This procedure was repeated several times in ten aged catalysts with the purpose of obtaining replicas for the statistical analysis.
5. Statistic Filter: Hotelling T2 The catalysts that are applied to this industrial process present close vectors of characteristics of activity (contents of nickel, vanadium and molybdenum). To establish the statistical importance in the differences of properties of whatever two catalysts the distribution T2 of Hotelling -multidimensional analogous of Student’s distribution(Johnson, 1998) is proposed. Additionally, it is necessary to evaluate if the differences among the repeated measurements taken for metals removal (Ni, V, Mo) in a catalyst sample are statistically smaller than the differences among the averages values for metals removal from other catalysts samples. Therefore, it is possible to compare simultaneously all the activity measurements (Ni, V, Mo) and to determine if there is any irregular data that must be reviewed or not taken into account.
6. Computer-aided Forecast for Catalytic Activity 6.1 Artificial Neural Network Method (ANN) The application of ANN (multilayer perceptron with retropropagation algorithm) is proposed to obtain neural network models and to predict the catalytic activity from all the selected parameters (Hoskins and Himmelblau, 1998). When the final model is obtained, the sequential importance of the physico-chemical variables in the catalyst activity can be determined from the sequential elimination of each variable, the error in the final prediction will be useful to attribute a place in the order of importance (Bartlet, 1994). The ANN should be considered as a mathematical tool, similar to linear regression analysis. The key feature of ANN over regression analysis is that ANN use non-linear mathematics and therefore can be used to model highly complex and nonlinear functions such as those in petroleum hydrotreating processes (Hilera, 1995).
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6.2 Statistics Methods The dependency of the catalytic activity from selected physico-chemical parameters must be calculated. For this, Brandon method is proposed to define the catalytic activity as a product of six different expressions. Each of these expressions is a function of each of the selected parameters, and its order depends on the sequential importance on the activity (Núñez et al, 2000). We also proposed a modification to Brandon’s method through its organization in a cyclic procedure where the last result will act as initial experimental data, and then the method will be run again, until reaching the lowest possible error. Likewise, forty one (41) traditional statistical models to establish the dependence between catalytic activity and physico-chemical properties (multiple lineal regressions with 1-6 parameters, regression by steps, polynomial with interactions, etc.) are proposed to compare the results with results from other proposed methods. 6.3 Fuzzy Logic Method A large part of qualitative data about catalysts selection presents some uncertainty, and often, hampers a strict mathematical statement (Dennis et al., 1997). The application of fuzzy logic in combination with other mathematical methods can promote the efficiency of catalyst activity prediction due to formalization of such knowledge (Qian et al., 1995). The algorithm for task decision includes: revealing parameters to determine catalyst activity (six measured properties), postulating universal multitudes, ranks, and specific vocabulary to determine belonging degrees (example: very low, low, medium, high, very high – porosity, area, etc.), obtaining a verbal description of the process and building a set of fuzzy relationships (i.e. if porosity is very high the activity is very low). Finally, calculation of the fuzzy composition for new input parameters, and defuzzify the prediction need to be done.
7. Results Based on the Hotelling-T2 method, specific software was developed, and the comparison of twenty-four hydrotreatment catalysts was attained. With a probability factor of 95%, it was possible to deduce that the database from the compared catalysts has sufficient reliability to assure its utilization in this research. In this way, the reliability of the methodology of introducing reference catalysts was demonstrated. 7.1 Artificial Neural Network Method (ANN) The “perceptron multilayer normal feed forward” network type was selected. Eight different tables with randomized data were organized. Eighteen (18) data sets were used for network training and six (6) data sets were used for testing or validation. Three different architectures were selected for network training, and a hidden layer with 3-5 nodes was defined (Quick-Propagation rulers and Dot-Product input functions were used). The normalized data sets were applied with the best table and the most efficient network architecture. After more than 2000 iterations and a very low forecast error the final network model was defined. The physicochemical properties for the chosen catalysts and forecast from Brandon method and ANN are shown in Table 1. Also, with the final model and the proposed methodology, the order of influence of the input variables on activity established by the application of ANN was porosity > surface area > pore diameter > equivalent size > initials Ni and Mo contents. 7.2 Statistics Methods Specific software for the application of Brandon method and multiple linear regression was developed. Comparative results (sum of quadratic errors) of some methods are
Computer-Aided Forecast of Catalytic Activity in an Hydrotreating Industrial Process 549 showed in Table 2. It can be observed that traditional statistical methods do not work for this type of forecast, and they should be combined with advanced computer-aided tools. The modified Brandon method yielded better results than original Brandon method, obtaining results that are in accordance with the real catalytic phenomenon. The equations confirm that the retention of metals is associated with an optimum porosity, high superficial area and pore diameter, and a small equivalent size. Additionally ANN provides a more exact prediction than those by other traditional prediction methods, and it is possible to establish isolated effects and dependences of the chosen variables. However, the effective learning depends on the supplied information, quality and quantity. With 24 data sets for 24 hydrotreatment catalysts the results are consistent, but more data sets are necessary if strengthening the generalization ability is desired. Table 1. Physicochemical properties of fresh catalysts and some forecasts for catalytic activity No. Cat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Equiv. Size (mm) 1.0 1.0 1.1 0.9 1.8 1.0 1.8 1.3 1.0 0.8 2.1 0.9 1.4 0.7 2.2 0.7 1.0 0.7 0.7 0.7 0.7 0.7 0.7 1.3
Pore Diam. (A) 118 94 132 116 85 86 80 128 99 131 133 140 141 116 133 116 137 116 116 116 116 116 116 113
Porosity (%)
Area (m2/g)
66.7 63.7 66.0 70.6 56.9 62.3 55.7 68.9 66.9 67.0 71.9 65.5 69.7 70.6 75.9 70.6 72.4 70.6 70.6 70.6 70.6 70.6 70.6 62.6
212 224 172 260 200 250 194 200 238 187 224 144 187 260 146 260 213 260 260 260 260 260 260 176
Mo Ni (wt.% (wt.%) ) 1.8 6.9 2.2 8.0 1.4 8.1 0.0 2.8 0.0 10.5 1.8 8.0 0.0 10.5 2.3 10.0 2.3 8.1 1.6 5.9 0.6 11.5 3.1 10.7 2.0 6.0 0.0 12.0 1.9 8.0 0.0 6.0 0.8 3.1 2.0 6.0 0.0 2.8 2.0 12.0 4.0 12.0 4.0 6.0 2.0 2.8 2.7 9.3
Activity Activity Experiment Brandon Method al 216,9 222 221,4 217 199,9 206 183,4 188 190,4 185 186,1 178 183,1 175 169,7 165 163,8 159 146,3 138 128,2 133 119,6 124 119,7 116 110,5 107 102,5 100 106,5 98 98,5 95 89,5 93 96,3 91 93,2 91 83,9 89 79,3 83 88,1 81 72,9 69
Activity ANN 219,8 207,3 198,4 185,5 195,5 180,7 173,5 165,0 159,1 138,5 133,1 124,1 115,9 113,8 100,3 97,9 95,3 99,8 91,9 92,7 88,9 81,6 81,1 70,6
7.3 Fuzzy Logic Method Quantitative ranks, fuzzy vocabulary and trapezoidal type belonging functions were defined for all the parameters. To formalize the correlation between parameters, expert assumed properties, which determine the catalyst activity. Nineteen rules from literature were obtained (Absi-Halabi et al., 1994; Bogdanor, 1986; Do, 1984; Furimsky et al., 1999; Howell et al., 1985; Nunez et al., 2000; Pazos et al., 1983; Pereira et al., 1987).
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Numerical values were referent to compare with experimental data. The fuzzy model was created by dialogue of the expert with the computer. Finally, the developed model predicts qualitatively the experimental activity of 22 out of 24 total data sets. Table 2. Sum of quadratic errors for some regression methods (training and test data) Multiple Regression Back Regression by steps Elimination 2
Σε 1. Training data (16) Σε22 . Test data (8)
Brandon Brandon Method Modified
ANN
20669,05
20844,28
19387,29
753,59
632,97
389,11
5633.27
6377.46
6121,94
826.90
740.96
376.22
8. Conclusions The proposed computer-aided strategy: Hotelling/Brandon-Method plus Statistical Models/Artificial Neural-Networks/Fuzzy-Logic, represent an effective methodology and a powerful computational tool for industrial catalytic investigations where uncertainty is high. Based on this methodology it was possible to predict the catalytic activity for 24 hydrotreatment catalysts through quantitative and qualitative models with a high reliability. Therefore, computer-aided forecast can be especially useful for decision-making about optimal textural and chemical parameters in catalysts for this complex process and it could also be helpful to provide appropriate recommendations for manufacture of a catalyst with selected properties.
References M. Absi-Halabi, A. Stanislauss, A. Qamra , 1995. “Hydroprocessing of vacuum residues: relation between catalyst activity and pore size distribution”, Fuel, 74 (8) 1211-1216. E. Bartlet, 1994. “Self Determination of input variable importance using neural networks”, Neural Parallel & Scientific Computation 2, Iowa State University: 103 – 114. J. Bogdanor, 1986. “Characteristics of a commercially aged Ni-Mo/Al2O3 hydrotreating catalyst”, Eng. Chem. Prod. Res. Dev., 25, 220-226. H. Dennis, H. Rouvray, 1997. “Fuzzy Logic in Chemistry”. Academic Press, 356p. D. Do, 1984. “Effect of pore size and concentration in large molecules”, AIChe J. 30: 849-854. E. Furimsky, F. Massoth, 1999. “Deactivation of HDT catalysts”, Catal. Today, 52 (4): 381-386 J. Hilera, J. Martínez, 1995. “Redes Neuronales Artificiales: Fundamentos, Modelos y Aplicaciones”, Addison Wesley Iberoamericana, Madrid, 1995, 388p. I. Hoskins, D. Himmelblau., 1988. “Artificial Neural Networks Models of Knowledge Representation in Chemical Engineering”, Comput. Chem. Eng. 12: (9/10) 881-890. R. Howell, C. Hung , K. Gibson, H. Chen, 1985, “Catalyst selection important for residuum hydroprocessing”, Oil Gas Journal,. 83 (30): 121-126 D. Johnson, 1998. “Métodos Multivariados Aplicados al Análisis de Datos”, Kansas State University, Int. Thompson editores, Soluciones Empresariales, México, 566p. V. Kafarov, F. Jiménez, M. Nuñez, In:. Vías para el Diseño de Plantas Químicas en Condiciones de Incertidumbre, Ed. Unniversidad de la Habana, Cuba, 2000. M. Núñez, Z. Pachón, V. Kafarov, D. Resasco. 2000. “Deactivation of Ni-Mo/Al2O3 catalysts aged in a commercial reactor during the HDT”, Appl. Catal. A, 5057: 1-10. J. Pazos, J. Gonzalez, A. Salazar, 1983. “Effect of catalyst properties and operating conditions on HDT high metals feeds”, Ind. Eng. Chem. Proc. Des. Dev., 22: 653-661. C. Pereira, R. Donelly, L. Hegedus, 1987. “Design of hydrodemetallation catalysts”. Catalyst Deactivation. Edit. Marcell Dekker. 315 p. Y. Qian, P. Tessier, 1995. “Application of Fuzzy Relational Modeling to Industrial Product Quality Control”, Chem. Eng. Technol., 18: 1-7.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A framework for modeling particle size effects in emulsion polymerization systems using computational fluid dynamics linked to a detailed population balance model Rebecca C. Elgebrandta, David F. Fletchera, Vincent G. Gomesa, Jose A. Romagnolia,b a
Department of Chemical Engineering, The University of Sydney, NSW 2006, Australia Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA, 70803
b
Abstract To improve modeling of emulsion polymerization systems at a reasonable computational cost, a hybrid-multizonal framework is being developed using the process simulation software gPROMS and the computational fluid dynamics (CFD) package FLUENT. The use of a detailed kinetic model in conjunction with CFD enables the incorporation of information about a number of additional phenomena that might affect the PSD, into the kinetic model. One phenomenon in particular is the shear dependence of coagulation which can now be treated in much greater detail. Additionally, information from the kinetic model, such as changes in the viscosity of the latex due to the evolution of the PSD that may affect the flow field is also passed to the CFD package. The details of the framework is presented, as is a preliminary study on the effect of the exchange flows between the zones and the effect of the shear rates on the PSD. Keywords: emulsion polymerization, hybrid-multizonal model, CFD, process simulation
1. Introduction Modeling and simulation of emulsion polymerisation is a challenging task because of the complex physico-chemical sub-processes existing within the multiphase process. The particle size distribution (PSD) is of major importance to product characteristics and a number of kinetic models have been developed in order to predict its evolution. These kinetic models assume perfect mixing within the reactor. However, in reality this is not valid, as the flow field in the reactor also plays an important role in the evolution of the PSD. Not only does it affect reactor homogeneity, it also plays an important part in reactor heat transfer and controls the coagulation behaviour. Additionally, the flow field alters the dynamic viscosity of latex in an emulsion polymerization reaction because of the non-Newtonian rheology. This effect is particularly strong for latices with high solid content. The effect of the kinetics, as well as the flow field, on the PSD in emulsion polymerization is thus of major interest to the polymer industry. While extensive models of either of the two processes are readily available, combined models are still in their infancy.
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In this paper we propose a hybrid-multizonal framework for modeling the evolution of the PSD in an emulsion polymerization reactor using a combination of kinetic and flow field modeling at reasonable computational cost. In order to reduce computational expense the proposed framework makes use of mixing fields within compartments, inside of which the flow properties are assumed to be uniform. This drastically decreases the number of regions to which the kinetic calculations need to be applied, while still providing a more comprehensive simulation of the system compared with using the kinetic model applied to the reactor alone. The changes in rheology caused by the changing PSD and shear are also taken into account by estimating the viscosity change depending on the PSD and the solid content.
2. Coupled framework for PSD simulation The initial framework enables the kinetic model in the process simulation tool gPROMS to access flow field information generated from the commercial CFD package FLUENT. In order to reduce the computational time a 2D mesh was generated and six computational zones were created (see Fig 1). Consequently, the kinetic model is divided into six corresponding sub-models, one for each CFD zone. The selection of zones is based on a rough estimation of the regions of high shear from an initial simulation. Communication occurs both between the CFD and the kinetic model, as well as between the sub-models within the kinetic model. The communication between the CFD package and gPROMS is carried out using the gPROMS Foreign Process Interface (FPI) in Excel and Visual Basic for Applications (VBA). The FPI works well in this context since it allows the kinetic model to both send and receive information before resuming its calculations. 2.1. The CFD model The CFD model was based on a 1L laboratory reactor used for emulsion polymerization at the Department of Chemical Engineering at the University of Sydney. A 2D axisymmetric geometry was created with the mirror plane located in the middle of the reactor, extending from the bottom, through the middle of the impeller and shaft, until reaching the top. The reactor operates in the turbulent regime and the turbulence model selected was the standard k-ε turbulence model due to its robustness. Since the kinetic model only updates the flow field at certain, preset times a steady state model is used. Due to the 2D geometry, the pitched-blade impeller could only be modeled as a spinning disk impeller. This will unfortunately result in much lower shear rates and exchange flows compared with those generated with the actual impeller set-up. 2.2. The kinetic model The kinetic model for emulsion polymerization was created in gPROMS. The model proposed by Zeaiter et al. [1, 2] has been further developed to include the coagulation event, as well as being adapted to work in a multizonal framework. The kinetic model is governed by three major population balance equations (PBE) for particle formation for three different types of particles: i) particles containing one
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polymeric radical (n1p) ii) particles containing no radicals (n0) and iii) particles containing one monomeric radical (n1m), as seen below: ∞
∂n0 (V , t ) = ρ ⎡⎣ n1p + n1m − n0 ⎤⎦ + kdM n1m − n0 (V ) ∫ B (V , V ') ⎡⎣ n0 (V ' ) + n1p (V ' ) ⎤⎦ ∂t 0
(1)
∞
+ ∫ B (V ', V − V ' ) ⎡⎣ n0 (V ' ) n0 (V − V ' ) + n1p (V ' ) n1p (V − V ' ) ⎤⎦dV '+ Qnin − Qnout 0
0
0
∂n1p ∂ = ρinit (V ) n0 − ρ (V ) n1p − ktr C p n1p − ( Kn1p ) + k1pC p n1M ∂t ∂V ∞
+ ∫ B (V ,V − V ') ⎡⎣ n0 (V ') n1p (V − V ') + n1p (V ') n0 (V − V ') ⎤⎦ dV ' 0
(2)
∞
-n1p ∫ B (V , V ') ⎡⎣ n0 (V ') + n1p (V ') ⎤⎦dV ' 0
−1 +δ (V − V0 ) × ⎜ ⎡⎣ IM jcrit ⎤⎦ k pjcrit , w Cw + ⎝
jcrit −1
∑ [ IM ] k 1= z
i
i e , micelle
[ micelle] ⎟ + Qin − Q out ⎠
n1p
∂n1M = − ( ρ + k1p C p + kdM ) n1M + keE [ E ] n0 + ktr C p n1p + Q inm − Q out n1 n1m ∂t
n1p
(3)
The kinetic model was modified to accommodate the exchange flows between the submodels in the hierarchical model by implementing the molar flows Qin and Qout. For these preliminary studies a simplified coagulation model was implemented with the coagulation constant, B, given by the following expression:
B=
2 G ( a1 + a2 ) 3 W0
3
(4)
where G is the shear rate, a is the particle radius of the colliding particles and W0 is the stability ratio. It should be noted that more sophisticated models of coagulation than equation (4) are required for quantitative studies of the PSD. The current treatment is more suitable for qualitative investigation of the coagulation phenomena and thus works well for establishing the proposed framework where more accurate models can be implemented if desired. The viscosity of the latex was estimated using the Dougherty-Krieger model (equation (5)) in combination with Farris’ stiffness factor (equation (6)) as previously suggested in [3].
⎛ φ ⎞ ηr = ⎜1 − ⎟ ⎝ φm ⎠
−[η ]φm
(5)
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η r = ∏ H (φ j )
(6)
j =1
2.3. Data communication Communication between gPROMS and the CFD package is carried out in the following main steps (see Fig 2). Initial exchange flow and turbulent shear rates are required to start the kinetic model. The stability ratio is calculated using Matlab and the results are printed to a file for the FPI to access. When the reaction has progressed for a certain amount of time the flow field, and thus the exchange flows between the zones, need to be updated. The FPI is provided with the current zero shear viscosities for the six zones in the reactor, which are used to launch the CFD software. After convergence of the flow calculations the exchange flows and the average turbulent shear are then returned to the foreign object which forwards the information to the kinetic model. Calculations are resumed until the next appropriate time for exchange flow and shear rate update when the described procedure is repeated.
Figure 1. The zones and the exchange flows in the reactor.
Figure 2. Flow diagram of the framework
3. Method Due to the detail and complexity of the kinetic model, some variables such as polymerisation reaction rate and concentration of desorbed radicals were held constant in the hierarchical model in order to simplify the calculations. However, the initial framework can still be considered to be quite complex, containing over 60 000 model equations. A preliminary investigation of the versatility of the framework was carried out by simulating a standard semi-batch emulsion polymerisation reaction of polystyrene resulting in a latex with a solid volume fraction of 0.24. The effects of zone-specific variables, such as the exchange flow rates and the turbulent shear rates were studied. Three cases with different exchange flows were studied: one case with zero exchange between the zones, one case with very low exchange flows, ranging between 0.001 and 0.016 kg/s, and a third case with higher exchange flows (0.020 to 1.600 kg/s). Two cases with different shear rates were investigated. In the first case the shear rate ranged from 5 s-1 in zone 1 to 31 s-1 in zone 4 (the impeller zone). In the second case the shear rates varied between 51 s-1 and 310 s-1.
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4. Results and discussion The PSD of the three cases with different exchange flows was investigated. For the case with zero exchange flows and the case with very low exchange flows no difference was detected. Between the two cases with exchange flows the difference was so small it could be neglected for this case. However, the communication of the viscosities and the exchange flows between Fluent and gPROMS, using the FPI was found to work very well, thus opening the way for study of more complex cases. The effect of the shear rates on the PSD was found to be quite considerable (see Fig 3-5). This large effect can be attributed to the qualitative coagulation model selected, where the coagulation rate coefficient is directly proportional to the shear rate. Again the communication of the average turbulent shear rate per zone from FLUENT to gPROMS was found to work smoothly, providing a means of studying the coagulation phenomena on both the overall PSD in the reactor, as well as in the different zones. As expected, the coagulation phenomena were found to result in larger particles and a lower molar particle concentration. Higher shear rates were also found to result in a lower molar concentration of particles due to increased coagulation. This trend was observed for the overall PSD, as well as between the different zones, highlighting that the selection of the zones is important as observed by e.g. Bezzo et al [4, 5] and Kresta et al [6]. It is also clear that a 3D model should be implemented due to the limited mixing that can be obtained with a spinning disk impeller as in the 2D model shown here. Consequently, a 3D model of a 1L laboratory reactor with a pitched-blade impeller is currently being implemented into the framework. The availability of kinetic data for each zone in the reactor present an exciting opportunity for better insight of what actually occurs locally in the emulsion polymerisation reactor while it is operating. As mentioned here, the effects of the viscosity, the exchange flows and coagulation can be studied. However, a number of other phenomena could be included, depending on preference, such as e.g. mixing of ingredients (especially when it comes to high viscosity cases), temperature gradients and the effect of reactor design on the overall reaction.
Figure 3. Overall PSD without coagulation included.
Figure 4. Overall PSD for low and increased shear rates.
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Figure 5. PSD for each zone in the reactor.
5. Conclusions The proposed framework using CFD in conjunction with a detailed kinetic model of emulsion polymerisation in gPROMS was found to provide an exciting opportunity to study a number of phenomena in an operating reactor. As mentioned here, the effects of changes in viscosity and in turn the exchange flows between the zones and the shear rates on the PSD can be investigated both as an overall property of the reactor, as well as in each individual zone. This provides a means to obtain a deeper insight into the operation of an emulsion polymerisation reactor, as well as the effects of its design and the various reaction events.
References 1. 2. 3. 4. 5. 6.
Zeaiter, J., A framework for advanced/intelligent operation of emulsion polymerisation, in Department of Chemical Engineering. PhD Thesis,2002, The University of Sydney: Sydney. Zeaiter, J., et al., Operation of semi-batch emulsion polymerisation reactors: Modelling, validation and effect of operating conditions. Chemical Engineering Science, 2002. 57(15): p. 2955-2969. Elgebrandt, R.C., et al., Analysis of shear-induced coagulation in an emulsion polymerisation reactor using computational fluid dynamics. Chemical Engineering Science, 2005. 60(7): p. 2005-2015. Bezzo, F. and S. Macchietto, A general methodology for hybrid multizonal/CFD models - Part II. Automatic zoning. Computers & Chemical Engineering, 2004. 28(4): p. 513-525. Bezzo, F., S. Macchietto, and C.C. Pantelides, Computational issues in hybrid multizonal/computational fluid dynamics models. AIChE Journal, 2005. 51(4): p. 1169-1177. Kresta, S.M., R. Krebs, and T. Martin, The future of mixing research. Chemical Engineering & Technology, 2004. 27(3): p. 208-214.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Pricing Utilities for Large-Scale Chemical Production Site Kentaro Hirata a, Pang Chan a, Haruo Sakamoto a, Chi-Wai Hui b, * Process Development and Design Laboratory, Process Systems Engineering and Production Technologies Field, MCC-Group Science & Technology Research Center, Mitsubishi Chemical Corp., 1, Toho-cho, Yokkaichi, Mie 510-8530, Japan b Chemical Engineering Department, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong a
Abstract Pricing utilities is one of the major important economic evaluations in the chemical production industry. Usually, company will evaluate its utility price several times per year for chasing the market condition. With the proper pricing strategy, the actual values of utilities can be reflected, which would allow the company to make correct business decision. This paper recommends a new procedure to develop a strategy for the chemical production site to set up utilities prices to reflect not only its real economic values or production costs but also to provide a better signal for business decisions. Using “Marginal Values Analysis” (MVA) technique is one of the effective ways for the pricing policy and this technique will be adopted for pricing. It is believed that with the proper utilities pricing, overall energy usage will be rationalized that generates merit to the company. Keywords: Utility price; Site-Model; Marginal Value Analysis; Optimization 1. Introduction Costing steam and electricity are the most important business strategy for economic evaluations in the chemical production industry. Experience in enhancing site efficiency can come through by evaluating the utility prices correctly [1]. Traditionally, utility cost was calculated based on the fuel cost, production demands and equipment constraints. However, the fuel cost and demands were subject to the market condition, and sometime they are difficult to predict. The marginal costs, afterwards, became the major applications of costing utilities. The marginal costs are calculated to reflect the production cost of a utility stream by tracing its corresponding path [2-3]. This approach however may not be able to provide correct results since a utility stream may have more than one-generation path. Therefore it is difficult to evaluate contributions for different paths during the marginal cost calculation. Instead of only calculating a marginal cost that reflects a utility production cost, Hui [4] proposed a marginal value analysis by representing it to be marginal profit and *
Author to whom correspondence should be addressed: [email protected]
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the product value of a stream. The site-model, an optimization tool, will then be employed to provide insights into the marginal value application for costing utilities. As the site-model includes all utility and material balances and interconnections inside the chemical site, this can overcome the localized constraints in the traditional approach. As a result, the true cost of utilities can be established using the modeling, optimization and MVA techniques. Eamples used for demonstration will be presented in this paper. Subsequently, utility prices vary with time of use (shifts) will be investigated in order to generate merit to the company. 2. Problem Definitions An example of utility plant together with production site illustrated in figure 1 is used in the case study. The target focuses on the utility plant selling utilities for the customers. Originally, the utility plant sells steam (High pressure steam, HPS and Low pressure steam, LPS) and electricity for five production plants (ETY, VCM, PVC, PP and PE). These five plants are regarded as the internal users because they are connected to the utility plant directly. Afterwards, there are three external users (Batch Plants A, B and C) joined into the site. Pipe connection is then established for supplying steam for them. However, external users are allowed to select their own electricity suppliers. In the basis of supply, the utility demand for the internal user must be satisfied first, and then followed by the external user. Thus, the selling price of utilities is separated into internal and external prices. External price would be the price after satisfying the internal. The utility plant includes two boilers (B1 & B2), two back pressured turbines (T1 & T2) and a condensing turbine (T3). As the total utility demands in the site increases greatly after introducing the external users. To resume the capability of the utility plant, new turbines and/or boilers may be essential, such that the utility price might be necessary to revise in the future. Fuel Gas VHP HP LP EL Fuel
Fuel
Utility Plant B1
ETY Plant
B2
VCM Plant
PE Plant
T1
T2
TX1
T3
PP Plant Purge
Internal Users External Users
Condensate
BFW
PVC Plant
: Storage Tank : VHP : HP : LP : Electricity
PLANT A
PLANT B
PLANT C
Fig. 1: Simple Utility Plant and Production Site.
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3. Site-Model Definitions Site-model is a multi-period linear programming model, which includes all units of the site and their interactions. It is hence possible to use one single model for optimizing all trade-offs in the site. The definitions of site-model are shown below. Indices: p m a t s r
Plant or unit Material (includes utility and manpower) Alternative (variable properties, e.g. input or output) Time period Shift in a time period, t Material or utility balance equation index
Sets: P M A T S R
Set of p Set of m Set of a Set of t Set of s Set of r
A variable in a site-model is defined by three indices (p,m,a). With the combination, users can recognize the variable’s corresponding (p,m,a) easily. Parameters: Er,p,m,a,t,s Lp,m,a,t,s Up,m,a,t,s SLt,s Cp,m,a,t,s
coefficient of variable at (p,m,a) in period t, shift s for equation r lower bound of variable at (p,m,a) in period t, shift s upper bound of variable at (p,m,a) in period t, shift s time length of shift s in period t cost/price of (p,m,a) in period t, shift s
Positive continuous variables variable of (p,m,a) in period t, shift s Fp,m,a,t,s Continuous variables operating cost in period t, shift s Profitt,s 3.1 Material and energy balance equations r ∈ R, t ∈ T , s ∈ S ∑ (Fp,m,a,t ,s × Er , p,m,a,t ,s ) = 0
(1)
3.2 Bounds of variables L p , m, a ,t , s ≤ Fp , m , a ,t , s ≤ U p , m , a ,t , s
p ∈ P, m ∈ M , a ∈ A, t ∈ T , s ∈ S
(2)
p ∈ P, m ∈ M , a ∈ A, t ∈ T , s ∈ S
(3)
r∈( p ,m ,a ),t , s
3.3 Profit calculation Pr ofitt , s =
∑ (F
p , m, a ,t , s
p, m, a
× C p , m , a ,t , s × SLt , s )
3.4 Objective function Maximize the total profit for all planning periods t and shifts. max ∑ Pr ofitt ,s
t ∈T , s ∈ S
(4)
t ,s
Equations (1) to (4) form a basic site-model and is solved by the solver XPRESS. 4. Example 4.1 Base case Base case shows the chemical site’s condition where there are only internal users incorporated with the utility plant, external users are excluded from the base case. The utility plant sells HPS, LPS and electricity for those 5 internal production plants,
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the prices are given in the table 1. The monthly production rates for 3 final products (PVC, PP & PE) are being fixed and then the selling price would be analysed by the marginal value analysis whether the price setting is appropriate or not. From the site model calculation, the annual total-site profit of the base case is 2182.18M¥. The marginal value (MV) of steam and electricity are studied and shown in figure 2. MV of all utilities varies regularly throughout the year, except in maintenance periods (Apr & Sep). During the normal operation periods, however, it is observed that HPS MV displays a strange pattern (MV, Day: 620, Night: 960, Mid-night: 910 ¥/T). Usually, the normal pattern should be D>N>M, because of larger product demand in D shift. Hence, it is difficult to conclude that the selling price of HPS in D shift is profitable or not profitable. By setting the electricity importation price in D shift to be zero, it found that the HPS MV becomes, Day: 960, Night: 910, Mid-night: 910 ¥/T, the normal pattern can now be obtained (figure 3). More importantly, less HPS generation in D shifts is discovered after setting to zero importation price. It can be explained that formerly there is not enough electricity for production in D shift, so the utility plant uses all of its ability to generate electricity. The steam balance is being altered. Excess HPS has been resulted. However, there is no extra turbine for utilizing these excess HPS. So, HPS becomes less worth. On the other hand, by setting the zero electricity importation price in D shift, the utility plant will import more electricity from outside. This can restore a bit steam balance by buying more electricity and generate less HPS. Thus, HPS becomes more valuable and MV of HPS increases.So, the original selling prices of utilities are actually higher than the production costs. The price setting is reasonable and brings benefit to the utility plant. Table 1: The utility prices for internal users. D 1000 800 12000 -14000
3500
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1000
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2000 0 D N M D N M D N M D N M D N M D N M D N M D N M D N M D N M D N M D N M
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0 D N M D N M D N M D N M D N M D N M D N M D N M D N M D N M D N M D N M
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HPS (Yen/Ton) LPS (Yen/Ton) Electricity (Yen/MW) EL Importation cost (Yen/MW)
EL
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Fig. 2: The MV of utilities. (left) Fig. 3: The MV of utilities with zero EL importation price in D shift. (right)
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4.2 Case 1 In this example, external users are included in the plant site with fixed production rates. The offered steam price would be a little bit higher than internal users as usually in-house steam generation is guaranteed. However, external users have right to select their own electricty suppliers. Supposing that the utility plant gives the electricity price with 5% discount comparing with the local electricity company (table 2), so that external users purchase steam and electricity from the utility plant. Thus, the plant takes the responsibility to supply all utilities to internal and external users. From the site-model calculation, the annual total-site profit is increased to 2525.25M¥. However, the MV of HP and LP again demonstrate the strange pattern in the shifts. This illustrates that the steam balance doesn’t in good condition due to the excess HPS and LPS generation. In addition, the MV of electricity in D & N shifts are higher than the selling prices for both users. This also indicates that the profit gained from selling electricity is not desired at these two shifts. Table 2: The new utility prices for internal and external users in case 1. HPS (Yen/Ton) LPS (Yen/Ton) Electricity (Yen/MW) EL Importation cost (Yen/MW)
Internal user prices D N M 1000 1000 1000 800 800 800 12000 6000 4500 -14000 -8000 -4000
External user prices D N M 1200 1200 1200 900 900 900 13300 7600 3800 -14000 -8000 -4000
4.3 Case 2 In order to improve this situation, a high pressure condensing turbine (HP-CT), TX1, is proposed to utilize the HPS and LPS surplus. After the site-model calculation, the generation capacity of turbine TX1 is determined as 8.53MW. It requires the capital cost of 2068.20M¥. The total profit increases to 2764.78M¥ by excluding the investment cost at this moment, it is about 9.48% more than the first case. The MV of electricity decreases to a low level, such that the selling price of electricity is higher than the production cost for both users. Thus selling electricity makes profit. Besides this, the MV of HPS & LPS reveals the normal pattern, i.e. D>N>M. However, the MV shows the production costs of HPS & LPS increased. This gives a signal for the company to adjust the selling prices of HPS & LPS in order to get a reasonable return. 4.4 Case 3 In this example, the steam prices of HPS & LPS will be revised. Moreover, it is decided that steam price should be included the capital cost of the new turbine, TX1, by assuming that the payback time equal to 8 years. The ROI is calculated as 11.6%. Under this policy, the new utility price can be able to reflect the equipment cost. Since the capital cost, payback time of TX1 and ROI are known, the annualized capital cost has been calculated as 410.3M¥ [5]. Based on the result of case 2, the additional annualized profit is 362.7M¥ (Net Present Value calculation is considered
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with time equal to 8 yrs). Hence, the original prices of utilities are actually not enough to cover the capital cost of TX1 within 8 years. Now, by revising the steam price for the both users (table 3), the site profit is calculated as 2888.82M¥. The additional annualized profit is 550.5M¥. Now the capital cost can be embraced into the new utility price and give rational benefit to the utility plant. Table 3: The utility prices for internal and external users in case 3. HPS (Yen /Ton) LPS (Yen /Ton) Electricity (Yen /MW) EL Importation cost (Yen/MW)
Internal user prices D N M 1500 1500 1500 1100 1100 1100 12000 6000 4500 -14000 -8000 -4000
External user prices D N M 1600 1600 1600 1200 1200 1200 13300 7600 3800 -14000 -8000 -4000
4.5 Case 4 Different from the electricity price, the steam price is currently the same at all shifts. To explore more opportunity on the plant profitability, steam price is being varied with the shifts. The steam price can be set as (D>N>M) or (D>M>N) or (N>D>M)…etc. Then investigating which option is the best pricing strategy from the utility plant’s point of view. The result summary is shown in the table 4. It was found that by setting the steam price as (M>N>D) for internal users will bring maximum benefit for the utility plant, while setting as (N>D>M) is more favorable if only external user is considered. Lastly, by setting the price as (M>N>D) is optimum if both users are taken into consideration. Table 4: Pricing strategy for steam prices vary with shifts. Profit (internal user only), M¥ Profit (external user only), M¥ Profit (Both users), M¥
D>N>M 2883.03 2888.84 2883.04
Steam prices vary with Shifts D>M>N N>D>M N>M>D M>D>N 2888.63 2883.56 2889.48 2894.20 2888.48 2889.18 2889.16 2888.47 2888.29 2883.91 2889.82 2893.85
M>N>D 2894.75 2888.81 2894.74
5. Conclusions The paper indicates the importance of costing utilities by using marginal value analysis. The site-model is adopted to solve the problems. The actual values of utilities can then be obtained easily and accurately in order to make correct business strategy. References [1] D. Cooper, Do you value steam correctly? Hydrocarbon Processing, July 1989, 44-47. [2] A.P. Rossiter, and S.M. Ranade. Marginal Costs Set the Scene for Profitable Operation and Investment in Pinch Retrofits. IChemE: 109, 283-301, 1998. [3] S.M. Ranade, S.C. Shreck, and D.H. Jones, Know Marginal Utility Costs. Hydrocarbon Processing: 68(9), 81-84, 1989. [4] C.W. Hui, Computers and Chemical Engineering, 24, (2000), 1023-1029. [5] M.V. Biezma and J.R. San Cristóbal, Investment criteria for the selection of cogeneration plants, Applied Thermal Engineering, In Press, 2005.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Optimal experimental design for the precision of a subset of model parameters in process development Aidong Yang, Elaine Martin, Gary Montague and Julian Morris Centre for Process Analytics and Control Technology School of Chemical Engineering and Advanced Materials, University of Newcastle, Merz Court, Newcastle upon Tyne, NE1 7RU, UK
Abstract Mathematical modelling is important for process development, but often requires a large amount of experimental effort to generate the necessary data. To reduce the effort, it is important to recognize that model parameters associated with transport phenomena may be less important than those of chemical kinetics in the early stage of development. This is because the characteristics of transport phenomena can change significantly between different process scales. Thus, the experimental effort in the early development stage can be reduced by applying optimal experimental design techniques which focus specifically on the precision of the subset of parameters pertaining to the chemical kinetics. This idea, not reported previously in the literature, is tested through a simulated case study based on a toluene nitration process model. It is shown that subset parameter oriented designs outperform their full set counterpart in terms of achieving better precision of the chemical kinetic parameters with the same number of experiments, or requiring fewer experiments to achieve the same level of precision. Keywords: parameter estimation, experimental design, process development
1. Introduction In chemical process development, mathematical models play a key role in process scaleup. However, mathematical modelling requires the undertaking of a significant number of experiments to generate the necessary data. Since the development time is naturally linked to the experimental effort, reducing the number of necessary experiments is desirable for speeding up the development process. A chemical process typically involves chemical reaction(s) and various transport phenomena. Within the context of scale-up, it has been recognized that the “intrinsic” kinetics of chemical reactions are similar between scales, whilst this is typically not the case for transport phenomena (Atherton, 1999). Consequently, it has been proposed that the chemical kinetics is characterised through the undertaking of experiments at smaller scales, whilst transport phenomena are investigated at scales that are more closely aligned to those of commercial production (e.g. Mayer, 2002). However, this strategy of separation is not appropriate in some cases due to the difficulty in breaking the couplings between the chemical reaction, the mixing, and the transport of mass and energy (e.g. Atherton, 1999). When developing such models, modelling at smaller scales would be required to address both the chemical kinetics and transport phenomena, although that part of the resulting model which accounts for the transport phenomena may not be accurate at larger scales. Consequently to ensure all experiments performed are relevant, it would be desirable to limit the effort expended in the early stages of development but ensuring that the chemical kinetics are satisfactorily characterized. For those parameters describing the transport phenomena, it is more
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relevant to address the issue of precision later in the scale-up process with experiments being performed at larger scales. To investigate this approach, an application of model-based experimental design for parameter estimation is considered (Atkinson & Donev, 1992; see also Chung et al, 2000; Asprey & Macchietto, 2000; Issanchou et al., 2003). Of specific interest in the analysis is the estimation of a number of parameters, some of which are associated with the chemical kinetics, whilst others are related to the modelling of the transport phenomena. The focus of this paper is to demonstrate how optimal experimental design techniques can be applied to estimate the chemical kinetic parameters (as a subset of all parameters) to a high precision, whilst simultaneously estimating all parameters. Such techniques have, to date, only been applied to a very limited extent in chemical engineering (cf. Hunter et al., 1969; Atkinson & Bogacka, 2002) and not been examined in the context of modelling for the scale-up of chemical processes. In Section 2, the optimal design theory applied in this work is outlined. A case study on the modelling of a toluene nitration process is presented in Section 3, as an example to demonstrate the aforementioned idea. Some concluding remarks are given in Section 4.
2. Theory of Optimal Experimental Designs Optimal experimental designs operate upon certain criteria which can be computed according to the selected design points and the models being investigated. In this section, the design criteria for parameter precision and their computation as involved in this work are briefly introduced. More comprehensive descriptions can be found in Atkinson & Donev (1992), for example. An optimal design criterion is typically a function of the information matrix (M) that corresponds to a model and to a particular experimental design. For a model of p parameters, M is a pxp symmetric matrix. More specifically for a linear model, M is the inverse of the variance-covariance matrix of the parameter estimator. This property holds asymptotically for nonlinear models. In this paper, two design criteria are considered. The first one, A-Optimality, minimizes the sum of variances of the parameter estimates, which can be represented as the trace of the inverse of the information matrix M: (1) min Tr ( M −1 ) , ξ
ζ represents an experimental design. The second one, As-Optimality, considers the precision of a subset of parameters whilst pursuing A-Optimality. Computationally, it minimizes the sum of the variances of the subset of parameters to be precisely estimated. Without losing generality, assume the first s parameters of a model are to be precisely estimated, then the objective function for an A-optimal design can be written as: s
min ∑Vi ,i , ξ
V = M −1 ,
(2)
i =1
ζ represents an experimental design, Vi,i is the ith diagonal element of V - the inverse matrix of M. The decomposition of the information matrix M is now discussed. Consider a chemical process with multiple responses to be observed and whose observations are described by the model:
yiu = η i ( xu ,ψ ) + ε iu ,
(3)
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where yiu is the ith observed response for experimental condition xu , i=1,…,m, u=1,…,n; ψ is a set of model parameters to be estimated; and ε iu is the random observation error. Typically it is assumed that the responses are independent. For nonlinear models, optimal experimental design is usually performed in a sequential manner, due to the dependency of the information matrix M on the parameter values (Atkinson & Donev, 1992). The sequential procedure is initiated with an initial guess for ψ. This enables the computation of M and then an optimal experimental point according to the design criterion (such as A- or As-optimality). After the designed experiment has been performed, ψ is updated using the experimental data. This process continues until the precision of ψ becomes satisfactory. In such a sequential design, an experimental design ζ appears as a single design point to be carried out exactly once, and the information matrix M can be composed as follows: n
M = ∑ Fu Σ u Fu , T
−1
(4)
u =1
where -
-
M is the information matrix corresponding to n sequentially designed experimental points;
⎧ ∂η ( x ,ψ ) ⎫ Fu = ⎨ i u ⎬ ⎩ ∂ψ l ⎭i =1,...,m;
;
(5)
l =1,.., p
and -
∑u-1 = { σ u }, ij
i , j = 1,..., m
is the inverse of the variance matrix of observations
th
recorded at the u experimental point. From Eq. (4), the following formula can be derived: T
−1
M u +1 := M u + Fu +1 Σ u +1 Fu +1 .
(6)
This formula can be used to update the information matrix at each step in the sequential design.
3. Case study: Toluene nitration process modelling Nitration of toluene is a liquid-liquid reaction process that takes place in a stirred batch reactor and which involves mass transfer between the organic phase and the aqueous phase. The detailed mathematical model can be found in Zaldivar et al (1995, 1996) and D’Angelo et al (2003), and thus is not presented here. In this case study, two reaction kinetics parameters (A and E) and one mass transfer parameter (C) are to be estimated simultaneously:
k ' = A exp(− E / RT ),
(7)
d 32 (8) = C (1 + 2.0ϕ )We −0.6 . Da In Eq. (7), k’ is the intrinsic reaction rate constant, A is the frequency factor, E is the activation energy. In Eq. (8), d 32 is the Sauter mean diameter of droplets, Da is the
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diameter of the stirrer, ϕ is the fraction of the organic phase, We is the Weber number, and C is a regression parameter. The value of C depends on the particular implementation of the nitration process (Zaldivar et al, 1996; cf. also Quadros & Baptista, 2003). It is assumed that at the early stage of process development, the experimental effort required is only that necessary for precisely estimating A and E, although C is unknown and has to be estimated simultaneously. To undertake the simulated case study, the mathematical model of the toluene nitration process was implemented in gPROMS (PSE, 2004). Simulations were performed to generate data, which, after the addition of random noise, are used as the measurements. In each simulated experiment, four variables were manipulated within the specific ranges (cf. Table 1), and measurements of the composition of the organic phase were recorded. Four preliminary experiments, designed by equally dividing the range of each manipulated variable, were performed to generate the data required to obtain an initial estimate of each parameter prior to the optimal experimental designs. Two sets of sequential experimental designs (A- and As-optimal) were then performed, using the optimal design algorithms realized in gPROMS according to the theory presented in Section 2. The results for these two sets of design, each containing five sequential experiments, are presented in Table 1. The corresponding parameter estimation performance is shown in Table 2. Note that the parameters in this table are marked as A’, E’, and C’ as opposed to A, E, and C, since a transformation was applied to the three parameters to allow for them to be of the same order of magnitude. Table 1. Results of Sequential experimental design: A-Optimality and As-Optimality Number of experiments
4+1 4+2 4+3 4+4 4+5
A-Optimal As-Optimal A-Optimal As-Optimal A-Optimal As-Optimal A-Optimal As-Optimal A-Optimal As-Optimal
Reaction temperature (K) (298-318) 318 318 318 298 318 318 298 298 318 318
Manipulated variable Initial amount Mass of of Toluene H2SO4 (kg) (1.0 – 1.1) (mole) (2.0 – 2.6) 2.0 1.1 2.6 1.0 2.6 1.0 2.0 1.1 2.0 1.1 2.6 1.0 2.0 1.1 2.0 1.1 2.0 1.1 2.6 1.0
Point in time for measurements (s) (1800 – 10800) 2070 10800 10800 10800 2705 10800 10800 10800 2523 10800
Table 2 shows that the As-optimal design results in improved precision for the two chemical kinetics parameters, A’ and E’ when these two parameters are preferentially selected in terms of precision. From another perspective, to achieve the same level of precision of A’ and E’, fewer experiments are required when only a subset of parameters are targeted. For example, the first three experiments designed by As-Optimality are already sufficient to yield estimates of A’ and E’ that are of comparable precision to those obtained with all five experiments designed by A-Optimality. Clearly, the former design gives a worse estimate for C’. However, as argued earlier, the loss of precision of C’ at this stage is acceptable since the value of C’ fitted for the scaled up process can be very different from the value fitted for the current experiments.
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The rationale as to why the As-optimal designs outperform A-optimal designs is because, unlike the former, the latter generally yields a compromise in terms of the precision of all the individual parameters to achieve overall optimality. This can be made evident from the pattern observed in Table 1. More specifically, longer reaction times before measurements are taken leads to more information for enhancing the precision of A’ and E’, thus they are dominant in the As-optimal design. In contrast, the A-optimal design alternates between short and long reaction times to balance the need to enhance the precision of all parameters. Table 2. Results of parameter estimation following A- and As-Optimal designs. Number of data sets 4+1
4+2
4+3
4+4
4+5
Parameter
Results of estimation (estimated value +/- 95% conference interval)
A’ E’ C’ A’ E’ C’ A’ E’ C’ A’ E’ C’ A’ E’ C’
With A-Optimal design 5.821 +/- 0.2795 2.039 +/- 0.08595 3.755 +/- 0.3647 5.868 +/- 0.1896 2.053 +/- 0.05904 3.807 +/- 0.2732 5.844 +/- 0.1818 2.046 +/- 0.05667 3.633 +/- 0.197 5.877 +/- 0.1125 2.056 +/- 0.03459 3.641 +/- 0.1905 5.873 +/- 0.1104 2.055 +/- 0.03397 3.617 +/- 0.1607
With As-Optimal design 5.886 +/- 0.2197 2.058 +/- 0.06769 3.926 +/- 0.6671 5.907 +/- 0.1353 2.065 +/- 0.0409 3.946 +/- 0.6196 5.872 +/- 0.1088 2.054 +/- 0.03326 3.809 +/- 0.547 5.86 +/- 0.09085 2.051 +/- 0.02748 3.802 +/- 0.5363 5.849 +/- 0.0805 2.048 +/- 0.02454 3.758 +/- 0.5061
4. Conclusions and future work Successful process modelling with less experimental effort can contribute to the speedup of process development. In this paper, it has been argued that, when chemical kinetic parameters and transport phenomena parameters are to be estimated simultaneously, in the early stage of process development, and when the latter vary significantly as the realization of the process changes with different scales, optimal experimental designs that focus on the precision of only chemical kinetics parameters are better choices than designs targeting all parameters. This idea has been verified by a subset parameter oriented design, namely As-optimal design, through a case study on the modelling of a toluene nitration process. Savings in experimental effort have been observed. In the future, the benefits of subset parameter oriented design methods will be further examined through work on evaluating the effect of nonlinearity on the optimal design criteria, and incorporating Bayesian design methods across different stages of process development.
Acknowledgements The authors acknowledge the financial support of the EPSRC GR/R64407/01 “Vertical Integration of Product Development and Manufacturing”.
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References Asprey, S.P., Macchietto, S., 2000. Statistical tools for optimal dynamic model building. Comput. Chem. Engng, 24, 1261-1267. Atherton, J. H., 1999. Chemical Aspects of Scale-up. In: W. Hoyle (Ed.), Pilot Plants and Scaleup of Chemical Process II. The Royal Society of Chemistry, Cambridge, UK. Atkinson, A.C., Bogacka, B., 2002. Compound and other optimal designs for systems of nonlinear differential equations arising in chemical kinetics. Chemometrics and Intelligent Laboratory Systems, 61, 17–33. Atkinson, A.C., Donev, A.N., 1992. Optimum Experimental Designs, Oxford Univ. Press, New York. Chung, S. H., D. L. Ma, R. D. Braatz, 2000. Optimal model-based experimental design in batch crystallization. Chemometrics and Intelligent Laboratory Systems, 50, 83–90. D’Angelo, F.A., Brunet, L., Cognet, P., Cabassud, M., 2003. Modelling and constraint optimisation of an aromatic nitration in liquid–liquid medium. Chem. Engng. J., 91, 75–84. Hunter, W.G., Hill, W.J., Henson, T.L., 1969. Designing experiments for precise estimation of all or some of the constants in a mechanistic model, Canadian J. Chem. Engng, 47, 76–80. Issanchou, S., Cognet, P., Cabassud, M., 2003. Precise parameter estimation for chemical batch reactions in heterogeneous medium. Chem. Eng. Sci., 58, 1805 – 1813. Mayer, T., 2002. Scale-up of polymerization processes. Current Opinion in Drug Discovery & Development, 5(6), 960-965. Quadros, P.A., Baptista C.M.S., 2003. Effective interfacial area in agitated liquid-liquid continuous reactors. Chem. Eng. Sci., 58, 3935-3945. PSE, 2004. gPROMS Advanced User Guide, Process Systems Enterprise Ltd., 23.02.2004. Zaldivar, J.M., Molga, E., Alós, M.A., Hernández, H., Westerterp, K.R., 1995. Aromatic nitrations by mixed acid: slow liquid–liquid reaction regime. Chem. Eng. Process. 34, 543– 559. Zaldivar, J.M., Molga, E., Alós, M.A.,Hernández, H., Westerterp, K.R., 1996. Aromatic nitrations by mixed acid: fast liquid–liquid reactions. Chem. Eng. Process. 35, 91–105.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
The complex distillation column network systematic optimization by mathematical programming Seungjune Choi, Hosoo Kim, Chonghun Han, En Sup Yoon Seoul National University, School of chemical and biological engineering 56 Shillim-9dong Kwanakgu, Seoul, 151-744, Korea
Abstract We propose a new approach to minimize total operation cost of distillation column network where different types of complex columns exist. A new optimization framework for complex distillation column network was presented, and the proposed approach has been applied to the industrial process. The proposed approach is composed of two-level optimization procedure instead of solving overall equations at one time. We can obtain economic benefits in the optimized design using this systematic approach. Keywords: Distillation column, Optimization, Mathematical Programming,
1. INTRODUCTION With increasing attention to energy saving in distillation columns, many research papers have been published on the distillation column sequencing problem and optimization of unit distillation column. They can be categorized by two approaches. One is thermodynamic approach and the other is systematic approach. Yeomans and Grossmann presented a nonlinear optimization model for the optimal synthesis of heat integrated distillation sequence. By use of pinch analysis and mathematical programming which uses the states task network or state equipment network superstructure representations. Allgor et al. presented screening models for reaction/distillation networks, which simultaneously consider aspects of process synthesis, design and equipment allocation in order to derive rigorous lower bounds on the manufacturing cost. Noda et al. proposed an optimal structure for batch distillation column for energy conservation and verified the results by pilot scale test that is composed of online estimator and optimizer. Barttfeld et al. examined the performance of the different representation model of mathematical programming and general disjunctive programming to determine the configuration and operation condition of distillation column such as number of trays, feed and product locations and energy use in the separation. Lang and Biegler proposed a nonlinear programming formulation for unit tray optimization that is usually MINLP. Integer variables were replaced by continuous variables by the use of differentiable distribution function. In this paper, we propose a multi-agent modeling approach to minimize total operation costs of distillation column network where different types of columns exist. The proposed approach is composed of multi agents instead of solving overall equations at one time. At the main agent, optimal distillation column load is determined, and at the sub-unit agent the optimal operation of unit column is determined. A new
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optimization framework for real time distillation column network was presented, and the proposed approach has been applied to the industrial process.
2. DISTILLATION COLUMN NETWORK OPTIMIZAITON USING MATHEMATICAL MODELING Task allocation is the problem of assigning responsibility and problem-solving resources to an agent. Minimizing task interdependencies has two general benefits regarding coherence: First, it improves problem-solving efficiency by decreasing communication overhead among the problem-solving agents. Second, it improves the chances for solution consistency by minimizing potential conflicts. In the second case, it also improves efficiency because resolving conflicts can be a time-consuming process. The objective of the column network optimization is not only the minimization of the each cost of the distillation column network, but also determination of overall optimal condition. In order to determine the optimal operation condition, bi-level agent modeling approach is proposed. [Figure 1] At the upper agent, optimal column load distribution is determined by screening model and at the lower agent model, unit distillation column optimization by MINLP modeling. Because the benefits of the column load distribution by screening model is larger than the lower level optimization benefit due to the operation constraints such as product specification or environmental constraints, bi-level optimization is a robust approach instead of finding optimum solution satisfying all the constraints at once. 2.1. The screening model for column load distribution optimization As mentioned, bi-level agents have the concept of screening models for column load distribution optimization at upper level. The screening models yield a rigorous lower bound on the cost of production, providing both design targets and a valid way in which to prune or screen discrete alternatives (process structures and equipment configurations) that cannot possibly lead to the optimal solution.(Allgor, 1997) The models consider changes to the process structure, the operation of the tasks, and the allocation of equipment simultaneously. In addition, these models embed aspects of the process synthesis not considered in previous research dealing with process design. However, they do not provide a detailed process design, so they must be used in conjunction with techniques that consider the dynamics of process in detail such as the multistage dynamic optimization formulations used to address the performance subproblem. The process network optimization problem may be formulated as the following mixed integer problem. N
Min ∑ OCN ( X i , v, y ) X i ,v , y
(1)
N =1
Objective function is to maximize the total operation cost benefit by optimal load distribution. X i represents the flow rates of interconnections in the distillation column
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network. N represents the number of distillation columns. OC N is the operation cost at Nth distillation column, respectively. v typically denotes intermediate material states of initial conditions for tasks. y is a set of parameter that can only take binary values; this define the assignment of equipment to tasks. Subject to: (1) Material balance for each column NFN
∑ DC i =1
i,N
= DN + BN + S N for N = 1...NC (2)
(2) Capacity of each column
CapMin , N ≤ CapN ≤ CapMax , N
(3)
(3) Operation range of each interconnection steam
Fint, Mini ≤ Fint, X i ≤ Fint, Maxi for i = 1...NI
(4)
(4) Operation range of reboiler and condenser
RBMin , N ≤ RBN ≤ RBMax , N for N = 1...NC (5) CON Min , N ≤ CON N ≤ CON Max , N for N = 1...NC (6) (5) Environmental & Product Specification constraints
TCN ≤ TCMax , N BC N ≤ BCMax , N
(7), (8)
The drawback of these models is that the models do not consider the detailed operation of the tasks, so the model solutions do not correspond to designs that can be implemented directly. Instead, the screening model provides targets for the detailed design of actual process.(Allgor and Barton, 1997) And it also enable the derivation of a rigorous algorithm to address the mixed integer optimization formulation of distillation network process. Column Network Global Optimization Framework
Column Optimizat ion #1
Column Optimizat ion #2
Column Optimizat ion #n
Figure 1. Bi-level modeling optimization framework
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2.2. The unit column optimization modeling Consider a distillation column with N trays, including the condenser and the reboiler. The stages are numbered bottom upward so that the reboiler is the first tray and the condenser is the last (Nth) tray. Only the total condenser and kettle-type reboiler case is considered the other cases can be dealt with similarly. For definiteness, only two feeds are considered. Let I={1,2,…,N} denote the set of trays and let R={1}, C={N}, and S={2,3,…,N-1} denote subsets corresponding to the trays in the reboiler, in the condenser, and within the column, respectively. Let Fd1 and Fd2 denote the feeds. Let c denote the number of components in the feeds, and let J = {1,2,…,c} denote the corresponding index set. Let
F k , T fk , p kf , v kf , z kf and h kf , k = 1, 2 denote, respectively, the molar flow rate, the temperature, pressure , the vapor fraction, the vector of mole fractions and the molar specific enthalpy of the corresponding feeds. Let pi denote the pressure prevailing on tray i. It is assumed that
preb = p1 , pbot = p2 , ptop = pN −1 and pcon = pN are given, although one may treat them as variable to be determined, if desired. Then p1 ≥ p2 ≥ ... ≥ pN , and for simplicity, let
p kf ≥ pbot , k = 1, 2 L
L
Let Li , xi , hi and f ij denote the molar flowrate, the vector of mole fractions, the molar specific enthalpy, and the fugacity of component j, respectively, of the liquid leaving L
L
tray i. Similarly Vi , yi , hi and f ij denote the corresponding quantities for the vapor. Let denote the Ti temperature prevailing on tray i. Then
fijL = f ifL (Ti , pi , xi1 , xi 2 ,..., xic ) fijV = f ifV (Ti , pi , yi1 , yi 2 ,..., yic ) hiL = hiL (Ti , pi , xi1 , xi 2 ,..., xic )
(9)
hiV = hiV (Ti , pi , yi1 , yi 2 ,..., yic ) where the functions and/or procedures on the right-hand sides depend on the thermodynamic model used.
3. APPLICATION EXAMPLES The proposed framework has been applied to an dehydration column network process. TA (Terephthalic Acid) is one of the most important raw materials in chemical industries. TA is produced by reacting p-xylene and air in acetic acid. At the solvent dehydration section, water is removed and dehydrated solvent is recycled to the reactor to be reused. Solvent dehydration section is composed of several distillation columns and usually about 40-50% of total steam consumption is made at this section. Figure 2 shows the overall column network process that is composed of five distillation columns (three conventional columns and two azeotropic columns).
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Table 1 Adjustment Results [ton/hr] Interconnected Before Column After Column Change Stream Load Change Load Change X1 4.18 0.00 -1.48 X2 10.99 15.50 4.51 X3 0.00 0.00 0.00 X4 3.90 5.00 1.10 X5 2.98 3.00 0.02 X6 6.36 0.00 -6.36 X7 0.00 0.00 0.00 X8 20.00 20.50 0.50
This distillation column network is composed of distillation columns of different types, sizes. Feed stream entering columns have different characteristics such as concentration, temperature, pressure, status (liquid/vapor) and efficiency. Usually, there exist interconnections between plants to efficiently use the capacity of equipments in dehydration system. The minimization of the total operation costs of the distillation column network with satisfying operation constraints is the objective of this problem. Flow rates of interconnection streams between plants have been optimized by proposed approach. Distillation column network optimization model was developed by the screening model using the information on the sensitivity of interconnection streams to objective function. Table 1 shows the optimization results. Table 2 The Results of Distillation Column Load Change [ton/hr] Before Column After Column Change Load Change Load Change Column #1
26.46
23.44
-3.03
Column #2
48.21
53.46
5.24
Column #3
43.52
43.02
-0.50
Azeo Col. #1
25.72
23.50
-2.22
Azeo Col. #2
20.00
20.50
0.50
To minimize the total reboiler steam consumption with satisfying operation constraints, optimum interconnection streams were determined, the result was applied to the process. The column load of each distillation column by the change of interconnection stream flow rates is shown in Table 2. Table 3 The Results of Steam Consumption Before After Change Change Change Column #1
20.41
13.49
-6.9
Column #2
55.99
60.83
4.84
Column #3
30.02
29.81
-0.21
22.42
21.38
-1.04
15.28
15.61
0.33
144.12
142.12
-3.00
Azeo. Col. #1 Azeo. Col. #2 Saving
S. Choi et al.
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Table 3 shows the steam consumption change by applying optimization result to the process. This result is meaningful in that the steam consumption saving was achieved without additional investment such as equipment modification or installation to the process. The benefit from this steam consumption saving is about 40 million dollar/year. X1
Col#1
X2
Decanter
X3 Azeo#1
X4
Col#2
X5
X6
X6* X7
Decanter
Col#3
X8
Azeo#2
Figure 2. Dehydration column network process
4. CONCLUSION In this paper, we have proposed a novel framework for distillation column network optimization using mathematical modeling and applied this approach to an industrial process. The proposed approach is a bi-level optimization model. At upper level agent, the screening model is used for optimal column load distribution. We can decide not only column load distribution, but also feed stream allocation for relevant unit column. At lower level, unit distillation column optimization is solved by rigorous modeling. The results show that about 5% steam consumption saving can be achieved without any additional investment to the existing process such as equipment modification or installation.
References Yeomans, H.:Grossmann, I.E. Nonlinear disjunctive programming models for the synthesis of heat integrated distillation sequences, Comp. Chem..Eng. 1999, 23, 1135-1151 Allgor, R.J.; Evans, L.B.; Barton, P.I. Screening models for the batch process development Part 1. Design targets for reaction/distillation networks, Chem. Eng. Sci. 1999, 54, 4145-4165. Lang, Y-D; Biegler, L.T. Distributed stream method for tray optimization, AIChE J. 2002, 48, 582-595.
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Wendt, P.L; Garcia, H.A.; Wonzy, G. Optimal operation of distillation processes under uncertain inflows accumulated in a feed tank, AIChE J. 2002, 48, 1198-1211
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Modelling and Simulation of Coal and Petcoke Gasification in a Co-current Flow Reactor *
Elida M. López, Vicente Garza and Joaquín Acevedo
Department of Chemical Engineering, Instituto Tecnológico y de Estudios Superiores de Monterrey, Garza Sada 2501, Monterrey, NL, 64849, Mexico
Abstract A mathematical model is developed for the simulation of the gasification process in a reactor where a carbon slurry is fed together with oxygen and water vapour. The model considers several heterogeneous and homogeneous reactions to estimate flows and composition of the exit gases, including combustion, gasification and hydro-pyrolysis of carbon. Mass and heat-transfer phenomena are described for the particular physical arrangement, including bulk film diffusion processes with variable particle size and heat transfer by radiation from the hot region downstream to the cooler entrance region. Preliminary results are in good agreement with experimental data from a pilot plant, including exit composition and temperatures, and ignition and highest-temperature points. An analysis of the main operational parameters is given, which could be used for the final design of the plant. Keywords: carbon gasification, petcoke gasification, gasifier simulation, mathematical modelling.
1. Introduction Coal and petcoke gasification has gained increasing attention in recent years as an alternative source of energy largely because of rising oil and gas prices. Integrated gasification combined cycle power projects and other applications are growing worldwide. The gasification process transforms coal into gases (mainly CO2, CO and H2), typically utilizing fluidized or moving bed reactors. In 1963, Davidson and Harrison presented a mathematical model to simulate the behaviour of fluidized bed reactors with a two-phase theory, separating solid free bubbles that flow through the bed and an emulsion where solid particles are suspended in the interstitial gas. Based on these ideas, Gordon and Amundson (1976) presented a mathematical model for the combustion of coal in a non-isothermal fluidized bed, considering first-order heterogeneous reactions for the combustion and the gasification of C, and a second order reaction for the oxidation of CO. Weimer and Clough (1981) separated the bubble phase in two, to allow for an entrance (jet) region where gas and solids are in contact and introduced the water-gas shift reaction and the oxidation of CO and H2 in this region. The authors also considered three heterogeneous reactions (oxidation of C and gasification of C with CO2 and H2O) and one homogeneous (watergas shift reaction) occurring both in the bubble phase and in the interstitial gas of the emulsion phase. Mass and energy balances were developed for each phase plus an energy balance for the solids, allowing also a variable particle size through the bed. Heat transfer between jet-bubble, bubble-interstitial gas and jet-solid was evaluated, as well as heat-exchange by radiation between particles. *
corresponding author. E-mail: [email protected]. Tel: (52)81-8158 2034
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Recently Nagpal et al. (2005) presented a moving bed gasifier model for petcoke/char where the solids are characterized by two different species: fixed and volatile carbon. The scheme allows the separation of combustion and gasification processes that fixed carbon undergoes, while volatile carbon is divided into a gas producing fraction and a tar forming fraction; in turn, tar also volatilizes to CO, CO2, H2, with a production of soot. The authors mentioned that volatile content of petcoke is typically low, and thus its effect on the production and composition of syngas is small. The application that motivates this work is from the steel industry, where syngases (CO and H2) can be used as reduction agents of iron in the first part of the process. Currently, these gases are obtained by using natural gas in a reactor where other processes are integrated. Gasification would then be performed in a co-current flow reactor where solids (coal and/or petcoke) would be fed at the top-centre of the reactor with a mixture that may contain water, oxygen and vapour. Combustion would be controlled by the ratio of C to O2 fed in order to obtain the desired levels of syngas. The aim of this work is then to build a robust model, flexible enough to analyse the different feeding options and operational conditions, and their effect on the flow and composition of exit gases, helping to reduce the experimental work and some of the uncertainty in the design process.
2. Mathematical model The mathematical model consists primarily on the species (i) mass balance and the energy balance for both the solid (p) and the gas (g) phase, which are integrated through the reactors length. The species concentration (Ci) and the temperature profile (Tg, Tp) change in the axial direction (z) assuming uniform conditions radially. Further details about the model are described in the following four sections, while a summary of the most important equations that define the model is given in Table 1. 2.1. Reaction model The reaction model considers combustion, gasification and hydro-pyrolisis of C to produce CO, CO2, H2. Three heterogeneous reactions (oxidation of C, and its gasification with CO2 and H2O) and two reversible homogeneous reactions (oxidation of CO and the water-gas shift) are included; formation of H2S and NH3 can be easily estimated by stoichiometric balances, according to the obtained conversion of C, following the model of Nagpal et al. (2005). Particular kinetic data for these reactions (Arrhenius’ ko, j , Ej) is used to account for different reactivity of carbon in diverse coals and petcoke. A list of sources for these parameters is also given in Nagpal et al. (2005). 2.2. Mass transfer model The reactions described in the previous section are modified, and even controlled under some conditions, by diffusion processes. The reaction rate models have then to be modified to consider these resistances through mass-transfer coefficients. For the heterogeneous reactions occurring at the surface of the solid particles, e.g. combustion of C, an effective reaction constant can be expressed as a sum of two resistances involving the kinetic constant (Kr,j) and the mass-transfer coefficient (Km). The mass transfer coefficients can be estimated by Sherwood’s number (Sh) using typical correlations in terms of Schmidt’s (Sc) and Reynolds’ (Re) numbers. 2.3. Heat transfer model In the same way, heat transfer by convection particles-gas (QConv-PG) and gas-reactor walls (QConv-GW) can be estimated through Nusselt’s number, without much effort.
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Table 1. Mathematical model for the co-current flow gasifier
dmi
Mass and Energy Balances
dz
= ∑ν i , j R j
∑ mi Cpi
− Q Rad − PG ± Q Rad − z − Q =0 dz Conv − PG R1: C + H2O → H2 + CO ; R2: C + CO2 → 2CO R3: C + ½ O2 → CO ; R4: CO + ½ O2 → CO2 ; R5: CO + H2O ↔ H2 + CO2
R j = ke ∏ Cin ;
Mass Transfer
Sh = 2 + 0.6(Sc )
Radiation Heat Transfer Particle velocity
Conv − PG
dT p
Rate of reactions
Convective Heat Transfer
+ Q Rad − PG ± Q Rad − z + Q
dz
+ QConv −GW + Q Rxn = 0
∑ mi Cpi
Reactions involved
dTg
E − j 1 1 1 ; k r = k 0j e RT = + ke kr km 1
Nu = 2 + 0.6(Pr ) Nu = 0.027(Pr )
1
3
1
3
(Re )
3
1
(Re )
2
1
2
(Re )0.8
; k m = Sh D ; h= ; h=
Nu k g Nu k g
dp dp dr
; particle – gas ; particle - gas ; gas – walls
Q Rad -PG = ε Aσ ( Tp4 - Tg4 ) ; particle - gas Q Rad - Z = ε Aσ ( Th4 - Tc4 ) Δz
; hot zone – cold zone Lz dv p 3C D ρ (v g − v p ) | v g − v p | = ; C D = 24 1 + 0.14 Re 0.70 Re dz 4ρ p v p d p
(
)(
)
Heat lost to the surroundings by natural convection is evaluated in a similar way, however, this loss was negligible for the actual reactor. Heat transfer by radiation is accounted for in two separate ways. Radiation from solid particles to gas (QRad-PG) is evaluated at a specific point (length) in the reactor according to Stefan-Boltzman’s law. A second term however (QRad-Z) is needed to estimate radiation from the “hot zone” of the reactor, radiating back to the entering feed, “cold zone”, in order to ignite the mixture. The model considers that this cold region receives heat by radiation from downstream at an assumed average temperature (Th), according to Stefan-Boltzman’s law. The difference between absorption constants of solid particles and gas implies that this heat is mainly received by the solid particles, thus creating a temperature difference between them, affecting mass and heat transfer. As the particles reach their ignition temperature and combustion starts, temperature rises abruptly marking the beginning of the hot region. This hot region, with the produced gases and the remaining solids, loses heat by radiation to the cold region at an assumed average temperature (Tc). Since the temperature profile is not known a priori, the solution of the model requires initial estimates that must eventually converge and satisfy the overall energy balances.
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2.4. Momentum transfer model Reactants enter through a nozzle located at the top of the gasifier. According to field experiments, it is then assumed that the mixture gradually increases its cross-sectional area until it comes in contact with the gasifier wall, thus having a conical shape from the entrance to this point. After reaching the wall, the mixture follows the cylindrical geometry of the gasifier. Gas velocity (vg) along the reactor length is calculated by continuity taking into account changes in cross-sectional area and density (ρ). Jet expansion, i.e. the gradual increase in cross-sectional area, has a strong effect on the gas velocity lowering it as it expands. Solid particle velocity (vp) is adjusted based on a drag-model, making use of a drag coefficient (CD) calculated as a function of the Reynolds number. Results indicate that, for most of the reactor, the gas has a tendency to slow the solid particles until the relative velocity between them becomes zero. All velocity profiles are assumed one-dimensional varying only in the axial direction.
3. Numerical Results The model was used to simulate the operating conditions of a pilot plant, for different feed mixtures. Some of the most important analyses required included the effect of the amount of water used in the slurry to feed the solids, oxygen requirements and its effect of exit gas compositions, ignition points and maximum temperature for different types of solids. Typical model results include concentration and temperature profiles along the length of the reactor as seen in Fig. 1. The model predicts a peak in temperature for both the gas and the solid near the entrance to the reactor.
Figure 1. Concentration and temperature profiles for a typical simulation run.
This peak is due to the carbon combustion reaction and it is accompanied by a sharp increase in the concentration of CO2. At this point, combustion rate is controled by diffusion of oxygen to the particle surface and not by the rate of the reaction. Once O2 is totally depleted and temperature is still relatively high, gasification reactions become more important and they consume an important amount of water vapour and the
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previously formed CO2. These reactions are endothermic and their effect is to lower the temperature of the reactor. Finally the water-gas shift reaction plays an important role in determining the final composition of the gas species mostly occuring forward to form H2 and CO2 until an equilibrium is reached. Results presented in Fig. 1 were obtained from a 3 m reactor with inlet conditions as presented in Table 2. In the same Table 2, a comparison is made between the exit gas composition obtained from the pilot plant and from the mathematical model. Table 2. Results from a pilot plant run and the mathematical model. Inlet Conditions Carbon slurry (m3/h) Slurry temperature (C) 3
O2 (m n/h) 3
Water vapour (m n/h)
0.086
Coal (kg/hr)
47
25
Particle diameter (μm)
70
74
O2 temperature (C)
25
0
Temperature (C)
220
Exit Gas Composition CO2 (Pilot Plant)
0.34
CO2 (Model)
0.32
CO (Pilot Plant)
0.38
CO (Model)
0.42
H2 (Pilot Plant)
0.27
H2 (Model)
0.25
Several parametric runs were made to study the effect of changing operating conditions and feeds. The results of these runs were in general agreement with pilot plant experiments in terms of placement and length of ignition/combustion sections and temperature profiles; maximum discrepancies in compositions were around 20%. For example, Fig. 2 shows the effect of changing the inlet ratio of H2O to O2. The amount of syngas produced and the carbon conversion is maximized for low H2O/O2 ratios, however, most of the syngas in this case is CO. Increasing the H2O/O2 ratio reduces the carbon conversion and total syngas production but increases the amount of H2. The desired operating point could then be dependent on the requirements of a given application in terms of the syngas quality. Another set of runs was also made to determine the effect of O2 in the feed as shown in Fig. 3. For low O2 conditions the proportion of H2 in the exit gas is maximized but the carbon conversion is relatively low. As expected at high O2 values a large proportion of CO2 is obtained accompanied by a complete conversion of carbon.
4. Conclusions A model was developed to predict gas compositions, temperatures and fuel conversion in a co-current carbon/petcoke gasification reactor. The model was used to simulate the conditions of a pilot plant, providing some insight regarding the phenomena that take place in the reactor. The model can potentially be used to determine the optimum operating conditions for specific process requirements. Finally, the model shows that this type of reactor may indeed be used as a gasifier and can compete with other technologies such as fluidized beds and moving bed reactors. With additional validation, the model could also be used as a design tool to determine the necessary reactor length, since generally the combustion reaction takes place very near the
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entrance but gasification will require some minimum length to occur to the desired degree.
Figure 2. Effect of varying Water-vapour to Oxygen ratio in the feed.
Figure 3. Effect of varying percentage of O2 in the feed.
Current work also includes the incorporation of Nagpal’s volatilization model to predict tar formation and further analysis to determine the conditions or applications for which this type of reactor would be most useful. More complex developments will include a more efficient numerical integration scheme that allows a bidimensional model with radial dispersion.
References Davidson, J. F., and D. Harrison (1985) Fluidization. Academic Press. Gordon, A. and N. R. Admundson (1976) Modelling of fluidized bed reactor – IV, Chem. Eng. Sc., 31,5, 1163-1178. Nagpal, S., T. K. Sarkar and P. K. Sen (2005) Simulation of petcoke gasification in slagging moving bed reactors, Fuel Processing Technology,, 86, 617-640. Weimer, A. W. and D. E. Clough (1981) Modeling a low pressure steam-oxygen fluidized bed coal gasifying reactor, Chem. Eng. Sc., 36, 3, 549-567.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Simulation of (electro)chromatography by means of CFD Dirk-Uwe Astratha, Thomas Schneiderb, Wolfgang Arlta a
Lehrstuhl für Thermische Verfahrenstechnik, Friedrich-Alexander University Erlangen-Nuremberg, Egerlandstr. 3, 91058 Erlangen, Germany b Fachgebiet Thermodynamik und Thermische Verfahrenstechnik, Technische Universität Berlin, Straße des 17. Juni 135, 10623 Berlin, Germany
Abstract Computational Fluid Dynamics is employed to develop models of (electro)chromatographic processes that take the radial coordinate into account. The liquid chromatographic models are based on the results of computed tomography experiments that proved the packing structure of the invastigated columns to be heterogenous. In the context of electrochromatography, CFD is used to evaluate the potential of the technique to be scaled up to columns sizes that have not yet been used in practice. Keywords: Liquid Chromatography, Electrochromatography, Computational Fluid Dynamics, Computed Tomography
1. Introduction High Performance Liquid Chromatography (HPLC) has proved to be a versatile unit operation for the gentle separation of substances which are difficult to separate by more common techniques like distillation or extraction. Consequently, its importance for the production of highly purified products in the fields of pharmaceuticals, fine chemicals and life sciences has increased rapidly during the last couple of years. Several models have been developed for the simulation and scale up of chromatographic separations starting from the results of analytical method design. The majority of these models are based on one-dimensional (spatially) differential mass balances. Furthermore it is assumed that the model parameters (e.g. the porosity) are constant throughout the column. Because of this, these models lack the ability to account for gradients other than axial concentration, temperature or velocity gradients. Within the scope of our work the commercial computational fluid dynamics (CFD) code Star-CD was used for the development of more-dimensional models of chromatography that offer an opportunity for a more accurate description of chromatographic processes.
2. Computational Fluid Dynamics Computational Fluid Dynamic methods rely on the numerical integration of the underlying partial differential equations representing the problem at hand. The numerical integration schemes transform the system of partial differential equations (PDE system) into a system of algebraic equations which may be solved computationally. There is a variety of methods for the numerical integration of the partial differential equations (e.g. Finite Difference, Finite Element, Spectral methods, etc.). A detailed discussion of the Finite Volume method employed by the commercial CFD code Star-CD that was used in our work would go beyond the scope of this paper and may be found elsewhere [1].
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In order to simulate chromatographic separations by means of rigorous models, the continuity equation as well as the equations of conservation for the species present
∂Ci ∂Q + F ⋅ i + v ⋅ ∇Ci = Dij ∇2Ci ∂t ∂t
(1)
have to be solved numerically. For the implementation of the accumulation term of the stationary phase and the underlying adsorption equilibrium, the original modelling capabilities of the CFD-code must be extended via Fortran user coding [2]. The equation of motion is commonly replaced by the so called Darcy equation in order to describe pressure driven fluid flow within a porous medium.
v=−
κ (∇P − ρg ) μ
(2)
Herein the permeability is a function of the external porosity and the particle diameter as given by the Blake-Kozeny equation.
d P2 ε3 κ= 150 (1 − ε )2
(3)
If the separation process is not isothermal the equation for the conservation of energy has to be solved as well. It is worth noticing that in cases where the column is either not isothermal or the concentration dependency of the material properties cannot be neglected, the differential equations are coupled.
3. Chromatography and Computed Tomography 3.1. Motivation Chromatographic columns that are packed with so called slurry methods offer a rather homogenous core region surrounded by a denser and less permeable region in the vicinity of the column wall [3]. In our work we used x-ray computed tomography as a non-invasive measurement technique to examine the uniformity of the column and determine input parameters for a 2D CFD-Model. 3.2. Computed Tomography measurements X-ray computed tomography is a non-invasive measurement technique that allows to monitor progressing tracer fronts in situ. During the transition of the object the x-rays are attenuated following Lambert-Beer’s law. The intensity of the attenuated beams is detected to obtain a projection of the object. The projection data can be used to reconstruct an image file consisting of an array of CT-numbers (CT). CT is defined as a dimensionless attenuation coefficient in terms of SI-Units. For a porous object, CT is given as the weighted mean of the mobile (MP) and the stationary phase (SP). During a breakthrough experiment MP1 is displaced by MP2. The saturation S is given by the volume fraction of the corresponding mobile phase. During the breakthrough CT is given by
CT = ε ⋅ [S MP1 ⋅ CTMP1 + S MP 2 ⋅ CTMP 2 ] + [(1 − ε ) ⋅ CTSP ]
(4)
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585
Consequently the local saturations S during a breakthrough can be determined from three CT images representing either saturated and the transient state.
Figure 1: CT-images of a Potassium-Iodide/Methanol solution replacing pure Methanol inside a chromatographic column filled with a polydisperse ODS phase (C18; dP=40-63µm).
3.3. CFD-Model The additivity of the retention time and the variance allows to determine permeability and dispersion maps of the column from the local saturation histories. In our work we subdivided the column into thirty sections (three axial sections enframed by the monitoring positions and ten radial annuli, respectively). The parameters of these sections were implemented into a two dimensional model of the column. To ensure numerical accuracy, time step and spatial mesh width were chosen to give Courant- and Dispersion-/Diffusion-Numbers close to unity. 3.4. Results The experimental results given in Figure 2 are intra-column saturation profiles of a Potassium-Iodide solution replacing pure Methanol recorded at the two most downstream monitoring positions inside a chromatographic column filled with a polydisperse ODS phase. Potassium-Iodide was chosen as the tracer due to the relatively high atomic mass of Iodine that allows for good contrast in the CT-images, the use of a non-polar ODS phase should minimize interactions between the ionic tracer and the stationary phase. The experimental results are compared with a) two independent fits of the equlibrium dispersive model (EDM) and b) intra-column breakthrough curves computed with the StarCD model. It can be seen that the common equilibrium dispersive model (1D; uniform parameter distribution) is unable to account for the tailing in the recorded saturation histories. This phenomenon is frequently observed in practice. Due to the non-homogeneity of the packing the part of the front close to the wall falls behind during the migration process thereby causing the band tailing. On the other hand the StarCD model (2D, axial and radial parameter distribution) accounts well for the prolongated breakthrough of the band rear showing that a more sophisticated modelling results in an enhanced predictivity.
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Figure 2: Comparison of experimental (CT) and simulated (EDM, StarCD) intra-column breakthrough-curves at two different axial positions inside an ID=50mm; L=350mm chromatography column packed with a polydisperse ODS phase (C18; dP=40-63µm). PotassiumIodide/Methanol solutions replacing pure Methanol.
4. Electrochromatography 4.1. Motivation Nowadays, capillary electrochromatography (CEC) is a routine analytical scale separation technique [4,5] comparable to High Performance Liquid Chromatography (HPLC), with the important difference that the flow of the eluent through the chromatographic bed is not induced by applying a pressure difference across the column length but by means of electroosmosis [6,7]. To evaluate the feasibility of scale-up, Joule Heat generated by the electrical current through the column and its effects on the separation have to be examined. Joule Heat causes radial as well as axial temperature gradients in the separation columns that on their part cause dispersion of solutes, e.g. via the temperature dependent velocity profile, molecular diffusion and adsorption to the column packing. 4.2. CFD-Model In this context, CFD is used to solve the coupled steady state energy- and momentumbalance equations for column sizes that have not yet been used experimentally. The simulated systems consist of the chromatographic bed itself, which is modeled using StarCD’s built in equations for porous media, and the column wall. The boundary conditions are: zero pressure at inlet and outlet (since no external pressure gradient is applied), slip at the inside of the column wall (since electroosmotic flow is generated at the column wall as well), constant temperatures at the inlet (Tin) and at the outside of the column wall (Twall) and zero temperature gradient at the outlet (thermally developed flow). The basic balance equations for momentum end energy are extended by source terms via user coding, e.g. for the energy balance equation (steady state):
Simulation of (electro)chromatography by means of CFD
ρc p v ⋅ ∇Ti = λeff ∇ 2Ti + κ eff (Tm ) E 2
587 (5)
For the energy balance equation (Eq. 5), the source term includes the effective electrical conductivity κeff of the system, dependent on the systems mean temperature, and the Electrical field strength E. The magnitude of the source terms and their dependence on parameters like the electric field strength and the properties of the eluent are derived from experiments with analytical scale columns, for which the effects of Joule Heat are negligible over a wide range of operating conditions. While the source term in the momentum equation is made dependent on the local temperature, the source term of the energy equation depends on the average temperature in the fluid/porous medium section of the columns (assumption of a homogeneous electrical conductivity and therefore a homogeneous electrical field strength). The calculations allow for predictions of macroscopic temperature and seepage velocity profiles in larger diameter columns and of the dependence of these profiles on parameters like the electrical field strength. 4.3. Results Results presented here are for an eluent (25mM Tris(hydroxymethyl)-aminomethane, pH 8.0, diluted 1:19 v/v in Acetonitrile) that was found to be ideal for scale up for the reason that it features a high flow velocity (in interaction with the reversed-phase packing material used in the experiments) while allowing only moderate electrical current and therefore moderate Joule Heat. Figure 3 shows the radial profiles of temperature and seepage velocity for several axial positions in a 5mm ID and 7.5mm OD column. It can be seen that the thermal as well as the hydrodynamic entrance lengths are in the order of four inner column diameters.
Figure 3: Radial temperature and seepage velocity profiles at several axial positions for the eluent described in the text, Tin = Twall = 293.15K, E = 30kV/m, ID 5mm, OD 7.5mm
Figure 4 shows the dependence of the maximum temperature and the average seepage velocity in the column on the electrical field strength E. The nonlinear dependence of the maximum temperature is expected due to the fact that the dependence of the source term in the energy balance equation on the electric field strength is quadratic. The nonlinearity of the seepage velocity’s dependence on field strength however is due to the different temperature fields in the column at different field strengths and the fact that the source term in the momentum equation is dependent on temperature.
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The results given in the figures show that, with a careful choice of the eluent (ionic strength, organic modifier), acceptable flow rates are generated without producing excessive temperature gradients in columns that are significantly larger than those currently used on the analytical scale (typically 100µm ID or less). However the dimensions are still much smaller than typical columns in preparative separations.
Figure 4: Maximum temperature and average seepage velocity in the columns for different field strengths. Other conditions as in figure 3.
References [1] H.K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics - The finite volume method, Longman, Edinburgh Gate, UK, 1995 [2] H. Boysen, G. Wozny, T. Laiblin, and W. Arlt, CFD simulation of preparative HPLC columns with consideration of nonlinear isotherms, Chemical Engineering & Technology 26 (2003) 651-655 [3] G. Guiochon, T. Farkas, H. Guan-Sajonz, J.-H. Koh, M. Sarker, B. Stanley, and T. Yun, Consolidation of particle beds and packing of chromatographic columns, J. Chromatogr. A 762 (1997) 83-88. [4] K.D. Bartle and P. Myers (eds.), Capillary Electrochromatography, The Royal Society of Chemistry, Cambridge, UK, 2001. [5] Z. Deyl and F. Švec (eds.), Capillary Electrochromatography, Elsevier, Amsterdam, 2001. [6] R.J. Hunter, Zeta Potential in Colloid Science – Principles and Applications, Academic Press, London, 1981. [7] A.S. Rathore and A. Guttman (eds.), Elektrokinetic Phenomena – Principles and Applications in Analytical Chemistry and Microchip Technology, Marcel Dekker, New York, 2004.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Modeling of heat transfer processes in particulate systems Zoltán Sülea, Csaba Mihálykóa, Béla G. Lakatosb a
Department of Mathematics and Computing, bDepartment of Process Engineering University of Veszprém, Egyetem Street 10, Veszprém, 8200, Hungary
Abstract A population balance model, taking into account the particle-particle and particle-wall heat transfer by collisions is presented for modelling heat transfer processes in fluidsolid systems. The spatial distribution of the temperature is described by the compartments-in-series with back-flow model. An infinite hierarchy of moment equations, describing the time evolution of the moments of particle temperature in cells is derived that can be closed at any order of moments. The properties of the model and the effects of parameters are examined by numerical experiments using the moment equation model. The simulation results indicate that the population balance model provides a good tool for describing the temperature inhomogeneities of the particle populations in particulate systems, and can be used efficiently for analysing the heat transfer in fluidsolid energy conversion processing. Keywords: Heat transfer, Fluid-solid systems, Population balance model, Moment equation model, Simulation
1. Introduction In modelling heat transfer in fluid-solid processing systems, five interphase thermal processes are to be considered: the fluid-particle, fluid-wall, particle-particle, particlewall and wall-environment. In systems with intensive motion of particles, the particleparticle and particle-wall heat transfers occur through interparticle and particle-wall collisions. Extensive experimental and theoretical work has been published on wall-bed and fluid-particle heat transfer processes, but studies of the effects of the interparticle collisions on the heat transfer processes in multiphase systems, especially examinations of the direct particle-particle heat transfer, are rather scarce. Delvosalle and Vanderschuren [1] developed a deterministic model for describing heat transfer between particles, Molerus [2] derived a model for heat transfer between the wall and particles. These models were applied by Mansoori et al. [3] in a four-way interaction Eulerian-Lagrangian model, computing the interparticle contact heat conduction in turbulent heat transfer in gas-solid flows. Burgschweiger and Tsotas [4] used an age distribution model in modelling fluidized bed drying. Mihálykó et al. [5] derived a stochastic model for particle-particle heat transfer, starting from a simple kinetic model with random parameter. Lakatos et al. [6], developing a general population balance model of interactive populations of disperse systems extended this model for spatially distributed systems, coupling the population balance equation with the axial dispersion model of flow of particles. However, the particle-wall interactions that seem to be important in describing heat transfer processes in continuous processing systems have not been taken into account. In the present paper, the population balance model is extended to describe also the wall-particle heat transfer processes by collisions, taking into account the fluid-solid,
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wall-fluid and wall-environment heat transfer processes as well. The spatial distribution is described by a compartment model. The moment equations and their numerical solution are shown, and the properties of the model are analysed by simulation.
2. Mathematical model Consider a continuously operated fluid-solid energy conversion system in which the fluid-solid suspension flows turbulently and the effects of the particle-particle and particle-wall collisions and heat transfer are significant. Particles of different temperatures, described by the population density function nin(Tp,t), are fed continuously with constant volumetric flow rate qp, while the fluid flows in with volumetric flow rate qf and inlet temperature Tf,in(t). Heat exchange occurs between the fluid, particles and the wall, as well as between the system and the environment through the wall. The main assumptions concerning the system are as follows. 1) The particles are of constant size and are not changed during the process. 2) The system is operated under stationary hydrodynamic conditions, and the influence of thermal changes on the hydrodynamics is negligible. 3) The heat transfer between the fluid and particles, wall and fluid, as well as the wall and environment are continuous processes, characterised by the heat transfer coefficients βpf, βwf and βwe, respectively. 4) The interparticle heat transfer occurs by collisions, and is described by the random variable ξ1 ∈ [0,1] with probability density function b1 [5,6]. 5) The particle-wall heat transfer also occurs by collisions that is characterised by the random variable ξ 2 ∈ [0,1] with probability density function b2 [7]. 6) There is no heat source inside the particles. 7) The heat transfer by radiation is negligible. 8) The spatial variation of temperature along the unit is described by a compartments-in-series with back-flow model as it is shown in Fig.1.
(1 + R) q
q
(1 + R) q
1st cell
2nd cell
(1 + R) q Kth cell
…
Rq
Rq
q
Rq
Fig.1. Compartments-in-series with back-flow model; q – volumetric flow rate, R – back-flow ratio
Under such conditions, the mathematical model of the system is formed by a mixed set of partial integro-differential and ordinary differential equations. Population balance equations, describing the variation of temperature distribution of the particle population: ∂nk (T p , t ) ∂t −
=−
(1 + Z k R )q p V
(
∂T p
to
+
(1 + S k )q p V
nk −1 (T p , t ) +
Rq p V
nk +1 (T p , t ) −
1 ⎛ T p − p1 zTw ⎞ 1 nk (T p , t ) − k 2 nk (T p , t ) + k 2 nk ⎜⎜ dz + , t ⎟⎟b2 ( z ) p z p1 z − − 1 1 1 ⎠ 0 ⎝
(1)
∫
k − k1nk (T p , t ) + 1 M 0; k
subject
)
∂ [ K p T f ; k (t ) − T p nk (T p , t )]
the
T p max 1
⎛ 2(T p − S )
∫ ∫ nk ⎜⎜⎝
T p min 0
initial
z
⎞ 2 + S , t ⎟⎟nk ( S , t )b1 ( z ) dzdS ), k = 1,2...K , t > 0 z ⎠
conditions
nk (T p ,0) = n0 (T p ), k = 1,2...K ,
where
n0 (T p , t ) = nin (T p , t ) and n K +1 (T p , t ) ≡ 0 . Here nk(.,.) denotes the population density function of particles by means of which
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591
n k (T p , t )dT p expresses the number of particles having temperature in the interval (T p , T p + dT p ) at the moment of time t in the kth compartment (cell), while M 0;k de-
notes the number of particles that is constant in each cell (N), what agrees with the condition 2). V denotes the volume of a compartment, k1 is the intensity of interparticle collisions, and k 2 stands for the intensity of collisions between the particles and the wall. p1 =
mwC w and m p C p + mwC w
p2 =
m pC p m p C p + mw C w
where m
denotes mass and C denotes heat capacity. Indices: f – fluid, p – particle, w– wall, in – input. The auxiliary symbols: S1 = 0, S K = 1 , Z 1 = Z K = 1 , S l = 1 , Z l = 2 , l = 2, ..., K − 1 were introduced for the sake of compact notation of the model. Heat balance equations for the fluid phase:
dT f ;k (t ) dt
=
(1 + Sk )q f V
T f ;k −1(t ) +
Rq f V
T f ;k +1(t ) −
(1 + Z k R)q f V
T f ;k (t ) − (2)
T p max
−
∫ K f (T f ;k (t ) − Tp (t ))nk (Tp , t )dTp − K w (T f ;k (t ) − Tw;k (t )),
k = 1,2...K , t > 0
T p min
subject to the initial conditions T f ;k (0) = T f ,0;k , k = 1,2...K . Here T f ;0 (t ) = T f ,in (t ) and T f ; K +1(t ) ≡ 0 . Heat balance equations for the wall:
dT w;k (t ) dt
(
T p max 1
−V ⋅ k2
)
= V ⋅ K w' T f ;k (t ) − Tw;k (t ) − Ve ⋅ K we (Tw;k (t ) − Te (t ) ) −
∫ ∫ (
) (
)
p 2 Tw;k (t ) − T p n k T p , t z b2 ( z ) dz dT p
(3) k = 1,2...K , t > 0
T p min 0
subject to the initial conditions Tw; k (0) = Tw,0; k , k = 1,2...K where index e denotes the environment. In Eqs (1)-(3) parameters β pf a pf β wf a wf β wf a wf β pf a pf β a Kw = , K w' = , Kp = , Kf = , K we = we we ρfCf ρfCf mwC w m pC p mwC w are aggregates of the heat transfer coefficients and the corresponding contact areas a wf , a pf , a we , fluid density ρ f , heat capacities and masses. The first term on the left hand side of Eq.(1) denotes the rate of accumulation of particles having temperature (Tp,Tp+dTp) in the kth cell. The first term on the right hand side describes the change of the number of particles with temperature (Tp,Tp+dTp) due to the fluid-particle heat transfer. The second, third and fourth ones describe, respectively, the variation of the population density function because of input and output of particles in the kth compartment, the next two terms describe the variation of temperature distribution of particles due to the particle-wall collisions heat transfer while the last two term describes the variation of temperature distribution of particles due to the direct heat transfer between the particles by collisions. In Eq.(2), the first three terms on the right hand side describe the variation of the temperature due to the inflow and outflow of fluid, the next term describes the overall
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heat transfer between the fluid and the particles population, and the last term represents the continuous heat transfer between the fluid and the wall of the kth cell. Finally, in Eq.(3) the first two terms on the right hand side describe the fluid-wall and the wall-environment heat transfers in the kth compartment, while the last term represents the variation of the wall temperature due to the particle-wall collisions. The details of the derivation of model (1)-(3) will be presented elsewhere [7].
3. The moment equation model Introducing the moments of the population density function of particles, expressed as 1
∫
M I ; k (t ) = T pI nk (T p , t )dT p , mI ; k (t ) = 0
M I ; k (t ) M 0; k
, I=1,2,…, and k = 0, 1...K + 1
(4)
we can derive an infinite set of the moment equations of the system. Indeed, multiplying both sides of Eq.(1) by TpI and integrating from Tp,min to Tp,max, after some suitable transformations we get the following system of ordinary differential equations: ⎡1 I ⎤ = IK p ( M I −1;k (t )T f ;k (t ) − M I ;k (t )) + k1 ⎢ ∑ M i;k (t )M I −i;k (t )bi(,1I) − M I ;k (t )⎥ dt ⎣ N i =0 ⎦ I i ⎡ ⎤ ( 1 + S R ) q ⎛i⎞ ⎛I⎞ k p M I ;k −1 (t ) + k 2 p1 ⎢− M I ;k (t ) + ∑ ⎜⎜ ⎟⎟bi( 2) ∑ ⎜⎜ ⎟⎟(−1) i − j Twj;k M I − j ;k (t )⎥ + i j V i =0 ⎝ ⎠ j =0 ⎝ ⎠ ⎣⎢ ⎦⎥ dM I ;k
+
Rq p V
M I ;k +1 (t ) −
(1 + Z k R)q p V
(5)
M I ;k , k = 1,2...K , I = 1,2,..., t > 0 1
I −i
i
1
⎛ I ⎞⎛ z ⎞ ⎛ z ⎞ M I ;k (0) = M I ,0;k where bi(,1I) = ⎜⎜ ⎟⎟⎜ ⎟ ⎜1 − ⎟ b1 ( z )dz and bi( 2) = z i b2 ( z )dz . i ⎝2⎠ ⎝ 2⎠ 0 0⎝ ⎠ Eqs (5) form an infinite hierarchy, but, due to the linear nature of the rate of changes of the particle temperature, this set of equations can be closed at any order. In order to have a closed set of heat balance equations of the energy conversion system we need only the first order moment equation, while the second and higher order moments can be used to characterize the temperature distribution of particles. The set of the recursive differential equations (5) was closed at the second order moment, analysing in this way the total heat balance of the particulate system and the variance of the temperature of
∫
(
particles, expressed as σ k2 = M 2;k / M 0;k − M 1;k / M 0;k
∫
)2 .
4. Simulation results The heat transfer properties of the system were examined by computer simulation solving the set formed by the balance equations (2) and (3) coupled with the first and second moment equations from the hierarchy (5) in the case of K=3, subject to the corresponding initial conditions by an ODE solver of MATLAB. In simulation, the basic values of the constitutive parameters were chosen as [4]: V = 1.5 m 3 , V e = 1 m 3 , M 0;k ≡ 2× 10 8 ,
q p = 1.03 × 10−3 m3 / s ,
C p = 944 J/kg/ D K ,
β pf = 10 W/m 2 / D K ,
k1 = 720 s −1 , k 2 = 10 s −1 , C w = 464.73 J/kg/ D K , a pf = 1.02 × 10 −5 m 2 , b1( 2) = 10 −5 ,
Modeling of Heat Transfer Processes in Particulate Systems awf = 4.84 m 2 ,
ρ f = 0.94 kg/m 3 ,
q f = 0.51 m 3 / s ,
mw =
593
C f = 1008.3 J/kg/ D K ,
β wf = 5 W/m 2 / D K ,
min,1 (t ) ≡ 20 D C ,
= 190.0 kg ,
T f ,in (t ) ≡ 120 DC ,
Te (t ) ≡ 20 D C , m p = 3.2 × 10 −6 kg , awe = 4.84 m 2 , β we = 5 W/m 2 / D K .
The initial values were: m1,0;k = 20 D C , Tw,0;k = 20 D C , T f ,0;k = 20 D C , k = 1, 2, 3. The transients of the fluid and wall temperature, as well as of the mean temperature of particles induced by a step change of the input fluid temperature Tf,in=120 oC are presented in Fig.2 for K=3 and R=3. It is seen that the particles make heated entirely in the third cell of the system under the present environmental conditions. However, this does Tf;k(t) m1;k(t) Tw;k(t)
t
Fig.2. Variation of the temperature of fluid (-), mean temperature of particles (*) and temperature of the wall of system (+) as a function of time (R=3)
not mean the total homogenization of the temperature of particles as it is shown in Fig.3 for different values of parameters βpf and k1. Naturally, increasing intensity of interparticle collisions reduces the temperature dispersion but increasing the fluid-particle heat transfer increases the dispersion of the temperature of particles in the 3rd cell. σ k 2 (t )
+ - k1 = 700, β pf = 1 ○ - k1 = 700, β pf = 10 □ - k1 = 1000, β pf = 1 × - k1 = 1000, β pf = 10
1
2
3
Cell no
Fig.3. Effects of the fluid-particle heat transfer coefficient and the intensity of interparticle collisions on the variance of temperature of particles in steady states
Variation of the steady state values in the three cells, i.e. along the processing unit are presented in Fig.4 as a function of the back-flow ratio R. The diagram shows that the system at R≈40 becomes approximately perfectly stirred. The system at R=0 proves to be most efficient thermally although in this case large temperature
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differences arise between the cells. The back-flow of fluid and particles reduces the efficiency of the thermal process. Tf;k(t) m1;k(t)
1st cell 2nd cell
3rd cell
R
Fig.4. Variation of the temperature of fluid (-) and the mean temperature of particles (o, ×, •) as a function of R (K=3)
5. Summary A population balance model was developed for describing heat transfer processes in fluid-solid processing systems using a compartment model to describe the spatial distribution of the temperature in a unit. Both the particle-particle and particle-wall heat transfer are modelled by collisions with random parameters making possible this way to characterise the temperature distribution of particles. The population balance equation developed was transformed into a set of the ordinary differential equations for moments, and the properties of the system were studied by simulation. The results revealed that the intensity of interparticle collisions play significant role in reducing the temperature dispersion of particles, while increasing fluid-particle heat transfer acts inversely. The simulation indicated that the population balance model can be used efficiently for analysing the heat transfer in fluid-solid energy conversion processes.
Reference 1. Delvosalle, C. and Vanderschuren, J., 1985, Gas-to-particle and particle-to-particle heat transfer in fluidized beds of large particles. Chemical Engineering Science, 40, 769-779. 2. Molerus, O., 1997, Heat transfer in moving beds with a stagnant interstitial gas. International Journal of Heat and Mass Transfer, 17, 4151-4159. 3. Mansoori, Z., Saffar-Avval, M., Basirat-Tabrizi, H., Ahmadi, G. and Lain, S., 2002, Thermo-mechanical modeling of turbulent heat transfer in gas-solid flows including particle collisions. International Journal of Heat and Fluid Flow, 23, 792-806. 4. Burgschwieger, J. and E. Tsosas, 2002, Experimental investigation and modelling of continuous fluidized bed drying under steady-state and dynamic conditions. Chemical Engineering Science, 57, 5021-5038 5. Mihálykó, Cs., Lakatos, B.G., Matejdesz, A. and Blickle, T., 2004, Population balance model for particle-to-particle heat transfer in gas–solid systems. International Journal of Heat and Mass Transfer, 47, 1325-1334. 6. Lakatos, B.G., Mihálykó, Cs. and Blickle, T., 2006, Modelling of interactive populations of disperse systems. Chemical Engineering Science, 61, 54-62. 7. Süle, Z., Mihálykó, Cs. and Lakatos, B.G., Population balance model of heat transfer processes in particulate systems (to be published).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A comprehensive investigation on high-pressure LDPE manufacturing: Dynamic modelling of compressor, reactor and separation units Prokopis Pladis, Apostolos Baltsas and Costas Kiparissides Chemical Engineering Department and Chemical Process Engineering Research Institute Aristotle University of Thessaloniki, P.O.Box 472, 54006 Thessaloniki, Greece
Abstract A comprehensive mathematical model is developed for the simulation of high-pressure Low Density Polyethylene (LDPE) plants. Correlations describing the thermodynamic, physical and transport properties of the ethylene-polyethylene mixture are presented and compared with experimental data. Energy balances around the compression units are derived to calculate the energy requirements. A detailed kinetic mechanism is proposed to describe the molecular and structural developments of the free-radical polymerization of ethylene. Based on the postulated kinetic mechanism, a system of differential mass balance equations are derived for the various molecular species, total mass, energy and momentum in the polymerization system. Simulation results show that the proposed mathematical model can be successfully applied to the real-time prediction of reactor temperature profile and polymer melt index. Moreover, model predictions are compared with industrial measurements on reactor and coolant temperature profiles, reactor pressure, conversion, and final molecular properties for different polyethylene grades. Finally, various equations of state (e.g., Sako-Wu-Prausnitz, SAFT, PC-SAFT) are employed to simulate the operation and phase equilibrium in the flash separation units. Keywords: Modeling, LDPE, Compressor, Reactor, Separation, Phase Equilibrium.
1. Introduction Low Density Polyethylene (LDPE) is used in a large number of applications (e.g., packaging, adhesives, coatings, and films), as a result of the wide range of molecular and structural properties. LDPE has been commercially produced in high-pressure reactors for more than 4 decades. Two reactor technologies (i.e., tubular and autoclaves) are employed in the high-pressure polymerization of ethylene. The polymerization of ethylene is typically carried out at high temperatures (120-320ºC) and pressures (15003000 bar). Thus, in the presence of a mixture of initiators (e.g., peroxides, azo compounds), ethylene can be polymerized via a free-radical mechanism. A large variety of LDPE grades is usually produced from a single reactor line, (e.g., with different polydispersity, long chain branching and density, 0.915-0.935 g/cm3). A generic flow diagram (Figure 1) of the high-pressure ethylene copolymerization process can be described as follows: Fresh ethylene, after the primary compression, is mixed with the recycled ethylene and comonomer (e.g., vinyl acetate, methyl acrylate, ethyl acrylate, methacrylic acid, etc.), that is then pressurized to the desired reactor pressure in the second compression stage. Polymerization of the monomers is
595
P. Pladis et al.
596 Ethylene
Coolant
Peroxides
Reactor Zone 1
Peroxides
Coolant
Coolant
Reactor Zone N
...
Feed
Primary Compressor
Peroxides
Reactor Zone 2
Products
Cooler
Valve
Secondary Compressor Coolant
Coolant Side Feed 1
Side Feed 2
Wax Separator
..
Coolant
HP Separator
Side Feed N-1
Cooler
HP recycle Valve LP Separator
Wax Wax Separator
Cooler
LP recycle
Wax Dryer
Extruder
Silo Polyethylene
Figure 1. Schematic representation of a high-pressure LDPE tubular reactor process.
initiated by adding a mixture of chemical initiators (e.g., organic peroxides). The monomer conversion per reactor pass can vary from 15 to 35 %. The separation is performed in two successive stages. In the first stage, the let down valve drops the pressure of the outlet reactor stream to 150-300 bar. The ethylenepolyethylene mixture entering the high-pressure separator is split into a polymer rich liquid phase (containing 70-80% per weight) and an ethylene rich gas phase (containing ethylene and small amounts of wax). The polymer rich liquid phase from the bottom of the high-pressure separator is directed to the low-pressure separator. In the second stage, the pressure of ethylene-polyethylene mixture entering the low-pressure separator is further reduced to about 1.5 bar. The ethylene gas leaving the low-pressure separator is directed to the primary compressor and is mixed with fresh ethylene feed. The liquid bottom stream leaving the low-pressure separator (containing very low concentration of ethylene) is sent to the extruder where the polymer is pelletized. Over the past 30 years a great number of papers have been published on the modeling of LDPE tubular reactors (Kiparissides et al., 2005). However, most of the published studies are limited to the modeling of the polymerization reactor. As a result, there are only a few publications that deal with the description of the modeling of the high- and low-pressure separation units. The development of a comprehensive mathematical model for the high–pressure LDPE process should include detailed modeling of the following process units: a) the monomer(s) compression unit, b) the polymerization reactor, and c) the product separation system. In this study, the thermodynamic, physical and transport properties of the reaction mixture at the various stages of the process are calculated by using a number of equation of states. In addition the energy requirements of compressor units is calculated. A comprehensive mathematical model for the design and simulation of high-pressure
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597
LDPE reactors is presented. The predictive capabilities of the proposed mathematical model are demonstrated by direct comparison of the model predictions with literature experimental measurements and industrial data covering a wide range of operating conditions. Finally, the calculation of phase equilibrium and the dynamic operation of high and low-pressure separator units is discussed. The ethylene-polyethylene phase equilibrium is calculated using various equations of state (e.g., Sako-Wu-Prausnitz, SAFT, PC-SAFT). The dynamic model of the separator is able to predict deviations from the theoretical phase equilibrium state as it has been observed in real plant data.
2. Modeling of LDPE Plant Units Compressor Units. The accurate modeling of primary and secondary compressor units are essential in LDPE production plants. In the primary compressor system, the pressure is raised from about 1.5 bar to about 260 bar. In the secondary compressor system, the pressure of the compressed monomer(s) and solvent(s) is raised to the reactor feed operating conditions (2400 - 2700 bar).The compression of gases is accomplished in high-pressure reciprocating compressors. To account for the temperature increase after a compression stage the energy balance calculations around the compressor unit should be derived. From the steady-state energy balance around the compressor unit, we obtain for the initial (1) and final conditions (2), respectively: ΔH = H 2 − H 1 = − Ws Normally, the inlet conditions (T1,P1) and the discharge pressure P2. are known. Thus we know only H1 and H2 and Ws are left as unknowns. In a compression process the isentropic work is the minimum shaft work required for compression of a gas from a given initial state to a given discharge pressure: (ΔH )s = − Ws (isentropic) In a non-ideal operation the compression efficiency is defined as follows: η=
Ws (isentropic ) (ΔH )s = Ws ΔH
Compression efficiencies are usually in the range 70 to 80 percent. The compressor efficiency is used to determine the actual enthalpy change and therefore the actual temperature at the compressor outlet. For the thermodynamic calculations SAFT equation of state was employed. Tubular Reactor Units. Polymers made by free-radical polymerization are typically mixtures of macromolecules with different molecular structural characteristics (e.g., copolymer composition, chain length, and short and long chain branching frequencies). Since the molecular features of the produced polymers are directly related to their enduse properties, control of the polymer chain microstructure during polymerization is of profound importance. This presupposes a thorough understanding of the polymerization kinetics. In the present study, a comprehensive kinetic mechanism is postulated to describe the free-radical polymerization of ethylene. The elementary reactions considered are summarized in Table 1 (Kiparissides et al, 2005; Pladis and Kiparissides, 1998). The kinetic constants are taken by Kiparissides et al. (2005). The predictive capabilities of the mathematical model were examined by simulating the operation of an industrial high-pressure LDPE tubular reactor. Figures 2 - 5 illustrate some representative simulation and experimental results of the industrial LDPE tubular polymerization reactor. In Figure 2, scaled reactor temperature profiles are plotted for three homopolymer polyethylene grades (A, C, E). The number of temperature peaks (three) corresponds to the respective initiator injection points. The continuous lines
P. Pladis et al.
598 Table 1 : Kinetic Mechanism of Ethylene Polymerization Initiator(s) decomposition k ∗ I i ⎯⎯→ 2R
;
di
i = 1, 2, ..., N i
Chain initiation reaction
R∗
+ M
k ⎯⎯→ R1 I
Thermal initiation k
thj 3M ⎯⎯→ R1
Propagation k
Rx + M
p ⎯⎯→ R x +1
Transfer to Monomer tm R x + M ⎯k⎯→ Dx + R1
Transfer to CTAs k
tsij R x + S k ⎯⎯→ D x + R 1 , k = 1, 2 …, Ns
Transfer to Polymer (LCB) k tpij
R x + D y ⎯⎯→ D x + R y Intramolecular Chain Transfer (SCB)
Rx
k ⎯⎯→
Rx
bi
β scission of secondary and tertiary radicals k β ,k β '
R x ⎯⎯⎯→ D =x −1 + R 1 β scission of internal radicals k R x + D y ⎯⎯→ D x + R z + D =y− z B
Termination by combination k tcij
R x + R y ⎯⎯→ D x + y Termination by disproportionation k tdij
R x + R y ⎯⎯→ D =x + D y
represent model predictions (obtained through the on-line parameter estimator modulus of the software) while the discrete points represent the experimental temperature measurements. It is apparent that the model predictions are in a very good agreement with measured temperatures. In Figure 3-5, ethylene conversion, number average molecular weight, and long chain branching per 1000 carbon atoms are plotted with respect to the reactor length for Grades A, C, E. In all cases, the predicted final properties are in a good with the experimental measurements. Separator Units. To accurately predict the performance of the flash separators, a study of the thermodynamic phase equilibrium behavior of an ethylene/polyethylene mixture was undertaken. The phase equilibrium in the separator units is of major importance because it determines the residual amounts of monomer and other gases in the polymer leaving the high- and low-pressure separators and, on the same time, determines the flows and compositions of streams in the LDPE plant.
A Comprehensive Investigation on High-Pressure LDPE Manufacturing
599
30
1.00
25
0.90 0.80 0.70 Grade A Grade C Grade E Grade A exp. Grade C exp. Grade E exp.
0.60 0.50
0.40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Relative Reactor Length
Figure 2. Predicted vs measured temperature profiles (Grades A, C, E).
Ethylene Conversion
Relative Reactor Temperature
1.10
20 15 Grade A Grade C Grade E Grade Aexp. Grade Cexp. Grade E exp.
10 5
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
Relative Reactor Length
Figure 3. Predicted vs experimental ethylene conversion profiles (Grades A, C, E).
The separation of LDPE from the unreacted monomer and solvents is carried out in a two-stage process downstream the tubular reactor (Buchelli, 2004). In the first stage, the pressure of the reactor outlet stream is reduced to 260 bar and then is directed at the inlet of the high-pressure separator. The polymer rich liquid phase from the bottom of the high-pressure separator is directed to the low-pressure separator. In the second stage, the pressure of ethylene-polyethylene mixture entering the low-pressure separator is further reduced to about 1.5 bar. In Figure 6 the Molecular weight distributions (MWD) of the vapor and liquid phases as well as the NAMW calculated at 1500 bar using the Sako-Wu-Prausnitz equation of state. As it can be seen the polyethylene of the vapor phase consist of polymer with lower molecular weights compared with the polymer in the other phase. Figure 7 depicts the effect of separator pressure on the number average molecular weight of the polymer that is distributed in the two phases.
3. Conclusions It is well known that the dynamic behaviour of the complete plant can be completely different from the behaviour of the reactor due to the various recycling streams and different time-scaled process units (Cervantes, 2000). Grade transition operation is essential in continuous polymer plants because many grades of polymers are produced from the same process. The reduction of the amount of off-specification polymer during the grade transition operation is important for the economical operation of continuous polymer plants. The development of a comprehensive mathematical model for the high– pressure LDPE process should include detailed modeling of the following process units: a) the monomer(s) compression unit, b) the polymerization reactor, and c) the product separation system as well as accurate predictions of the thermodynamic and transport properties of the fluid at the various stages of the process. In the present study a comprehensive mathematical model for the design, simulation of industrial highpressure LDPE plants was developed. Various equations of state and correlations for the predictions of physical, thermodynamic and transport properties of the reaction mixture were calculated and each of the basic process units were successfully modelled.
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References A. Buchelli, M.L. Call, A.I. Brown, C.P. Bokis, S. Ramanathan, and J. Franjione, 2004, Nonequilibrium Behavior in Ethylene/Polyethylene Flash Separators, Ind. Eng. Chem. Res., 43, 1768. A. Cervantes A, S. Tonelli, A. Brandolin, A. Bandoni , L. Biegler , 2000, Large-scale dynamic optimization of a low density polyethylene plant, Comp. Chem. Eng., 24, 983 C. Kiparissides, A. Baltsas, S. Papadopoulos, J. Congalidis, J. Richards, M. Kelly, and Y. Ye, 2005, Mathematical Modeling of Free-Radical Ethylene Copolymerization in High-Pressure Tubular Reactors, Ind. Eng. Chem. Res., 44, 2592. P. Pladis and C. Kiparissides, 1998, A comprehensive Model for the Calculation of Molecular Weight and Long Chain Branching Distribution in Free-Radical Polymerizations, Chemical Engineering Science, 53, 18, 3315.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Sensitivity Analysis in the Simulation of Complex Solids Processes D. Schwier, A. Püttmann, E.-U. Hartge, G. Gruhn, J. Werther Institute of Solids Process Engineering and Particle Technology, Hamburg University of Technology, D-21071 Hamburg, Germany
Abstract In the present work a method to quantify the range of uncertainty of selected stream values and to detect the most influential parameters within a given flowsheet has been developed for the flowsheeting of solids processes. By using heuristic optimization methods the range of uncertainty of relevant stream values can be determined. Distributions of attributes of solids can be handled and shape and location of distributions as well as characteristic parameters like the Sauter diameter can be predicted. Furthermore the most influential parameters for the units or the models in the range of uncertainty can be detected by investigating the difference quotients in different operating points. With this information the time required to improve the accuracy of prediction of the simulation results can be reduced significantly. As an example the simulation of a hydrocyclone is presented. The influence of the parameters of an empirical separation model on the solids mass flow collected with underflow is demonstrated and the range of the product particle size distribution depending on uncertain parameters is examined. Keywords: flowsheet simulation, sensitivity analysis, optimization
1. Introduction Flowsheet simulation is the numerical solution of material and energy balances and the determination of the intensive properties of variable structures and substances on the basis of coupled mathematical models for the different process steps. The use of software tools for flowsheet simulation is state of the art for processes involving just fluids [1]. The success of the flowsheet simulation of processes in chemical engineering always depends on the accuracy of the parameter values, which describe the state of the process. Even small inaccuracies can have a large effect on the relevant stream values within the flowsheet. However, not all parameters will have a decisive influence on relevant stream values. One purpose of the sensitivity analysis in flowsheet simulation is therefore to detect those influential parameters, which determine the accuracy of the results of the simulation. The second purpose is to determine the lower and upper bounds of the output stream values depending on the uncertain parameters. There are various commercial software tools for the flowsheet simulation of fluids processes available (e.g. AspenPlus [2], Pro/II [3], gProms [4] etc.). For a description of processes involving solids distributed parameters like the particle size need to be considered. However, the stream structure of the simulation tools for fluid processing is insufficient to handle those distributed attributes. Therefore, in recent years the software tool “SolidSim” has been developed by a group of German research institutes [5, 6] that
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contains a stream structure which permits the handling of distributed parameters and even of distributed attributes of these latter parameters. Another major difference between solids and fluids processes is the accuracy of the models available. While there are very precise physical models available for the prediction of fluid operations like distillation or liquid-liquid extraction, for typical solids operations (e.g. screening or grinding) only empirical or semi-empirical models are available. To ensure a sufficiently precise simulation result usually miscellaneous empirical coefficients have to be determined for the sequence of different unit operations. But in general not all values of the product streams or the streams between the different operations are decisive. In order to reduce the effort to determine the necessary parameters for the various models it is useful to detect those parameters which have the largest effect on the performance of the model. For this task a method for a sensitivity analysis has been developed and implemented that allows detecting of those parameters which exert the major effects on the important stream values.
2. Sensitivity Analysis in Flowsheet Simulation For the flowsheet simulation in process engineering different types of parameters have to be distinguished [7], namely • product values (y): They are stream values (e.g. mass flow, temperature etc.) which characterize the product streams. In sensitivity analysis the range of these parameters has to be determined. • feed values (x): They are defined like the product values, but are valid for the streams which enter the system. Variabilities of these parameters are investigated in flexibility studies of processes. For a sensitivity analysis these values are considered as constant. • structure parameters (c): The set of structure parameter describes the interconnections between the different process units. Changes in the set of the structure are made in the synthesis of a process. This step has to be finished before a flowsheet simulation can take place. Therefore the structure parameters are invariant for the sensitivity analysis. • design parameters (d): They specify the design of the system. Because of inaccuracies in the manufacturing process or of changes during the operation these parameters are considered as fluctuating. Since the performance of solids processes is often largely depending on the design parameters they are considered in the sensitivity analysis. • model parameters (p): They are part of the model definitions for the different process units. They may depend on the numerical values of the feed, design or control parameters. When no experimental data are available for the process under consideration their values have to chosen empirically. However uncertain parameters may cause major deviations from reality in the results. Further uncertainties are caused by measuring and model errors. Due to the predominantly empirical models and the resulting large number of model parameters in solids processes the treatment of these latter parameters is the main area of the sensitivity analysis. • The mathematical model can be formulated as follows: M(y, x, u, d, c, p ) = 0 (2.1) with M as the vector of model equations. Local and global analyses are suitable for different applications: The local analysis provides a hierarchy of the influences of the
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parameters on the selected product values. In the case of the model parameters this information can be used to reduce the effort of experiments. By determining the most influential parameters in experiments the precision of the simulation can be improved decisively. The information about the influences of the design parameters can be used for a redesign of the process. The global analysis determines the accuracy of the simulation within the specified bounds for the uncertain parameters by giving upper and lower bounds of the selected stream values. 2.1. Global Analysis of Sensitivity To take uncertainties in flowsheet simulation into account interval methods are often used in chemical engineering [9]. Interval methods describe functional dependencies between interval values. Interval values are characterized by a lower and an upper interval bound, in a special case both bounds can match and the accordant value becomes a point value. The general formulation of an interval model is according to Colditz [8]: M(x, [y], [p], [d ], τ) = 0 (2.2) with
[p] = ([p1 ] , [p 2 ],..., [p i ] )T
(2.3)
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[p i ] = (p i ∈ R i
) [
]
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(2.4) The design parameters (d) are defined analogously. Since in the present work the flowsheet simulation of steady-state processes is considered only the time τ is not considered in the present sensitivity analysis. For the interpretation of the interval models methods of optimization are used. The enclosure of the target values is determined by minimization and maximization as a function of the uncertain parameters. In this case the product parameter yi is the target value of the optimization problem: ⎤ ⎡ [y i ] = ⎢ min {y i M(y, p ) = 0}, max {y i M(y, p ) = 0}⎥, i = 1, 2,..., m p ∈ [ p] ⎦⎥ ⎣⎢p ∈ [ p] (2.5) with the uncertain model parameters p (or alternatively the design parameters d) this leads to minimal enclosure of the bounds for yi. For the application of this optimization problem to solids processes the structure of the software tool has to be taken into account. The different process units do not reveal the model equations which are implemented. They just receive a set of input parameters and provide the according output values. Therefore a solution of the above optimization problem with methods of mathematical programming [10] is not possible but methods of numeric programming have to be used instead. In the present work two different metaheuristic methods were selected: The first one is the Tabu Search method [11]. Based on a random start combination of the uncertain parameters the neighborhood solutions are calculated and the best ones are selected. This procedure is continued until a previously defined stop criterion. To enhance efficiency and escape local optima a list of the already inspected combinations is present (tabu-list). The second one is the Simulated Annealing method [12] where a random start solution is calculated and a new parameter combination is selected at random. If this provides a better solution value the combination is selected in any case.
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Worse solutions are accepted with a certain probability which is high at the beginning and decreases during the process. Thus the solution can avoid local optima. 2.2. Local Analysis of Sensitivity The local analysis of sensitivity investigates the effects of changes of one parameter on the characteristic product stream values in a specific operation point while all other parameters are held constant [7]. The simplification of eq. (2.1) provides y = y(q ) (2.6) with q as the vector of the model parameters p or the design parameters d. In order to quantify the influence of the parameter qj on the stream value yi the logarithmic sensitivity function [7] is used, ∂y i q j q j ∂ ln y i (q ) y E i (q ) = ⋅ = qj ∂q j yi ∂ ln q j (2.7) Equation (2.7) provides a dimensionless coefficient for the percentage change of the product value if the uncertain parameter is changed for one percent. Because the model equations are not directly available for the calculation of the derivative is performed numerically by a difference quotient.
( )
2.3. Treatment of distributed properties in the sensitivity analysis Distributions of solid attributes like particle size are usually composed of several discrete values. Since in solids processes these attributes usually are important for the properties of the products, the sensitivity analysis has to be applicable to distributed parameters. Basically two alternatives are possible: Either a separate analysis for each interval of a distributed property has to be carried out or a distribution is represented by suitable parameters. The first alternative causes a large numerical effort. Furthermore it cannot be ensured that the bounds of the separate intervals of the distribution are connected to one specific parameter combination for each bound. An effective method to characterize the bounds of the distribution with one combination of parameters each is to use coefficients which describe the position and shape of the distribution. In the present work the position of the distribution has been characterized by the median value (xd,50) and the steepness of the distribution by the ratio Ψ of the 25- and 75-values respectively, of the cumulative distribution, x d,25 (2.8) ψ= x d,75
3. Example: Sensitivity Analysis for the Performance of a Hydrocyclone Hydrocyclones are commonly used apparatuses for solid-liquid separation or classification in the liquid phase. The procedure for the local sensitivity analysis was tested with the empirical hydrocyclone model by Plitt et al. [13, 14]. By using this algebraic model which contains four parameters F1-4 the sensitivity analysis could be tested. As an example the influence of the four model parameters on the mass of solids recovered in the underflow is demonstrated here. The results of the calculation are displayed in Table 1.
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Sensitivity of the solids mass in underflow Logarithmic sensitivity Hierarchy of influence coefficient -0,226 1-
Increasing F1 increases cut size F2 Increasing F2 0,097 2+ increases sharpness of separation F3 Increasing F3 0,006 4+ increases pressure drop F4 Increasing F4 -0,032 3increases mass flow of suspension at underflow Table 1: Results of the local sensitivity analysis of the hydrocyclone model of Plitt et al. [13, 14] (the sign with the hierarchy number indicates an increase (+) or decrease (-), respectively, of the solids mass collected in the underflow) Tab. 1 shows that the mass of solids at the underflow is influenced by all four parameters to a different extent. According to Table 1 the strongest effect on the mass of solids in the underflow is due to parameter F1 which controls the cut size. An increasing cut size causes fewer solids to be collected in the underflow. The second largest influence has F2, which also effects the separation function directly. The parameters F4 and F3 only have indirect effects via the sharpness of separation. With more precise parameters F1 and F2 a significant enhancement of the accuracy of the simulation can be achieved. Relatively seen, the influences of F4 and F3 could be neglected for the determination of the solids mass at the underflow. In a global sensitivity analysis the ranges of the selected product stream values “median xd,50” and “steepness Ψ” were determined depending on the ranges of uncertainty of the parameters F1 to F4 which are allowed to vary between 0.5 and 3. The results of the sensitivity analysis are presented in fig. 1.
Figure 1: Range of the particle size distribution of the product in the hydrocyclone underflow depending on uncertain parameters F1-4 Figure 1 shows the range of the resulting distributions in the grey marked area. The median value starts with a minimum of 76 μm and ends with a maximum of 111 μm.
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Both distributions shown in fig. 1 have been chosen according to the set of parameters for the minimum median value and the maximum median value, respectively. The parameter Ψ of the distributions varies from 0.23 to 0.33. It is recognizable that the combination of median value and steepness encloses the area between the shown distributions.
4. Conclusions It could be shown that the proposed method for sensitivity analysis using optimization methods is able to detect the most influential unit parameters concerning a stream value. The ranges of distributions of solids can be treated as well as concentrated parameters like a mass flow.
References [1]
W. D. Seider, J.D. Seader, D. R. Lewin: Process design principles: synthesis, analysis and evaluation. John Wiley & Sons, 1999
[2]
Aspen+: www.aspentech.com, 2005
[3]
Pro/II: www.simsci-esscor.com, 2005
[4]
gProms: www.psenterprise.com, 2005
[5]
E.-U. Hartge, M. Pogodda, C. Reimers, D. Schwier, G. Gruhn, J. Werther: SolidSim – A new system for the flow sheet simulation of solids processes, Proc. 7th World Congr. Chem. Engng., Glasgow, UK, July 10-14, 2005
[6]
E.-U. Hartge, M. Pogodda, C. Reimers, D. Schwier, G. Gruhn, J. Werther: SolidSim – Ein Werkzeug für die Fließschemasimulation von Feststoffprozessen, Aufbereitungstechnik 2006, in print
[7]
S. Weiss (Ed.): Verfahrenstechnische Berechnungsmethoden, Teil 6: Verfahren und Anlagen, VCH Verlagsgesellschaft Weinheim, 1987
[8]
S. Colditz: Untersuchungen zur Flexibilität und Parameterunsicherheit bei verfahrenstechnischen Prozessen, Fortschr. Verfahrenstechn. Reihe 3, VDI-Verlag Düsseldorf, 1997
[9]
G. Gruhn, G. Fichtner: Anwendung intervallmathematischer Methoden in der Verfahrenstechnik, Chemie-Ingenieur-Technik, 21 (1997), Supplement, pp 187-192
[10] I.E. Grossmann, J.A. Caballero, H. Yeomans: Mathematical programming approaches to the synthesis of chemical process systems, Korean J. Chem. Engng., 16 (1999), 4, pp. 407-426 [11] M. Gendreau: An introduction to tabu search, in: F. Glover, G.A. Kochenberger (Eds.): Handbook of metaheuristics, Kluwer Dordrecht, 2002 [12] S. Kirkpatrick, C.D. Gellat, M.P. Vecchi: Optimization by simulated annealing, Science 220 (1983), pp. 671-680 [13] L.R. Plitt: A mathematical model of the hydrocyclone classifier, CIM Bulletin, 80 (1976), pp. 39-50 [14] L.R. Plitt, A. Broussaud, C. Conil: An improved method of calculating the water-split in hydrocyclone, Minerals Engng., 3 (1990) 5, pp. 533-535
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Identification of Parametric and Structural Models Based on RTD theory via GAMS Package S.Hocine*, L.Pibouleau, C.Azzaro-Pantel, S.Domenech Laboratoire de Génie Chimique - UMR 5503 CNRS/ INPT-UPS BP 1301 - 5, rue Paulin Talabot - 31106 TOULOUSE Cedex 1 - France
Abstract In the context of the development of a computer-aided design tool for the control and the resolution of dynamic MINLP problems, we present a safety study of ventilated nuclear enclosures. Previous work on this problem used stochastic approaches. We have solved it using a deterministic method based on a modification of the Branch & Bound method included in the GAMS Package. The problem involves carrying out a parametric and structural identification, starting from an initial superstructure, according to the systemic approach. It includes a significant number of linear and bilinear constraints. In order to avoid certain numerical difficulties during the resolution, two different models representing the problem are presented. Keywords: Optimization, structural and parametric identification, MINLP, ventilation enclosure, systemic approach.
1. Introduction To prevent and detect chemical and radiological hazards in industrial premises, the validation of a proper ventilation system is required. The research on air distribution in ventilated rooms traditionally involves full-scale or scale-model experiments, computational fluid dynamics (CFD) tools and residence time distribution (RTD) approaches. The solution adopted here is based on the well-known RTD widely used in chemical engineering to model non-ideal flows, called here “systemic approach”. A superstructure involving the set of all the possible solutions corresponding to the physical reactor is defined, and the model has to be selected among this superstructure according to the comparison between its simulated response to a stimulus, and the experimental response. The superstructure is defined as a combination of elementary systems, representing ideal flow patterns (Levenspiel, 1972). The structure and parameters of the model are simultaneously optimized in order to fit an RTD experimental curve with a minimum number of elementary units, which constitutes a key point for future control purposes of the process. The problem is a dynamic constrained MINLP (Mixed Integer Non Linear Programming) involving both binary variables (representing the presence or absence of elementary units) and continuous variables (volumes, concentrations, flow rates and time). In previous works, only software tools performing the optimization of a little number of parameters for models with given structures are available (Thereska, 1998 – Leclerc et al., 1995). This study is an alternative to the previous works of Laquerbe (1999) and Brienne et al. (2005). In the former case, stochastic methods (simulated annealing and genetic algorithm) are implemented and the optimality of the solutions is not guaranteed. The solution found by the latter is based on the Laplace transform, and *
Author to whom correspondence should be addressed: [email protected].
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due to simplifying assumptions, the obtained solution is not optimal. To circumvent these difficulties, the SBB solver of the GAMS package that implements a branch and bound algorithm is used (Gams, 2004). The main advantage of this solver is that no initialisation of the variables is required, which was identified as a critical and large time consuming stage in Laquerbe (2000). From the industrial application presented in the paper, the proposed method proves to be more efficient than the previous ones in terms of convergence and ease of implementation.
2. General basics 2.1. Systemic approach In order to fit as well as possible an experimental RTD curve, the elementary building blocks of the model are made up of ideal flows, i.e. perfect mixed and plug flow. In perfect mixed flow, the residence times are firstly unspecified and an instantaneous homogenisation of the fluid in any point is considered. Classical CSTR (continuous stirred tank reactor), constitutes an example of this model in the reactor domain. Another type is the plug flow reactor (PFR) characterized by a single residence time for all the particles of the fluid. 2.2. RTD theory The RTD curves make possible to analyze complex processes depending only on the flows present in the system. These curves are classically obtained by tracing. There are two classical types of stimuli. 2.2.1 DIRAC impulsion: inducing an injection of a unit quantity of tracer in an interval of time as weak as possible. This disturbance is quite delicate to reproduce numerically. 2.2.2 Unit step disturbance: carried out by two methods, the former ‘positive’ consists in imposing a constant emission of the tracer; the latter ‘negative’ is to stop the tracer injection. In a qualitative way, the total information collected on the flow is the same in the two cases. In this study, for simplicity reasons, only a unit step disturbance is used. 2.3. RTD for ideal flows For both types of the models, the RTD are given in the following table. Table 1. RTD of ideal flows
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⎡ ⎛ F in C out = C in × ⎢1 − exp ⎜⎜ − ⎢⎣ ⎝ V cstr ⎧ ⎛ V pfr ⎞ ⎪ C out = 0 for t < ⎜⎜ F ⎟⎟ ⎪ ⎝ in ⎠ ⎨ ⎛ V pfr ⎪ C out = C in for t ≥ ⎜⎜ ⎪⎩ ⎝ F in
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3. Presentation of the MELANIE enclosure A real ventilated room, the laboratory enclosure called MELANIE, of volume Vtot = 100m3, is presented. It is possible to obtain various air flows and residence time distributions either by changing the position of blowing and exhaust openings (near the floor or near the ceiling), or by modifying the exhaust flow rate (Espi, 1998).
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Fig. 1. MELANIE enclosure The configuration studied here corresponds to an exhaust flow rate equal to 1100m3/h (Ftot) at 20°C. The modelling was carried out by using the experimental response of this system to a release of helium impulse used as tracer gas. The experimental curve, constituted of 451 points, obtained during 900 seconds, is shown in Fig. 2.
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4. MELANIE modelling In order to find the model for the MELANIE enclosure related to the RTD curve obtained by a positive unit step disturbance, a variant of the superstructure of Brienne et al. (2005), shown in Fig. 3, is presented. The discrete (binary) variables (Yb,j) of the dynamic MINLP translate the existence or not of unit operations and flows. The continuous variables are related to flows (Fb), concentrations (Cb,j(t)) and volumes (Vb,j). The index b represents the number of branches (1≤b≤8), while the index j represents the reactor number; it indicates also the input and output concentrations of the reactor j (0≤j≤5). Index t represents time. In addition to classical volume, input node, output node, reactor existence and flow rate constraints, the problem formulation involves the following equations. Balance on CSTR reactors - these equations have the following form (example for CSTR(1,2)):
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Balance on PFR reactors – two formulations have been implemented and are given for example for PFR(1,1) (in relations (2) and (3), the value k = 100 was chosen):
()
⎛ C (t) ⎞ ⎡ ⎤ C 1(,t1) = ⎜⎜ 0 ⎟⎟× ⎢ π + A tan ⎛⎜ k *⎛⎜ t − V1 , 1 ⎞⎟ ⎞⎟ ⎥ 1 2 F π ⎝ ⎠ ⎝ ⎠ ⎦ ⎠ ⎣ ⎝ (t ) × sigmoid C 1(,t1) = C in
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The problem dimensions in terms of variables and constraints vary according to the number of time points (equidistant points) chosen in the range [0, 900], as shown in Table 2. Independently of the number of time points, all the problems involve in addition to continuous variables, 8 flow rates variables and 41 volumes variables. The problem consists in extracting from the superstructure, shown in Fig. 3, the model allowing the specified output concentration to be reached, with a minimum number of streams and elementary units. The problem is solved with the objective of minimizing the following objective function F: n
(t ) (t ) Min F = ρ × ∑ ( C out , mod − C exp, mod ) 2 + i =1
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Fig. 3. Superstructure of the MELANIE enclosure
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5. Numerical results According to the superstructure of Fig. 3, the results obtained for the two PFR formulations and for three numbers of time points (i) are reported in Table 3. Numerical tests were also carried out for 360 discretisation points, but the results were the same as the ones obtained for 180 points. For comparison purposes, the result previously obtained by Laquerbe (1999) is shown in table 4. Table 3. Results according to the two PFR formulations
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9
Quadratic deviation =6.81 10-7
Quadratic deviation =3.7 10-6
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Quadratic deviation =7.91 10-3
Quadratic deviation = 1.42 10-2
Quadratic deviation = 3.93 10-2
Quadratic deviation = 4.5810-2
180
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Quadratic deviation =0.113
For 9 points in the time space, the solutions obtained by the two formulations are very different, but this difference tends to vanish when the number of points increases. However, the quadratic deviation tends to increase with the number of points. The
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solutions obtained with the sigmoid formulation for 45 points and with the inverse tangent for 180 points are very close, so it can be assumed that a good local solution may exist near these solutions. With the inverse tangent formulation, the lag of the PFR concentration at the discontinuity point is slightly greater than the one obtained with the sigmoid formulation. Compared with the solution found by Laquerbe (1999), the two methods presented here give simpler structures with a better quadratic deviation. Let us recall that in the work of Laquerbe(1999), the solution is obtained by means of a two level procedure. At the upper level a genetic algorithm (GA) define the process structure (i.e. the integer variables) and at the lower level a NLP (package from IMSL library) is solved to optimize the operating conditions of the process units. However, in this problem, the GA converges always towards a quite poor solution for the process structure, giving thus a significant quadratic deviation. This failure in the GA convergence is probably due to a poor coding of the process variables and quite inefficient genetic operators (particularly the cross-over) and parameters.
6. Conclusion A real ventilated room is modelled in order to fit an experimental RTD curve as well as possible, by implementing a deterministic approach based on the GAMS package. The results are better than those previously obtained by Laquerbe (1999) by means of a stochastic algorithm. From the presented example, the proposed method, based on the use of the SBB solver, is more efficient than the previous ones both in terms of solution quality and ease of implementation. In particular, variable initialisation is not required for SBB, even though the initialisation step was a very hard task in Laquerbe (1999) approach. References J.-P. Brienne, L. Montastruc and I. Nikov, Methodology for characterization of residence time distribution in inverse fluidized beds, 1st South East European Congress of Chemical Engineering, Belgrade (Serbie et Montenegro) 25-28 September 2005. E. Espi, 1998, Prévision des transferts de contamination en cas d’incendie dans un local ventilé, PhD thesis, INP Toulouse.
GAMS, 2004, A user’s guide, GAMS Development Corporation. C. Laquerbe, J. C. Laborde, S. Soares, P. Floquet, L. Pibouleau ans S. Domenech, 2000, Synthesis of RTD models via stochastic procedures : simulated annealing and genetic algorithms, Computers and Chemical Engineering, 25(9-10),1169-1183. C. Laquerbe, 1999, Modélisation des écoulements dans un local ventilé par une approche systémique, PhD thesis, INP Toulouse. J.P. Leclerc, C.Detrez, A. Bernard and D. Schweich, , 1995, DTS : Un logiciel d’aide à l’élaboration de modèles d’écoulement dans les réacteurs, Revue de l’Institut Français du pétrole, vol. 50, n° 5, p. 641-656. O. Levenspiel, 1972, Chemical reaction engineering, 2ième Edition, Wiley Int. J. Thereska, 1998, L’application des radiotraceurs dans les unités industrielles – Bilan et perspectives, 1er Congrès Français Traceurs et méthodes de traçage, Nancy, 3-4 Novembre 1998, Récents progrès en génie des procédés, Volume 12, n°61, p. 1-8.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Hybrid modeling for continuous production of bioethanol E. Ccopa Rivera, Ivana Mantovaneli, Aline C. da Costa and R. Maciel Filho Faculty of Chemical Engineering, State Universtiy of Campinas, Campinas, SP, P.O. Box 6066, 13081-970, Brazil
Abstract Changes in the operational conditions are quite common in plants of alcoholic fermentation; they occur not only due to the variations in the quality of the raw material but also due to variations of dominant yeast in the process. Thus, it is clear that the main difficulty in model-based techniques for definition of operational strategies, control and optimization, is the problem of obtaining a reliable model. This work proposes the use of hybrid models for continuous production of bioethanol plants. The hybrid model comprises first a system of differential equations describing the mass and energy balances of the process. Complementary to the mass and energy balances, the kinetic rates of the biological system are obtained by neural network sub-models. Multilayer perceptron neural networks for an Extractive Alcoholic Fermentation Process and Functional Link Networks for an Alcoholic Fermentation Process with Multiples Stages were used for modeling the kinetic. The results show that typical non-linearities present in this kind of bioprocesses are adequately represented. Agreement between predicted and operating data was very good. The models are reliable enough to be used for further optimization and control studies. Keywords: Bioethanol, Hybrid modeling, Multilayer Perceptron Neural Network, Functional Link Network.
1. Introduction A potential substitute for petroleum in Brazil and in several countries may be biomass, particularly sugar cane. The sugar cane industry keeps the greatest commercial energy production in the world with ethanol and the almost complete use of sugar cane bagasse as a fuel. Brazilian annual production capacity of ethanol is currently in 18 billions of liters and, with the increase in demand, the idea is to multiply this capacity through the installation of new factories and optimization of the operation of the existing ones. Thus, there is an intensified interest in the study of all the steps involved in ethanol production. Among the main problems related to the alcoholic fermentation process is the lack of robustness of the fermentation in the presence of fluctuations in the quality of the raw material and modifications in microbial metabolism. These lead to changes in the kinetic behavior with impact on yield, productivity and conversion of the process. The lack of robustness can be corrected by adjustments in the operational and control parameters of the process when fluctuations occur. In order to accomplish this, it is important that a mathematical model be available to aid in the decision making, mainly when the difficulties of monitoring the key process variables (concentrations of biomass, substrate and ethanol) are taken into account. However, the operational changes described make the prediction of the dynamic behavior of the process with a single model difficult, as they lead to changes in microorganism kinetics. Thus, it would
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be of great advantage to have a mathematical model that could be easily adapted to changes in operational conditions. A way to deal with this problem is to use hybrid neural models. These models are expected to perform better than “black-box” neural network models (pure ANN models), since generalization and extrapolation are confined only to the uncertain parts of the process and the basic model is always consistent with first principles. Besides, significantly fewer data are required for their training. Several authors have applied hybrid modeling for bioprocesses (e.g. Pischogios and Ungar, 1992; Zorzetto et al., 2000) In particular Harada et al. (2002) and Costa et al. (2001) dealt with the hybrid modeling of the alcoholic fermentation process. The objective of this work is to investigate the use of hybrid models to describe continuous production of bioethanol in order to obtain a fast and easy model to be used for on-line implementation. Hybrid models have been developed based on first principles (physical knowledge) associated with Artificial Neural Networks (ANN). Multilayer Perceptron Neural Networks (MLPNN) were used for an Extractive Alcoholic Fermentation Process and Functional Link Networks (FLN) were used for modeling the kinetic of an Alcoholic Fermentation Process with Multiples Stages. Hence, in this work a tool is developed to build up hybrid models with relatively smaller amount of data compared to the “black-box” approach required for process identification exploring the benefits of two different ANN structures. 1.1. Hibrid Models Psichogios and Ungar (1992) were among the first who proposed the serial approach for hybrid modeling of bioprocess. The kinetics relations are modeled by a static mapping of process states to biomass specific rates. Thus, the function to be learned by the ANN is static, while the process dynamics are represented by the differential equations of the model. As noted by Psichogios and Ungar (1992), the hybrid approach provides better predictions than “black box” modeling of bioprocesses with a pure neural network approach. In this work, it was used two different ANNs: MLPNN, which are able to approximate generic classes of functions, including continuous and integrable ones, and FLN, which have good non-linear approximation ability with the advantage that the estimation of their weights is a linear optimization problem. The use of the MLPNN structure has been proved to be a good approach (Zorzetto et al, 2000), however there is a large incentive to use ANN structures that are quicker to be identified and easier to be implemented, such as the FLNs. This is particularly interesting for control and on-line optimization applications. 1.2. Multilayer Perceptron Neural Network (MLPNN) A MLPNN consists of three types of layers: an input layer, an output layer and one or more hidden layers. Each layer may have a different number of neurons, and even a different transfer function. An appropriate architecture would help to achieve higher model accuracy. It has been shown that all continuous functions can be approximated to any desired accuracy, with a network of one hidden layer of sigmoidal f (X) = 1/(1 + exp − X ) hidden units (nodes) and a layer of linear output nodes. Such structure was used in this work. Eq. (1) describes mathematically a MLPNN. M
N
j =1
l =1
y1 = F1[∑ W1 j f j (∑ w jl X l + w j 0 ) + W10 ]
(1)
In Eq. (1), wj,l and W1,j specifies the adjustable parameters; i.e., the weights and biases. X is the input variable of the network.
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1.3. Functional Link Network (FLN) The FLN was developed by Pao (1989) to increase the computing power of neural networks. The input-output relationship of the FLN is represented by Eq. (2). M
yi (X e ) = ∑ wij h j (X e ) , j =1
1≤i≤m
(2)
In the general structure of an FLN the hidden layer performs a functional expansion on the inputs which maps the input space, of dimension n, onto a new space of increased dimension, Mz(Mz>nz). In this case, the expansion results, hj(Xe), are a series of monomials of Xe. The output layer consists on m nodes, each one, in fact, a linear combiner, and each wij is an FLN weight. In these networks, a non-linear functional transform or expansion of the network inputs is initially performed and the resulting terms are linearly combined. More details about the FLNs can be found in Costa et al. (1999).
2. Building up the models This section shows the models development through the hybrid approach for a continuous extractive alcoholic fermentation process and a continuous alcoholic fermentation process with multiples stages. 2.1. Case study 1: Extractive alcoholic fermentation process High concentrations of ethanol inhibit the fermentation process, particularly when a fermentative medium with high concentration of substrate is used, as is the case in the majority of the industrial processes. Taking these into account, a process of fermentation combined with a vacuum flash vessel was considered, which selectively extracts ethanol from the medium as soon as it is produced. The process is composed of four interlinked units: fermentor (ethanol production unit), centrifuge (cell separation unit), cell treatment unit and vacuum flash vessel (ethanol–water separation unit). It was shown that this scheme presents many positive features and better performance than a conventional industrial process. The extractive alcoholic fermentation process is shown in Figure 1. A detailed description of the process and mathematical model can be found in Costa et al. (2001). FLR, SLR, XLR, PLR, TLR CO2 F0, S0, T0
air F, SF, XF, PF, TF
FV, PV, TV FE, S, XE, P, T
FR, SR, XR, PR, TR
F L A S H
FL FERM.
FLS F, S, X, P, T
CENTRIFUGUE
Fc1
FW, TW T R E A T.
Fc, S, Xc, P, T FP
air
Figure 1. Extractive alcoholic fermentation (Costa et al., 2001).
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2.2. Case study 2: Alcoholic Fermentation Process with Multiples Stages F0
FW 1 2
3 4
Cooler
Cooler
5
Cooler Cooler
water, Fa
FS FR
Cooler
acid Purge Stream
Heavy Phase
Centrifuge FV Light Phase
air Surge Tank
Figure 2. Alcoholic fermentation process with multiples stages (Andrietta and Maugeri, 1994).
The fermentative process is illustrated in Figure 2. The system is a typical large-scale industrial process made up of five continuous-stirred tank reactors (CSTR) attached in series and operating with cells recycle. Each reactor has an external heat exchanger whose objective is to keep the temperature constant at an ideal level for the fermentation process. This process operating conditions are real conditions of typical industrial distillers in Brazil. More details about this process are given by Andrietta and Maugeri (1994). 2.3. Training and Validation The training was performed using a full factorial design 22+star configuration, involving the variables S0 (kg/m3) and T0 (oC). Both variables had their values changed every 10 hours (this period was sufficient to reach the steady state) within the ranges 90 kg/m3 to 270 kg/m3 (S0) and 28°C to 40°C (T0) for Case 1 and 150 kg/m3 to 210 kg/m3 (S0) and 28°C to 35°C (T0) for Case 2. Thus, representative data containing the input and output signals of the process was generated. For validation, the disturbances were also a sequence of steps with periods of 10 hours within the operational interval 120 kg/m3 to 230 kg/m3 (S0) and 30°C to 38°C (T0) for Case 1 and 162 kg/m3 to 198 kg/m3 (S0) and 28°C to 32°C (T0) for Case 2. For the current cases, the specific rates of microbial growth were modeled with an ANN with four inputs (concentrations of biomass, substrate and ethanol, and temperature). 2.4. Quality of Prediction The quality of prediction of the proposed hybrid models was given by the residual standard deviation (RSD), Eq. (3), which provides an indication of the accuracy of the predictions. 1 n RSD= (∑ ( yi - yPi ) 2 )0.5 n i =1
(3)
In this equation yi is the ‘real’ value (calculated by the deterministic model), yPi is the value predicted by the hybrid model and n is the number of points. The magnitude of the RSD changes depends upon the magnitude of the variable to be predicted. It is worthwhile mentioning that it is easier to analyze the RSD as a percentage of the average of the real values yi (Atala et al., 2001), Eq. (4). RSD(%) =
RSD yi
× 100
(4)
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3. Results and Discussions
40
72
37
69
34
66
3
P5 [Kg/m ]
3
P [Kg/m ]
The developed hybrid models described well the simulation data. Figures 3(a) and Figure 3(b) for Case 1 and Case 2, respectively, show the results which indicate the good performance of the hybrid model for ethanol concentration. Table 1 for Case 1 and Table 2 for Case 2, shows the prediction results of the hybrid model for concentrations of biomass, substrate and ethanol, and temperature. The range of deviations were from 0.019% to 0.649% and 0.002% to 1.973% for Case 1 and Case 2, respectively. Deviations below 10% are considered acceptable regarding bioprocess engineering (Atala et al., 2001). These results show that this model is reliable enough to be further used for optimization and control studies.
31 28 25
(a)
63 60
0
20
40
60
80
(b)
Time [h]
57
0
20
40
60
80
Time [h]
Figure 3. Process output (solid line) and synthetic data of the hybrid model (dashed line): (a) Case study 1; validation data for ethanol concentration, P, into the reactor. (b) Case study 2; validation data for ethanol concentration, P5, of last (fifth) reactor. Table 1. Residual Standard Deviation for Case study 1 Output variable
RSD (%)
Biomass (X)
0.019
Substrate (S)
0.649
Ethanol (P)
0.359
Temperature (T)
0.042
Table 2. Residual Standard Deviation for Case study 2 Reactor (i)
RSD(%) Biomass (Xi)
Substrate (Si)
Ethanol (Pi)
Temperature (Ti)
1
0.0037
0.151
0.0323
0.0087
2
0.0021
0.309
0.0147
0.0049
3
0.0020
0.754
0.0126
0.0100
4
0.0024
1.429
0.0149
0.0125
5
0.0023
1.973
0.0143
0.0126
4. Conclusions Bioreactors operating in continuous steady state are quite difficult to model, since their operation involves microbial growth under constantly changing conditions. Hence, there is a need for the development of simple though realistic mathematical descriptions
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of alcoholic fermentation process, which can be used to develop optimal operating strategies. This work demonstrated that hybrid model coupling deterministic representation with universal approximators (MLPNN) or linear combination of expanded input variables (FLN) was able to describe the fermentative processes with good performance. The residual standard deviations (RSD%) showed low values. Therefore, the developed models may be used to predict system performance and to design process controllers. It is worthwhile mentioning that the proposed approach for hybrid modeling is able to deal with disturbances in transient phases of the process and different yeasts since the identification procedures be carried out adequately with a suitable set of data representative of the process.
References Andrietta, S.R. and Maugeri, F., Optimum design of a continuous fermentation unit of an industrial plant for alcohol production. Advances in Bioprocess Engineering, 1994, 1 47-52. Atala, D.I.P., Costa, A.C., Maciel, R. and Maugeri, F., Kinetics of ethanol fermentation with high biomass concentration considering the effect of temperature. Applied Biochemistry and Biotechnology, 2001, 91-93(1-9) 353-366. Costa, A.C., Henriques, A.W.S, Alves, T,L.M, Lima, E.L. and Maciel Filho, R., A hybrid neural model for the optimization of fed-batch fermentations. Brazilian Journal of Chemical Engineering., 1999, 37 125-137. Costa, A.C., Atala, D.I.P., Maugeri, F. and Maciel Filho, R., Factorial design and simulation for the optimization and determination of control structures for an extractive alcoholic fermentation. Process Biochemistry, 2001, 37 125-137. Harada, L.H.P., Costa, A.C. and Maciel Filho, R., Hybrid neural modeling of bioprocess using functional link networks. Applied Biochemistry and Biotechnology, 2002, 98-100(1-9) 10091024. Pao,Y. H. Adaptive Pattern Recognition and Neural Networks. Addison-Wesley Publishing Company, California, 1989. Psichogios, D.C. and Ungar, L.H., A hybrid neural network-first principles approach to process modeling. American Institute of Chemical Engineers Journal, 1992, 38 1499–1511. Zorzetto, L.F.M., Maciel Filho, R. and Wolf-Maciel, M.R., Process modelling development through artificial neural networks and hybrid models. Computers and Chemical Engineering, 2000, 24 1355–1360.
Acknowledgements The authors acknowledge FAPESP and CNPq for financial support.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
PREDICTION AND ESTIMATION TECHNIQUES FOR MODELING PERVAPORATION PROCESS Mario E.T. Alvarez, Elenise B. Moraes, Maria R.W. Maciel Separation Process Development Laboratory (LDPS). Chemical Engineering School. State University of Campinas, ( Unicamp), CP 6066, ZIP CODE 13081-970, CampinasSP, Brazil.
Abstract In this work, a new mathematical model was proposed for pervaporation process, considering a predictive model for the diffusion coefficient. Through the model equations, a simulator called PERVAP was developed to evaluate the performance of the pervaporation process. For the model validation, the separation of the azeotropic mixture ethanol/water was studied through simulation using a hydrophilic membrane, polyetherimide (PEI), using experimental data obtained from literature. The prediction of the diffusion coefficient was carried out considering the free-volume theory and the parameters were determined from pure component and polymer properties. Group contribution method was also used to predict the binary interaction parameters between the components and the polymer. The proposed model presented good agreement with experimental data, allowing its application for separating other azeotropic systems and, also, for studying the effects of process variables. Keywords: Pervaporation, Modeling, Simulation, Estimation Techniques.
1. Introduction Pervaporation is an effective process for separating azeotropic mixtures, close-boiling point compounds, and mixtures consisting of heat-sensitive materials. Pervaporation process is a separation technique based on a selective transport of permeate through a dense layer membrane associated with an evaporation of the permeate by lowering the partial pressure of the permeants downstream applying vacuum. Therefore, pervaporation differs from the other membrane processes such as reverse osmosis, ultrafiltration, microfiltration, electrodialysis and gas permeation, because phase change occurs during the separation. Pervaporation holds a great potential for utilization by the traditional chemical industry and by emerging areas such as environmental and biochemical engineering, using dense membranes to separate different types of mixtures. The advantages of the proposed model in comparison with the existing models present in the open literature are in the predictions of the binary interaction parameters between the components and the polymer, of the diffusion coefficient and of the activity coefficient by UNIFAC group contribution. These make the proposed model more predictive and independent of experimental data for calculating these parameters. The unique required parameter to be determined experimentally by sorption experiments is the internal activity coefficient in the membrane. In this work, a mathematical model was proposed for the pervaporation process in order to separate binary mixtures, based on the solution-diffusion mechanism, where the prediction
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of diffusion coefficient was carried out using the free-volume theory [1,2]. The freevolume parameters were determined from pure component viscosity data and the binary interaction parameters between the components and the polymer were determined using a group contribution method. Simulations were carried out for separating the mixture ethanol/water, using the PERVAP simulator, which was developed in this work, based on the model proposed. The used membrane is of polyetherimide (PEI), which has preferential permeation to water (hydrophilic). To validate the simulation results, experimental data from the literature [3] were used. The effects of the downstream pressure on the selectivity and on the flux were analysed for this system.
2. Mathematical model for Pervaporation Process According to the solution-diffusion mechanism, the mass transport of the permeant components through the membrane can be described by Figure 1. Liquid feed in upstream
Boundary layer
Permeate vapor in downstream
Membrane
Cim,1
Ci,f
where, Ci,f , Concentration of component i in the feed (mol/m3) m
Ci ,1 , Concentration of component i
in the membrane in the upstream (mol/m3) m
Ci , 2
Direction of the liquid feed
δbl
Ci,2 Permeate flux
A z
m Ci , 2
, Concentration of component i
in the membrane in the downstream (mol/m3) Ci,2 , Concentration of component i in the downstream (mol/m3) δbl , boundary layer thickness (m) A , membrane thickness (m)
Direction of the permeate flux
Figure 1. Permeation and concentration profiles of the permeant i through the membrane
2.1. Transport in the boundary layer Considering Figure 1, at steady state, according to the first Fick’s law, the solution of the general equation of the molar flux of solute i through the boundary layer is expressed as [4]: Di , f Pe Ci , f exp( Pe) − Cim,1 (1) Ji = δ bl exp( Pe) − 1 where Pe is the Peclet number; Di,f is the diffusivity coefficient of component i in the liquid feed, which can be estimated by Wilke and Chang [5] equation. If either the membrane presents a high selectivity to a component or the feed side operates in a turbulent regime, the convective flux in the feed can be ignored [6]. Considering that the system operates in a regime of Pe<<1, then, Eq. 1, can be reduced to a simpler expression (Eq. 2): Di , f (2) Ji = Ci , f − Cim,1
δ bl
(
)
where the concentration of component i in the feed phase (Ci,f) can be solved expressing the driving force in terms of activities. 2.2. Transport in the membrane At steady state, the diffusive flux of each component, considering the interfaces of the membrane with the feed phase and with the permeate phase, is defined as:
Prediction and Estimation Techniques for Modeling Pervaporation Process
(
621
)
Dim m (3) Ci ,1 − Cim, 2 A m where Di is the diffusivity coefficient of the component i in the membrane. In the permeation process, the permeants define a concentration profile through the membrane, from the feed phase up to the permeate phase. Taking into account the permeation process, the concentration of permeants i and j in the membrane in the downstream is described by the following equations: xi , f α p Cim,2 = (4) m γ i 1 + (α − 1)xi , f Ji =
[
C mj ,2 =
]
( 1 − xi , f ) p Pi sat
[
(5)
]
γ mj 1 + (α − 1 ))xi , f Pjsat
where: p=P2/ Pi sat is the partial pressure, P2 is the downstream pressure (kPa), Pi sat is the saturated vapour pressure (kPa), α is the selectivity, xi,f is the feed mole fraction of component i, γ im and γ mj are the activity coefficient in the membrane for component i and j, respectively. Solving Eqs. 2 and 3, using Eqs. 4 and 5, Eq. 6 is obtained: ⎛ ⎞ Dim ⋅ Di , f ⎜ x γ − xi , f α p ⎟ Ji = i , f i , f ⎜ ⎟ 1 + (α − 1)xi , f ⎟⎠ ( Dimδ bl + Di , f A )γ im ⎜⎝
(6)
The activity coefficient in the membrane, γ im , is a experimental parameter that is determined from the solubility of the component i in the membrane [7], which is m expressed by the internal molar concentration in the membrane, Ci , and the activity coefficient in the feed is determined by UNIFAC method, γi,f, as described by the following equation (Eq. 7): γ i , f xi , f γ im = (7) Cim Considering that the mass transport in the feed phase does not present resistance over the boundary layer, then, it can be assumed that the thickness of the phase of the boundary layer, δbl, can be negligible. So, Eq. 6 becomes: xi , f α p ⎞⎟ Dm ⎛ J i = i ⎜⎜ xi , f γ i , f − (8) ⎟ 1 + ( α − 1 )xi , f ⎟⎠ Aγ im ⎜⎝ m
where, the diffusion coefficient of component i in the membrane, Di , can be predicted by the free-volume theory described by the following equation (Eq. 9) [1,2]: ⎛ ⎞ ⎜ ⎟ ∗ ∗ ˆ ˆ ω V ξω V + E − ⎛ ⎞ ⎜ 1 1 2 2 ⎟ (9) Dim = D0 (1− φ1 )2 (1− 2χφ1 )exp⎜ ⎟ exp − ⎝ RT ⎠ ⎜ K11 ω K − T + T + K12 ω K − T + T ⎟ ⎜ ⎟ 1 21 g1 2 22 g2 γ ⎝ γ ⎠ where, D0 is a constant pre-exponential factor; E is the energy required to overcome attractive forces from neighbouring molecules; γ is an overlap factor for free-volume; Vˆi∗ is the specific critical hole free-volume of the component i required for jump; ω1 is
(
)
(
)
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the component weight fraction; ω2 is the polymer weight fraction; ξ is the ratio of critical molar volume of solvent jumping unit to that of polymer jumping unit; K11 and K21 are the solvent free–volume parameters, K12 and K22 are polymer free-volume parameters; φ1 is the component volume fraction; χ is the component/polymer interaction parameter, which was determined in this work by the group contribution method [8]; Tgi is the solvent glass transition temperature; T is the temperature and R is the gas constant. 2.3. Selectivity Determination The selectivity, for the pervaporation process, can be obtained from flux equations for the components i and j, which can be expressed by Eq. 10:
αi,j
⎛ xi , f α i , j p ⎞⎟ Dimγ mj ⎜⎜ γ i , f xi , f + ⎟⎟ 1 − xi , f ⎜ 1 + − 1 x α i , j i , f ⎝ ⎠ = sat ⎛ ⎞ 1 − x pP i, f i ⎟x D mjγ im ⎜⎜ γ j , f x j , f − ⎟⎟ i , f sat ⎜ P 1 + − 1 x α j i,j i,f ⎠ ⎝
( (
[ (
) )
(
) (10)
) ]
2.4. Determination of permeate composition The composition of component i in downstream, yi,2, can be determined considering the permeate total flux, which is the sum of the permeate component partial fluxes, i.e., JT = Ji + Jj, as can be seen in Eq.11: yi , 2 =
Ji Ji + J j
(11)
Using this modelling, the PERVAP simulator was developed in order to carry out simulations for existing and new processes.
3. Results and Discussions Simulations were made, using the PERVAP simulator, for separating the azeotropic mixture ethanol/water in the PEI membrane. The results were validated with experimental data from the literature [3]. The operating conditions were: temperature 310.15 K; downstream pressure 0.133 kPa; membrane film thickness 160 μm. 3.1. Determination of free-volume parameters Free-volume parameters were determined from viscosity data and temperature of pure components and the binary interaction parameter (component/polymer) was predicted using the Group Contribution Equation of State for Polymer Solutions [8]. The specific critical hole free-volume was estimated by the additive method of atomic constant of “Sugden” [9]. These parameters are presented in Table 1. Table 1. Free-volume parameters estimated and predicted used in the diffusion coefficient prediction [10] * K21- Tg1 Do E K11/γ Vˆ1 χ ξ (cal/mol) 3 3 (K) (cm2/s) (cm /g) (cm /gK) Ethanol 0.985 0.312×10-3 111.80 11.6×10-4 0.043 0.124 0
Water
1.071
2.180×10-3 -152.29 8.55×10-4 0.053 0.035
0
Free-volume parameters of the polymer (PEI) were estimated and their values are: Vˆ2* = 0.804 cm3/g, K12/γ = 6.93×10-4 cm3/g K and K22- Tg2 = -509.9 K. With the parameters (for ethanol, water and polymer), it is possible to calculate the diffusion coefficients in the membrane for ethanol (i) (8.616×10-4 m2/h) and water (j) (3.779×10-4 m2/h). The activity
Prediction and Estimation Techniques for Modeling Pervaporation Process
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coefficients in the membrane were calculated from experimental data of flux and the composition, where γ im = 2.670 m3/mol and γ mj = 0.211 m3/mol. Results from simulations are shown in Figures 2 to 7. simulated experimental data ethanol water
2
Permeate flux (mol/m h)
10 8 6 4 2
simulated experimental data
40
Selectivity, αWa,Et
12
30 20 10 0
0 0,0
0,2
0,4
0,6
0,8
1,0
0,0
Feed ethanol mole fraction
0,6
9
simulated experimental data
7
2
0,4 0,2
0,2
0,4
0,6
0,8
1,0
simulated ethanol water
8
0,6
0,0 0,0
0,8
Figure 3. Selectivity vs feed ethanol mole fraction (P2=0.133 kPa, T=310.15 K)
Permeate flux (mol/m h)
Permeate ethanol mole fraction
0,8
0,4
Feed ethanol mole fraction
Figure 2. Permeate flux vs feed ethanol mole fraction (P2=0.133 kPa, T=310.15 K) 1,0
0,2
1,0
Feed ethanol mole fraction
Figure 4. Permeate ethanol mole fraction vs feed ethanol mole fraction
6 5 4 3 2 1 0
1
2
3
4
5
Downstream pressure (kPa)
Figure 5. Influence of downstream pressure on the permeate flux (xet=0.415, T=301.15K)
According to Fig. 2, it can be observed that the permeate flux of ethanol tends to increase, increasing the feed ethanol mole fraction, while the water flux decreases. However, the ethanol flux is higher than the water flux for ethanol mole fraction above 0.95. In Fig. 3, it can be seen that when the feed ethanol mole fraction increases above 0.6, the selectivity increases, indicating a better water separation through the membrane, preferentially when it has a solution with high ethanol concentration in the feed, as for example in the azeotropic point (0.9 ethanol mole fraction at 101.33 kPa). Fig.4 shows the variation of the ethanol mole fraction in the permeate with the ethanol mole fraction in the feed, indicating that the azeotropic point was eliminated. It can also be observed that the model presents a good agreement with experimental data, as shown in Figures 2, 3 and 4. In Fig. 5, it can be verified that higher is the downstream pressure, lower is the permeate flux, for feed ethanol mole fraction equal to 0.415. The water flux in the permeate is always higher than the ethanol flux. The rate of ethanol flux by water flux decreased of 0.18 (at 0.133 kPa) to 0.17 (at 4.5 kPa) (Fig. 5), indicating a small decrease
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in the ethanol flux in relation to the water flux, when the pressure increases. This causes an increase in the selectivity of water, which can be proved when the selectivity behavior is analyzed with the pressure, according to Fig. 6. Fig. 7 shows a decrease of the permeate flux with the increase of the pressure for all ethanol mole fractions. In the case of pressure equal to 3 and 4 kPa, it was observed an increasing of the fluxes until reach a maximum, and then they decrease, indicating a higher diffusion of the component through the membrane with the increase in the ethanol mole fraction. For mole fraction above 0.6, the total flux tends to decrease (for all studied pressures) and converges to the same point, increasing the ethanol concentration. Total permeate flux (mol/m2h)
4,25
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Figure 6. Influence of the downstream pressure on the selectivity
12 11 P2 = 0.133 kPa 10 P = 1.0 kPa 2 9 8 P2 = 2.0 kPa 7 P = 3.0 kPa 6 2 5 P = 4.0 kPa 2 4 3 2 1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
Feed ethanol mole fraction
Figure 7. Total permeate flux variation with feed ethanol mole fraction
4. Concluding Remarks The proposed model is indicated for dense membranes that are typically used in pervaporation. So, in order to validate the model, PEI membranes had been used, showing good agreement with the experimental data, confirming their applicability for this type of process. In the present work, it was studied the separation of ethanol/water mixtures in PEI membrane, which is characterized by presenting a high affinity for water removal at the permeate side and enrichment of ethanol at the retentate one. The good performance of the model also allows its application for the study of the effect of the process dependent variables, as for example, the downstream pressure and the feed composition on the separation. Moreover, it can be verified that it was possible to break the azeotropic point. The developed software is also useful to study other water-organic azeotropic system.
References [1] J.S. Vrentas and J.L. Duda, J. Polym. Sci. Part B: Polym. Phys., 15 (1977) 403. [2] J.S. Vrentas and J.L. Duda, AIChE J., 25 (1979). [3] R.Y.M. Huang and X. Feng, Separation Science and Technology, 27 (1992) 1583. [4] P.K. Ten and R.W. Field, Chemical Engineering Science, 55 (1999) 1425. [5] C.R. Wilke and P. Chang, AIChE J., 1 (1955) 264. [6] R. Gref, Q.T. Nguyen and J. Néel, Separation Science and Technology, 27 (1992) 467. [7] J-P. Brun, C. Larchet, R. Melet and G. Bulvestre, J. Membr. Sci., 23 (1985) 257. [8] M.S. High and R.P. Danner, Fluid Phase Equilibria, 53 (1989) 323. [9] R.N. Haward, J. Macromol. Sci. Rev. Macromol Chem., C4 (1970) 191. [10] M.E.T. Alvarez, Ph.D. Thesis, State University of Campinas (UNICAMP), Brazil, (2005).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Model discrimination and parameter estimation through sensitivity analysis Mauricio Sales-Cruz, Rafiqul Gani CAPEC, Department of Chemical Engineering, Technical University of Denmark, DK2800 Lyngby, Denmark
Abstract In this paper, a systematic procedure to support the building of process models through the use of a computer-aided modeling system, called ICAS-MoT, is presented. Specifically, the step-by-step procedure allows the decomposition of the model solution into several sub-problems: first, a sensitivity analysis helps in the parameter discrimination and gives the order of the parameters to be estimated, then the parameters are estimated using either least squares method or maximum likelihood method, and finally the applicability of the model parameters is evaluated by determining the confidence limits of the estimated parameters. The application of these computer-aided tools is highlighted through a complex kinetic parameter estimation problem. Keywords: computer-aided modeling, model discrimination, parameter estimation, sensitivity analysis.
1. Introduction Commonly, model-based optimization problems are related to (kinetic) parameter estimation, optimal start-up/shutdown operation, time scheduling, reactor design, optimizing process control, analysis of dynamic systems and so on. To solve these problems, several alternative models are often proposed to explain the same data, and objective criteria are needed to choose among models. The alternative models may, in some cases, be nested. Nested models are constructed such that a simpler model can be obtained from a more complex model by eliminating one or more parameters from the more complex model. Thus the model discrimination problem may be reduced to the appropriateness of adding or removing some of the related parameters. The question of increased model complexity with or without additional model parameters needs to be investigated with respect to computational cost, accuracy and parameter uncertainty. In general, the more parameters contained in a model, the less reliable are the parameter estimates. Criteria to select among models must weigh the trade-off between increased information and decreased reliability. Moreover, the model refinement is an iterative procedure where, experimental and expert guided process of adding, deleting, or modifying assumptions until a model that satisfactorily explains the data is obtained is in general a difficult and time consuming task. The objective of this paper is to highlight the model parameter estimation and model discrimination features within the computer-aided modeling system called ICAS-MoT (Sales-Cruz and Gani, 2003). Specifically, model refinement and model parameter discrimination are done through a sensitivity analysis, together with a statistical analysis, which are tools included in ICAS-MoT. As a case study, the propane aromatization on zeolite catalyst is considered. The large number of reactions and reactive species involved gives a better process description but complicates the process
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modeling as reported by Katare et al. (2004a; 2004b) and Lukyanov et al. (1995). To refine the model, a sensitivity analysis is performed by testing different values of the parameters in a systematic manner so that the less sensitive parameters can be discriminated/eliminated. Numerical optimization methods (i.e. least square or maximum likelihood method) are used to estimate the parameters and statistical information is obtained to determine the reliability of the model parameters.
2. Model construction strategy For building and discriminating nested models, the first step involves specifying a mathematical formulation of a detailed model that describes the process. Then three stages should be followed: Stage I: model parameter discrimination, the strategy uses this model to perform parameter sensitivity analysis before any data are collected, in order to determine whether or not the model parameters can be identified so that some of them can be discriminated, breaking down the parameter estimation problem into sub-problems; Stage II: model refinement, the strategy involves designing experiments for parameter estimation (using least squares or maximum likelihood method) and model adequacy checking to determine whether or not the proposed model can be refined; and Stage III: model validation, experiments are designed for improving the precision of the parameters within that model, eventually arriving at a final, statisticallyverified model formulation.
3. Computer-aided modeling system There are many programs for fitting nonlinear models to experimental data, and the use of these software is now widespread. In this case, the model discrimination procedure (i.e. the model screening, comparison and improvement) is done systematically through our computer-aided modeling system, called ICAS-MoT (Sales-Cruz and Gani, 2003) that allows performing the following modeling steps in a fast, reliable and efficient way: (1) model formulation, (2) model solution, (3) sensitivity analysis, (4) model parameter optimization, and (5) statistical analysis of results. An important feature of ICAS-MoT is that the model developer does not need to write any programming codes to enter the model equations. Models are entered (imported) as text-files or XML-files, which are then internally translated. In the model analysis step ICAS-MoT orders the equations into lower triangular form (if feasible), generates the incidence matrix, verifies the degrees of freedom, and checks for singularity. After this interactive model analysis, the appropriate solver for the model equations is selected together with a corresponding strategy of solution. Then the sensitivity analysis option allows testing different values of the parameter, which can help in the model parameter discrimination task. For model parameter regression (Englezos and Kalogerakis, 2001), the least square method (by minimizing the sum of the square errors between the experimental values and the predicted ones) or maximum likelihood method (by maximizing the probability distribution function of the estimation error) can be used. For the statistical analysis of results, ICAS-MoT performs analysis of variance, correlation factors calculations, F significance test and confidential intervals of the parameters estimated (Montgomery and Runger, 1999).
4. Case study: Kinetic model for propane aromatization 4.1. Process description Propane aromatization is a complex heterogeneous reaction that involves many
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individual reactions species and many more chemical reaction (one species can participate in many reactions). Katare et al. (2004a, 2004b) have developed a reduced kinetic model by lumping all the isomers of the same carbon number into a single component. The schematic mechanism of the reactions considered in the kinetic model is shown in Fig. 1.
C3H8
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Fig. 1. Reaction pathways for propane aromatization over HZSM-5 (Katare et al., 2004a).
4.2. Model formulation In this work, the detailed model for propane aromatization was taken from Venkatasubramanian (2005), who using information about the relative rates of the reactions, stability of the intermediates, quantum chemical calculations and empirical correlations from the literature lumped the reactions into 33 reaction families, so as to reduced the number of model parameters to 13. The reaction rate model where the kinetic model is embedded, consists of a differential-algebraic equation system with 29 algebraic equations (representing the mass balance in the active sites) and 31 ordinary differential equations (representing the concentration of the chemical species in the gas phase), the 13 kinetic model parameters found in the model are classified in this paper into four parameter types (as shown in Table 1). Table 1. Model parameters classified according type.
Parameter k0 k1 k3 k5 k6 k10 k7 k8 k9 k2 k4 k11 k12
Parameter Description
Protolysis of carbonium ions Carbonium ion desorption Carbenium ion desorption Beta-scission Aromatization Carbonium ion dehydrogenation Alkane adsorption Hydride transfer reactions Olefin adsorption Increase in adsorption enthalpy for alkanes with carbon number Increase in activation energy for alkanes desorption with carbon number Increase in activation energy for carbonium ion dehydrogenation with carbon number Entropy factor to determine the equilibrium between beta-scission and oligomerization
Reference value 200.0 11.3x104 853.0 25.6x104 6.7x107 94.0 47.0 100.0 8923.0 6.0
Type a a a a a a b b b c
6.0
c
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c
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d 3
a: First order rate constants (in terms of mol/g/h), b: second order rate constants (in m /g/h), c: Energy terms (in KJ/mol), and d: the entropy term (normalized by the universal gas constant).
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628 The generic form of the mathematical model is as follows: dx = f ( x, y , θ, t ) ; dt
g ( x, y ,θ, t ) = 0
(1)
Where vector x represents the concentration of the chemical species in the gas phase, vector y represents the concentration in the catalyst sites and vector θ represents the kinetic model parameters. 4.3. Sensitivity analysis A sensitivity analysis is performed to evaluate the effect of each parameter, by perturbing them with respect to their reference value (given in Table 1) and computing the output response in terms of the percentage of change of the C3H8 composition (the main reactive component in the reaction). Alternatively, plots of the analytical derivatives would also provide the same information. Figures 3 and 4 show the results of this sensitivity analysis: (a) parameters k12 (Fig. 4b) and k6 (Fig. 3a) are the least sensitive (i.e. the change in output response is less than 1%); (b) parameters k5 (Fig. 3a), k8 (Fig. 3b) and k9 (Fig. 3b) are also not very sensitive as changes in output responses vary by up to 2.0%; (c) parameters k2 and k11 (Fig. 4a) are lightly more sensitive as changes output responses vary by up to 5%, (d) parameters k1 (Fig. 3a), k4 (Fig. 4a) and k10 (Fig. 3a) are sensitive as changes in output responses vary by up to 15%, (e) parameters k0 (Fig. 3a) and k7 (Fig. 3b) are very sensitive as changes in output responses vary by up to 20%, and finally, (f) the most sensitive parameter is k3 (Fig. 3a), as changes in output response varies by up to 100%. Based on the above analysis, the model is rather insensitive (as changes in output responses within 5 %) to seven parameters k12, k6, k5, k8, k9, k2 and k11, but highly sensitive to the other six parameters (k1, k4, k10, k0, k7 and k3).
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Fig. 4. Sensitivity analysis: (a) energy parameters and (b) entropy parameter.
4.4. Strategy for parameter estimation and model validation Based on the above analysis, the following strategy for model parameter regression is proposed for stage II (i.e. model parameter estimation). Using available experimental data, the parameter identification problem is split into six sub-problems following the order of sensitivity of each parameter: (1) first all the parameters are fixed at their reference value, then the least sensitive parameter k12 (and also the only one parameter of type d) is regressed using the least squares minimization method; (2) then, taking this new value of the parameter k12, the following four less sensitive parameters (k5, k6, k8, k9) are regressed; (3) with these five parameters (k12, k5, k6, k8, k9) regressed then the parameters corresponding to the energy terms (k2 and k11) are regressed; (4) with these seven new values (k12, k5, k6, k8, k9, k2 and k11), the parameters k1, k4 and k10 are regressed; (5) having these ten parameters regressed (k12 , k5, k6, k8, k9, k2, k11, k1, k4 and k10), then the parameters k0 and k7 are regressed; and (6) finally, with these twelve parameters regressed, the most sensitive parameter k3 has to be regressed. For the model refinement, once all these parameters have been identified, a simultaneous regression must be done but with a reduction in the parameter bounds to determine whether or not any of the estimated (step by step) model parameter changes with purpose of selecting the best model parameters. And lastly, for stage III (model validation), experiments must be designed in order to improve the precision of the parameters estimated, eventually arriving at a final statistically-verified model formulation. Following the aforementioned strategy of solution for stage II, the regressed parameters are presented in Table 2. Afterwards, when the simultaneous regression of all parameters was done, as it was expected, only the regressed values of parameters k6 and k12 (the least sensitive ones) did not change. That means that the model could be reduced by eliminating these two parameters (k6 and k12). To verify the accuracy of the model parameters, the following statistics of the regression was calculated: correlation coefficient = 0.972549, standard error of the estimates = 13.612893 and F significance = 5.206348e-002. In addition the 95% confidence limits were calculated and are reported in Table 2. It can be noted that the most uncertain parameters also have the widest bounds, the need for which is a topic of further investigation.
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Parameters
Bounds
k0 k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12
102 ≤ k0 ≤ 107 104 ≤ k1 ≤ 1010 6 ≤ k2 ≤ 14 103 ≤ k3 ≤ 109 6 ≤ k4 ≤ 12 103 ≤ k5 ≤ 108 107 ≤ k6 ≤ 1013 10-3 ≤ k7 ≤ 102 10-3 ≤ k8 ≤ 102 10-1 ≤ k9 ≤ 104 102 ≤ k10 ≤ 108 2 ≤ k11 ≤ 6 18 ≤ k12 ≤ 25
Step by step value 217.8642 132579.0 6.82674 1025.72 8.648333 238174.3121 6x107 4.7773 125.87432 7863.5726 93.3911 3.78924 18.0
Simultaneous value 5.6472x105 69122.67 6.0 1.1277x107 12.0 255194.49 6x107 46.107 99.7586 8922.80 125.475 6.0 18.0
±95% CI* 59680.8614 7305.03699 0.6340933 1191778.36 1.2681866 26969.5194 6340933 4.87268997 10.54271 942.981283 13.2604761 0.6340933 2.4389342
* Confidential interval of the simultaneous regression
4.5. Other case studies The ICAS-MoT system for model discrimination and parameter estimation has also been tested with all the problems reported by Katare et al. (2004a). In all cases, better optimized parameters were obtained (detailed results can be obtained from the authors).
5. Conclusions In this paper, a computer-aided procedure for parameter discrimination and parameter estimation of nested models has been presented, illustrating the procedure with a complex case study related to the propane aromatization kinetics. In general, a systematic model building (i.e. model parameter discrimination, model refinement and model validation) permits a substantial saving in experimental (time and cost) effort. Specifically the parameter model regression was done based on a sensitivity analysis, obtaining helpful information about the parameter identifiability. Based on the model parameter discrimination results, additional experimental data are being selected for final model validation. Once the model is ready the ICAS-MoT system also provides a customized simulator of the process for future design/optimization studies.
References Englezos, P., Kalogerakis, N., 2001, Applied Parameter Estimation for Chemical Engineers, Marcel Dekker, New York. Froment, G.F., 1975, AIChE J., 21 (6), 1041-1057. Katare, S., Bhan, A., Caruthers, J.M., Delgass, W.N., Venkatasubramanian, V., 2004a, Comp. Chem. Eng., 28, 2569-2581. Katare, S., Caruthers, J.M., Delgass, W.N., Venkatasubramanian, V., 2004b, Ind. Eng. Chem. Res., 43, 3484-3512. Lukyanov, D.B., Gnep, N.S., Guisnet, M.R., 1995, Ind. Eng. Chem. Res., 34, 516-523. Montgomery, D.C., Runger, G.C., 1999, Applied Statistics and Probability for Eng., Wiley, NY. Sales-Cruz, M. and R. Gani, 2003, Computer-Aided Chemical Engineering, vol. 16: Dynamic Model Development, Eds. S.P. Asprey and S. Macchietto, Elsevier, Amsterdam.
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Venkatasubramanian, V., Laboratory for Intelligent Process Systems (LIPS), Purdue University, personal communication, January, 2005.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Solving MINLP Containing Noisy Variables and Black-Box Functions Using Branch-and-Bound Eddie Davis and Marianthi Ierapetritou Rutgers University, 98 Brett Road, Piscataway, NJ 08854, USA
Abstract Since existing MINLP solvers such as DICOPT require the existence of explicit deterministic equations, they are unable to address systems containing noise and unknown model equations. In this paper we propose a new algorithm based on a Branch-and-Bound main structure to solve MINLP problems involving noise and blackbox models, where a response surface method developed in our earlier work is applied to solve a noisy NLP at each node of the branch and bound tree. The algorithm is applied to an example problem to further clarify the steps of the proposed approach. Keywords: Optimisation, MINLP, Noisy Functions, Black-Box Models
1. Introduction Process design optimisation requires information regarding system behaviour in the form of mass and heat balances and/or phase equilibrium relations. When exact constitutive relations are unavailable due to lack of adequate information, optimisation of these “black-box” systems can be achieved using surrogate models. Process stochasticity arising from inaccurate measurements of process outputs complicates model development and may cause gradient-based optimisation algorithms to terminate prematurely by getting trapped in artificial local optima. As a result, there is a need for the development of robust algorithms capable of optimising systems where the exact model is unknown or involves uncertainty. In an earlier work, we developed two approaches for the optimisation of noisy and/or black-box models involving only continuous variables, using response surface models to represent the unknown physical relationships (Davis et. al., 2005). The form of the response surface is a low-order polynomial created by the input-output response mapping, which is then used in an optimisation algorithm. We proposed two algorithmic procedures where in the first one we utilize direct search methods, changing to steepest descent once the optimum region has been located, while the second one uses sequential quadratic programming (SQP) depending upon whether improvement in the objective function is obtained by applying a quadratic approximation of the local response surface over the global region. For functions whose optimum exists near the bounds of the design region, SQP provides faster convergence as the ability to move in a better derivative-based overall direction quickly overcomes the slower movement of sequential simplices. These ideas are extended in this paper towards the solution of noisy nonlinear programs involving discrete decisions (MINLP) since many process synthesis, design, and operations problems can be modelled as integer programming problems due to choices of process units, operating conditions, or task assignments. However, seldom is all the information available that is required to build a deterministic analytical model. In general, MINLP solution approaches such as Branch & Bound (B&B) (Ravindrawn & Gupta, 1985, Fletcher & Leyffer, 2001), Outer Approximation (OA) (Duran &
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Grossmann, 1986), and Generalized Benders Decomposition (GBD) (Geoffrion, 1972) split the overall problem into easier NLP and MILP subproblems, that provide a converging sequence of upper and lower bounds. In B&B, an NLP is formulated at the first node from the MINLP by relaxing the integrality constraint for all binary variables. New subproblems are created by sequentially branching on binary variables. Values of prior binary variable assignments yielding the best objective solution are retained until the optimum is found. Multivariable branching (Achterberg et al., 2004) and parallel branch-and-bound methods (Gendron & Crainic, 1994) accelerate convergence, but the problem may still be computationally very expensive. The OA algorithm relies upon linearization of the objective function and constraints to reduce problem complexity, but assumes differentiability and certain convexity conditions. Furthermore, the computational cost associated with the master MILP increases as the linearizations accumulate with an increasing number of iterations. In GBD, the master MILP is formulated using the dual information of the relaxed NLP problem. Compared with OA the master MILP problem of GBD is less computationally expensive, but more iterations may be required. An alternative outer approximation approach was recently suggested by our group for certain classes of convex problems (Goyal and Ierapetritou, 2004) which is based on the ideas of simplicial approximation to provide an outer approximation of the feasible region leading to a solution of only one MILP problem. Other approaches include the Extended Cutting Plane method (Westerlund & Pettersson, 1995) that successively linearizes the most violated constraint at the predicted minimiser, generating a sequence of nondecreasing lower bounds and the Generalized Disjunctive Programming (Raman & Grossmann, 1994) where the constraints are written in terms of logical operators to reduce the computational complexity. There are also many variants of the above methods which exploit special problem structure. With the exception of B&B, a limitation of the remaining algorithms is that they assume differentiability of the objective function and constraints, a condition that is not satisfied if the problem is noisy, involves black-box models and/or uncertainty. In the next section, we propose using the basic ideas of the B&B framework to solve MINLP containing noisy variables and black-box functions, applying the method we have previously developed to solve noisy NLP problems for the solution of subproblems at each node.
2. Proposed Approach 2.1. Response Surface Methods for Solving Noisy/Black-Box NLP A response surface is an input-output mapping based on a set of sampling points generated to cover the range of interest in terms of input variables. It corresponds to a local approximation of system behaviour and is built whenever noisy input-output data are the only information available, as opposed to an explicit model. The response surface functionality is comprised of basis functions whose coefficients are determined through least squares. Optimisation commonly occurs via steepest descent, in which the new iterate is the minimiser of the response surface over the local region (Myers & Montgomery, 2002). Due to the simple functionality, analytical derivatives are obtained quickly, thereby alleviating the costs associated with expensive numerical gradients. Since a response surface is determined at each iteration, this can lead to many sampling points being required. Furthermore, optimisation can be slow because local steps are taken towards an optimum. Thus, we developed two algorithms in order to reduce the computational complexity of the standard response surface methods (Davis et al., 2005).
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In our first approach, direct search is employed in order to move towards the vicinity of the optimum, followed by steepest descent to refine the candidate solution. First, sampling points are determined at the vertices of a hyper-rectangle. After obtaining the noisy outputs, we move to the best point in terms of the objective function value. Initially, this results in movement along diagonals, but once the center point produces the best objective value, this indicates that we have found a region containing a local optimum. Additional sampling points are obtained, and a response surface is fitted and minimised using a steepest descent method. The current solution is improved by minimizing a response surface centered around it. The size of the reduced subregion is determined as the norm of the previous and current minimisers. Since a model is not updated at every iteration, and fewer points are required for the direct search portion of the algorithm, the computational requirements are reduced. However, a limitation of this method is that only local steps are taken towards an optimum. We address this issue in our second method by optimising response surfaces using SQP over the entire feasible region, leading to possible global movement towards the optimum. At each iteration in SQP, a quadratic program (QP) is built, consisting of a quadratic approximation of the objective and linearized constraints. In our approach, the response surface serves as the objective function for the QP, which is then minimised over the entire feasible region. If sampling at the predicted optimiser produces a better solution, this point becomes the next iterate. Otherwise or for the case that the predicted minimiser falls outside the feasible region, the new iterate is the best minimiser of the current sampling set. The tradeoff is one additional function call per iteration balanced against larger steps taken towards the optimum. The limitation of both these methods is that they assume that all variables are continuous. In order to extend these approaches to deal with discrete decisions, we propose to utilize the approach presented in the next section that builds around a B&B framework. 2.2. Proposed Approach for Solving Noisy/Black-Box MINLP Based on the approaches presented in the previous section to solve NLP problems with noisy variables and/or black-box parts of the model, in this section, we propose a new approach when integer variables are involved based on the basic B&B structure. The problem formulation addressed is as follows:
ªmin f1 (x, y, z1 )+ f 2 (x, z2 )º « s.t. h(x, y, z ) = 0 » 1 « » « ī(x, z2 )=0 » « g(x, y, z ) d 0 » 1 « » L U « x dxdx » « » n x « » « y {0,1} p » « » q+w ¬ z {z1 , z 2 } ¼
(P1)
where x is the set of n bounded continuous inputs, y is the set of p binary variables, and z is the set of output variables, partitioned into subsets z1 q and z2 w depending on whether the relationships to the inputs are known or unknown, respectively. Explicit model equations are stated in terms of the equality and inequality constraints h(x,y,z1)
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and g(x,y,z1), respectively, and the unknown equations describing the noisy/black-box portion of the model are represented using the function *(x,z2). Formulating the MINLP in this way allows us to address systems whose model equations are BB and/or noisy. The proposed approach follows the basic steps of a Branch and Bound scheme. At the first node, a relaxed form of (P1) is solved by allowing the binary variables to be continuous. If this relaxation proves infeasible, (P1) is also infeasible; otherwise, the solution is (x,y,f). If an integer solution is found, this solution is accepted as the best possible for (P1), and the algorithm terminates. On the other hand, if fractional values exist for the binary variables, a LB is established and branching occurs on a subset of the infeasible binary variables, creating two new MINLP. One of the subproblems is selected and the solution of its NLP relaxation for the remaining binary variables is obtained. The LB/UB is updated depending upon whether the optimum improves upon the current respective value and is infeasible/feasible in terms of binary variables (Floudas, 1995). New NLP subproblems are formulated only at nodes for which the corresponding LB is lower than the current UB. Node selection rules are applied in determining the next NLP to solve, and the solution procedure continues of updating the upper and lower bounds until the list of remaining subproblems is empty, or the difference between the UB and LB falls below a prespecified tolerance G1. Each LB or UB is within some tolerance G2 of the true deterministic value due to the noisy outputs z2, and because the relaxed NLP solution at each node is obtained using an approximation of the black-box function *(x,z2). After conducting replicated sampling at a set of nominal input conditions, the value of G2 can be obtained from the variance in the outputs. Since the objective is a function of the outputs, the amount of propagated noise in the objective may exceed G2, so we require G1 > G2; otherwise, it is possible that the stopping condition |UB – LB| < G2 may never be satisfied. The algorithm is shown in Figure 1. Choose a candidate subproblem, solve relaxed NLP at node k, get (x,y,f)k
Obtain G2, pick G1 (G1 > G2), Set node index k = 1, (fLB ,fUB)k = (-f,+f)
Yes
fUB,k+1 = min(fUB,k , fk|yk feasible) fLB,k+1 = max(fLB,k , fk |yk infeasible)
Any remaining candidate subproblems? No No
|fUB,k+1 - fLB,k+1| < G1? Yes
Branch if fLB,k+1 = fk, formulate new MINLP subproblems, k = k + 1
Terminate. fMINLP,opt = fUB,k+1, Minimiser is (x,y)k
Figure 1. Noisy MINLP Branch-and-Bound Algorithm In order to solve the relaxed NLP at any node, we apply the RSM algorithms described in Section 2.1 Since the x-z2 space is considered to be described by a black box model, convexity cannot be guaranteed, and the global optimum may be missed. To avoid finding a suboptimal solution, we apply the RSM algorithm to a set of random
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initial feasible points to diversify search, although this cannot ensure that the global optimum will be attained. Multiple simulations are then conducted at the optimiser to obtain an expected optimal objective function value, which serves as a surrogate LB or UB of the objective function.
3. Results and Discussion We apply our method to an example problem from Floudas (1995) modified to include a black-box function, as follows: min y1 + y2 + y3 +5z2 ½ ° ° 4 2 2 2 ° s.t. ī( x1 ,x2 , z2 )= z2 ( 8x1 -8x1 +1)( 2x2 -1)- N (0,ı )=0 ° ° ° 3x1 y1 y2 d 0 °° °° 5 x2 2 y1 y2 d 0 ® ¾ ° ° d x +0.1y +0.25y 0 1 2 3 ° ° ° ° 2 x1 y1 y2 y3 d 0 ° ° 0.2 d x1 , x2 d 1 , yi =^0,1` , i=1...3 ¯° ¿°
(P2)
The unknown function in the noisy MINLP is simulated according to *(x1,x2,z2) where V = 0.01. Since MINLP solvers such as DICOPT require explicit deterministic equations for the objective and constraints, it is not possible to obtain a standard in terms of CPU time in order to assess the competitiveness of our method. Whenever 0-1 specifications are made to the binary variables, these values are substituted into the explicit constraints as part of problem preprocessing in order to obtain new bounds for the continuous variable space, thereby ensuring that the RSM algorithm does not conduct search in an infeasible region. At each node, our RSM algorithms were applied to ten random initial points in order to avoid convergence to a suboptimal solution for the relaxed NLP; ten additional replicate experiments were performed at the predicted optimiser to obtain the expected solution. The tolerance G1 was set at 0.05 and further search was terminated if either the difference between the best UB and LB fell below this value, or we exhausted the list of candidate subproblems. Due to infeasibility in the relaxed NLP subproblems at two of the nodes the algorithm terminated since we ran out of new possible subproblems. At the final node, the noisy NLP optimiser converged to the vicinity of the deterministic solution (x1,x2) = (0.35,0.2). The B&B tree for the example problem is presented in Figure 2 and additional solution information is shown in Tables 1 and 2. (x1,x2,y1,y2,y3,fLB,fUB) = 1 (0.2,0.2,0.912,0.813,0.475,-0.993,+ f) y1 = 0
y1 = 1
2
3
Inf.
y3 = 0 Inf.
4
(0.2,0.2,1,1,0.2,-0.971,+ f) y3 = 1 5
(0.35,0.2,1,1,1,-0.971,2.337)
Figure 2. Branch-and-Bound Tree for the MINLP Example
E. Davis and M. Ierapetritou
638 Table 1. Results for the illustrating example Node
(x1,x2,y1,y2,y3)
1 2 3 4 5
(0.2,0.2,0.912,0.813,0.475) Infeasible (0.2,0.2,1,1,0.2) Infeasible (0.35,0.2,1,1,1)
Noisy MINLP fLB fUB -0.993 +f -0.993 +f -0.971 +f -0.971 +f -0.971 2.337
CPU Time (s) MIP NLP Total 0.047 0.105 0.152 -----0.042 0.127 0.169 ----0.037 0.114 0.151
Table 2. Results for the deterministic MINLP Node 1 2 3 4 5
Deterministic MINLP fUB fLB (0.2,0.2,0.912,0.813,0.475) -0.987 +f Infeasible -0.987 +f (0.2,0.2,1,1,0.2) -0.987 +f Infeasible -0.987 +f (0.35,0.2,1,1,1) -0.987 2.356 (x1,x2,y1,y2,y3)
CPU Time (s) MIP NLP Total 0.047 0.098 0.145 -----0.042 0.116 0.158 ----0.037 0.105 0.142
4. Conclusions and Future Work In this paper we presented a Branch-and-Bound algorithm to address problems involving integer and noisy variables, as well as parts of the model that are not explicitly known. At each node, a noisy NLP is solved using response surface methods developed in our earlier work. The proposed algorithm is demonstrated using a simple example. We will continue to look for ways to improve the efficiency of this method and also develop algorithms that can address noisy MINLP of special problem structure in order to enhance computational performance.
References L. Biegler et. al., 1997, Systematic Methods for Chemical Process Design, Prentice-Hall, NJ. A. Brook et. al., 2004, GAMS: A User’s Guide, GAMS Development Corporation, Wash. D.C. E. Davis et. al., 2005, Adaptive Optimization of Noisy Black-Box Functions Inherent in Microscopic Models, ESCAPE-15, 193. M. Duran & I. Grossmann, 1986, An Outer-Approximation Algorithm For A Class of MixedInteger Nonlinear Programming, Math. Prog., 36, 307. R. Fletcher & S. Leyffer, 1996, Solving Mixed Integer Nonlinear Programs by Outer Approximation, Math. Prog., 66, 327. C. Floudas, 1995, Nonlinear and Mixed-Integer Optimization, Oxford University Press, N.Y. B. Gendron & T. Crainic, 1994, Parallel Branch-and-Bound Algorithms: Survey and Synthesis, Op. Res., 42, 6, 1042. I. Grossmann, 2002, Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques, Opt. & Eng., 3, 227. A. Geoffrion, 1972, Generalized Benders Decomposition, J. Opt. Th. & Appl., 10, 4, 237. V. Goyal & M. Ierapetritou, 2004, Computationl Studies using a Novel Simplicial Approximation Based Algorithm for MINLP Optimization, Comp. & Chem. Eng., 28, 9, 1771. R. Myers & D. Montgomery, 2002, Response Surface Methodology, John Wiley & Sons, N.Y. R. Raman & I. Grossmann, 1994, Modeling and Computational Techniques for Logic Based Integer Programming, Comp. & Chem. Eng., 18, 563. T. Westerlund & F. Pettersson, 1995, A Cutting Plane Method for Solving Convex MINLP Problems, Comp. & Chem. Eng., 19, S131.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Modelling and Simulation of High Pressure Industrial Autoclave Polyethylene Reactor Érico Caliania, Marcello Cavalcantib, Fabiano A.N. Fernandesc, Liliane M.F. Lonaa a
Universidade Estadual de Campinas, Faculdade de Engenharia Química, Caixa Postal 6066, Campinas – SP, Brazil b Politeno Indústria e Comércio S/A, Camaçari – BA, Brazil c Universidade Federal do Ceará, Departamento de Engenharia Química, Campus do Pici, Bloco 709, 60455-760 Fortaleza – CE, Brazil
Abstract High-pressure technology for polyethylene production has been widely used by industries around the world. The high-pressure autoclave reactor model developed in this work is based on a non-isothermal dynamic model, where PID control equations are used to maintain the operation at the unstable steady state. Kinetic mechanisms to describe the polymerization rate and molecular weight averages are presented. The model is capable of computing temperature, concentration gradients and polymer characteristics. The model was validated using industrial data, presenting good representation of the behavior of the autoclave reactor. Keywords: polyethylene, autoclave reactor, modeling.
1. Main Text Low density polyethylenes are used in a large variety of applications. Generally they are produced in either autoclave type or tubular reactors. This work develops a mathematical model for the production of low density polyethylene in an industrial high-pressure autoclave reactor. The system modeled in this work consist of a series of two autoclave reactors with a length to diameter ratio of 15:2. Mixing in both reactors is provided by a shaft running down the center of the reactor with several impeller blades. The mixing pattern in the high-pressure reactor makes it behave more like a continuous stirred tank reactor (CSTR) rather than a tubular reactor. In the first reactor a baffle is placed near the end of the reactor as to reduced the backmixing of the product at the last part of reactor 1. Heat transfer through the walls is limited, so that the reactor is essentially adiabatic and cooling is provided by the inflow of cold monomer. The inflow of initiator at several points down the reactor provides control of the temperature which may vary down the length of the reactor. A scheme of the modeled high-pressure reactor is shown in Figure 1. Ethylene free radical polymerization mechanism and kinetics has been outlined by Zabisky et al. (1992) and the outlines of the mixing model for the high-pressure autoclave reactor has been proposed by Chan et al. (1993). The autoclave reactor presents several complications such as nonideal mixing, presence of unstable steady states, possibility of gel formation and reaction in two phases (monomer rich phase and polymer rich phase).
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Figure 1. Industrial high-pressure reactor.
2. Mixing Model The mixing pattern in an autoclave type reactor tends to be of a recirculating nature. The effect of mixing on reactor performance is very important, especially since an imperfectly mixed vessel requires more initiator per unit of polymer produced than does a more perfectly mixed reactor under the same conditions (Georgakis & Marini, 1982). The initiator tends to decompose near the feed points, and not in the bulk of the reactor, thus not promoting as much polymerization as if the initiator was uniformly distributed throughout the reaction mixture. The temperature gradients down the reactor also suggest imperfect mixing (Chan et al, 1993). In order to account for the imperfect mixing in the reactor, the autoclave reactor can be subdivided into several sections which can be represented by a series of small reactors consisting of a CSTR segment followed by a plug-flow segment to account for the temperature gradients down the reactor. This plug-flow segment can be transformed into a series of small volume CSTRs in order to avoid solving partial differential equations. To account for the back mixing promoted by the impeller blades, each main CSTR segment of the reactor is allowed to recycle part of its volume back to the previous CSTR main segment (Figure 2). The model developed for the reactor is a dynamic model and includes temperature controller equations to maintain the operation point at the desired steady state. This is needed because the industrial reactor normally operates at an unstable steady state in which the operation can either cause the temperature to rise or cool down the reactor until no polymerization occurs. The mass balance for a species in a volume segment of the reactor is given by:
dN i,S dt
rec rec = Fifeed ,S + Fi , S −1 + Fi , R − Fi , S − Fi ,S + ri , S ⋅ Vi , S
(i, S ≥ 1)
(1)
The plug-flow segments have no feed streams nor have any stream leaving as recycle to other segments. To investigate the effect of the macromixing parameters on the reactor fluidodynamics, two main parameters are defined: volume fraction of the CSTR segment to the total volume of the section (θ) and the recycle ratio (β). These parameters have to be estimated for each reactor and for each section in the reactor.
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Larger θ denotes the more the section resembles an ideal CSTR, while larger β denotes higher axial mixing of contiguous sections. θ=
β=
VS,CSTR
(2)
VS,TOTAL
Q Srec
(S ≥ 1)
Q S + Q Srec
(3)
Figure 2. Mixing model.
The energy balance for the reactor requires to account for the inflows, outflows, recycles and the reaction in each segment. The reactor was assumed to be adiabatic and cooling is supplied by cold monomer feed. Heat generation is considered to come from the propagation reaction only.
(ρ
S
⋅ CpS ⋅ VS
) dTdt
S
( ) ) − ρ ⋅ Q ⋅ Cp (T ) − ΔH ⋅ r ⋅ V
(
= ρSfeed ⋅ QSfeed ⋅ CpSfeed ⋅ TSfeed − Tref + ρS−1 ⋅ QS−1 ⋅ CpS−1 ⋅ TS−1 − Tref
( ⋅ (T
rec rec rec + ρrec R ⋅ Q R ⋅ Cp R ⋅ TR − Tref
− ρSrec ⋅ QSrec ⋅ CpSrec
rec S
− Tref
S
S
S
S
S
− Tref
)
) (4)
S
Temperature control is done by manipulating the initiator feed based on the actual temperature in some measured segments of the reactor and on the temperature set point. The controller applied to the reactor was a continuous proportional-integralderivative type and 5 controllers were used to control the initiator feed into the segments 2, 3, 4, 6 and 8.
⎡ ⎤ τ Δt ΔFI = K c ⋅ ⎢(E n − E n −1 ) + ⋅ E n + d ⋅ (E n − 2 ⋅ E n −1 + E n −2 )⎥ τi Δt ⎣ ⎦
(5)
Ethylene free radical polymerization mechanism and kinetics has been outlined by Zabisky et al. (1992) and Chan et al. (1993) for a two phase kinetic mechanism where a monomer and a polymer rich phase exist in the reaction mixture. Herein, a homopolymer that presents only one phase in the reactor (monomer rich phase) was studied and therefore the momentum equations to account for the molecular weight of the polymer were adapted from Zabisky et al. (1992) for a one phase kinetic
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mechanism. The kinetic mechanism considers the initiation of radical by thermal decomposition of the initiator, chain propagation, termination by combination and disproportionation, transfer to monomer and to polymer, β-scission of terminal radicals and backbiting. The moments for live and dead polymers are given by:
dY0 = 2 ⋅ f ⋅ k d ⋅ [I] − k tc ⋅ Y02 − k td ⋅ Y02 dt
(6)
dY1 = 2 ⋅ f ⋅ k d ⋅ [I] + k p ⋅ [M] ⋅ Y0 − k tc ⋅ Y0 ⋅ Y1 − k td ⋅ Y0 ⋅ Y1 dt + k fp ⋅ (Y0 ⋅ Q 2 − Y1 ⋅ Q1 ) + k fm ⋅ [M] ⋅ (Y0 − Y1 ) + k db ⋅ (Y0 ⋅ Q 2 − Y1 ⋅ Q1 )
(7)
dY2 = 2 ⋅ f ⋅ k d ⋅ [I] + k p ⋅ [M ] ⋅ (2 ⋅ Y1 ⋅ Y0 ) − k tc ⋅ Y0 ⋅ Y2 − k td ⋅ Y0 ⋅ Y2 dt + k fp ⋅ (Y0 ⋅ Q3 − Y2 ⋅ Q1 ) + k fm ⋅ [M] ⋅ (Y0 − Y2 ) + k db ⋅ (Y0 ⋅ Q3 − Y2 ⋅ Q1 )
(8)
dQ 0 Q = (fluxo total )i ,S0 + 0,5 ⋅ k tc ⋅ Y02 + k td ⋅ Y02 + k fp ⋅ (− Y0 ⋅ Q 2 + Y1 ⋅ Q1 ) dt + k fm ⋅ [M ] ⋅ Y0 − k db ⋅ Q1 ⋅ Y0
dQ1 Q = (fluxo total)i,S1 + k tc ⋅ Y0 ⋅ Y1 + k td ⋅ Y0 ⋅ Y1 + k fp ⋅ (− Y0 ⋅ Q 2 + Y1 ⋅ Q1 ) dt + k fm ⋅ [M ]⋅ Y1 − k db ⋅ Q 2 ⋅ Y0
(
)
dQ 2 = (fluxo total )iQ,S2 + k tc ⋅ Y0 ⋅ Y2 + Y12 + k td ⋅ Y0 ⋅ Y2 + k fp ⋅ (− Y0 ⋅ Q 3 + Y2 ⋅ Q1 ) dt + k fm ⋅ [M ]⋅ Y2 − k db⋅ ⋅ Q 3 ⋅ Y0
⎛ Q2 Q 3 = ⎜⎜ ⎝ Q1 ⋅ Q 0
(
⎞ ⎟⎟ ⋅ 2 ⋅ Q 0 ⋅ Q 2 − Q1 2 ⎠
)
(9)
(10)
(11)
(12)
The kinetic parameters used in the simulations are based on the data published by Zabisky et al. (1992) and Chan et al. (1993). The full dynamic mathematical model was comprised of 308 ordinary differential equations to calculate the material balance of all components, the energy balance and the population balance (via method of moments). The model was solved using a 5th order Runge-Kutta integration method with variable integration step.
3. Results Industrial production recipes from Politeno (Brazil) were used for simulation. Validation of the model were done by comparing the values predicted by the model with observed industrial steady-state values for monomer profile, initiator flow rates, temperature profile and final product characteristics. The industrial reactor was divided into eight sections to be simulated. The volumes of the reaction sections were 16.8,
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13.7, 12.8, 13.3, 16.8, 8.3, 15.3 and 3.0% of the total volume of the reactor. The monomer feed distribution assumed that 12.5, 25.0, 50.0 and 12.5% of the total monomer feed entered the first segment of sections 1, 2, 3 and 5 respectively. The initiator was fed into the first segment of sections 2, 3, 4, 6 and 8. The reactor operated at pressure of 1600 atm and the temperature of the ethylene feed was of 353 K. The temperature at the third segment of sections 2, 3, 4 and 6 and 8 were controlled at 513, 513, 537, 513 and 531 K, respectively. Figure 3 shows a representative result of the variation of temperature (controlled variable) at reactor startup. Temperature control is a key factor for the operation of the autoclave reactor and it is very important that the controller manages to control the temperature within a short time spam. The PID controller implemented in the model and the control parameters found were able to control the temperature within a few residence time showing the efficiency of the model and it control system.
Figure 3. Dynamic response of an internal segment of the reactor (Segment 12) The mixing parameters, θ and β, are a second key factor for the model and must be throughly studied. Several simulations were done as to find the best set of mixing parameters for the industrial autoclave reactor that is being simulated. Figure 4 shows the effect of the recycle ratio β on the reactor behaviour. Comparing the profiles obtained in Figure 4 to actual reactor profiles have showed that none of the profiles obtained with simulations that were carried out with constant recycle ratios (all sections using the same recycle ratio) have displayed a satisfactory fit to the actual reactor data. Analyzing the configuration of the reactor and the fraction of ethylene feed in each section, we could establish that not all sections would display the same recycle ratio and that this parameter has to carefully set for each reactor design. For the design shown in Figure 1, the recycle ratio of section 4 (towards section 3) was set to zero since the baffle between sections 3 and 4 minimizes the recycle and backmixing of the reaction mixture. The recycle ratio of section 3 (towards section 2) was increased since the flow rate of ethylene fed into section 3 is larger and as such a better mixing can occur near this feeding point. The best configuration found for the mixing parameters of the model was: volume fraction of the CSTR segment to the total volume of the section (θ) of 0.70 for all sections; and recycle ratio (β) of 0.15, 0.30, 0.15, 0.15 and 0.05 for the sections 2, 3, 6, 7 and 8, respectively. A comparison with industrial data is shown in Figure 5. The results found in Figure 5 are quite good and the largest relative error of the predicted
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values was less then 5% and as such we can state that the model truly represents the industrial reactor behavior.
Figure 4. Effect of the recycle ratio on the monomer concentration (a) and the temperature profiles (b) for the first segment (CSTR segment) of each section of the reactor (θ = 0.70; each section comprised of 1 CSTR segment and 2 PFR segments).
Figure 5. Comparison of temperature profiles between the proposed model and actual industrial data for polyethylene homopolymerization.
4. Conclusions A comprehensive model describing the high-pressure autoclave reactor for polyethylene production was developed, accounting for the mixing pattern in the reactor, the mechanistic polymerization reaction and temperature control. The model was proven satisfactory and has fitted homopolymer recipes for initiator flow rates, temperature profiles and final polymer characteristics.
Acknowledgments The authors wish to acknowledge Politeno and Finep for the financial support for this project.
References Chan, W.M., Gloor, P.E. & Hamielec, A.E. AIChE J., 39, 111-126 (1993). Georgakis, C. & Marini, L. ACS Symp.Ser. 196, 591-602 (1982). Hulburt, H. M., Katz, S. Chem. Eng. Sci., 19, 555 (1964). Zabisky, R.C.M., Chan, W.M., Gloor, P.E. & Hamielec, A.E. Polymer, 33, 2243 (1992).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Energy saving in distillation columns: the Linde column revisited Giorgio Soavea, Laura Pellegrinib, Davide Barbattib, Nicolò Susanib, Susi Bonomib a b
via Europa 7, S. Donato Milanese, Italy Politecnico di Milano, piazza Leonardo da Vinci 32, 20133, Milano, Italy
Abstract Linde double column is a smart device to separate gas mixtures without energy consumption at condenser and reboiler. The sections operate at different pressures to achieve a thermal integration in the middle of the whole tower, allowing energy savings that in the case of air separation are total. Calculations performed using Aspen Hysys® and Icarus® show that the same distillation scheme can be applied to other systems but sometimes it is necessary to add an auxiliary cooler that limits the advantage of the proposed scheme. Keywords: distillation, energy saving, Linde column.
1. Introduction Linde double column is a resourceful method to separate mixtures of gases reducing heat and cold consumptions: the main difference between a normal column and this equipment is that Linde column consists of two sections (columns), one on the other, that work at different conditions. The bottom column, that works at high pressure, is used to produce reflux for itself and for the second one; the pressure difference between the two columns is such that the heat produced in the reflux condensation of the bottom column can be used as heat source in the top column reboiler. The main variable to be calculated is the amount of reflux to the bottom column that can define the purity of the final products and the thermal charges needed in the exchangers. An important case where Linde column is widely used is air separation, because critical proprieties of oxygen and nitrogen do not allow normal distillation; this process is an ideal case, because there isn’t need to heat or to cool with auxiliary fluids. This complete energy saving can be attained only because air components have a particular combination of relative volatilities that fits very well for Linde system.
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This paper will show how such a layout can be successfully applied to the separation of various hydrocarbon mixtures achieving important energy savings; sometimes however an external cooler is needed, which reduces in some way the economic advantage.
2. A simple example In this paragraph it will be explained how to decrease, and, if possible, how to eliminate, on the basis of the Linde scheme, the condenser and reboiler thermal duties, that are the main source of operating costs for every kind of distillation. The explanation will refer to a particular system, chosen for its simplicity, containing only two light hydrocarbons, ethane and propane: it is a binary mixture whose components have the appropriate relative volatilities to make the separation so easy that the two columns have few ideal stages (~30). This statement is correct because the boiling points, calculated at atmospheric pressure, differ from one another of 47 K: for ethane the normal boiling point is 184.5 K, whereas it is 231.1 K for propane. Three different cases were studied with Hysys®, corresponding to an ethane molar fraction in the feed of 50%, 60% and 70%. It can be remarked that the energy requests in the three cases have a different behaviour, because the richer in ethane is the mixture the easier the separation and the lower the energy required to keep the plant in operation. It is worth while underlining that there are only two heat exchangers in Linde configuration requiring another fluid, i.e. the reboiler of the high-pressure column and the condenser of the low pressure one; their duties can be substantially decreased and ideally brought to zero using the process simulator in the way reported here below.
Log(reboiler duty) [kJ/h]
Decrease of the reboiler duty 1.00E+07 FEED COMPOSITION
1.00E+06 1.00E+05
ethane 50%
1.00E+04
ethane 60%
1.00E+03
ethane 70%
1.00E+02 1.00E+01 1.00E+00 0.1
0.2
0.3
0.4
0.5
0.6
bottom 1 - ethane molar fraction
Fig. 1: Decrease of the reboiler duty versus bottom ethane content for various feeds.
At first we will discuss how to decrease the duty of the high-pressure reboiler: it can be cut down by increasing the molar fraction of ethane into the bottom product of the high pressure (column 1). Approaching a well-fixed concentration of ethane, which
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depends on the feed composition, the duty of the reboiler falls down to zero very quickly, as shown in Fig.1. That is to say that there is no need to have a bottom product very rich in propane (xB1), because the separation is accomplished in the low-pressure column (column 2); for example, examining the feed with 70% of ethane, 58% of ethane as molar percentage into the bottom product is required in order to eliminate the reboiler, whereas almost all the propane present in the system is forced to stay in the bottom too, since the head product contains ethane at 99.9%. While in case of high-pressure reboiler the problem has been solved with the simulator trying different values of the assigned variable (i.e. ethane molar fraction in the bottom) to reduce its duty, condenser thermal duty has been directly set to zero obtaining different outlet compositions in the three cases under study. The result of this “no-duty” separation is not always satisfying, as can be seen in the following report (Table 1): CASE
xD1
xD2
xB2
0.58
0
0
0.9987
0.0056
Ethane = 60% 0.999 0.478
0
0
0.9566
0.0644
Ethane = 50% 0.999
0
0
0.8639
0.1426
Ethane = 70% 0.999
xB1
0.44
reboiler duty [kJ/h] condender duty [kJ/h]
Table 1: Ethane molar fraction in distillates (xD1, xD2) and bottoms (xB1, xB2). Only in the case of 70% ethane feed a very good separation is achieved without energy supply, like air separation. A purity even better could be reached for a mixture ethane-propane which the most volatile component has a content higher than 70%. In the case of ethane content less than 70% the engineer has to look for the best set of thermal duties that the plant needs to perform the required separation. In any case the adoption of Linde double column assures an important energy saving.
3. The industrial case In the separation of ethane from propane the thermal integration is simply performed in the process flow diagram built up in Hysys® environment by means of two energy streams connecting the high-pressure condenser with the low-pressure reboiler, so that the reflux split is directly calculated by the simulator. Handling other mixtures the problem may arise of how to regulate the reflux streams, i.e. the system presents an additional degree of freedom. This happens in another binary system, whose components are ethane and ethylene; it is known that the separation of the C2 fraction is much more difficult than ethane-propane one because the relative volatility is very low, around 1.2; this is proved by the great energy consumption (Peters and Timmerhaus, 1980) and by the height (not less than 100 theoretic stages) of the towers used for this separation. The process flow diagram for C2 separation by means of Linde process is a more complex one, as shown below in Fig. 2:
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Fig. 2: Hysys® process flow diagram for ethane-ethylene separation.
It has been built up in Hysys® simulation environment using two columns which have only the reboiler: at first the thermal duty of the high pressure exchanger is minimized, whereas in the second column (low pressure) the thermal integration is simulated by linking the duty of an external cooler (thermal integrator) to the low pressure reboiler. Following the route of the streams it can be seen that distillate D1 is condensed in two stages and then is split in order to create the liquid refluxes for the columns; before entering the low pressure column the bottom B1 flows into an exchanger (D2–B1 exchanger) which allows to recover some energy from distillate D2 gas. The final products are two gaseous streams coming out from the low pressure column: bottom B2 gas and distillate D2 out. Some details about the columns are shown in Table 2 (ethane and ethylene are respectively 15.24% and 84.76% on molar basis in the feed): TRAY FEED TEMPERATURE TOP NUMBER ENTRANCE [K] PRESSURE [bar]
BOTTOM PRESSURE [bar]
COLUMN 1
40
tray 35
196 – 202
4
4.2
COLUMN 2
60
tray 35
172 – 191
1.2
1.44
Table 2: Double column features.
The determination of the best reflux split is possible with Hysys®: fixing the step size and lower and upper bounds for the independent variable, so that outputs are calculated and registered every time that inputs are changed, the optimum problem can be solved. If the variable is the split ratio, defined as the ratio between the molar flow of the reflux to high pressure column and the molar flow of D1 liquid (equal to the sum of refluxes), the simulator is able to determine his optimum value (about 0.65), i.e. the value in correspondence to which the system requires less energy to condense the stream D1 and in the meantime the products present a high purity, reaching 99.95% of ethylene in distillate D2 and 100% ethane in bottom B2. Anyway there are some residual thermal duties that cannot be removed without making the separation worse.
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In order to quantify the obtained results the value of the energy consumed according to the Linde approach has been compared with the consumptions for the single tower (100 ideal stages, i.e. 60+40) scheme, with the same operating conditions and the same purity in the outlet streams. Two levels of working pressure have been considered: 4 bar and 20 bar. From the comparative analysis it results that the adoption of the Linde tower allows large energy savings in both the cases (Table 3): 4 BAR TOWER
20 BAR TOWER
Heat Saving
53,1%
70,5%
Cold Saving
58,4%
71,8%
Table 3: Savings (percentages) adopting Linde double column.
The calculation refers to the power values expressed in kJ/h, obtained by Hysys® during the research for the optimum, but this comparison does not face the thermal level of the exchange: the operating costs associated to heat removal at about 200 K are very different from those for the same operation at 270 K; this is the main reason for using another simulator, Icarus®, that can perform a complete profitability analysis taking into account both fixed and operating costs. In the case under study the distinguishing feature is the cost of the cooling and heating fluids: for Linde column and the 4-bar single column the cooler works at about 163 K, so liquid ethane at atmospheric pressure is necessary as coolant, with a cost of about 20$/GJ (Turton, 2003); on the contrary heating has a very lower cost, because supplying heat at 213 K is not a problem (it is assumed to be free of charge). For the single column working at 20 bar it is necessary a coolant at about 233 K, e.g. liquid propane, whereas the reboiler process fluid has a temperature little below 273 K, and so water can be used as heat source. Under these hypotheses Icarus® can collect data from Hysys® and estimate costs (Table 4): Capital costs (M$)
Operating costs (M$/year)
Utilities costs (M$/year)
LINDE
6.851
4.543
3.000
4 BAR TOWER
7.382
8.944
7.222
20 BAR TOWER
8.589
8.670
6.909
®
Table 4: Estimated costs from Icarus .
Savings are evident, because capital costs for Linde tower are lower and in the meantime operating cost are almost halved; it is useful to analyse the utilities cost term, because it constitutes the main part of the operating costs and more or less represents the same percentage of operating cost for all the cases studied.
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4. Conclusions After studying different systems, we can assert that Linde tower allows important energy savings in separation of mixtures that present the following features: 1. binary system are preferred, because it is easier to have the appropriate temperature difference needed for the thermal integration between the sections; in case of multicomponent mixture it is better having two prevalent components, so that the system can be similar to a binary one; 2. even if usually separation is promoted by high relative volatility, for Linde device this shouldn’t be too different, otherwise the integration becomes more difficult; 3. the most volatile components should have higher content than the others. The combination of all the three options is summed up in the two previously quoted mixtures, i.e. air and ethane-propane with ethane content higher than 70%; this is the reason why it is possible to perform a separation without external exchangers, which means cutting down the operating costs. Anyway, whereas the air separation by Linde method is really applied, it is not the case of ethane-propane, since it does not represent a separation of industrial interest. At the end we can say that the we have proved the profitability of Linde concept applied to C2 separation, but we should extend the analysis considering that usually ethane-ethylene separation is achieved in a tower operating over 20 bar, along a distillation train into oil refinery plants and that the industrial coolant is the head product, after expansion, of the subsequent column.
References Aspen HYSYS® 3.2 Release (2004), User’s Guide, Aspen Technology. Aspen HYSYS® 3.2 Release (2004), Steady State Modelling, Aspen Technology. Aspen ICARUS® 12.0 Release (2002), User’s Guide, Aspen Technology. Aspen Icarus Process Evaluator™ 12.0 Release (2002), User’s Guide, Aspen Technology. Peters M. S. and K. D. Timmerhaus, (1980), Plant design and Economics for Chemical Engineers. 3rd edition, Mc Graw-Hill, USA. Soave, G. and J.A. Feliu, Saving energy in distillation towers by feed splitting, Applied Thermal Engineering, 22, 889-896, 2002. Turton R., R. C. Bailie, W. B. Whiting and J. A. Shaeiswitz,(2003), Analysis, Synthesis, and Design of Chemical Processes, Second Edition, Prentice Hall PTR, New Jersey.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Computer-aided modeling for hydrodesulfurization, hydrodenitrogenation and hydrodearomatization simultaneous reactions in a hydrotreating industrial process F. Jiménez1, V. Kafarov1, M. Nuñez2 1 2
Industrial University of Santander, Tel. +57-76344746, Bucaramanga, Colombia Colombian Institute of Petroleum, Piedecuesta, Colombia
Abstract The purpose of this research is to utilize computer aided modeling as a tool for the investigation of simultaneous hydrodesulfurization (HDS), hydrodenitrogenation (HDN) and hydrodearomatization (HDA) reactions in industrial reactors. This study is based on the modeling of an industrial process as a mean to predict the quality of products during the hydrotreatment of a vacuum gas oil (VGO) using a commercial catalyst (Ni-Mo/γAl2O3), including the three selected reactions (HDS, HDN, HDA), and utilizing data obtained at a pilot-plant, where catalytic experiments were carried out under typical industrial conditions. Analytical techniques such as nuclear magnetic resonance (NMR), gas chromatography coupled with high performance mass spectrometry (GC-MS), ultraviolet-visible spectrometry (UV-VIS), simulated distillation (SimDis), analysis of saturates-aromatic-resin (SAR), and standard tests (ASTM) were used for the construction of a database. A suitable numerical program with a user-friendly interface was written for this simulation. The simulated results show a good agreement with experimental data obtained from the pilot plant. Keywords: Industrial reactor, Computer aided modeling, Simultaneous reactions, HDS, HDN, HDA.
Introduction and Theoretical Basis Catalytic hydrotreatment (HDT) is one of the most important processes in the petroleum refining industry, where on the base of catalyst sold per year, it ranks third just after exhaust and catalytic cracking catalysts (Prins, 2000). The main purpose for HDT is to remove contaminants such as sulfur, nitrogen, metals, and oxygen, among others because of technical and environmental reasons. Typical industrial catalysts for HDT contain molybdenum and cobalt or nickel supported on alumina and their use in trickle bed reactors (TBR) for the HDT process is well-known and widely used because of its comparative advantages with respect to other fixed bed reactors. For the purpose of modeling, the hidrodesulfurization (HDS) process has been usually selected because of the broad and mature research, and the special consent in topics such as reactivity, kinetic, and effect of main operation parameters. However, models for simultaneous HDS, HDN and HDA are rather the exception in the literature. Moreover, the research of the mutual inhibitions among the sulfur, nitrogen and aromatic molecules is less precise and in some cases results are somewhat confusing. Likewise, the processing of heavy oils in the refinery industry has showed the necessity for more precise analytical techniques for use with heavy oil fractions such as vacuum
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gas oil or heavier. One of the purposes of this work is to confront some of the remaining challenges concerning the modeling and simulation of the hydrotreatment of vacuum gas oils. The strategy used in this work for the computer-aided modeling is presented schematically in Figure 1. 1.1 HDT Pilot Plant, Selection of Operation condition, Feed type and Properties.
1.2 Experimentation to detect inhibition effects
1. EXPERIMENTAL DATABASE
1.3 Experimentation for kinetic measurement
2.1 Preliminary Selection of Analytical Techniques
2.3 NMR, GC-MS, UVVIS, SIMDIS, SAR, ASTM for basic, nonbasic nitrogen, and total sulphur for VGO’s HDT
2.2 Exploratory Test for final selection
3.2.1 HDS Reactions
3.1 Sequential Design of Experiments
3.2 Models Discrimination
3.2.2 HDN Reactions 3.2.3 HDA reactions
2. ANALITYCAL DATABASE
3.3 Selected Models
3. KINETIC MODEL OF HDT SIMULTANEOUS REACTIONS
3.4 Parameters Estimation
4.1.1 Construction of Balance Equations 4.1 Reactor Model Construction
4.1.2 Thermodynamic & Transport properties
4. MODEL OF INDUSTRIAL REACTOR
4.2.1 Main Assumptions 4.2.2 Numerical Solution 5.1 Simulation of Process Performance
5.2 Inhibition Effects
5. SOFTWARE WITH A USER-FRIENDLY INTERFACE FOR INDUSTRIAL PROCESS SIMULATION
Figure 1. General Scheme of Investigation
1. Experimental database 1.1 HDT pilot plant, selection of operation conditions, feed type and properties All hydrotreatment tests were made in a pilot plant (Colombian Institute of Petroleum) using the catalyst in their original size (commercial Ni-Mo/Al2O3, trilobe shape, equivalent diameter of 1.8 mm, length 4.1 mm). The reactor has an inside diameter of
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1.9 cm and a length of 73.5 cm. The ranges for operating conditions were T=330-390 °C, P=5-10 MPa, liquid hourly space velocity (LHSV)=1-3 h-1, gas/oil ratio=4.0-6.5. The liquid feed was a vacuum gas oil (VGO) from a crude oil of intermediate character (naphtenic/parafinic, 27API, 0.8% sulfur) 1.2 Experimentation to detect inhibition effects A vacuum gas oil severely hidrotreated (10 MPa, 370 °C, LHSV=1 h-1, H2/VGO=6.5) was selected as the matrix feed. Six pure reactive: dibenzotiophene, acridine, carbazole, anthracene, naphtalene and tetralin (all purchased from Fluka) were added to the matrix according to a factorial experimental design. Thirty-two additional tests were made for this purpose. 1.3 Experimentation for kinetic measurement For the estimation of kinetic parameters we used previous experimental results obtained in the pilot plant with three vacuum gas oils, involving 46 tests with different operational conditions. Twelve more additional experiments were carried out according to the sequential design of experiments.
2. Analytical database Exploratory test and selection of analytical techniques After a broad review of available analytical techniques to study the reaction and internal aromatic transformation, a selection was made: gas-chromatograph mass spectrometry (GC-MS) for aromatic families distribution (including sulfur families), ultravioletvisible spectroscopy (UV-VIS) for determination of aromatics families, nuclear magnetic resonance (NMR) for total aromatics content and others, and standard methods (ASTM) for the determination of total sulfur, basic nitrogen, and non-basic nitrogen in medium and heavy oil fractions. Additional analysis included simulated distillation (SimDis) and saturate-aromatic-resin (SAR) analysis for the hydrotreated samples.
3. Kinetic Model for HDT Simultaneous Reactions 3.1. Sequential design of experiments The sequential methods include the use of mathematical models and require the handling of n-dimensional matrixes of differential equations, taking advantage of the information and insight obtained from the previous experiments to select the settings of the independent variables for the next experiment in an optimal way (Froment & Bischoff, 1990). Two types of sequential methods for optimal design have been developed: for optimal model discrimination, and for optimal parameter estimation . 3.2. Models discrimination The model discrimination is made selecting an experimentation zone with equally spaced intervals. The design criterion is based upon the divergence among rival models. The setting of operating variables for the nth experiment is that value which maximizes the divergence among the rival models. A set of nine different models of kinetic equations of the Langmuir-Hinshelwood and Power type were taken from the literature (Broderick and Gates, 1981; Cheng et al, 2001; Medjell et al, 2001; Cotta et al, 2000; Girgis and Gates, 1991; Froment et al, 1994; Avraam and Vasalos, 2003; Tsamatsoulis and Papayannakos, 1998; Van Hasselt et al, 1999). A user-friendly computational program was developed, which supplemented with a statistical analysis, discriminated among the nine models and
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minimized the volume of the confidence interval. The selected kinetic models for HDS hydrogenation and hydrogenolysis were the ones proposed by Broderick and Gates (1981). For HDN and HDA the expressions from Avraam and Vasalos, 2003 were selected. Results obtained with this program are in concordance with data reported in the literature.
4. Model of Industrial Reactor 4.1. Reactor model construction A set of several reactor models were evaluated: Cheng et al, 2004; Lange et al, 2004, Avraam and Vasalos, 2003; Chowdhury et al, 2002, Korsten and Hoffman, 1996; Medjell et al, 2001; Chen et al, 2001; Cotta et al, 2000; Froment et al, 1994; Tsamatsoulis and Papayannakos, 1998; Van Hasselt et al, 1999. 4.1.1. Main assumptions The following assumptions were made for the mathematical model derivation: onedimensional heterogeneous model (no radial gradients) with both gas and liquid in plug flow, operating in steady state and isothermally. Reactions occur in liquid phase when in contact with the solid phase, without liquid evaporation or vapor condensation. Main reactions are HDS, HDN and HDA (mono, di and tri-aromatic). Catalyst wetting is complete, the liquid volume in the reactor remains constant, and there is no catalyst deactivation. 4.1.2. Balance equations According to the above assumptions, two three-phase reactor models, reported by Korsten & Hoffman (1996) for liquid and gas phases, and by Froment et al. (1994) for solid phase were selected, combined and used (Jiménez et al, 2005). 4.1.3. Thermodynamic and transport properties A specific numerical package was developed for the evaluation of thermodynamic and physico-chemical properties of hydrocarbon compounds and mixtures, and for calculation of interfacial mass transfer rates, using the effective diffusivity method and gas-liquid equilibrium at the interface, using information reported in literature. (Rodriguez & Ancheyta, 2004; Avraam and Vasalos, 2003; Ronze et al, 2002; Pooling et al, 2001; Cai et al, 2001; Khadilkar et al, 1999; Korsten and Hoffman, 1996; Goto and Smith, 1975) 4.1.4. Numerical solution A specific numerical package was also developed for integration of the model’s equations based on the orthogonal collocation method for solid phase, and RungeKutta-Gill for gas and liquid phase, using information reported in literature. (Rice et al, 1995; Villadsen and Michelsen, 1978). Results obtained with this program are in concordance with data reported in the literature. 4.2. Trickle bed reactor simulation performance The impact of the main operational variables on the reactor performance can be summarized as follows: to improve the sulfur and nitrogen conversion three procedures can be chosen: increase temperature, decrease space-velocity, or increase pressure, as evidenced in Figures 2 and 3 where the model-predicted product basic nitrogen and sulphur content were compared with the pilot plant data for temperature and pressure. In a similar manner the concentrations profiles for selected compounds along the reactor are showed in Figure 4. In this figure as the reactor length increases the monoaromatics concentration increases, whereas sulfur, nitrogen and di+tri aromatics
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concentration decrease, the H2S concentration increases rapidly reaching a maximum value at a medium reactor length and then slowly decreasing.
Conversion, X
1
Sulphur
0,8 0,6
Basic Nitrogen
0,4 0,2 0 330
350
370
390
Reactor Temperature (C) Figure 2. Simulated and experimental variation of temperature vs. sulfur and basic nitrogen.
Figure 3. Simulated and experimental variation of reactor pressure vs. sulfur and basic nitrogen.
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Dimensionleess
concentration (C/C0)
0.8 0.7
H2S
0.6 0.5
Tri and Di Aromatics
0.4 0.3
Sulfur
0.2
Basic Nitrogen
0.1 0.0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Dimensionless reactor lenght (z/z 0)
Figure 4. Simulation of selected compounds along the reactor
4.3. Inhibition Effects In this work inhibitions effects on HDS reactions by low and high concentrations of aromatic molecules such as anthracene, naphthalene, and tetraline were detected. Similar inhibition effects by basic nitrogen (acridine) and non-basic nitrogen (carbazole) were observed to some extent. In a similar manner inhibition effects on HDN by sulfur and aromatics were also observed.
References D. Avraam, I. Vasalos, 2003. “HdPro: a mathematical model of trickle-bed reactors for the catalytic hydroprocessing of oil feedstocks”, Catalysis Today, 79–80: 275–283. D. Broderick, B. Gates, 1981. “Hydrogenolysis and hydrogenation of dibenzothiophene catalized by sulfided CoO-MoO3/Al2O3”. AIChE Journal, 27 (4): 663 -672. Z. Cheng, X. Fanga, R. Zeng, B. Han, L. Huang, W. Yuan, 2004. “Deep removal of sulfur and aromatics from diesel through two-stage concurrently and countercurrently operated fixed-bed reactors”, Chem. Eng. Sci, 59: 5465–5472 J. Chen, Z. Ring, 2001. “Modeling and Simulation of a Fixed-Bed Pilot-Plant Hydrotreater”, Ind. Eng. Chem. Res, 40: 3294-3300. R. Chowdhury, E. Pedernera, and R. Reimert, 2002, “Trickle Bed Reactor Model for desulfurization and Dearomatization of diesel”, AICHE Journal, 48 (1) 126-135. R. Cotta, M. Wolf-Maciel, R. Maciel Filho, 2000. “A cape of HDT industrial reactor for middle distillates”, Computers and Chemical Engineering, 24: 1731 – 1735. G. Froment, G. Depauw, V. Vanrysselberghe, 1994. “Kinetic Modeling and Reactor Simulation in Hydrodesulfurization of Oil Fractions”, Ind. Eng. Chem. Res., 33 (12): 2975 – 2988. G. Froment, 2004. “Modeling in the development of HDT processes”, Cat. Tod, 98: 43–54 G. Froment, K. Bischoff. 1990, “Chemical Reactor Analysis and Design” 2nd. Edition. John Wiley & Sons, 664p. M. Girgis, B. Gates, 1991. “Reactivities, Reaction Networks, and Kinetics in High – Pressure Catalytic Hydroprocessing”, Ind. Eng. Chem. Res., 30 (9) 2021-2058 F. Jimenez, M. Nunez, V. Kafarov, 2005. Study and modeling of simultanous HDS, HDN and HDA on vacuum gas oil hydrotreatment. Comp. Aided Chem. Eng., 619-624 M. Khadilkar, P. Mills, M. Dudukovic, 1999. “Trickle-bed reactor models for systems with a volatile liquid phase”, Chemical Engineering Science, 54: 2421 – 2431
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H. Korsten, U. Hoffman, 1996. “Three Phase Reactor Model for Hydrotreating in Pilot Trickle Bed Reactors”, AICHE Journal, 42 (5) 1350-1360 R. Lange, M. Schubert, W. Dietrich, M. Grunewald. Unsteady-stae operation of trickle bed reactors. Chem. Eng. Sci., 59, 5355-5361. E. Matos, R. Guirardello, 2000. “Modeling and Simulation of HDM and HDS process with transient catalytic efficiency”, Brazilian Journal of Chem Eng, 17 (2) 1-14 T. Medjell, R. Myrstad, J. Rosvoll, P. Steiner, 2001. “A new kinetic model for HDS of oil products”, Studies in surface science and catalysis 133.Elsevier Science B.V. 189-194. B. Poling, J. Praustnitz, J. O’Conell, 2001. “The properties of Gases and Liquids”. Mc. Graw Hill, International Edition, 5a Ed, 865p. R. Rice, D. Do, 1995. Applied Mathematics and Modeling for Chemical Eng. John Wiley & Sons. M. Rodriguez, J. Ancheyta, 2004, Modeling of HDS, HDN, and the Hydrogenation of Aromatics in a Vacuum Gas Oil Hydrotreater, Energy & Fuels, Vol 18, No. 3, 789 – 794 F. Shiraishi, T. Hasegawa, 1995. Accuracy of the numerical solution of a two-point boundary value problem by the orthogonal collocation method. Journal of Chemical Engineering of Japan, vol 28 no 3, pp 316-323 D. Tsamatsoulis, N. Papayannakos, 1998. “Investigation of intrinsic HDS kinetics of a VGO in a trickle bed reactor with backmixing effects”. Chem. Eng. Sci, 53 (19): 3449–3458. B. Van Hasselt, P. Lebens, H. Calis, C. Van Den Bleek., 1999. “A numerical comparison of alternative three-phase reactors with a conventional TBR”, Chem. Eng. Sci, 54: 4791- 4799. J. Villadsen, M. Michelsen, 1978. “Solution of differential equation models by polynomial approximation” Prentice Hall Inc,. 344p.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Modelling and Dynamic Simulation of Thermal Stresses in Brazed Plate-Fin Heat Exchanger F. Picard
a b ∗
, D. Averousb , X. Jouliaa and D. Barreteaua
a
Laboratoire de G´enie Chimique, LGC,UMR-CNRS 5503, INP-ENSIACET, 118 route de Narbonne, F-31007 Toulouse Cedex 4,France b
Nordon Cryog´enie SA, 25 bis, rue du fort, B.P. 87, F-88194 Golbey Cedex, France The thermal behaviour of heat exchangers is studied with accuracy by developing a mathematical model in 2D. It represents the system in the cross section of the flow direction by taking into account all the parting sheet, bar and fin characteristics (geometries, thermal parameters). Temperatures are calculated on each point on the whole stack, in particular next to the bar wall. This represents the first step towards the study of global and local mechanical stresses. 1. INTRODUCTION Plate-fin heat exchangers (PFHEs) are widely used in process industries such as gas processing and petrochemical industries including cryogenic applications : industrial gas and hydrocarbon separation are the most common, but many processes use this type of equipment, including recovery of natural gas liquids, helium refrigerators and liquefiers, hydrogen purification, ammonia and ethylene processes, nuclear engineering and syngas production. Maximizing thermal efficiency with high availability is the main purpose for such cryogenic processes. PFHEs promote exchange between many streams simultaneously (cases with more than 12 streams are common) allowing to save energy very efficiently. However any perturbation on a stream entering the PFHE is propagated to the other streams and, as a result, has an influence on all duties of its immediate surroundings. The comprehension of local and global thermal stresses within PFHEs constitutes a primary objective to reduce the possibility of component damage or failure during operation. Because components in PFHEs are relatively closely and rigidly connected, a large local metal temperature differences may cause significant thermal stress, which could result in local mechanical failure or damage. Though PFHEs are very tolerant of large steady-state stream-to-stream temperature differences, they may be damaged if subjected to transient or continuously unsteady operating conditions producing excessive thermal stressing [1,2].
∗ This work has been performed with the financial and technical support of Nordon Cryog´enie and the Association Nationale de la Recherche Technique (ANRT).
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Thus, a dynamic simulator will provide an estimation of the constraints with respect to time and allows one to check the consistency of the operating policy, especially in the start-up and shutdown conditions. Furthermore, dynamic simulation of PFHEs provides operating information essential for process availability [3–5]. 2. MODELING AND NUMERICAL ASPECTS A brazed aluminium plate-fin heat exchanger consists of a block (core) of alternating layers (passages) of corrugated fins(figure 1). The layers are separated from each other by parting sheets and sealed along the edges by means of side bars, and are provided with inlet and outlet ports for the streams. The block is bounded by cap sheets at the top and bottom [6].
Figure 1. Brazed Plate-Fin Heat Exchanger.
Figure 2. Cross Section perpendicular to the main flow : description of the bar-wall.
The dynamic model in 2D will allow to study the temperature in any point of the PFHE in the cross section of the flow direction, including plates (cap-sheets, parting sheets), bars and joints, in accordance with flow operating conditions. Figure 2 represents the main characteristics of the system which have been taken into account in this program. Some assumptions have to be used in order to define rigorously this model. -Insulation losses through cap sheets and side bars are integrated;
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-Thermal capacities and overall heat transfer coefficients are assumed to be constant; -Thermal exchanges within the stack are treated with conduction in the direction of the width and the stacking; -Fluid properties are considered uniform over the layer. Generally speaking, process modeling is based on conservation law for energy. Only heat balances are considered in this model. Heat is directly transferred between adjacent passages through the separating plates and by conduction through the fins. Due to transversal and longitudinal conduction, temperatures are strongly linked. The general descriptive equation , for a given layer and plate i (except for the first i=1 and last plate) :
−ρCVi ∂z
∂Tip ′ f p p p s = hi Aef i ∂z (Ti − Ti ) + hi ηi Ai ∂z Ti − Ti+1 ∂t ′ f p p p s + hi−1 Aef i−1 ∂z (Ti − Ti−1 ) + hi−1 ηi−1 Ai−1 ∂z Ti − Ti−1 ∂Tp ∂Tp z+∂z z + λi lei − λi lei ∂z ∂z z
(1)
z+∂z
The two first terms of the right hand side correspond respectively , to the convection and conduction with the lower layer and plate and the two next ones with the upper layer and plate. The last terms are the widthwise conduction. Equations for bars are given by taking into account all the general conduction terms in accordance with directions of the width and stacking, deeply linked to the boundary conditions in particular for the side of the bar in contact with the fluid. 3. CASE STUDY The advantage of the thermosiphon is to provide a natural convective mode of circulation with a high mass flow rate without the need for an external driving system subject to possible failure. It consists of short cooling loops connected in parallel between a lower and an upper manifold filled with the coolant. Cool-down from -17◦ C to -30◦ C will be carried out with Ethylene gas at 28 bar, tapped from the main compressor output. The gas will first be circulated through a heat exchanger cooled by liquid propylene. We will study the dynamic response of the heat exchanger during the starting time until the steady-state condition is reached (table 1). In this paper, we are interested in the warm starting of the PFHE : all temperatures are initialized at the warmest one (temperature of Ethylene in this case), and then the calculation begins with operational conditions, i.e. by taking into account the coldest fluid (Propylene) from t=0s. In this way, we will study the dynamic response of the heat exchanger, in particular the distribution and the evolution of the temperatures within the whole stack. Of course, The same approach can be applied to trip cases that can lead to thermal fatigue.
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Table 1 System specifications. Passage Identity Fluid Phase flow Temperature (◦ C) Number of passages
A Ethylene V →L −17 / − 30 55
B P ropylene L → L/V −40 54
T (°C) -35
-30
-25
-20
-15 0
0
10
20
30
40
-15
T (°C)
-20 -25
|Tmax-Tav| Tmax
-30 -35
Tmoy Tmin
|Tmin-Tav|
0,2
50 14 12 10 8 6 4 2 0
t (s)
x (m)
0,4 0,6 0,8 1 1,2
t (s)
Figure 3. Characteristic temperatures of the PFHE during the starting .
Figure 4. Temperature distribution of the Wall-Bar during perturbation.
Figure 3 represents the average (T av), minimum (T min) and maximum (T max) temperatures for the entire PFHE calculated on the middle of each sheet during the first 50s of the perturbation. The response of the start is not immediate and the temperature difference beetwen the maximum and average temperatures occurs at a peak just after the beginning of the starting (≈ 3s). In this case, the maximum temperature corresponds to the cap sheet ones while the average temperature is calculated by taking into account all the kind of sheets in the exchanger. In fact, we can assume without any doubt that the average temperature is closely linked to the evolution of all the parting sheets within the PFHE. Thus, we can note parting sheets need only 3s to reach the steady-state condition whereas the response of the cap sheet is much slower, due to its boundary conditions and its bigger thickness. In figure 4 we see the distribution of the temperatures on the bar-wall from the top to the bottom. Each temperature is the average one of each element (plate or bar) in accordance
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with the abscissa of the stacking pattern. The most part is nearly at constant temperature even if we can observe some very soft variations due to the boundary conditions of each bar which is in contact with a defined fluid. On each extremity (≈ 0,1m on each side i.e. 15% of the total height in this case), the temperature tends towards the cap sheet temperature limit and 100s are necessary to reach the steady-state condition. Finally, by comparing these time reponses with ones obtained in the last paragraph, we can observe the bar-wall and the cap sheets thermal inertia seems to be not negligible on the global behaviour of the PFHE. In this study, the two cap sheets and the bar-wall represent respectively ≈ 5% and ≈ 18% of the global mass of the PFHE.
0,05
0,1
0,15
T (°C)
-20 -25 t (s)
-30 W-B
-35
x (m)
Figure 5. Temperature distribution in the cap sheet during the perturbation.
0 -15 -17 -19 -21 -23 -25 -27 -29 -31 -33 -35
0,05
0,1
T (°C)
0 -15
0,15
t (s)
W-B
x (m)
Figure 6. Temperature distribution in the fifty fourth parting sheet during the perturbation.
Figures 5 and 6 show, respectively for the cap sheet and for the parting sheet positioned just in the middle of the exchanger, the complete response of the temperature distribution on their whole lengh. For each curve, we can distinguish three different thermal zones : the side bar deeply linked to the bar-wall, the large length focused on the middle of the sheet, facing the side bar, and the thermal transition area linking the two others. For a parting sheet, the steady state is quickly reached (≈ 3s) while the bar side needs 100s to be stabilized. Because of the small area in contact with the bar-wall and its large length subjected to the fluid convection, the temperature profile is constant on its whole length. Only a small length of the parting sheet is useful to offset the temperature differentials with the bar-wall. It is also interesting to note that the temperature difference at the wall-bar/plates side interface is continuous during the first seconds of the perturbation, avoiding in fact risks of failure and damage depending on the value. For the cap sheet, we can see a slower thermal response on the whole length (≈ 50s in the middle of the sheet) and its non-negligible thermal influence on the bar-wall temperature distribution. The bar-wall temperature tends to the average temperature T av when the steady-state condition is reached. But the inertia of the two cap-sheets slows down fully this evolution.
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This program is very useful to study thermal stresses, in particular next to the barwall and the cap-sheets, by taking into account most of the PFHE features. But these results have to be linked closely with mechanical engineering to study global stresses and resulting metal fatigue in accordance with operating conditions. In this way, this new tool provides important information for a good design of the PFHE in order to improve the availability of the components.
Greek symbol Subscript Abbrevations
Aeff As C e h T Tp t V x ρ i B−W
efficient surface area between a plate and a passage (m2 .m−1 ) half secondary surface of fins (m2 .m−1 ) metal specific heat (J.Kg−1 .K−1 ) plate thickness (m) heat transfer coefficient of stream (W.m−1 .K−1 ) fluid temperature (K) plate temperature (K) time (s) volume of metal per passage (m−3 .m−1 ) length (m) density (kg m−3 ) plate index Bar − wall
REFERENCES 1. P. Carter, T.J. Carter and A. Viljoen, Failure analysis and life prediction of a larger, complex plate fin heat exchanger, Engineering Failure Ananlysis, South Africa, 1996. 2. C. Tolergard, O. B. Andersen, Failure in aluminium plate-fin heat exchanger, ESOPE, 2001 3. D. Averous, Simulation and modeling of plate-fin heat exchanger in complex configurations, PhD Thesis, France, 1998. 4. H. Pingaud, Steady and unsteady simulation of plate-fin heat exchanger, PhD Thesis, France, 1988 5. J.M. Le Lann and A. Sargousse, Panoply of DAE integrators dedicated to solve complex chemical engineering problems, internal report, France, 1998. 6. Alpema, The standards of the brazed Aluminium Plate-Fin Heat exchanger Manufacturers Association, Texas, 2000.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
ReDrop - An Efficient Simulation Tool for Describing Solvent and Reactive Extraction Columns. Mehmet Altunok, Tobias Grömping, Andreas Pfennig Lehrstuhl für Thermische Verfahrenstechnik, RWTH Aachen University, 52056 Aachen, Germany, www.tvt.rwth-aachen.de
Abstract In the ReDrop algorithm individual drops are followed along their way through an extraction column. This simple and efficient way to solve drop-population balances also for complex situations, taking e.g. lifetime of the drops as well as reactions into account, can be regarded as a Monte-Carlo integration of the balance equations. ReDrop has been applied to accurately describe the transient and steady-state behavior of pulsed columns with sieve trays, regular and random packings. The agreement with results of pilot-plant scale experiments is excellent, also for a technical system. Keywords: Solvent Extraction, Reactive Extraction, Drop-Population Balance, Extraction Column, Modeling.
1. Concept of ReDrop Solvent-extraction columns can be modeled based on drop-population balances by solving the transient behavior with these balances. For the various property coordinates describing the condition of each drop distributions have to be defined which are usually divided into classes. Typically classes with respect to drop size, concentration of transfer component and height position in the column can to be accounted for. If any further property of a drop needs to be considered in the simulation, a new distribution with respect to this property has to be considered, increasing the dimensionality of the problem. This leads to problems in depicting the column behavior for slightly more complex situations realistically, since the effort to solve the set of non-linear integrodifferential equations increases rapidly. It is e.g. known that contact time between the phases has a significant influence on mass-transfer rate. This effect is usually neglected in directly solving drop-population balances. To avoid these problems and being able to extend the simulations also to reactive extraction, we have developed the ReDrop model (Henschke 2004, Weber et al. 2005, Bart et al. 2005). The idea behind the ReDrop algorithm (representative drops) is to explicitly simulate the behavior of a discrete number of individual drops in a solvent or reactive extraction column with or without pulsation. Usually 1000 drops per meter of column height is a representative choice. For each drop only individual properties have to be stored. Thus adding a variable in the description of the drop conditions only adds a single value for each drop, the dimensionality of the problem remains unchanged. Thus this approach appears to be well suited to be extended to reactive extraction, where e.g. a variety of components have to be accounted for with their compositions. The ReDrop approach can also be viewed as a Monte-Carlo method to solve the drop-population balances.
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A further advantage of the ReDrop approach is that the program code can be written in a modular fashion, where each block has a clear physical meaning. This is illustrated in Fig. 1, where the algorithm of ReDrop is shown schematically. In the inner drop loop the effects acting on the drops are accounted for. These effects fully describing drop behavior in the column are drop sedimentation, mass transfer, as well as drop breakage and coalescence. The influence of swarm behavior and that of the internals on the drops has to be taken into account appropriately in the models as well as axial dispersion of the continuous phase. After each drop has been handled properly the information is then collected for each height element in the next step of the time loop. Thus ReDrop allows to perform transient simulations of an extraction column. initialization and data input divide column into height elements drop feeding
time loop
for each drop: drop loop • determine height element • drop velocity • mass transfer • handle drops leaving the column • splitting and coalescence for each height element: • new concentration • backmixing • new hold up
next time step
Fig. 1: Schematic representation of the ReDrop algorithm. The behavior of the drops concerning the basic effects acting on the drops is modeled with engineering models, the parameters of which have been determined from experiments in lab-scale measuring cells with a small amount of substance (Henschke, 2004, Weber et al., 2005). These experiments are necessary, since the behavior of the drops strongly depends on the composition of the system with respect to trace components. These components, which may e.g. be surface-active, determine coalescence, sedimentation, and mass transfer. The models have to be expressed in a way allowing application of the Monte-Carlo scheme. Thus e.g. drop coalescence as well as drop splitting are accounted for as individual events which are described with the corresponding probabilities which then are evaluated for each individual drop at any given time step by comparing the probability with a random number. If e.g. drop splitting will take place in a next step the number of drops into which the mother drop will split is determined as well as their size distribution. Here again random numbers are used for determining the number of daughter drops. The parameters in all these models have either been taken from
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literature directly or have been determined specifically as to account for the stochastic nature of the ReDrop model (Henschke, 2004). An advantage of the ReDrop model is that the attributes of the drops need not to be separated into classes of the corresponding values, since the individual values for all individual drops are used. Thus no classes for concentration or drop diameter need to be defined, rather the individual values of the roughly 1000 drops per meter of column height are used.
2. Lab-cell Measurements The lab cell for determining mass-transfer rate is shown in Fig. 2 (Henschke und Pfennig 1999). The height of the cell is approximately 30 cm for the cell shown without internals. A second cell is available which is larger by a factor of roughly 2.5, which can be equipped with packings. The flow of continuous phase in the position shown is from top to bottom. If the heavier phase is to be dispersed, the cell can be mounted upside down. A drop of dispersed phase with known volume is introduced into the cell through a nozzle with a computer driven syringe. At a certain position in the conical part the rising drop meets a point where sedimentation velocity matches the counter flow of continuous phase, thus the drop is levitated. After the flow of continuous phase has been switched off for a short period of time the drop rises further until it can be withdrawn from the collecting funnel again with a computer driven syringe. Determining how the composition of the drops has changed from inlet to outlet of the cell gives quantitative information on mass-transfer rate as function of contact time between the phases and drop diameter, where the diameter is varied in the relevant region between 1 mm and 4 mm. Usually some hundred drops have to be collected to obtain a reliable composition analysis. If only physical extraction takes place in systems of water-like viscosity, the extraction is essentially completed after 60 s, if the viscosity increases or chemical reactions are superimposed, extraction of a single drop can take several minutes. From the experimental results in the single-drop measuring cell these effects can directly be seen and accounted for appropriately in modeling. For the experiments also in the other cells in which sedimentation velocity and coalescence rates are determined as little as 10 liters of original system have to be supplied to obtain proper results. For a given system all necessary measurements are performed in roughly two weeks. Thus the approach via lab-scale measuring cells and ReDrop simulation saves experimental time and effectively also the cost for extractor design. Also based on the lab-scale experiments a variety of different column types can be modeled. While today the appropriate column type is chosen mostly based on experience usually in a rather early stage of process design, this new approach allows to easily optimize different column types with the simulations based on the same lab-scale experiments. Then the decision for the optimal column type can be made on the ground of these simulations based on objective reasoning, e.g. cost functions.
3. Applications The ReDrop algorithm has first been developed for pulsed sieve-tray columns (Henschke, 2004) and then was extended to columns with regular or random packings (Bart et al., 2005). The appropriate lab-cell measurements were used to derive the corresponding models describing the individual drop behavior with respect to the above-mentioned basic effects. The ReDrop simulations were then validated for a variety of systems, mostly EFCE standard test systems, which have been defined in the past, but also for technical systems (Henschke, 2004, Bart et al., 2005, Weber et al.,
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2005). The predictions for the standard test systems agreed to approximately 10% with the experimental data available over the entire relevant region of operating variables. The technical example has been supplied by Bayer Technology Services and INEOS Phenol, the lab-scale experiments were performed and the results of the ReDrop simulations compared with those of pilot-plant scale experiments. This comparison is shown in Fig. 3. It can be seen that the concentration profile along the column can be predicted with high accuracy for two examples of column conditions. Also the flooding point has been predicted an agreed to roughly 10% with the experimental values.
Fig. 2: Lab cell for measuring mass-transfer rate of drops. ReDrop simulations not only allow prediction of steady-state column behavior but can also predict the onset of flooding inside the column due to the transient nature of the simulations. Thus ReDrop allows also to inherently describe the limits of operability making separate flooding correlations obsolete. In Fig. 4 the column behavior is characterized for the system n-butyl acetate + water where acetone is the transfer component. The column was a 3m column with sieve trays of 39% open tray area and holes of 4mm diameter. It can be seen that all details of the operation can be modeled for a wide range of operating conditions. This diagram can then be used to choose the optimal operating conditions and the optimal design of the column with respect to the individual flow rates and thus effectively the phase ratio.
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normalized Phenol concentration
1 technical system vorg/vaq = 2 0,1
40%
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lo o of f
80
%
o
din
o f fl
g
od
ing
1E-3
0,0
0,2
0,4 0,6 normalized column length
0,8
1,0
Fig. 3: Application of ReDrop to extraction of technical system in a column with regular packing.
4. Conclusions It has been shown that ReDrop can account for a variety of drop attributes in solving the drop-population balances, namely drop diameter, concentration of drop, vertical position of the drop, and life time of the drop. Based on the appropriate measurements in lab cells the behavior of columns with and without pulsation can be described for the following types of internals: sieve-trays, random, and regular packings. ReDrop can easily be extended to account essentially for an arbitrary number of attributes characterizing each drop. Also it has been shown, that the transient state of the column can well be represented, especially inherently including the limit of operability, namely the flooding point. These results show that ReDrop is efficiently able to perform detailed simulations of extraction columns with different internals. The full transient behavior including flooding can be described based on experimental information obtained on lab scale with a relatively small amount of original substances.
5. Current Work The ReDrop algorithm is today extended to include reactive extraction fundamentals, thus being able to include reaction equilibria and kinetics. This also shows the advantages of the ReDrop algorithm as compared to population-balance modeling.
6. Acknowledgements This project has been funded as a part of the AiF project 40 ZN (Bart et al., 2005) and partly also by the German Science Foundation.
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Fig. 4: Operating conditions for the system butyl acetate + acetone + water. In the diagram various iso-lines are shown as function of the specific flow rates of both phases. These include the flooding limit as well as liquid load expressed in percentage of flooding, the phase ratio of dispersed to continuous phase, the number of theoretical stages in a column of 3 m height as well as the holdup of dispersed phase expressed in %.
References H.-J. Bart, D. Garthe, T. Grömping, A. Pfennig, S. Schmidt, J. Stichlmair, 2005, Vom Einzeltropfen zur Extraktionskolonne - Anforderungen und neue Entwicklungen, final project report AiF 40 ZN 1+2+3, http://www.tvt.rwth-aachen.de/frame3_5.htm, November 14, 2005. M. Henschke, A. Pfennig, 1999, Mass-Transfer Enhancement in Single-Drop Extraction Experiments, AIChE J. 45(10), 2079-2086 M. Henschke, 2004, Auslegung pulsierter Siebboden-Extraktionskolonnen, Shaker Verlag, Aachen, 2004 M. Weber, W. Bäcker, T. Grömping, A. Pfennig, 2005, Anwendung der neuen Auslegungsmethode für Extraktionskolonnen auf ein technisches Beispiel, 589. DECHEMAKolloquium am 10.03.2005, DECHEMA-Haus, Frankfurt am Main
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Numerical Simulation of Micro Roughness Effects on Convective Heat Transfer Stephan Scholl and Wolfgang Augustin Institute for Chemical and Thermal Provess Engineering, Technical University of Braunschweig, Langer Kamp 7, D – 38106 Braunschweig, Germany
Abstract During the induction phase of crystallization fouling a micro structured roughness is generated at the heat exchanger surface. If this is not accounted for in the extraction of fouling resistances from experimental data it leads to negative apparent values for the fouling resistance. In a numerical simulation study the effect of micro structured roughness on single phase convective heat transfer has been investigated. Two different models with respect to flow simulation and roughness geometry definition have been applied. It was found that under these conditions a maximum increase of the fluid side heat transfer coefficient by 20 … 50 % compared to the plain surface is achieved for a surface roughness coverage of 10 … 20 %. This is combined with physically meaningful assumptions for surface coverage progression during the induction period. The negative apparent fouling resistances can then be explained qualitatively and quantitatively in satisfactory agreement with experimental results. Keywords: Fouling, micro roughness, convective heat transfer, heat transfer enhancement, simulation
Introduction
fouling resistance
[10-3 m2 K / W]
The built-up of undesired material layers on heat and mass transfer surfaces, such as heat exchangers or membranes, is referred to as fouling. As for heat exchanger fouling five major types of fouling are distinguished: biological fouling, crystallization fouling, deposition fouling, reaction fouling and corrosion fouling. CaSO4
III
Exper. Simul.
II
I
Exper. Simul.
Exper. Simul.
Induction phase
Fig. 1
Fouling resistance of an aqueous CaSO4 solution for different salt concentrations on stainless steel [Brahim 2003]
tind time
[h]
The present study focuses on an effect that is frequently found in crystallization fouling. Fig. 1 shows typical fouling curves for CaSO4 aqueous solutions. An electrically heated plain metal surface is placed in a slightly under- or supersaturated solution. Constant heat flux is applied and bulk phase and surface temperature are monitored. For salt concentrations of cf = 2,42 g/l and cF = 2,89 g/l the fouling layer built-up starts right
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away quantified through the fouling resistance Rf. This is referred to as the growth phase. For cF = 2,22 g/l no significant fouling resistance appears over a time span of approx. 450 h. This is referred to as induction period. During the induction period first crystal nuclei are generated and deposited at the heat transfer surface. Following the induction time tind ≈ 450 h the crystal growth phase proceeds.
1. Surface roughness modeling Fig. 2 depicts three Scanning Electron Microscope pictures taken ex situ during the induction phase at three different times. It may be seen that individual crystals form on the surface generating an increasing coverage of the heat transfer surface. Due to the inverse solubility of the investigated salts the highest saturation and supersaturation exists at the hot surface. Additionally, from energetic considerations it can be shown that nucleation and crystal growth will preferably start at grain boundaries of the metal surface as these spots result in a minimum nucleation energy required [Augustin 1992].
Fig. 2
Ex-Situ SEM pictures during the induction period of CaCO3 on stainless steel.
Fig. 2 suggests that the single layer coverage of the virgin surface proceeds without a significant built-up of second and third layers, at least in the initial phase of the induction period. The investigations were therefore concentrated on a maximum single layer coverage of the surface. Coverage was quantified in terms of the fractional coverage θ, θ = covered portion of the surface . full plain surface
(1)
With respect to surface roughness θ = 0 and θ = 1 are identical situations. As roughness elements two different geometries were taken: In a first approach a random distribution of pyramid-base-shaped roughness elements were placed on the surface. Starting from a plain surface the coverage was increased continuously reflecting the progression of crystallization fouling. In the second approach the surface was covered with cube-shaped elements in a user defined way. Different distinct surface coverages ranging form θ = 0.05 to θ = 0.25 were looked at. The characteristic length of the cube elements was 500 µm. This was also the height of the pyramid-shaped elements and could be interpreted as Rz = 500 µm roughness parameter of the surface.
2. Quantification of fouling effect on heat transfer The fouling effect on heat transfer is quantified through the fouling resistance Rf. The total heat flux density q = Q/A is given by q = Q/A = k (T∞, hs - T∞, cs) .
(2)
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k is the overall heat transfer coefficient and T∞,hs, T∞,cs are the bulk fluid temperatures at the hot side and at the cold side.The overall heat transfer coefficient summarizes all relevant individual mechanisms for heat transfer from hot side to cold side. For nonfouling conditions it reads s 1 1 1 = + w + k o α hs λw α cs
(3)
αhs and αcs are the individual heat transfer coefficients on hot and cold side, resp. sw is the wall thickness and λw is wall thermal conductivity. In fouling conditions the actual overall heat transfer resistance is given by the corresponding resistance of the clean surface plus the fouling resistance Rf(t): 1 1 = + R f (t ) . k ko
(4)
Typically as flow conditions and physical properties of the fluid do not change significantly, k0 will be set constant for each experimental run. All fouling related effects on heat transfer will therefore fall into an apparent Rf(t). For curves III and II in fig. 1, Rf(t) shows an immediate increase while for curve I Rf(t) is negative throughout the whole induction period. As a negative fouling resistance is not meaningful in a physical sense this effect is generally explained by the micro structural roughness of the partially covered surface [Augustin 1992, Müller-Steinhagen 2002, Fahiminia 2005]. While the effect of macroscopic structures on heat transfer may well be accounted for, the calculation of the fluid side heat transfer coefficient does not take into account the enhanced head transfer due to this randomly distributed micro roughness. An increase in αcs in reality which is not reflected in modeling does show in an apparent negative fouling resistance Rf(t). The induction phase is therefore also referred to as roughness control phase [Fahiminia 2005]. It was the purpose of this study to investigate numerically the effect of micro structured roughness on the convective single phase heat transfer coefficient and to relate this to the findings in the framework of crystallization fouling during the induction phase.
3. Numerical simulation For the numerical simulation of the micro roughness effect on single phase convective heat transfer two different models were applied. In both models dimensions of the flow cell were length x width x height = 20 x 15 x 10 mm. The heat transfer portion of the surface was placed central in the bottom surface with dimensions 10 x 5 mm. The physical properties of liquid water with full temperature dependence were used for the fluid. A fully developed flow profile with an average flow velocity of w = 0.25 m/s at T = 300 K bulk temperature was set as left hand inlet condition. This corresponds to a channel Reynolds number of Re = 5020. Wall temperature was set constant at Twall = 320 K. In both models the maximum height of the roughness elements – which corresponds to the monolayer fouling height – was 500 µm. Model 1, pyramid-based shaped
Model 2, cube shaped
Fig. 3 Pyramid-base shaped roughness elements in simulation model 1 and cube shaped elements in model 2
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In model I fluid flow was modeled with a Reynolds-Averaged-Navier-Stokes (RANS) approach. This implies a steady-state flow situations at all local positions of the simulation volume and averaged turbulence parameters. Transient conditions were introduced for the fouling layer built-up. At the start of a calculation sequence a random selection of grid cells were identified as growth elements. These were grown at a rate of ∆h = 25 µm per time step. After 20 time steps the selected cells have grown to the maximum height of 500 µm, see fig. 3 left. This terminates a calculation sequence and a new one is initiated by the next random selection of growing cells. The local heat transfer coefficient was calculated from an enthalpy balance for each cell. For the roughness elements the identical physical properties as for the base material, i. e. stainless steel, were applied. As the roughness also generates an increase in the geometric heat transfer surface the overall heat transfer coefficient on the cold side αcs was calculated without and with accounting for this effect. Numerical simulation was performed on the commercial code Fluent 6.1 with approx. 410,000 grid cells.
heat transfer coefficient α [W/(m2 K)]
αmax = 11600 ... 12100 W/(m2 K) arithm. average acc. for area increase final average
αmax α0 = 8220 W/(m2 K)
α0
Fig. 4 Increase of heat transfer coefficient as a function of moved grid cells in model 1. Without and with accounting for increased heat transfer surface
≈ 1,5
no. of moved grid cells
Fig. 4 depicts a typical result for the average fluid side heat transfer coefficient as a function of the number of “grown” grid cells. For the plain surface with no roughness an average heat transfer coefficient of α0 = 8,220 W/(m2 K) is obtained. Each simulation symbol represents the final status at the end of a simulation sequence with a roughness height of 500 µm. The heat transfer coefficient increases significantly with increasing number of roughness cells. The red symbols represent α0-values with the plain surface as reference area while the blue symbols take into account the increase of the heat transfer area due to roughness. Maximum values for αrough are found for a roughness coverage of 10 … 20 % of the original surface. Maximum ratio of αrough/α0 was found to be 1.4 … 1.5. In model 2 surface roughness was introduced as user-supplied geometry definition. Cubes with a characteristic dimension of 0.5 x 0.5 x 0.5 mm were placed on the surface, see fig. 3 right. Fluid flow was described with Large Eddy Simulation (LES) accounting for stochastic turbulent fluctuations. Time step was ∆t = 0.5 ms and local heat transfer coefficients were integrated over 500 time steps. The total number of grid cells was 750,000 making use of the symmetry at the center line. Fig. 5 shows the percentage increase of the fluid side heat transfer coefficient for model 2. Again an increase of αrough/α0 up to a ratio of 1.25 may be seen with a maximum at 15 % surface coverage. It may also be seen that a disadvantageous positioning of
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increase of heat transfer α/α0 [%]
roughness elements for the 10 % coverage case shows in the roughness effect on the heat transfer coefficient.
Fig. 5 Increase of heat transfer coefficient as a function of coverage in model 2
coverage Θ [%]
Numerous simulation runs were performed with variations of geometric, physical property as well as operational parameters. It was generally found that the fluid side heat transfer coefficient increases in the order of 20 … 50 % due to micro roughness of the surface. A maximum value of αrough/α0 was obtained for a surface roughness coverage around 10 to 20 %.
4. Roughness and fouling As a generalized approximation function for the increase of the fluid convective heat transfer coefficient with respect to surface coverage the following formulation was used [Bronstein 1980]
α rough = 1 + a θb exp (c θ) αo
(5)
a, b an c are adjustable parameters which allow to fit eq. (5) to the simulated (or experimental) findings of αrough = f(θ). Based on the above simulation results parameters were set such a maximum heat transfer enhancement of 50 %, i. e. arough/a0 = 1.5, was obtained for a surface coverage of θ = 0.2, see fig. 6.
α rough/α 0 [-]
1,6
Fig. 6
1,4 1,2 1 0
0,2
0,4 0,6 0,8 coverage Θ [-]
Approximation of microstructure roughness effect on the increase of heat transfer coefficient. Curve according to eq. (5).
1
Eq. (5) and fig. 6 quantify the increase of convective fluid heat transfer coefficient due to micro structured roughness caused by fouling. Finally, this is combined with the generation of an additional heat transfer resistance due to the built-up of an insulating fouling layer. The true heat transfer coefficient under roughness conditions at a given time t is obtained from α(t) = (αrough/α0)(θ(t)) α0 .
(6)
As the actual progression of surface coverage with time during the induction phase is not known yet, three different characteristic curves have been assumed, see fig. 7 left.
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At a given time t the surface coverage is obtained from fig. 7 left. Eq. (5) is applied to calculate the increased heat transfer coefficient at this coverage. From eq. (6) the true heat transfer coefficient under roughness conditions can be obtained and from that the true overall heat transfer coefficient k = f(αrough). This is compared to the overall heat transfer coefficient for the clean surface as determined without accounting for roughness effect. Finally, this can be transferred into an apparent fouling resistance through
0,0008
0
20
apparent fouling
Θ(t)-1 Θ(t)-2 Θ(t)-3
40
60
time [h]
80
100
2
1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0
(7)
resistance [m K/W]
coverage
[-]
Rapp = 1/krough - 1/k0 . Θ(t)-1 Θ(t)-2 Θ(t)-3
0,0006 0,0004 0,0002 0 -0,0002 -0,0004 0
20
40
60
80
100
time [h]
Fig. 7 Progression of surface coverage during the induction phase (left) and apparent fouling resistances (right) for three different coverage progressions It may be seen from fig. 7 right that this results in the typical fouling curves as obtained from crystallization fouling experiments, see fig. 1. Depending on the progression of surface coverage with fouling deposition during the induction period an apparent negative fouling resistance is found. The numerical value of the apparent fouling resistances is well within the range typically extracted from fouling experiments.
Conclusions Based on numerical simulations of micro roughness effects on convective single phase heat transfer it is found that a maximum increase of the heat transfer coefficient of 20 to 50 % compared to the plain surface can be obtained. The highest increase shows at a surface coverage of 10 to 20 %. Combining this with different assumptions for fouling progression during the induction phase the typically observed negative apparent fouling resistances during the induction period are found. Future investigation will focus on the experimental verification of the micro roughness effect on convective heat transfer as well as the coverage progression during the induction phase.
References I. N. Bronstein, K. A. Semendjajew, 1980, Taschenbuch der Mathematik.Verlag Harry Deutsch, Thun. F. Brahim, 2003, Numerische Simulation des Kristallwachstums auf wärmeübertragenden Flächen (Fouling). Dissertation TU Braunschweig. W. Augustin, 1992, Verkrustung (Fouling) von Wärmeübertragungsflächen. Dissertation, TU Braunschweig. H. Müller-Steinhagen, 2002, Verschmutzung von Wärmeübertragungsflächen. VDI-Wärmeatlas, 9th Ed., Springer Verlag, Berlin Heidelberg. F. Fahiminia, A. P. Watkinson, N. Epstein, 2005, Calcium Sulfate Scaling Delay Times under Sensible Heating Conditions. Proceedings of ECI Conference "Heat Exchanger Fouling and Cleaning – Challenges and Opportunities", Kloster Irsee/Germany.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Classical Models of Secondary Settlers Revisited R. David a , A. Vande Wouwer a , P. Saucez a , and J.-L. Vasel b a Service
d’Automatique - Service de Math´ematiques, Facult´e Polytechnique de Mons, 31 Boulevard Dolez, 7000 Mons, Belgium
b Service
Assainissement et Environnement, Universit´e de Li`ege, 185 Avenue de Longwy, 6700 Arlon, Belgium Secondary settlers, which ensure the separation of the activated sludge from the treated effluent, are described by distributed parameter models taking temporal and spatial variations into account. Over the years, a number of mathematical models have been proposed, including Kynch, Tak´acs and Hamilton models. The objective of this study is threefold: (a) to highlight the influence of the model formulation on the numerical solution procedure, (b) to propose a modified expression of the settling velocity, which avoids sharp spatial variations reported in the literature, and (c) to develop a numerical solution procedure following a method of lines strategy rather than the classical ”tanks-in-series” approach. Keywords: Secondary settler, Distributed parameter model, Method of lines 1. INTRODUCTION The performance of the activated sludge process (in a wastewater treatment plant) strongly depends on the performance of the secondary settler, which separates the sludge from water by gravity sedimentation. For process optimization, it is therefore of great interest to understand the dynamic behavior of the sedimentation process. In the past several decades, several one-dimensional models of sedimentation have been discussed. One of the pioneering works in this area is due to Kynch [2], who derived a basic mass balance partial differential equation (PDE) from Stoke’s law. However, this simple model appears to be quite challenging in terms of numerical solution due to the existence of steep moving fronts and sharp spatial transitions. In order to alleviate these problems, Tak´acs and coworkers [5] proposed a discretization of the settler into several layers (following a standard ”tank-in-series” formulation), associated with constraints on the material fluxes between layers, and a mathematical formulation of the settling velocity in terms of the concentration of solid particles. Tak´acs model is certainly one of the most popular nowadays. Some variations of the PDE have been proposed, such as the introduction of spatial dispersion in order to smooth off sharp gradients [1]. Recent simulation studies [3], [7] highlight the difficulties associated with the numerical simulation and parameter estimation of these mathematical models. The purpose of this paper is to analyze the origin of the reported difficulties, and to develop a numerical simulation tool based on a slightly modified formulation of the settling velocity and a method of lines solution procedure of the mass balance PDE, as implemented in the MATLAB toolbox MATMOL [6]. This paper is organized as follows. The next section introduces the basic one-dimensional model of sedimentation, the mathematical formulation of the settling velocity due to Tak´acs, and the classical spatial discretization into n layers (”tanks-in-series” formulation). Section 3 presents numerical simulation results obtained with Kynch, Tak´acs and Hamilton models and discusses potential improvements.
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Section 4 proposes a modification of the expression of the settling velocity, implying that the settling velocity vanishes at a finite value of the solid concentration (in contrast to Tak´acs law), and discusses the influence of this modification on the classical models. Section 5 develops a numerical solution procedure following a method of lines strategy. Finally, Section 6 draws some conclusions. 2. SECONDARY SETTLER MODELLING Clear water
Qf = qf A
QW = qW A
s
0
zf
Sludge zL
QS = qS A
z
The secondary settler is a tank which separates the treated effluent (QW ) from the activated sludge (QS ) thanks to the effect of gravity (see Fig. 1). For simplicity, a simple geometry is considered in this study (tank of constant section A and depth zL , fed at the rate Q f at the level z f ).
Figure 1: Settler sheme 2.1. Mass balance equation The considered mathematical models are based on a mass balance PDE for the solid particles (in concentration C) settling down with a sedimentation (or settling) velocity νs :
∂ C ∂ (Fs +C q) + = C f q f δ (z − z f ), ∂t ∂z
(1)
where Fs = C νs is the sedimentation flux and q is the hydraulic velocity: qS in the lower part of the settler (z ≥ z f ) and −qW in its upper part (z ≤ z f ); δ is the Dirac impulse function. 2.2. Settling velocity A fundamental parameter in the model formulation is the [5] settling velocity νs . Tak´acs and coworkers proposed the following double-exponential law: νs = max 0, min ν0 , ν0 e−rh (C−Cmin ) − e−r p (C−Cmin ) . The velocity first increases due to the gravity acceleration exerted on the solid particles (influence of r p ), reaches a maximum (ν0 ) and then decreases as the particles are hindered by the other ones (formation of the sludge, influence of rh ). ν0 is the theoretical maximum value of the velocity and Cmin the minimum concentration needed for settling. 2.3. ”Tanks-in-series” formulation The traditional ”tanks-in-series” formulation amounts to a discretization of the mass balance PDE with a first-order finite volume method. The settler is spatially discretized into n layers, on which a mass balance is calculated. The numerical concentration values represented in the next figures correspond to the middle of each layer. The flux Fs,i between two adjacent layers i and i + 1 is equal to νs,i Ci , where νs,i = νs (Ci ). The mass balance equations can be written as follows (see also [7]): First layer: Δz C˙1 = qW C2 − qW C1 − Fs,1 i-th layer above the feed level (2 ≤ i ≤ m − 1): Δz C˙i = qW Ci+1 − qW Ci + Fs,i−1 − Fs,i Feed level (layer m) : Δz C˙m = q f C f − qW Cm − qS Cm + Fs,m−1 − Fs,m i-th layer below the feed level (m + 1 ≤ i ≤ n − 1): Δz C˙i = qS Ci−1 − qS Ci + Fs,i−1 − Fs,i Last layer (tank’s bottom, layer n) : Δz C˙n = qS Cn−1 − qS Cn + Fs,n−1
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3. NUMERICAL SIMULATION STUDY The system of mass balance ordinary differential equations (ODEs) can be solved using the solver ODE15s of MATLAB [4]. The numerical values of the parameters used in this study are taken from [7]. The initial conditions correspond to a settling tank filled with clear water, i.e. Ci = 0 for i = 1, ..., n. 3.1. Model of Kynch Equation (1) is the PDE model which was originally proposed by G. Kynch in 1952 [2]. Numerical simulation results are presented in Fig. 2. A steep concentration front moves through the settler and, after some time, the predicted spatial concentration profile becomes uniform in the lower part of the settler, with the exception of a sharp spatial transition at the outlet boundary, as observed in [3]. This sharp spatial transition at the outlet boundary is not physical, but results from the discretization scheme. The shape of the spatial concentration profiles can be explained by considering the fourth ”tanks-in-series” F −F equation at steady-state (C˙i = 0): Ci = Ci−1 + s,i−1qS s,i . The concentration profile is uniform since the term
Fs,i−1 −Fs,i qS
vanishes in steady state (as the sedimentation flux is constant). At the outlet boundary, F
the last ”tanks-in-series” equation at steady-state (C˙n = 0) gives: (Cn − Cn−1 ) = s,n−1 qS . The observed discontinuity results from the second term (sedimentation flux from the previous layer). 3.2. Model of Tak´acs Tak´acs [5] introduced constraints on the gravitational flux Fs , as an ad-hoc procedure applicable to the ”mixed-tanks-in-series” formulation, based on the comparison between Fs values for successive layers: - in the upper zone of the settler (above the feed level): Fs,i = min (νs,i Ci , νs,i+1 Ci+1 ) if Ci+1 > Ct , Fs,i = νs,i Ci if Ci+1 ≤ Ct , where Ct is a threshold concentration (Ct = 3000 g/m3 in [7]); - in the lower zone of the settler (below the feed level): Fs,i = min (νs,i Ci , νs,i+1 Ci+1 ). The simulation results are more realistic (accumulation of sludge at the bottom of the settler), but there is an interplay between the model formulation (conditions on Fs ) and spatial discretization (number of layers), as shown in [7]. Consequently, the predicted concentration profiles change when the number of layers n is increased, and there is no convergence to the desired physical profiles, but to the concentration profiles predicted by Kynch model! (see Fig. 3). This makes the model of Tak´acs very delicate to use in practice as an ad-hoc choice of a number of layers and of the numerical values of the model parameters has to be made a priori. 3.3. Model of Hamilton Hamilton and coworkers [1] proposed to introduce a term accounting for material dispersion in Kynch 2 q) model: ∂∂Ct + ∂ (Fs∂+C − D ∂∂ zC2 = C f q f δ (z − z f ). The ”tanks-in-series” formulation is modified accordz ingly. The dispersion term introduces back-mixing effects, which smooth off the traveling waves and give realistic concentration profiles (see Fig. 4). To understand how translatednumerically, this is consider the second ”tanks-in-series” equation at steady-state: Ci = Ci+1
D qW + Δz D qW +2 Δz
+Ci−1
D Δz
D qW +2 Δz
+
Fs,i−1 −Fs,i D . qW +2 Δz
The value of Ci is influenced by Ci+1 (as in Kynch model), but also by Ci−1 . In Kynch model, the coefficient of Ci+1 is equal to 1 while in Hamilton model the influence of neighboring layers is weighted by coefficients, whose sum is equal to 1.This explains the smooth aspect of the concentration profiles along the settler. Numerical results converge for increasing numbers of layers, in contrast to Tak´acs results. 3.4. Summary and discussion The mathematical simplicity of Kynch model contrasts with the difficulty associated with its numerical solution. A discretization of the spatial domain into n layers - following a standard ”tanks-in-series” approach - yields unrealistic steady-state profiles, with a sharp spatial transition at the outlet bound-
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ary. The numerical procedure suggested by Tak´acs to alleviate this problem has the drawback that the constraints on the sedimentation flux are intimately related to spatial discretization. The concentration profiles change dramatically when the number of layers is increased, and ultimately turn back to the profiles predicted by Kynch model. The number of layers recommended by Tak´acs (n = 10) is insufficient to ensure convergence and leads to numerical simulation programs in which the issues of numerical accuracy (solution convergence) and model identification (estimation of the physical model parameters from experimental observations) cannot be separated (which should be the case). The model proposed by Hamilton introduces a dispersion term, which smooths off the spatial gradients and facilitates the numerical solution procedure (constraints on the sedimentation fluxes are no longer required), but this model does not include a clear definition of the outlet boundary condition, which are required for a second-order (in space) equation. These observations suggest the following: - the best way to clearly separate the model formulation and the numerical solution procedure would be to define an appropriate mass balance PDE and its associated initial and boundary conditions, and to solve the resulting set of equations using an advanced numerical procedure such as the method of lines; - the definition of an outlet boundary condition implies the consideration of a vanishing settling velocity, which would, following Tak´acs law, lead to infinite concentrations (indeed the second exponential in Tak´acs law vanishes for infinite concentrations). This latter point is addressed in the next section. 4. MODIFIED EXPRESSION OF THE SETTLING VELOCITY 4.1. A new formulation The original law implies that for νs = 0, C is infinite. Physically, the sedimentation velocity should vanish at a finite concentration Csat , representing the concentration at which the sludge forms a solid aggregate. The second exponential in the original law is thus replaced by a parabola. Here, as an example value, Csat = 11685g/m3 , which corresponds to the maximum value obtained with Kynch equation (see Fig. 2, t = 1000h) and is of the same order of magnitude as the maximum concentration reported in [5], and validated with experimental data. The first part of the sedimentation law is reduced to a simple exponential law with one parameter rp (set to 1.4945 10−3 m3/g in the study). The new sedimentation law is given by: ⎧ νs = max 0, min ν0 , ν0 er p (C−Cmin ) − 1 for C < Cν0 ⎪ ⎪ ⎪ 4 ⎨ sat ) νs = (C−C for Cν0 ≤ C ≤ Csat 4 (C −Csat ) ν 4 04 ν ⎪ ⎪ 0 ⎪ ⎩ νs = 0 for C > Csat
(2)
Cν0 is a concentration value corresponding to ν0 and is the boundary value between the exponential law and the parabolic law. 4.2. Impact on the previous models The results obtained with the models of Kynch and Hamilton are slightly modified, but remain basically the same. The biggest change is obtained with Tak´acs model, which now gives results comparable to those obtained with Hamilton’s model . Furthermore, the numerical results now converge as expected with increasing n, see Fig. 5. A sharp, unrealistic transition is observed at the top of the settler though, which is the consequence of the discretization method and the condition on the solid flux. This anomaly is in fact also present in the numerical results obtained with Hamilton model, even though the introduction of the diffusion term softens the resulting curves. However, the saturated settler should not present this drastic reduction in concentration at the clear water outlet. Numerically, this reduction can be explained by the first Hamilton ”tanks-in-series” equation at steady-state C (˙1 = 0),
Classical Models of Secondary Settlers Revisited Fs,1 D W + Δz
which shows that: C1 = C2 − q always be smaller than C2 .
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, i.e., the first layer is only influenced by the next layer, and C1 will
5. METHOD OF LINES SOLUTION The method of lines is a standard numerical solution procedure for PDE models, which proceeds in two separate steps: (α ) the spatial derivatives are first approximated using finite difference, finite element or finite volume methods; (β ) the resulting semi-discrete (discrete in space - continuous in time) equations are then integrated in time using an ODE or DAE solver. The model of Hamilton, with the modified settling velocity, is solved using the Method of Lines. To this end, the settler is decomposed into two zones, I and II, which are delimited by the feed point. Indeed, the feed flow rate is a pointwise input, which introduces a discontinuity in the concentration profile. To avoid this spatial discontinuity, the source term in (1) is replaced, through the system decomposition into two zones, by a boundary condition. The mass balance equations for these two zones can be written as: 2 ∂ (−qW CI +Fs,I ) ∂ CI − D ∂∂ zC2I = 0 ∂t + ∂ zI ∂ CII ∂t
∂ (q C +F
I
+ S ∂IIzII s,II − D ∂∂ zC2II = 0 II Boundary and continuity conditions are defined at the top, feed and bottom levels of the tank: ∂ CI - for z = 0: ∂ zI = 0 - for z = z f : q f C f + (νs,I − qW )CI − (νs,II + qS )CII = 0 - for z = zL : νs,II = 0 Initial conditions are given by: CI (t0 , z) = 0, and CII (t0 , z) = 0. Spatial grids are defined in each zone, with n1 and n2 uniformly distributed grid points, respectively. The spatial derivatives are approximated using higher-order finite difference methods, implemented as differentiation matrices in a MATLAB library called MATMOL, [6]. In particular, the first-order spatial derivatives (convective terms) are approximated with five-point biased upwind FD schemes method, and the second-order spatial derivatives (dispersive terms) are computed with five-point centered schemes. The resulting system of differential-algebraic equations (DAEs) is integrated in time using the MATLAB solver ODE15s. The resulting concentration profiles are physically coherent and no anomaly is observed. Convergence is ensured for small numbers of grid points (n1 = 76 and n2 = 76 in Fig. 6), and the computational load is very reasonable (a few seconds on a standard PC). )
2
6. CONCLUSIONS Most of the difficulties experienced in the numerical simulation of secondary settlers result from the interplay between model formulation (e.g. constraints on the sedimentation flux) and the discretization of the spatial domain into finite volumes (classical ”tanks-in-series” approach). In the present study, this problem is circumvented by focusing attention on: (a) the definition of a system of well-posed mass balance PDEs and their associated initial and boundary conditions, (b) their solution using an advanced numerical procedure such as the method of lines. In addition, the influence of the formulation of the sedimentation velocity is highlighted. Particularly, at high (but finite) solid concentrations, the settling velocity should vanish, whereas classical laws only vanish asymptotically (i.e. at infinite concentration). The proposed model and simulation tool provide the dynamic evolution of the concentration profiles with accuracy and efficiency (a few seconds CPU on a standard PC).
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4. 5. 6.
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3
C [g/m ] C [g/m ] C [g/m ]
7.
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t = 0.25 h
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t = 0.5 h
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t = 0.75 h
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t = 1 h
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0
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t = 1.5 h
3
2. 3.
J. Hamilton and R. Jain and P. Antoniou and S.A. Svoronos and B. Koopman and G. Lyberatos. Modeling and pilot-scale experimental verification for predenitrification process, J. Environ. Eng. - ASCE 118(1), 1992, pp 38-55. G.J. Kynch. A theory of sedimentation, Trans. Faraday Soc. 48, 1952, pp 166-176. I. Queinnec and D. Dochain. Modelling and simulation of the steady-state of secondary settlers in wastewater treatment plants, Wat. Sci. Tech. 43(7), 2001, pp 39-46. L.F. Shampine, I. Gladwell and S. Thompson. Solving ODES with MATLAB, Cambridge University Press, 2003. I. Tak´acs, G.G. Patry and D. Nolasco. A dynamic model of the clarification-thickening process, Wat. Res. 25(10), 1991, pp 1263-1271. A. Vande Wouwer, P. Saucez and W.E. Schiesser. Simulation of Distributed Parameter Systems using a MATLAB-based Method of Lines Toolbox - Chemical Engineering Applications, Industrial Engineering and Chemistry Research , 43, 2004, pp 3469-3477. L. B. Verdickt and J. F. Van Impe. Simulation analysis of a one-dimensional sedimentation model, Preprints of the 15th triennial IFAC World Congress, Barcelona, Spain, 2002.
C [g/m ]
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Figure 2. Dynamic evolution of the concentration profile Figure 3. Changes in the steady-state concentration pro-
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predicted by Kynch model with n = 30.
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profile obtained with Tak´acs model for increasing n (from 10 to 100).
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Figure 6. Dynamic evolution of the concentration profile obtained with the method of lines solution procedure (with n1 = 76 and n 2 = 76).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
An approach to implicit modelling for complex process optimization X. G. Yuan, W. Z. An, Y. J. Liu, Y. Q. Luo and C. J. Liu State Key Laboratory of Chemical Engineering and Chemical Engineering Research Center, Tianjin University,Tianjin 300072, P.R.China
Abstract The difficulties for the complex process optimization come not only from the combinatorial nature and the nonlinearity of the problems, but also from how the problems could be formulated. The complex process is defined in this paper as that a model capturing the behavior of the process is not available, whereas the description of the process could come at lower-scale level. The situation of complex so defined could be encountered in cases for process synthesis and multi-scale modelling. This paper presents a randomized search strategy for tackling this difficulty. Within the proposed search strategy, an explicit formulation including objective function as well as constraint equations is replaced by ‘right-away’ evaluations of the criteria. The behavior of a randomized search methodology and an example of application of the approach on optimization of thermally coupled complex distillation column system is demonstrated. Keywords: complex process, process modeling, process optimization, Mixed-integer Nonlinear Programming, randomized search.
1. Introduction Many chemical process synthesis problems could be formulated as a Mixed-integer nonlinear programming (MINLP) problem (Duran and Grossmann, 1986). MINLP formulation is theoretically perfect for representing these optimization problems as the latter could be fitly characterized with mixed (integer and continuous) variables. In general, an algorithmic procedure for MINLP solution calls for a properly formulated problem: a linear or nonlinear but convex (preferable for a stable solution) objective function, and a set of liner or nonlinear constraint equations satisfying some convexity conditions. However, in many cases, chemical processes under consideration are complex. The economical/technical evaluation of a process system may not be accomplished by an explicit equation but can start with some elementary subroutines, and in some of these routines there may exist iterative procedures. For such complex problems, the traditional optimization methods relying on gradient information of the criterion equation may have extreme difficulties and even fail. In recent years, the success of the applications of randomized search techniques, typically Simulated Annealing (SA) and Genetic Algorithms (GAs) has demonstrated their ability in solving the problems for complex process optimization. Several characteristics embedded on the stochastic nature of the approaches have led to the success. The one the mostly mentioned in the literatures is that convergences to locale optima could be effectively avoided by either a mechanism for accepting or rejecting a solution according to statistic probability, or by evolutions of a population of solutions. Another characteristic that makes a significant contribution to the success but has not been sufficiently recognized is that these methods rely on no other than the criteria values and their stochastic information, and it does not interfere how the criteria values have been evaluated. The purpose of this paper is to demonstrate how this character is important
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and could be benefited to develop robust, randomized search based methods to solve complex process optimization problems. The ideas are illustrated by developing a randomized search algorithm and solution of problems of either mathematical and chemical process optimization.
2. Complex process synthesis Synthesis is the main body of the preliminary process design activity, which is important for generating a profitable process. The market competition renders the profits of some chemical processes, especially for bulk productions marginal. In this context, more accurate economic evaluation of the process during the selection of the best one from alternatives is needed. The hierarchical strategy (Douglas, 1988) for process design is to guide the selection of the alternatives by adding successively process flowsheet details from a inside layer for chemical reaction to a intermediate layer for separation system synthesis and then to an outer layer for heat recovery network synthesis. This strategy could be known as adequate for using more accurate evaluation models. However, if we take into account the interactions between the layers that could be ignored by a traditional single pass application of this strategy, more detailed analysis may inevitably slow dawn the feedback of the final result to adjust the inner layers, and this will make any improvement of the design very expensive. Powerful search strategies have been developed to cope with the interactions between the layers by considering the selection of the process alternatives as an optimization problem. Most typical of all is the MINLP approach based on a superstructure representation (Duran and Grossmann, 1986). An MINLP formulation can be given as Min f(x, y) (1) g(x, y) ≤ 0 h(x, y) = 0 x∈X, y∈Y where the continuous variable vector x represents the continuous decisions and the integer variable y represents the flowsheet structure. f is the objective function and g and h are the vectors for the inequality and equality constraint equations respectively. For a direct solution of the optimization problem with a mathematical programming approach, problem (1) should be usually carefully modelled with the explicit forms for the functions of f, g and h. However, with the practical availability, explicit formulation of these costs is usually a non-trivial work, and usually should be started from some elementary modules which are not necessarily explicit functions. For example, the capital cost of a heat exchanger is a function of the heat exchange area needed, which could be expressed in terms of the process variables (temperatures, flowrate of process streams). However the capital cost has to be estimated by accumulating various direct and indirect costs associated with the equipment. Guthrie’s (1969) modular method is a factoring method for such a purpose and has been adopted to cost the investment of process equipment. By this method, various economic aspects are considered by a set of Modular Factors which themselves vary with a base cost associated with the heat exchange area. Fortunately, the behaviour of the capital cost so evaluated for heat exchangers could be acceptably approximated with a power law expressed as (Guthrie, 1969) s.t.
(2) Cost = α(A/A0)β where A is the heat exchange area, A0 is the area corresponding to a base cost, and α and β are constants. However, for many processes, the investment cost cannot be acceptably
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expressed in an explicit function as those in the case of heat exchanger. An example is the non-sharp distillation sequencing. For a non-sharp distillation column, the size of the equipment is not only the function of the capacity, but also effected by the “sharpness”, which is an optimization variable. The cost estimation could hardly be expressed in a power law as equation (2). To employ an MINLP approach, Floudas (1987) have developed a linear expression obtained by regressive analysis for the cost estimation for a non-sharp distillation column with the component flowrates, the sharpness of the key components and the operating pressure as variables. With this approximation, the synthesis of non-sharp distillation system could be performed but with about 10% more loss of accuracy for the cost evaluation. It could be estimated that to generate a profitable process, the offset of the economic criteria caused by the accuracy loss must be within the profit margin of the process. The case is even worse for the complex distillation system synthesis. Up to now, it has not been possible to develop an explicit expression capturing the costing behavior of a complex column, especially when the thermally coupled distillation configurations are considered. Whereas the economic estimation of complex column configurations could be performed step by step by adding-on various elementary cost estimated with corresponding approaches that have been well established. In fact, many chemical processes, we termed here as complex processes, have the same feature: a model capturing the behavior of the process is not available, whereas the description of the behaviour could come at lower-scale or elementary level, which can elaborate fine details of the process. For the process optimization with the environmental impact and/or the safety as a part of objective function, which constitutes a natural extension of traditional process optimization, it should not be expected to evaluate the criteria with an explicit model function. Generally, any evaluating cycles for multi-scale modelling within the chemical supply chain described similarly by a number of authors (see Grossmann (2004) for example) as shown in Fig. 1 could be characterized by the foregoing definition of complex processes.
Time scale
environment enterprise site plants process units single and multi- phase systems particles, thin films molecule cluster molecule Length scale
Figure 1. Generalized chemical supply chain
3. Randomized search: an explicit-equation-free approach Randomized search techniques [e.g. Simulated Annealing (SA) (Kirkpatrick, 1983), Genetic Algorithms (Gas) (Goldberg, 1989)] have been introduced for chemical process optimization as a robust approach that one can expect to survive in complex process engineering problems. Based on this technique, we have developed a randomized search based methodology for solving the problems of complex process optimization. The key to our approach is a “right-away” criterion evaluation that may undergo a complex evaluation cycle, within the randomized search scheme. Simulated Annealing, like several other randomized search methods, provides this scheme. As seen in Fig. 2, in a
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SA the search is guided by the stochastic information of the existing points of the evaluated value of the criterion, rather than the gradient information of a model equation (if any) of the criterion. It is then allowed to evaluate the criterion with fine level models, subroutines involving even iterative procedures, and an explicit overall model equation of the criterion itself is not necessary. SA as a randomized search method is approvable to converge to the global optimum for complex problems, but with the loss of computation efficiency. In fact, in the extreme end, a randomized search could come to an enumeration approach. Initialization Convergence check
End
Criteria evaluation Comparison of the current value to the current best one with Metropolis rule to decide the acceptance of the current solution or not Generation of a new solution
Figure 2. A Simulated Annealing schema
3.1. Simulated Annealing for complex distillation process optimization Based on randomized search strategy, a formulation for optimal synthesis of thermally coupled complex distillation column systems comprising simple columns, complex columns with side rectifier and/or side stripper as well as partially or fully thermally coupled (Petlyuk) columns with prefractionators (as shown in Fig.3) have been proposed (An, and Yuan, 2005a). A coding procedure is developed, so that the flowsheet structure can be represented and manipulated by a set of integer codes. Taking the recoveries of key components in the pretractionators and reflux ratios as continuous variables, a conceptual MINLP problem is formulated and solved with a SA algorithm. The procedure for computing the total cost of the complex columns includes from the thermodynamic models to the programs for column sizing and costing, and has been implemented for the use of the economic criteria evaluation. The solutions for an example problem for synthesizing a five-component separation system are illustrated in Fig. 3 (An, and Yuan, 2005b). It should be point out that such a problem has not been solved before in such a detail like in this presentation. 3.2. PSO Algorithm for Non-convex MINLP In this section, a newly developed randomized search technique ––an improved Particle Swarm Optimization (PSO) algorithm for solving non-convex MINLP problem for chemical process optimization is introduced and illustrated for the random search. PSO is a population based stochastic optimization technique developed initially b y Kennedy and Eberhart (1997), inspired by social behavior of bird flocking or f ish schooling. PSO shares many similarities with evolutionary computation techni ques such as GAs. The system is initialized with a population of random solutio ns and the search for the optima is performed with updating these solutions. The potential solutions, called particles, fly through the problem space by following
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the current optimum particles. Each particle adjusts its “flying” according to its own experience as well as those of other particles, and is driven towards the glo bal optima. This is operated by the following recursive representations xik+1=xik+vik+1 i
vik+1=c1r1(pik-xik)+c2r2(pok-xik)
(3),
(4)
i
where x k+1 and x k represent the new and the current positions of particle i respectively, vik+1 is the increment for the movement of particle i, pik and pok represent the current best positions for particle i itself and for all the particles respectively, r1 and r2 are the randomized parameters for balancing the experience of its own and that of the best particle with the constants c1 and c2. AB
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Fig. 3. Solutions for the synthesis of complex distillation column system. Here, an improved PSO algorithm – reduced space PSO is developed for solving non-convex MINLP problems with equality constraints, which are usually presented in the problems of chemical process optimization. The idea is that the variables are partitioned, and a reduced variable set is identified as the independent variables through analyzing and tearing equality constraints. Then the original problem is transformed into that without equality constraints, and with only the reduced variables. The performance of the proposed approach is illustrated with the following example problem. Min F(x1,x2,x3,y1,y2,y3,y4)=(y1-1)2 + (y2-1)2+(y3-1)2-ln(y4+1)+(x1-1)+(x2-2)+(x3-3)2 (5) s.t.
y1+y2+y3+x1+x2+x3 ≤5 y32 + x12+x22+x32 ≤ 5.5 y1 + x1 ≤1.2 y22 + x22 ≤1.64 y32 + x32 ≤4.25
y22 + x32 ≤4.64 y4 + x1≤ 1.2 y3 + x3 ≤2.5 y2 + x2 ≤1.8 y1, y2, y3, y4, ∈{0,1}
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0 -1 -2 4
4 2
0
0 -2
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(b) Solution locations for iteration 10
4
2
4
2
4
2 2
0
0 -2
2
0
0 -2
-2
-2
(c) Solution locations for iteration 100
(d) Solution locations for iteration 300
Fig. 4. Intermediate solutions for variables x1, x2 and x3 during the evolution for the example problem. Fig.4 gives some of the intermediate solutions during the evolution of the solutions before converging to the global optima.
4. Conclusion Many chemical processes could be identified as complex process whose behaviors could not be modeled as an explicit function, but could only be evaluated through detailed elementary analysis. It is this complexity that makes the solutions of many complex process synthesis problems difficult. The random search techniques have been found to be capable for dealing with such a complex problem. The key to the success of this technique is a “right-way” evaluation of the criteria value, without direct manipulating the model function for the criteria. This can make the solution procedure to be free from development of an explicit model function. The idea of this approach is not only valuable for solving the complex process synthesis problems, but also helpful for solving problems rising from multi-scale modelling within the concept of Chemical Supply Chain.
References An, W. and X. G. Yuan (2005a) Synthesis of Thermally Coupled Complex Distillation System Based on Stochastic Optimization --(I) Problem Formulation, Journal of Chemical Industry and Engineering (China) (in press) An, W. and X. G. Yuan (2005b). Synthesis of Thermally Coupled Complex Distillation System Based on Stochastic Optimization --(II) Illustrating examples, Journal of Chemical Industry and Engineering (China), (2005) (in press) Douglas, J. M., (1988). Conceptual design of chemical processes, McGraw-Hill, New York. Duran, M. A. and I. E. Grossmann (1986). A mixed-integer nonlinear programming algorithm for process systems synthesis, AIChE J., 32, 592-606 Floudas, C. A., (1987). Separation synthesis of multicomponent feed streams into multicomponent product streams, AIChE J. 33, 540-550 Goldberg, D. E., (1989). Genetic algorithms in search, optimization, and machine learning, Addison-Wesley Publishing C. Inc., New York Grossmann, I. E. (2004). Challenges in the new millennium: product discovery and design, enterprise and supply chain optimization, global life cycle assessment, Computers and Chemical Engineering, 29(1), 29-39 Guthrie, K. M. (1969). Capital cost estimating, Chemical Engineering, (March) 24, 114, Kennedy J. and R. C. Eberhart (1997). A discrete binary version of the particle swarm algorithm. In: Proceedings of Conference on Systems, 4104-4108, Man and Cybernetics, Piscataway, NJ Kirkpatrick, S., C. D. Gelatt and M. P. Vecchi. (1983). Optmization by simulated annealing, Science, 220 (4598), 671-680
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Structural design of polymers for membrane based separation processes using reverse simulation approach Soni Va., Abildskov Ja., Jonsson Ga., Gani Ra., Karayiannis Nb., Mavrantzas Vb. a
CAPEC, DTU, Lyngby 2800, Denmark, bICE-Forth, Patras 25600, Greece
Abstract In separation processes involving polymeric membranes one is interested in the design of new structured polymers that can match the desired degree of separation for the specified separation task. A reverse approach where the optimal polymeric membrane together with a list of candidate polymers all of which satisfy defined design (permeability) targets, is presented. The polymers are represented by their repeat units as well as their microscopic structure. The new method employs the permeability properties of a polymeric system, its relation to microscopic and/or chain conformation, as well as the separation task, to design the membrane based separation process as well as the polymeric membrane. Keywords: Polymer design, reverse design approach, membrane based gas separations, permeability.
1. Introduction The membrane separation technology is currently receiving increased attention and enjoys numerous industrial applications because of advantages such as, compactness and light in weight; low labor intensity; low maintenance (no moving parts); low cost for small sizes; appreciable energy savings; environmentally benign and clean technology with operational ease. It has the potential to replace conventional processes like distillation and other energy intensive processes to produce high quality products with greater flexibility in design or operation at partial capacity. The highlight of this work is simultaneous design of the membrane based separation process and of the polymeric membrane. This can be achieved by first defining the design targets through the properties of the polymer for the specified separation task (step 1), and then finding (designing) polymers that match the property targets (step 2).
2. Theoretical background 2.1. Problem Definition The basic design problem is to define the design parameters that affects any given membrane based separation and to identify which of these are important in order to design the separation process. Each design problem depends on variables such as the system properties (S), mixture inlet variables (I), outlet conditions (O) and membrane properties (P). Membrane properties in turn depend on the microscopic structural parameters (SP) of the polymer that is used as the membrane. The first step is to develop a process model of the membrane process in terms of these variables. Based on different design targets different cases could be considered: Case I - Both P and S to be determined: This kind of problem requires generation of alternatives in terms of S and P in order to achieve a desired degree of separation defined through (O) for any given
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inlet concentration (I) of the feed mixture. This means that the feasible alternatives for operational parameters like temperature, pressure, flowrates and polymer (which includes polymer repeat unit, number of repeat units and branching) must be found. Case II - Designing only system properties (S) for fixed P: Most of the cases of retrofit design of a process are included in this case. This essentially means that polymer and hence its properties are fixed so, in order to match a desired output for a given input we can only generate alternatives in terms of the operational parameters (S). 2.2. Reverse design approach In membrane based separation processes, using polymers as the separation media it is important to find or design the polymers that can assure the desired separation standards for a specified mixture. The commonly used forward approach (shown in Fig. 1), is essentially a trial and error procedure. A polymer is chosen/designed and the properties for the polymer-mixture pair are calculated using an appropriate polymer property model. Inserting this polymer model in the membrane process model the product purity is predicted and this procedure is repeated until the desired purity is obtained. This is an iterative procedure where for each polymer, all the steps need to be repeated. In contrast to the forward approach, a reverse approach is employed in this work (shown in Fig. 2) according to which the membrane process model is used to calculate the polymer properties (design targets) that are needed to achieve the desired separation and product purity while any number of polymer property models (including databases) are used to find the polymer structural design that matches the calculated design targets. In this way, the membrane process model does not need a polymer property model. On the other hand, many polymers may be designed (found) without having repeatedly to solve the membrane process model coupled with the corresponding polymer property model.
Fig. 1 Forward approach
Fig. 2 Reverse approach
The procedure requires that in the first step the target properties are calculated by solving the process model keeping product output fixed at desired values and property values as unknown variables. And then in the next step polymers corresponding to these properties are found. Note that in this approach, since property variables are the unknown variables, their solution does not need the constitutive model to be embedded in the process model, which is an advantage over the existing process as it avoids the trial and error procedure typical to forward approach. Designing (finding) polymers
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corresponding to target properties can be done mainly by three ways: I. Making an extensive literature search to find out polymers that matches the targets; II. Finding out monomer units using group contribution methods; III. Finding out microscopic structural parameters using property models. Advantages of this approach include a lot of saving of computational efforts as in the first step, where target values of the properties are calculated, there is no need to incorporate the property model, reducing thereby the problem complexity. Note that, permeability calculations in pervaporation process generally need diffusivity and solubility data. These calculations depend on the composition, temperature and pressure at each spatially discrete point of the membrane module. So, incorporating them in the membrane model could be fairly complicated.
3. Membrane based gas separations 3.1. Process Description Membrane gas separators are separation units which split a given gas stream into two product gas streams, a high pressure retentate stream and a low pressure permeate stream. The membrane provides a selective mass transfer layer. Due to difference in chemical potential the species permeate through the membrane material at different rates [1]. Pressurized feed gas is normally fed to the shell side and the components permeate at different rates to the fibre bore. The retentate gas that is depleted in fast permeating components is withdrawn at essentially the feed pressure. In this work a non-porous membrane is being considered. 3.2. Mathematical models 3.2.1 Membrane gas separation model Mass transfer in gas separation membranes: The mass transfer mechanism in the nonporous membranes is a combination of thermodynamically (solution) and kinetically (diffusion) controlling phenomena and the underlying model is called solution-diffusion model. According to this model, gas separation with membranes occurs via a pressure driven diffusion [2]. From thermodynamics, diffusion process occurs only if chemical potential ( μ k ) of diffusing species k, along the diffusion path is negative. The change of chemical potential across the membrane from the feed to the permeate is therefore given by, where the difference in fugacities provide the driving force for the transport of gas molecules across the membrane.
⎛ f Pk ⎞ ⎟ ⎝ f Rk ⎠
μ Pk − μ Rk = Rg T ln ⎜
(1)
Diffusion is a kinetic process and hence cannot be completely described by thermodynamics. The one-dimensional flux N Mk of component k through a nonporous polymer membrane is given by Fick’s first law (assuming constant diffusivity). Inserting the phase equilibrium relation describing the solution equilibrium, the equation becomes: N Mk = − Dk (c )
dck dl
⇒ N Mk =
Dk
δ
(c
Rk
⎛ Dk S k ⎝ δ
− cPk ) = ⎜
⎞ xφ P −yφ P ⎟ ( k Rk R k Pk P ) ⎠
The product of diffusivity Dk and solubility S k is called permeability.
(2)
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Material balance in membrane gas separators: Starting point of material balance is the continuity of component k; the balance is done over the membrane module (shown in Fig. 3). Inserting the expression for flux (Eq. 4) and writing the equation in dimensionless variables leads to, duk dX
⎛ z k + sk − u k φ p − u k φ p ⎞ = 0 Rk R Pk P ⎟ U ⎝ 1+ S −U ⎠
− Qk ⎜
(3)
Where,
Qk =
Pk Am PF nFT
nPk 0 , PP 0 nFk , PF
,X =
z L
, uk =
nPT nFT
,U =
nPT nFT
, zk =
N Mk
nFk nFT
, pP =
PP PF
, pR =
PR PF
NC
, U − ∑ uk = 0 k =1
nPk , PP nRk , PR
Fig. 3 Membrane module The boundary condition in dimensionless form is X = 0 : uk = sk Momentum balance in membrane gas separators: Applying symmetries of the system to the Navier-Stokes equation, and introducing dimensionless permeate pressure yields 8η Rg TLnFT dp pP P + U =0 (4) 4 2 dX π R PF n f Energy balance in membrane gas separators: Membrane gas separation is in general a non-isothermal process, which can be characterized by an isenthalpic expansion of the high pressure feed gas to the low pressure permeate conditions. A steady state integral energy balance around the entire membrane module assuming no work on the system yields Qout + H F ( TF , PF , nFk ) − H R ( TR , PR , nRk ) − H P ( TP , PP , nPk ) = 0 (5) In the case, where Joule-Thompson coefficient is very small, the separation can be considered isothermal. In that case TF = TR = TP and Qout can be calculated from the heat balance. 3.2.1Property models: The property of the polymer that determines the extent of the membrane based gas separation is the permeability of the gas-polymer pair (qK). Two types of property models have been considered in this work, one for each of the two methods given by II and III in section 2. Correlation and prediction of gas permeability through a group contribution method: As stated above the permeability coefficient Pk, is comprised of both kinetic and thermodynamic factors which in principle depend on different aspects of the gas/polymer pair, i.e. Pk = Dk S k
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Dk is the diffusion coefficient of the gas through a polymer and it varies from polymer to polymer to a much larger extent as compared to the solubility coefficient Sk. The most important factor on which the diffusion coefficient and solubility coefficient depends is the free volume of the polymer. And so it is reasonable to correlate the permeability coefficient to the free volume. Extensive work has been done [3] to show the utility of the expression in the following form: Pk = Ak exp ( − Bk / FFVk ) (6) Where, Ak and Bk are constants for a particular gas k. The fractional free volume, FFVk, has been defined as
( FFV )
[V − (V ) ]
=
0
k
(7) V Here, V is the volume of the polymer which is obtained from experimental measurement of the polymer density at the temperature of interest. The term V0 is the volume occupied by polymer chains. For any gas k:
(V ) 0
k
k
N
N
n =1
n =1
= ∑ γ kn (VW )n and V = ∑ β n (VW ) n
(8)
Where, γkn and βn represents a set of empirical factors [4]. Correlating the gas permeability with structural parameters: Developing such property models requires information on how properties like density, diffusivity, solubility, etc., vary as a function of polymer structure and architecture (e.g., length and degree of branching). Acquiring this information through molecular modeling approaches is particularly appealing if accurate data can be obtained and since experimental measurements can be time consuming or expensive. In general, a hierarchical modeling approach of three steps is followed: 1) Monte Carlo (MC) simulations to extract information about the density, radius of gyration and static structure factor (i.e. local packing of the polymer atoms) [5]; 2) Molecular Dynamics (MD) simulations to extract information about the diffusivity of the polymer chains and characteristic relaxation times; 3) Transition State Theory (TST) to extract information about the free volume, solubility, diffusivity and permeability of small gas molecules to a polymer matrix [6]. In this work, the structural, volumetric and dynamic properties of a rather simple polymer, those of polyethylene (PE), as a function of its molecular architecture, defined by branch length and branch frequency are used. Sufficient data about the equilibrium radius of gyration of linear and branched PE, the longest relaxation time, and the chain center-of-mass self-diffusion coefficient have been obtained in this case through a stateof-the-art Monte Carlo simulation algorithm. By analyzing these data using well established group contribution methods, closed-form analytical expressions have been developed capable of relating these properties to features of molecular structure and conformation of the polymer. For the case of PE considered here, the permeability of oxygen and nitrogen has been related to the molecular length (number of carbon atoms) of the main chain backbone. Enough data is available to generate property models for PE giving permeabilities as a function of NCarbon in the straight chain: PO PN
2
⎛ N - 67.42 ⎞ 7.66e - 13 with R 2 = 0.9989 ⎟ ⎝ 1.867e6 ⎠
= - ln ⎜ =
2
Carbon
⎛ N - 65.96 ⎞ 1.67 e - 12 with R 2 = 0.9984 - ln ⎜ ⎟ ⎝ 1.212e7 ⎠ Carbon
(9)
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3.3 Illustrative example The case study of air separation is illustrated in this work. Oxygen of more than 30% purity (for combustion) and more than 40% purity (for medical use) are required. Problem definition: The reverse problem to be solved requires the following information: Feed: 200 mole/s air; TF = 298.150C; PF = 1 bar Permeate: Desired Separation: XO2 = 50%; O2 recovery = 0.0038; Where, Recovery = moles of O2 in permeate/moles of O2 in feed Based on this information, it was found that the polymer that would do the desired separation should have the following permeabilities: PO2 > 4 barrer and selectivity > 4.5 Results: 20 polymers from literature; 13 polymers from group contribution method and 9 polymers generated through molecular modeling were selected and plotted as shown in fig. 4 (selectivity vs. permeability). The horizontal and vertical lines show the property targets and all the polymers in the shaded region match the target properties. Hence, the validation step was performed only for these polymers (shown in Table 1). It can be seen that the separation tasks are achieved for the polymeric membranes.
Fig. 4: All polymers on property target plots
% No. Polymer Purity 1 6FDA-6FpDA 54.24 2 6FBPA/TERE 50.68 3 HFPC 56.91 4 TMHFPC 55.95 5 TMHFPSF 56.86 6 TBHFPC 56.67 7 PE-78 59.02 8 PE-142 62.27 9 PE-500 63.77 10 PE-1000 65.69
Recovery 1.40E-02 1.04E-02 4.66E-03 7.10E-03 4.81E-03 1.09E-03 5.19E-03 3.46E-02 2.09E-02 1.73E-02
Method Literature Literature GC GC GC GC MM MM MM MM
Table 1: Validation for selected polymers 4. Conclusions A general algorithm for design of polymeric membranes that includes the molecular modeling as an option has been highlighted through a case study that gave very promising results. From the calculations for the case study, it can be seen that it is comparatively easier to formulate and solve the models for the reverse approach. Furthermore, no trial and error procedure is involved while making it more efficient, robust and having a wider application range. Current and future work is extending the design methodology to other polymeric membranes-based separation processes as well as generating new data for other polymers for the microscopic-macroscopic polymer models. References 1. R. Rautenbach and W. Dahm, 1986, J. Memb. Science, 28, 319-327 2. S. Tessendorf, R. Gani and M. Michelsen, 1999, Chem. Eng. Science, 54(7), 943-955 3. J. Park and D. Paul, 1997, J. Memb. Science, 123, 23-29 4. V. Krevelen, 1990, Properties of Polymers, Elsivier 5. N. Karayiannis, V. Mavrantzas and D. Theodorou, 2002, The American Physical Society, 88(10), 1-4 6. N. Karayiannis, V. Mavrantzas and D. Theodorou, 2004, Macromolecules, 37, 2978-2995
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Innovative flowschemes using dividing wall columns Michael A. Schultza, Dennis E. O’Brienb, Richard K. Hoehna, Charles P. Luebkea, Douglas G. Stewarta a
UOP LLC, Des Plaine, IL 60017 USA Jacobs Engineering, Chicago, IL 60606 USA
b
Abstract Over the last several years, UOP has been involved in the development of dividing wall column (DWC) technology. This has been a combined effort, drawing on resources from members in process design and development, engineering, control system design, technical services and operations, and tray design. This work has shown that innovative uses of DWCs (and the specialized DWC called the ‘split-shell column’) can lead to significant improvements in process flowschemes. Some examples of this will be described, including: 1) A process for distillate hydrocracking using a split shell column to efficiently remove a reaction byproduct from the process. 2) A generic multistage reaction/separation flowscheme with a single split shell column to provide the separation for all three reactor stages. 3) A detergents flowscheme that makes use of a ‘non-traditional’ DWC. In each case the DWC or split shell column is an integral part of the flowscheme, leading to savings in capital and operating costs, along with reductions in plot space and equipment count. Keywords: distillation, fractionation, dividing wall column, DWC
1. Introduction Although fractionation is a mature technology, it continues to be the dominant separation process in the refining and chemical industries. The dividing wall column (DWC) is a relatively recent innovation in the field of fractionation, even though the concept itself is not new (Petlyuk et. al., 1965). A sampling of the many papers written in this area includes topics such as the design((Amminudin et. al., 2001), (Dünnebier and Pantelides, 1999), (Fidkowski and Agrawal, 2001), (Hernández and Jiménez, 1999), (Nikolaides and Malone, 1988), (Triantafyllou and Smith, 1992)), analysis ((Annakou and Mizsey, 1996), (Carlberg and Westerberg, 1989), (Glinos and Malone, 1988), (Shah, 2002)), control((Abdul Mutalib and Smith, 1998), (Abdul Mutalib et. al., 1998), (Halvorsen and Skogestad, 1999), (Serra et. al., 2003)), and industrial applications of DWCs ((Kaibel et. al., 2004), (Jobson, 2005), (Schultz et. al., 2001), (Lestak and Collins, 1997), (Heida et. al., 2002)). Because of this extensive body of work, a lengthy introduction to DWCs is not necessary. In a typical case, three components or component groupings are separated in a DWC, as shown in Figure 1a. As discussed elsewhere (see, e.g., (Triantafyllou and Smith, 1992)) , a DWC can show a cost savings in both energy and capital costs due to a reduction in equipment cost and a more efficient separation. Typically, a 30% savings in both capital and energy costs is possible. Additional benefits result from the simplified flowschemes including a reduction in equipment count, a smaller plot space requirement, less drainage area, and a reduced flare load. A specialized type of DWC involves a column in which the dividing wall extends either all the way to the top (as shown in Figure 1b) or all the way to the bottom (as shown in Figure 1c) of the column. Consider the case shown in Figure 1b, where two different streams are separated. These streams have a common heavy key, but the light key in each stream is different. As a result, the upper portion of the column is separated by the dividing wall so that the light material on one side is segregated from the light material on the other side. A common bottoms section is used to recover the heavy product. Similarly, Figure 1c shows the case in which the upper section is common, recovering a single light product, while the bottoms section is separated by the dividing wall so that two bottoms products are produced. At UOP, these
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columns are referred to by the term ‘split shell columns’ in order to distinguish this type of design from the dividing wall column described above. A
A C
A B C
A
B
A
B C
B A B
C
c) Dividing Wall Column
C a) Split Shell Column With Divided Overhead Section and Common Bottoms Section
A C
B
C
b) Split Shell Column With Common Overhead Section and Divided Bottoms Section
Figure 1: Three types of Dividing Wall Columns
The split shell column differs from the dividing wall column in that no energy savings are realized when compared to a two column sequence for the same separation. However, because two columns are combined into a single column, there is often a capital cost savings, in addition to the other advantages mentioned above that are associated with a reduction in equipment count. As shown in Figure 1, these two types of columns, the DWC and the split shell column, can be used to separate either one or two feed streams into three different product streams. It is important to understand, however, that focusing simply on trying to find process applications that strictly fit these criteria will limit the number of innovative solutions that can be developed using these types of distillation columns. Instead, these concepts can be used as a starting point for finding additional applications for DWCs and split shell columns, leading to novel process flowschemes. This paper will describe three examples of the implementation of these ideas at UOP. Each uses a split shell column or dividing wall column in a different type of application. In the first example, a split shell column is used to remove a reaction byproduct from the process. In the next example, a split shell column is used for an interstage separation in a multi-stage reaction/separation configuration. Finally, an example will be shown in which a non-traditional dividing wall column is used to replace two distillation columns in the flow scheme.
Chapter 2 Example #1: Removal of Reaction Byproduct from Process UOP’s HyCycle UnicrackingTM process is a distillate hydrocracking process that is optimized to maximize production of high quality diesel fuel. In this example a split shell column is used to remove a reaction byproduct from the process. Figure 2 shows a simplified flowscheme of the separation section. A stripper column is first used to remove light ends from a reactor section effluent stream. The bottoms stream from this column is then sent to the product fractionator after being heated to the desired feed temperature. The product fractionator produces a high quality diesel product, a kerosene product, as well as an overhead liquid product that is sent to a naphtha splitter. The bottoms stream from this column is unconverted oil that is recycled to the reactor. A second effluent stream from the reactor section contains heavy poly nuclear aromatic (HPNA) species, a reaction byproduct which must be removed from the process. One way to do this would be to send this stream to a stripper in order to concentrate the HPNAs in a heavy stream. However, it is possible to integrate this striper into the product fractionator, as shown in Figure 2. This product fractionator is a type of split shell column as seen in Figure 1c, in which the upper section produces a single overhead product, while the bottoms section is separated by a dividing wall. The HPNA stripper feed is routed to the stripper side of the dividing wall, where the lighter components are stripped from this stream. The bottoms stream from the stripper side of the column, rich in HPNA components, is
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routed to an appropriate destination such as fuel oil or vacuum residue storage. The main bottoms product is unconverted oil that is recycled to the reactor section. In this way, the split shell column has been used to remove a reaction byproduct from the process. By integrating the HPNA stripper into the product fractionator, the capital costs have been reduced, and the flowscheme has been simplified as well. To Naphtha Splitter
To Kerosene Stripper To LPG Recovery To Diesel Stripper
Product Fractionator
Reaction Section Effluent
HPNA Stripper Feed
Stripper
Recycle Oil to Reactor
HPNA Rich Stream
Figure 2: Simplified HyCycle Separation Flowscheme
Chapter 3 Example #2: Interstage Separation Between Multi-stage Reaction/Separation Flowscheme A flowscheme will be described which makes use of a split shell column of the type shown in Figure 1b. In this case, the column is used as an interstage separation between a multi-stage reaction and separation flowscheme. The simplified reaction scheme for this process can be written as: 2A Æ B The feed stream consists of component A along with an inert component I, in a ratio of approximately 55/45. Because the relative volatility between A and I is quite low, separation of the two components by distillation is not economic. As a result, the inert component passes through the entire reaction train. Figure 3 shows a flowscheme that uses a distillation column after each reaction stage to recover the desired product B. Following each subsequent reaction and separation stage, the overhead product contains an increasingly greater concentration of I, as the A is reacted away. 11 A 45 I
25 A 45 I
55 A 45 I
Reaction Stage 1
25 A 45 I 15 B
Reaction Stage 2
15 B
11 A 45 I 7B
3A 45 I
3A 45 I 4B P-4
Reaction Stage 3
7B
4B
26 B
Figure 3: Multistage Reaction/Separation Flowscheme with Three Distillation Columns
Because each distillation column is recovering the same heavy component, the three columns could be combined into a single column, as shown in Figure 4. In this case, each of the overhead products consists of the same species, unlike the simple example shown in Figure 1b. However, the concentration of component A in each overhead product decreases with each subsequent reaction stage, while the desired product, B, is recovered in a single combined bottoms section. The overhead section of the split shell column is segregated so that the required purity of each overhead stream can be maintained as the streams
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are cascaded through the flowscheme. This design has a significant savings in capital costs, along with the other advantages associated with a reduction in equipment count. 25 A 45 I Reaction Stage 2
11 A 45 I 7B
3A 45 I
55 A 45 I
Reaction Stage 1
25 A 45 I 15 B
3A 45 I 4B
11 A 45 I Reaction Stage 3
26 B
Figure 4: Multistage Reaction/Separation Flowscheme With Single Split Shell Column
Chapter 4 Non-traditional DWC. Combining 2 columns into one The final example describes an application of a DWC in which two columns are combined into one. However, because the DWC described here makes use of some novel features, it is considered a nontraditional DWC. The UOP PEPTM process unit is an adsorptive separation process that removes C7+ aromatics from a desired C7+ olefin/paraffin mixture (the ‘C7+ olefins’) (Schultz et. al., 2001). The batch wise process uses two regeneration streams to purge the adsorbent beds and then desorb the C7+ aromatics. A previous design used two columns to separate the regeneration streams from the adsorbent beds as shown in Figure 7a. The first column separates the purge and desorbent material to the overhead stream while the bottoms stream contains the C7+ aromatics. This overhead product is then fed to the second column, where a second feed stream is sent as well. The second column recovers the purge material in the overhead and the desorbent and C7+ olefin/paraffin mixture in the bottoms stream. Purge Material Desorbent
Purge Material Desorbent C7+ Olefin/Paraffin
Purge Material
Purge Material
Purge Material Desorbent C7+ Olefin/Paraffin External Reflux
Purge Material Desorbent C7+ Aromatics
Purge Material Desorbent C7+ Aromatics
Desorbent C7+ Olefin/Paraffin External Reflux
C7+ Aromatics
C7+ Aromatics
Desorbent C7+ Olefin/Paraffin
a) Two Column Design
b) PEP DWC
Figure 5: PEP Fractionation--Comparison of Two Column Design to DWC
The current design, which has been implemented in at least five commercial units, makes use of a DWC that combines these two columns into one, as shown in Figure 7b. Because the separation is complicated in that the C7+ aromatics co-boil with the C7+ olefins, some novel features are required that distinguish this DWC from a traditional DWC. These include a novel trap tray and an external refluxing scheme to prevent the C7+ aromatic from mixing with the C7+ olefins. The DWC has a significant advantage over the previous design in that it shows an energy savings of approximately 50 % and a capital savings of approximately 15-25% when compared to the two column fractionation sequence.
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Chapter 5 Conclusions Three examples have been used in this article to show how dividing wall and split shell columns can be used to develop innovative flowschemes. In each case, the novel column serves the primary purpose of reducing the cost of the flowscheme (either capital cost or energy cost or both), but are also an integral part of the flowscheme. These examples also show that it is possible to develop designs that do not necessarily fit the design criteria of the standard DWC or split shell column configurations shown in Figure 1. The standard designs can be used as starting points for developing additional column designs that extend the idea of the dividing wall column.
References Abdul Mutalib, M. I.; Smith, R., 1998, Operation and Control of Dividing Wall Distillation Columns. Part 1: Degrees of Freedom and Dynamic Simulation. Trans IChemE, Vol. 76, Part A. pp. 308. Amminudin, K.A.; Smith, R.; Thong, D.Y-C; Towler, G.P., 2001, Design and Optimization of Fully Thermally Coupled Distillation Columns Part 1: Preliminary Design and Optimization Methodology, Trans IChemE,79 Part A, pp. 701. Annakou, O.; Mizsey, P., 1996, Rigorous Comparative Study of Energy-Integrated Distillation Schemes, Ind. Eng. Chem. Res., 35, pp. 1877. Carlberg, N. A.; Westerberg, A. W., 1989, Temperature-Heat Diagrams for Complex Columns. 3. Underwood’s Method for the Petlyuk Configuration. Ind. Eng. Chem. Res., 28, pp. 1386. Dünnebier, Guido; Pantelides, C.C., 1999, Optimal Design of Thermally Coupled Distillation Columns, Ind. Eng. Chem. Res., 38, 162-176. Fidkowski, Z. T.; Agrawal, R., 2001, Multicomponent Thermally Coupled Systems of Distillation Columns at Minimum Reflux, AIChE Journal, Vol. 47, No. 12, pp. 2713. Glinos, K.;Malone, M.F, 1988, Optimality Regions for Complex Column Alternatives in Distillation Systems. Chem. Engin. Res. Des., 66, 229. Halvorsen, I. J.; Skogestad, S., 1999, Optimal Operation of Petlyuk Distillation: Steady-state Behavior, Journal of Process Control, 9, pp. 407. Heida, B.; Bohner, G. Kindler, K., March 2002, Consider Divided-Wall Technology for Butadiene Extraction. Hydrocarbon Processing, pp. 50-B.
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Hernández, S.; Jiménez, A., 1999, Design of Energy-Efficient Petlyuk Systems, Comp and Chem Eng., 23, pp. 1005. Jobson, M., April 2005, Dividing Wall Distillation Comes of Age, The Chemical Engineer, pp. 30. Kaibel, G.; Miller, C.; Stroezel, M.; von Watzdorf, R.; Janse, H., 2004, Industrial Use of Dividing-Wall Columns and Thermally Coupled Distillation Columns. Chemie Ingenieur Technik, 76(3), pp. 258. Lestak, F.; Collins, Cyril, July 1997, Advanced Distillation Saves Energy and Capital. Chemical Engineering, pp. 72. Nikolaides, I. P.; Malone, M. M., 1988, Approximate Design and Optimization of a Thermally Coupled Distillation with Prefractionation. Ind. Eng. Chem. Res., 27, pp. 811. Petlyuk, F. B.; Platanov, V. M.; Slavinskii, D.M, 1965, Thermodynamically Optimal Method for Separating Multicomponent Mixtures. Int. Chem. Eng. ,5, pp. 555. Schultz, M. A.; Stewart, D. G.; Harris, J. W.; Rosenblum, S. P; Shakur, M. S.; O’Brien, D. E., 2001, Reduce Costs with Dividing Wall Columns, Chemical Engineering Progress, 98(5), pp. 64. Serra, M.; Espuña, A.; Puigjaner, L., 2003, Controllability of Different Multicomponent Distillation Methods, Ind. Eng. Chem. Res., 42, pp. 1773. Shah, P., 2002, Squeeze More Out of Complex Columns, Chemical Engineering Progress, 98(7), pp. 46. Triantafyllou, C.; Smith, R., 1992, The Design and Optimisation of Fully Thermally Coupled Distillation Columns. Trans IChemE, Vol. 70, Part A, 118-132.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
On the rapid development of new products through empirical modeling with diverse data-bases John F. MacGregora , Koji Mutekia, Toshihiro Uedab a
Department of Chemical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada b Mitsubishi Chemical Corporation, Yokkaichi, Mie, Japan
Abstract This research addresses the development of industrial products with specified final properties. There are basically three major degrees of freedom to control the final product properties: the selection of the raw materials, the selection of the ratios in which to use the selected materials, and the process conditions under which to manufacture the product. A new approach to modeling and optimization is presented that simultaneously takes into account all of these degrees of freedom for the development of new products. The approach involves building multi-block Partial Least Squares (PLS) models that pull together diverse industrial databases on previously made products and on the properties of the component materials used to make these products. The resulting models are then used in an optimization framework to select raw materials from much larger databases (including materials never previously used), to select the ratios in which to combine them and to select conditions under which to process them in order to yield a product with specified end properties at a minimum cost. The methodology has been successfully used for the development of several industrial products. Keywords: partial least square (PLS); mixture designs; polymer blends; optimization; design of experiments (DOE);
1. Introduction A new data-based modeling and optimization strategy that simultaneously takes into account both the selection of raw materials and the ratios in which to blend them is presented. The final goal is to speedily achieve target products having specified properties with minimum experiments and minimum total material costs ([1]). The databases generally available for this problem are depicted at Fig.1. The ( M × N ) R matrix contains the ratios of all the materials used in the formation of previous blends, the process blending conditions ( Z ) used during these blending experiments and the ( M × L) Y matrix containing L properties measured on the final blends. M is the number of previous blends, and N is the number of all the raw materials used in the blends. The data in R contains the fraction of each raw material used in the blend ( 0 ≤ ri , j ≤ 1, and ∑ ri , j = 1 ). The ( NN × K ) X DB matrix contains the information on all the available raw materials including both those used and not used in the past. K is the number of raw material properties and NN is the number of all the available raw materials. Most data on the X DB matrix can be often obtained from suppliers of the raw materials, while some additional measurements may have to be
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measured inside the blending company. The ( K × N ) XT matrix consists of the raw material property data that corresponds to the subset of materials used in the past experimental blends R . The X T matrix is a subset of the much larger raw material database of X DB . Raw material property
Exp No.
Process condition
Z (M × J )
Material
XT
Material
Raw Material Property
X
DB ( NN × K )
Database
(K × N ) Material
R (M × N )
Mixture Property
Y ( M × L)
Fig.1 Database available in industrial mixture design As an important feature of this data structure, there are no common dimensions over all the data matrices. The Z , R and Y matrices have one dimension in common (i.e. the blending material sample direction). X T has one dimension in common with the R matrix, but no dimension common to Y . That is, X T has the indirect relationship to Y , only through R . This kind of T-shape data structure can occur in situations when auxiliary data such as X T are obtained from a different source. The problem of finding the optimal selection of raw materials and their mixture ratios is something that has not been treated very efficiently in most industrial applications. Traditional approaches tend to treat the two problems (material selection and blend ratios) as separate steps. A set of raw materials is selected usually based on the experimenters’ best guess, and then a set of blending experiments (perhaps designed or trial and error) are run to see if the target properties can be achieved. If the results are not entirely acceptable another set of raw materials is selected and the process repeated. Such approaches lead to many blending experiments, a very inadequate investigation of the large number of possible materials that could be used, and a very long development time. It is the objective of this paper to present a methodology that can greatly reduce the development time for new products. The methodology simultaneously treats the problem of the selection of raw materials and their ratios by effectively exploiting the readily available data on past blending runs and on raw material properties (Fig.1).
2. Methodology 2.1. Mixture rules to combine the databases In the paper, the concept of the “ideal mixing rule” (Grassman et al [2]) is employed to combine the raw material property data matrix X T and the mixture ratio matrix R . Muteki and MacGregor [3] present a more general multi-block PLS approach for combining these matrices, but show that under assumptions that generally will hold in polymer blending problems, it essentially leads to this simpler blending rule approach used here. The mixing raw material property matrix X mix ( M × K ) is simply expressed as (1) X mix = R ⋅ X
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Clearly, properties such as the compound content (e.g. styrene mass percentage in styrene-butadiene rubber) follow this ideal blending rule exactly. It is also known that many polymer properties such as the weight average molecular weight approximately follow the ideal mixing rule (Grassman et al [2]). 2.2. Mixture PLS modeling using raw material property data information The traditional mixture models such as Scheffe model and mixture PLS models (Kettaneh-Wold [4]) represent only the relationship between mixture ratios matrix R and mixture property matrix Y . The model forms are generically expressed as (2) Y = f (R ) + ε In this work we use a new mixture modeling approach based on the mixing rules discussed above. The model is expressed as ` (3) Y = f ( X mix , Z) + ε where X mix (= R ⋅ X) as defined in Eq.(1). The mixture PLS model of Eq.(3) represents the relationship between the blend rule raw material properties ( X mix ) and the final product blend properties ( Y ).A Partial Least Square (PLS) regression model will be used to obtain the relationship in Eq.(3). The new mixture PLS models need the following two assumptions. The first is that the ideal mixing rules are approximately valid. The second is that the raw material properties available correlate well with the final blend properties ( Y ). This will be naturally satisfied in most industrial applications because the raw material properties ( X ) are measured because of their expected importance in any polymer blending. Muteki et al [5] demonstrate that the new mixture PLS model works well in practical industrial examples, unless one of above two conditions is severely violated. They also show several advantages of these new mixture PLS models: (1) they provide a direct interpretation of the effect of raw material properties on final blend properties, that is, what raw material properties affect what the mixture properties, something that traditional mixture models using only the R + Y matrices can not do; (2) the new mixture PLS models provide better estimates of the blend properties ( Y ) than the traditional mixture models in most industrial cases; (3) with new material grades which have never been used in the past, the new mixture PLS models can estimate final blend product properties by using only their raw material property data without implementing additional blending experiments; (4) introducing the X mix (= R ⋅ X) matrix (i.e. ideal mixing rule) opens the door to the new approaches to the design of mixture experiments (DOE) that can take into account both the effect of raw material properties and the effect of their blend ratios; and (5) the new mixture PLS models also open the door to the simultaneous selection of raw materials and their ratios in the optimization, as shown in this paper. 2.3. Optimization based on the new mixture PLS models In this section, we assume that the required data is available and a PLS model between X mix and the final blend Y has been built. The objective is now to use this model to simultaneously optimize the selection of the best raw materials and their blend ratios in order to achieve the desired mixture property vector y des with a minimum total
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raw material costs and a minimum number of raw materials. The formulation of optimization is expressed as NN
NN
j =1
j =1
Min ( ydes − xmix new ⋅ BPLS )T ⋅ W1 ⋅ ( ydes − xmix new ⋅ BPLS ) + w2 ⋅ ∑ rnew, j ⋅ c j + w3 ⋅ ∑ δ j rnew
Ideal mixing rule
s.t. xmix new = rnew ⋅ X DB k =K
(4)
SPEnew = ∑ ( xmix new − xˆmix new ) 2 ≅ 0 PLS model constraint Mixture constraint
Binary variable constraint
A
k =1 2 new, a
a =1
sa
Tnew = ∑ 2
NN
∑r j =1
new, j
τ
≤ const
= 1, 0 ≤ rnew, j ≤ 1
⎧1 r >0 δ j = ⎨ new, j =0 0 r , new j ⎩ rnew, j ≤ M j ⋅ δ j
The first term of the objective function is a weighted measure of the estimation error between the desired blend properties and the estimated blend properties through the mixture PLS model. The new raw material properties x mix new are estimated based on the ideal mixing rule between rnew and X DB . The second term in the objective function refers to the total raw material cost of the mixture. The third term in the objective function penalizes the total number of raw materials used to obtain the blend, since it is usually desirable to obtain the desired blend products using a minimum number of materials. Since the binary variable δ j is involved, this is a mixed integer quadratic optimization ([6]).
3. Industrial examples An real industrial example using this product development approach is presented in this section. The example feature the manufacture of thermo-plastic materials which are manufactured from mixtures of rubbers, polypropylenes, and oils. It can be assumed that the mixture property matrix Y is influenced by only the blend ratios and the properties of the raw materials, since the same manufacturing equipment was used and all processing conditions such as shear and extruder and mold temperatures were kept constant ( Z =constant in Fig.1). 3.1. Optimization (replacement of raw materials) We show an industrial example where the above PLS modeling and the optimization algorithm were used to design new materials. They both involve the problem of having to replace materials (rubbers) for business reasons, while maintaining the same final product blend properties, and doing so with minimum total raw material cost. In this case, the total number of raw materials (rubbers) was not penalized, and the objective function in the optimization (Equation (4)) consisted of only the first two terms error in the final product properties and the total raw material costs. Optimal product formulations for the target product (replacement of the rubber materials) were obtained by the optimization of equation (4). The results are shown in Table 1 for the product. (a) shows the materials and ratios of rubbers, oil and
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polypropylene used to make the existing product, and the total cost of this formulation, (b) shows the measured blend properties of the current product; (c) shows the new materials and their calculated ratios, obtained from our blend optimization approach, and the total cost of materials for the new product; (d) shows the property predictions (from the PLS model), the experimentally observed values of the final properties of the blend, and the quadratic distances of these properties from the target values ( y des − y ) T W1 ( y des − y ) . Existing product 2 Material + Mixture ratios used (a) Rub2
Rub6
Mixture ratios Cost
36.4 400
Final Blend Properties (b)
Oil 1 18.7 430
PP4 44.4 94.5
Y1 1 95
Observed
Y2 0.8878
Y3 32.4
Y4 0.5
Y5 1
Y6 3.9
Y7 100
75
266.77 (Total Cost)
New Product 2 New Materials + Mixture ratios (c) Rub1 Mixture ratios Cost
Final blend Properties (d)
Rub2 Rub_new2 O il1 PP3 17.67 24.75 8.83 47.93 0.82 400 400 430 94.5 95 253.7
Y1 Estimated Observed
Y2 0.8831 0.8869
Y3 30.92 28.90
Y4 0.37 0.50
Y5 0.80 1.00
Y6 3.17 4.70
Y7 97.33 100.00
78.74 0.528 81.50 0.510
(Total Cost)
(Quadratic error)
Table1 The result of optimization on target product (Rub1, Rub2, Rub6: grades from rubber materials that have been blended in the past, Rub_new2: a rubber grade that has not been blended in the past; Oil1: the grade of oil material; PP3-PP4: the grades of polypropylene materials, Y1-Y7: properties of the final blends) The raw materials that are to be replaced are Rub6 in the example (see (a) in Table 1). In the optimization solutions ((c) in Table 1), they are replaced by a selected ratio of another three rubbers (Rub1, Rub2 and Rub_new2) in the example. Note that Rub_new2 are rubber materials that had never been blended by the company in the past, but they were available in the database ( X DB _ rubber ). The other rubber materials such as the Rub2 had been often blended by the company in the past. This shows that the selection of raw materials has been truly implemented in the optimization. The effectiveness of this optimal material replacement solution can be seen by comparing the final blend properties of the existing material (targets for the new replacement blend) in (b) of Table1 with the estimated (from the PLS model) and measured values (from actual material products) in (d) of Table1. The predicted and actually achieved products had the seven properties very close to those desired. They considered the results to be within these replication errors on essentially all properties, and accepted both product development results as unqualified success. 3.2. Sensitivity analysis: Alternative blends with similar cost and similar properties Although only one optimal new blend product has been shown on each case in the previous section, there exists alternative blends with similar properties, similar costs and similar total number of raw materials. It is important to check the sensitivity of the blend optimization to small changes in the objective function, and look at alternative solutions to the blending problem provided by these sensitivity studies. As an example, four alternative blends (Case1-2: NLP, Case3-4: MINLP) obtained in this way for the target product are shown in Table2. These should be compared with the target properties in Table1 (b), and with the optimization solution actually used in Table1 (c) and (d).
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Table 2 Alternative solutions for target product: mixture results and costs and the final blend properties and quadratic errors in the predicted properties Case1 (NLP) Mixture ratios Cost
Rub5 Rub_new3 Rub10 Rub13 Oil 1 PP4 22.75 8.78 5.17 10.6 52.7 400 430 410 430 94.5
Y1 0 95 245.3325
Estimated
1.39 97 264.2908
Estimated
Y2 0.8811
Y3 27.27
Y4 0.204
Y5 0.5257
Y6 2.4046
Y7 96.615
80.676 1.4339
Case2 (NLP) Rub1 Mixture ratios Cost
Rub3 11.8 400
Rub6 32.6 400
Rub7 3.99 430
Oil 1 6.58 430
PP1 43.64 94.5
Y1 0.8868
Y2 Y3 Y4 Y5 Y6 Y7 33.8717 0.566 1.134 5.0275 99.7893 78.0299 0.1905
0.8862
Y2 Y3 Y4 Y5 Y6 Y7 33.4792 0.5194 1.0635 4.9125 99.3339 78.3977 0.1998
0.886
Y2 Y3 Y4 Y5 Y6 Y7 35.6614 0.7257 1.3396 5.2612 98.8368 75.9036 0.5193
Case3 (MINLP) Rub1 Mixture ratios Cost
Rub5 18.6 400
26.6 400
Rub_new4 Oil 1 PP4 9.3 44.2 430 94.5
Y1 1.3 95 262.559
Estimated
Case4 (MINLP) Rub1 Mixture ratios Cost
Rub3 27.5 400
Oil 1 24.3 400
PP3 44.7 94.5
Y1 3.5 95 252.7665
Estimated
The Case1 provides a more cost-effective blend than the optimal blend shown in Table1, but the quadratic error in the predicted properties is much larger, and is unacceptable for the researchers. Cases 2-3 provide more expensive blends than the optimal blend shown in Table1, but they achieve the target mixture properties more closely. The Case 4 provides an almost equivalent solution in terms of cost and quadratic error to the one actually implemented for the product in Table.1. The most significant feature is that all the blend conditions in Case 1-4 select very different combinations of raw materials. With such sensitivity results, the researcher can consider trade-offs between the closeness to the target properties, the total material cost, the total number of raw materials and other intangibles such as availability and reliability of suppliers, etc.
4. Conclusion An optimization approach to the development of new products with desired final properties is presented. The approach simultaneously considers the selection of raw materials, their required blending ratios and the process conditions. The methodology was applied very successfully to some industrial polymer blending problems involving the replacement of materials. However, the methodology is much more general, and should be applicable to a wide variety of product development problems, such as the development of new catalysts, food products, pharmaceuticals, etc.
References [1] K. Muteki, J. F. MacGregor, T.Ueda, On the rapid development of new polymer blends: The optimal selection of materials and blend ratios, submitted to Industrial Engineering Chemical Research (2005) [2] P.Grassmann, H.Sawistowski, R.Hardbottle, Physical principles of chemical engineering, Pergamon Press, New York, 1971 [3] K. Muteki, J. F. MacGregor, Multiblock PLS for L-shape data structures in mixture design, to be submitted to Chemometrics and Intelligent Laboratory Systems (2006) [4] N. Kettaneh-Wold. (1992) “Analysis of mixture data with partial least squares”. Chemometrics and Intelligent Laboratory Systems, Vol.14.pp. 57-69. [5] K. Muteki, J. F. MacGregor, T. Ueda, Mixture designs that include material properties, to be submitted to Chemometrics and Intelligent Laboratory Systems (2006) [6] I. Quesada, I.E. Grossmann, An LP/NLP based branch and bound aldorithm for convex MINLP optimization problems, Computer Chem Engng, Vol.16, No.10/11, pp.937-947, (1992)
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Conceptual design of reactive distillation flowsheets Guido Daniel,a Prashant Patil, a Megan Jobson a a
Centre for Process Integration, School of CEAS, The University of Manchester, PO Box 88, Manchester M60 1QD, United Kingdom
Abstract Economic and environmental reasons have lead to process intensifications in the process industries. Reactive distillation is the most prominent example. However, often it is a challenge to satisfy the requirement for sufficient catalyst volume for the reaction while providing the interfacial area needed for mass transfer. The combination of a reactive distillation column with a pre-reactor is a valuable alternative. This paper presents an approach to identify promising designs for such flowsheets and the optimum distiribution of the reaction extent between the pre-reactor and the reactive distillation column. The methodology uses a boundary value method for the design of the column; Chemical equilibrium is assumed. The column usually consists of one reactive core, two rectifying sections and one stripping section. The methodology will be demonstrated for an ETBE case study. Keywords: Reactive distillation, Conceptual design, Boundary value method
1. Introduction To exploit the potential of reactive distillation, methods have been developed for preliminary process design. Two major approaches exist for the generation of alternatives for a given reaction-separation problem. Mathematical optimisation methods have been developed [1-4]. However, they do not provide valuable and necessary insights into the process and are generally computationally intensive. Graphically-based methods overcome this problem [5-10]. Recently a methodology for reactive distillation columns based on a boundary value method (BVM) has been presented [11], which generates multiple designs without highly iterative procedures. As powerful as the design methods presented are, they often fail to address the practicality of the proposed columns. In heterogeneously catalysed reactive distillation columns one often needs to maximize the catalyst volume inside the column [12], especially in slow reaction systems. A flowsheet configuration to overcome these practical problems for reactive distillation columns is the combination with a prereactor. In industry, often a substantial part of the conversion is carried out in reactors upstream of the reactive distillation column [13], thus reducing the catalyst volume in the column. This paper presents a methodology to identify near-optimal flowsheet configurations for reactive distillation flowsheets involving a ‘finishing’ reactive distillation column and a pre-reactor. The methodology uses a graphically-based boundary value method for identifying the promising column designs for a chosen prereactor configuration. A cost function will be used to rank the feasible designs. Chemical equilibrium is assumed and systems with two degrees of freedom, according to Gibbs phase rule, are considered. The methodology will be illustrated for ETBE production.
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2. Flowsheet design 2.1. General considerations The flowsheet considered includes heat exchangers upstream and downstream of the pre-reactor to preheat the feed to the reactor temperature and to adjust the feed condition for the column, respectively. The reactor is assumed to be isothermal. An expansion valve depressurises the feed from the reactor pressure (to be chosen such that the reactor effluent is in the liquid phase) to the pressure of the reactive distillation column. It is assumed that in the pre-reactor, as well as on every reactive stage of the column, chemical equilibrium is reached. 2.2. Methodology The following assumptions enable us to use the boundary value method for the design of the reactive distillation column: • Feed flow rate and composition are fixed and given • The product purities are given • A reasonable range for the pre-reactor temperature is given (might be constrained due to catalyst operating temperature, etc.) • Pressure of the column and reactor are specified The design parameters of the flowsheet are the inlet temperature of the pre-reactor TR, the inlet temperature of the column TF and the design details of the column itself, namely the operating conditions reboil and reflux ratio, the number of non-reactive and reactive stages inside the column and the feed stage location. The algorithm to solve this design problem is shown in Figure 1. For a set of operating conditions of the prereactor the designs for the reactive distillation column are calculated based on boundary value methods. The costs for the flowsheets are calculated and the flowsheets are ranked accordingly.
Figure 1 Algorithm for the identification of near-optimal pre-reactor – reactive distillation flowsheets
The cost function for the flowsheets involves the annualised capital cost of the equipment and operating costs for the utilities used.
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3. Conceptual design of the reactive distillation column 3.1. Boundary Value Methods Design methodologies using Boundary Value Methods are based on the ideas first developed for reactive distillation columns by Barbosa and Doherty [5-6]. For fully specified product compositions a feasible design is characterised by a continuous composition profile through the column, starting from the products. The profiles can be calculated via tray-by-tray calculations for a given reflux (or reboil) ratio. However, all approaches for design of columns achieving chemical equilibrium have in common that they assume the feed is located within the reactive section. This assumption may lead to backward reaction inside the column, depending on the operating conditions [8]. In general the reaction extent is concentrated around the feed stage and it may happen that on many reactive stages nearly no reaction takes place. In order to identify feasible and economical designs for a given design problem, reactive stages should only be used when necessary due to their high cost. Thus a design methodology should incorporate information on the reaction extent during the calculation procedure. The only approach incorporating this information is that of Espinosa et al. [8]. However, their approach concentrates on the feasibility for such columns; it aims for the identification of a single design at minimum reflux condition and involves no process economics. 3.2. Structure of the finishing reactive distillation column This work presents a methodology to identify near-optimal designs for finishing reactive distillation columns. The main part of the reaction takes place in a pre-reactor upstream of the column. The task of the finishing column is to increase the overall reaction extent, while producing the desired products with the specified purity.
(b) ( a) Figure 2: (a) Different zones for finishing reactive disitllation columns; (b) Boundary condition and feasibility criterion ( ξ RD , ξ PR and ξTotal represent the reaction extent in the column, reactor and overall process, respectively) .
Such flowsheets represent the typical implementation of reactive distillation in industry [14]. The columns are normally structured as shown in Figure 1 (a). The column
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consists of four different zones serving different functions. This structure is especially suited for etherification reactions, including ETBE, MTBE or TAME production. In the non-reactive stripping section, the main product (e.g. ETBE) is purified from the reactants. In the non-reactive rectifying section II the product, entering the column with the pre-reacted feed stream is separated from the reactants. This zone is necessary to avoid backward reactions in the lower part of the reactive section. In the reactive section the remaining part of the reaction takes place. In the non-reactive rectifying section I the low-boiling inert components and other reactants are separated from the remaining product. 3.3. Design procedure for finishing reactive distillation columns The first step in the design procedure is to identify the product composition and the required reaction extent in the column. The input for the methodology is the composition and flow at the outlet of a pre-reactor. The definition of product purities for the bottom product and distillate enables the calculation of the overall mass balance of the column. Thus the reaction extent inside the column and the distillate and bottom product flow rates will be calculated. The overall reaction extent is the sum of the extent in the pre-reactor, ξ PR , and that of the column ξ RD . Once the purities for the bottom product and distillate are known, the column composition profiles can be calculated starting from the product compositions. To identify a range of feasible designs the profiles are calculated for several reflux and reboil ratios. Thermodynamic equilibrium is assumed on every stage. In the next step the starting point for the composition profile of the reactive core has to be determined. Here the distance to the reactive equilibrium surface is taken as a measure. If the distiance is below a given tolerance the point is accepted and the stage is tested for a minim reaction extent to be accomplished. The calculation of composition profiles proceed from the first stage of the reactive core for the corresponding reflux ratios. For the calculation of reactive profiles reactive and thermodynamic equilibrium is assumed. The calculation of the reactive core proceeds until the required reaction extent in the column is attained. This boundary condition (see Figure 1 (b)) can be formulated as:
ξTotal − ξ PR − ξ RD ≤ ε
(1)
At the stage where the boundary condition is satisfied, the non-reactive rectifying section II starts. The composition profiles can be calculated in the same manner as for the upper non-reactive rectifying section with the difference that the starting composition is that of the boundary condition and not that of the distillate. The nonreactive stripping profile is calculated starting from the bottom product for a range of reboil ratios. For a feasible design, a continuous profile throughout the column has to exist. The feasibility criterion is the intersection of the stripping section profiles and the profiles of the non-reactive rectifying section II (see also Figure 1 (b)). BVMs suffer from the high sensitivity of the composition profiles with respect to trace components in the products. To reduce this sensitivity a product region is defined for the bottom product and sets of composition profiles (manifolds) are used in the intersection search. This leads to a more robust design procedure.
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4. Illustrative example The proposed methodology has been used for design of a reactive disitllation flowsheet for the production of ETBE. The feed stream consists of a C4-stream containing the reactive component isobutene and an inert component n-butane as well as ethanol. The following reaction takes place over ion exchange resin catalysts as Amberlyst 15: Isobutene + Ethanol ↔ ETBE. The industrial flowsheet for ETBE and MTBE production are identical [15]. The reactants are fed to a pre-reactor (isothermal or adiabatic), where the feed nearly reaches chemical equilibrium. In the finishing reactive distillation column the remaining part of the reaction takes place and ETBE is produced at the desired purity. The range of reactor temperatures considered is 50-80 °C. The feed contains approximately 8 % excess ethanol, relative to to isobutene, to suppress side reactions and enhance the isobutene conversion. Table 1. Product and feed compositions for ETBE system Stoichiometric Boiling Distillate Components coefficients temp. at 8 bar Compositon TB υ xD [-] [°C] mole fr. Isobutene -1 60.8 5.9E-6# n-Butane 0 69.4 0.952# Ethanol -1 142.1 0.048 ETBE 1 156.7 4.1E-8# Flow rate [kmol/h] 98.68 # Specified values
Bottom product composition xB mole fr. 2.2E-4# 0.002 0.007# 0.991# 62.53
Feed composition xF mole fr. 0.278# 0.421# 0.301# 0.0# 223.2#
The reactor is modelled using Aspen PlusTM and the column feed stream is assumed to be in chemical equilibrium. The chemical equilibrium is described with an equation presented by Sundmacher et al. [16]. The column pressure is 8 bar. Table 1 shows the chosen purities for the system together with calculated values for distillate and bottom product. The chosen purities lead to an overall conversion of 99.97 % for isobutene; nearly pure ETBE is collected in the bottom product stream. At the top of the column a near-azeotropic mixture of n-butane and ethanol is produced. For this system the structure shown in Figure 1 (a) is needed to produce nearly pure ETBE since ETBE is not located on the reactive surface [9]. For the calculation of all thermodynamic properties an interface to Aspen Plus™ is used. The chosen range for the reflux and reboil ratio is 0.2 to 15. Energy balances are included during the calculation of composition profiles. Multiple column designs of various configurations are obtained for these specifications for each temperature of the reactor. Table 2 shows the best designs for a temperature grid of 10 °C based on total annualised cost. In Table 2 NRI denotes the number of non-reactive stages in rectifying section I, NRC the number of reactive stages in the reactive core, NRII the number of non-reactive stages in rectifying section II, NS the number of non-reactive stripping stages, R the reflux ratio and S the reboil ratio. It can be seen that the optimal temperature of the pre-reactor is around 60 °C. The results obtained can be used to initialise simulations with Aspen Plus™. Excellent agreement between the designs generated and the simulation results reveals the potential of the approach to generate and evaluate designs for a given reaction system systematically.
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Table 2. Best flowsheet designs based on total annualised cost TR NRI NRC NRII NS NTOT R S Cond. Duty Reb. Duty [°C] [-] [-] [-] [-] [-] [-] [-] [kW] [kW]
Total Cost [ €/yr]
50
1
7
6
9
23
1.63
1
1317
415
359000
60
1
7
7
7
22
1.88
1
1440
415
348000
70
4
7
7
9
27
1.99
1
1498
415
357000
80
4
7
4
5
20
2.34
2.84
1670
1180
400000
5. Conclusions This paper presents a methodology for the conceptual design of pre-reactor – reactive distillation flowsheets. These are advantageous in cases where practical issues favour the use of a pre-reactor. The approach avoids designs that give rise to reverse reaction in the column. The methodology identifies the optimal distribution of the reaction extent between column and pre-reactor. The approach has been illustrated for the production of ETBE.
Acknowledgements We acknowledge the financial support provided by the European Commision within the 6th Framework Programme, Project "INSERT – Integrating Separation and Reaction Technologies"; Contract-No: NMP2-CT-2003-505862.
References [1] Ciric, A. R. and Gu, D. (1994) AIChE J., 40, 1479-87. [2] Cardoso, M. F., Salcedo, R. L., de Azevedo, S. F. and Barbosa, D. (2000) Chem. Eng. Sci., 55, 5059-5078. [3] Jackson, J. R. and Grossmann, I. E. (2001) Comp. Chem. Eng., 25, 1661-1673. [4] Stichlmair, J. and Frey, T. (2001) Ind. Eng. Chem. Res., 40, 5978-5982. [5] Barbosa, D. and Doherty, M. F. (1988a) Chem. Eng. Sci., 43, 2377-89. [6] Barbosa, D. and Doherty, M. F. (1988b) Chem. Eng. Sci., 43, 1523-37. [7] Bessling, B., Schembecker, G. and Simmrock, K. H. (1997) Ind. Eng. Chem. Res., 36, 30323042. [8] Espinosa, J., Aguirre, P., Frey, T. and Stichlmair, J. (1999) Ind. Eng. Chem. Res., 38, 187-196. [9] Espinosa, J., Aguirre, P. A. and Perez, G. A. (1995) Ind. Eng. Chem. Res., 34, 853-61. [10] Hoffmaster, W. R. and Hauan, S. (2004) Chem. Eng. Sci., 59, 3671-3693. [11] Dragomir, R. M. and Jobson, M. (2005) Chemical Engineering Science, 60, 5049-5068. [12] Nava, J. A. O., Baur, R. and Krishna, R. (2004) Chem. Eng. Res. Des., 82, 160-166. [13] Subawalla, H. and Fair, J. R. (1999) Ind. Eng. Chem. Res., 38, 3696-3709. [14] Reusch, D., Janowsky, R. and Sakuth, M. (2001) In Ullmann's Encyclopedia of Industrial Chemistry. [15] Hamid, H., Ashraf Ali, M. and Editors (2004) Handbook of MTBE and Other Gasoline Oxygenates, Marcel Dekker, New York. [16] Thiel, C., Sundmacher, K. and Hoffmann, U. (1997) Chem. Eng. J. (Lausanne), 66, 181-192
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
LCA of a Spent Lube Oil Re-refining Process a
Tom N. Kalnesa, David R. Shonnardb, Andreas Schuppelc
UOP LLC, 25 E. Algonquin Rd., DesPlaines, Ill., USA, 60017-5017 Michigan Tech. Univesity, 1400 Townsend Drive, Houghton, M, USA 49931-1295 b Puralube GmbH, Hauptstrasse 30, Gebaude 27, D-06729, Troglitz, Zeitz,Germany b
Abstract Although re-refining of spent lubricating oils (used oils) has been practiced with varying technical and commercial success for over the past 50 years, a sustainable processing technology has yet to become widely accepted. Poor on-stream efficiency, inconsistent product quality, and careless management of feedstock contaminants and byproducts have often resulted in widespread environmental problems and poor economics. Environmentally-conscious design of processes and products is increasingly viewed as an integral strategy in the sustainable development of new refining and chemical processes. Life cycle assessment is becoming the preferred methodology for comparing the environmental impacts of competing processes. A life cycle analyses of a promising new re-refining technology, the HyLubeTM process, has been undertaken to quantify the intrinsic benefits of HyLube re-refining over the current practice of recovering used oils for fuel value. Keywords: lube, rerefining, HyLube, LCA, recycling
1. Introduction Others have studied used oil regeneration and consistently shown significant energy and emission benefits relative to virgin base oil production (Taylor 2001, Winberg 2002). However, the incremental benefit of used oil regeneration compared to used oil combustion for energy recovery is less clear (www.total 2003). In past studies, gross assumptions regarding combustion emissions, regenerated lube oil quality and regeneration byproduct management have made comparative results tentative in nature. Likewise, adequate characterization of the basic refining steps for conventional and synthetic base oil production were often limited by available data. For this reason, more complete life cycle assessments have been undertaken (Fehrenbach 2005) to determine the environmental impact of modern base oil regeneration facilities relative to other waste oil management practices. This paper focuses on the on a new and innovative rerefining technology, the UOP HyLube process. Puralube GmbH a subsidiary of Puralube Global recently commercialized the HyLube process. The first industrial scale plant, located in Troglitz/Zeitz, Germany, converts 220 metric tons per day of used oil into lube base oils and green fuels in an environmentally conscious manner. Key process features include (1) continuous processing of the entire used oil charge stock in a hydrogen-rich environment without use of intermediate oil storage, (2) efficient removal of contaminants such as chlorinated, sulfurous, and oxygenated organic compounds and polyaromatic hydrocarbons, (3) management of all sulfurous and odorous compounds to eliminate malodorous and toxic emissions and (4) production of deeply desulfurized base oil and distillate fuel products of consistent high quality. Table 1 provides a summary of the primary products produced by Puralube GmbH and their properties. A block diagram of the process is shown in Figure 1. Byproducts (fuel gas, fuel oil, and water) are managed to minimize environmental impact while maximizing energy recovery. The
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commercial performance data of the HyLube process at Puralube GmbH was used as the basis for this Life Cycle Assessment. Used oil feed to this unit includes a mixture of recovered motor oils, hydraulic oils, gear oils, and other paraffin-rich industrial oils. The data needed to characterize the production of virgin base oils and fuels as well as the emissions generated in the combustion of used oils and conventional fuels were taken from literature references. Hydrogen
Product API Gravity R E A C T O R
Viscosity @40 C
cSt
Viscosity Index Sulfur
Used Oil Feed
Heavy Lube
Diesel Oil
Fuel Oil Oil
34.4
34.0
32.8
38.8
21.8
15
28
56
3
105
0
118
121
<0.01
<0.01
<0.01
<0.5
<1
0
Water Content
wt-%
<0.01
Heating Value
MJ/kg
44.9
Fuel Gas Fuel Oil Diesel
Product Recovery Section
Medium Lube
wt-% <0.01
Color (D1500)
Fuel Oil Recovery Section
Light Lube
0.6
<0.01 41.8
Table 1. Quality of Lube products and Fuel byproducts from the commercial HyLube Process
Light Lube Medium Lube Heavy Lube Water to Treating
Figure 1. Block Diagram of the commercial HyLube process
2. Life Cycle Assessment Methods This study models the effects of re-refining in the HyLube process used oil generated in the European market. These effects are then compared to the impacts of burning the same oil in cement kilns as a substitute for the primary energy sources of either coal, heavy fuel oil (HFO) or natural gas (Nat Gas). ISO 14040 standards were followed in the study and SimaPro6.0 software was used for calculations. Figure 2 shows the material flows that are modeled for the case of combustion of used oil in cement kilns and for re-refining in the HyLube process. For combustion in a cement kiln, only the avoided flows of either coal, HFO, or natural gas are modeled. The scope of the impacts of these avoided flows is from cradle to storage. By doing this we are adopting as reference flows the production of lube base oils using conventional chemical processes. To satisfy cement kiln energy demands, we also include in the reference flows the production of coal, fuel oil, or natural gas using conventional extractive and processing methods and their combustion in cement kilns. This approach captures the essence of material reuse; the substitution of used oil for flows of virgin materials. Ecoprofiles from the EcoInvent database in SimaPro6.0 were used to develop inventories for these avoided cement kiln fuels. In our study, combustion emissions (CO2 and SO2) of used oil in a cement kiln are added and the emissions from combustion of the avoided fuel are subtracted (Fehrenbach 2005). For the HyLube™ case, when 1 kg of used oil is re-refined, 0.7 kg of base oil is produced as well as 0.15 kg of fuel oil, 0.1 kg of green diesel and water. We account for changes in the flows through refineries, petrochemical facilities and other extractive industries by contrasting them with the reference flows stated above. This is modeled by subtracting impacts of the avoided base and synthetic oil production, diesel
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production, and accounting for the avoided flow of fuel to a cement kiln. An inventory for the HyLube process was created by utilizing ecoprofiles from the Ecoinvent database within SimaPro6.0 for all of the necessary input materials and processes (water treatment) shown in Figure 2. An inventory for conventional base oil production that is avoided from refineries was created by using a residual oil ecoprofile from the Ecoinvent database in SimaPro6.0 plus material and energy consumption data for the additional base oil processing steps of solvent deasphalting, aromatic extraction, solvent dewaxing, and hydrofinishing (data from DOE sponsored study, Energetics Inc. 1998). An inventory for synthetic base oil production that is avoided from petrochemical plants was created by using an ethylene ecoprofile from the Ecoinvent database in SimaPro6.0 plus material and energy consumption data for the additional processing steps of alpha olefin (Alpha Olefins, 2004) production and polyalphaolefin production (High Performance Automotive Lubricants, 1995). The combustion emissions (CO2 and SO2) of used oil in a cement kiln are added but the emissions from combustion of the avoided fuel (HFO) are subtracted (Fehrenbach 2005). CRUDE OIL
SYNTHETIC OIL
BASE OIL From EU Refinery
Blend and Additize
Combustion of Waste Oil
Package and Sell
ENGINE SERVICE
Collection/ Transport Waste Oil
1
ReUse As Fuel*
Avoided flows: Coal, HFO, Nat Gas
CRUDE OIL
SYNTHETIC OIL
BASE OIL From EU Refinery
Blend and Additize
Re-refining of Waste Oil Avoided flow: HFO Package and Sell
ENGINE SERVICE
Collection/ Transport Waste Oil
ReUse As Fuel*
1 0.7 Base Oil
Water Treatment
Water
HFO
HyLubeTM Rerefining Process
NaOH Soda
H2
0.15
Green Diesel 0.1
Nat Steam, Gas electricity
Figure 2. Material flow diagrams for the utilization of 1 kg used oil (UO) by combustion in a cement kiln (upper) or as feed to the HyLube™ process (lower). The impacts of UO transportation are included in both cases. For the Combustion of used oil, only the avoided flows of coal, heavy fuel oil (HFO), or natural gas (Nat Gas) are modeled. In the HyLube process, the base oil produced has a high content of isoparaffins so that an estimated 0.2 kg synthetic oil can be avoided per kg WO.
3. Results / Discussion The results from this study will emphasize four categories; cumulative energy demand, climate change, acidification/eutrophication, and fossil fuel use. The reason for the focus on these impact categories is that the highest confidence in input data is for material and energy consumption, which related most closely to these categories. Data on emissions from cement kilns, especially for metals, has not been included in this study, and therefore toxicological effects are limited to fuels production steps. Figure 3 shows the cumulative energy demand for each alternative; used oil in cement kiln with
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displacement of coal, used oil in cement kiln with displacement of fuel oil, used oil in cement kiln with displacement of natural gas, and re-refining of used oil in the HyLube process. Each of the alternatives exhibit a savings of energy compared to the reference flow that was described in the LCA methods section above. The greatest savings of energy is achieved by re-refining of used oil (UO) in the HyLube process (-52.5 MJ/kg UO). This maximum benefit is realized by avoiding the production of virgin base oil, fuel oil and diesel in a refinery, and synthetic oil(s) in petrochemical plants. Producing base oils by conventional solvent refining methods and synthetic base oils by ethylene oligomerization to polyalfaolefin (PAO) require an estimated 58.2 and 95.6 MJ/kg of cumulative energy demand respectively. The next best alternative is combustion of UO with displacement of fuel oil, followed by displacement alternatives of natural gas and then coal. The maximum difference in energy savings among these alternatives is only 16%, and therefore all alternatives exhibit a very positive benefit in “unweighted” energy savings compared to the reference flows. UO Combustion; Displace Coal
UO Combustion; Displace HFO
UO Combustion; Displace Nat Gas
UO Re-refining
Figure 3. Comparison of used oil (UO) alternatives based on cumulative energy demand. HFO=heavy fuel oil, Nat Gas = natural gas
UO Combustion; Displace Coal
UO Combustion; Displace HFO
UO Combustion; Displace Nat Gas
UO Re-refining
Figure 4. Comparison of UO alternatives based on single environmental score in SimaPro6.0 using Ecoindicator 99(H).
A single environmental impact score for each UO alternative is shown in Figure 4. The EcoIndicator 99 (Hierarchist) method in SimaPro6.0 was used for aggregating eleven impact categories into one score. The impact categories in SimaPro are carcinogens, respiratory inorganics, respiratory organics, climate change, radiation, ozone layer, ecotoxicity, acidification / eutrophication, land use, minerals, and fossil fuels. The dominant category of impact for each case is fossil fuel consumption, the next most important categories are respiratory inorganics and climate change, with minor contributions from the remaining categories. Coal consumption is weighted less than petroleum or natural gas in the EcoIndicator 99 method, as seen in the relatively small benefit for the case of UO combustion with coal displacement. Re-refining of UO is the best case followed closely by UO combustion with either fuel oil or natural gas displacement. Based on our data sources, we believe that the highest accuracy predictions of impact are for climate change, acidification / eutrophication, and fossil fuel consumption. These impact categories are presented in Figure 5. The greatest benefit for climate change is for the case of UO combustion with coal displacement. This is logical because coal has the highest emission of CO2 per unit of fuel energy for any of the fossil fuels in this study, and therefore when coal is displaced by a less carbon intensive fuel, lower greenhouse emissions occur. The next best case is re-refining, followed by UO combustion with fuel oil displacement. The least desirable alternative for climate
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change is UO combustion with natural gas displacement. When UO displaces natural gas in cement kiln combustion, a higher rate of CO2 release occurs, but this increase is almost exactly compensated for by avoiding emissions from natural gas production. For the category of acidification / eutrophication, re-refining of UO is the best alternative. The least desirable alternative for this category is UO combustion with coal displacement. For fossil fuel consumption, the best alternative is re-refining of UO followed closely by UO combustion with displacement by either fuel oil or natural gas. The least desirable alternative for fossil fuel consumption is the alternative using coal displacement and UO combustion.
UO Combustion; Displace Coal
UO Combustion; Displace Nat Gas
UO Combustion; Displace HFO
UO Re-refining
Figure 5. Comparison of used oil (UO) alternatives based on specific environmental categories. All results were generated using SimaPro6.0 and the Ecoindicator 99(H) weighting methodology.
Climate Change
Acidification / Eutrophication
Fossil Fuels
The results of this study, as shown in Figures 3-5, indicate that re-refining UO in the HyLube process is more environmentally acceptable than UO combustion in cement kilns. Categories of environmental impact that were emphasized in this study are fossil fuels consumption, acidification / eutrification, and an aggregated single environmental score where weightings of all environmental categories occurs according to the Ecoindicator 99 (H) methodology. In the category of climate change UO re-refining is second only to UO combustion with displacement of coal, however it appears less and less likely for economic reasons in a situation of increasing petroleum and natural gas prices that coal will continue to be displaced. In addition, the reserves of coal are far greater than petroleum and natural gas and therefore, it is more sustainable to displace petroleum or natural gas when UO is combusted in industrial processes such as cement kilns. In this instance, re-refining of UO is much superior with regard climate change impacts.
4. Conclusions Conventional base oil production methods are energy intensive, consume a diminishing fossil fuel resource, and place a large burden on the environment. The current trend toward increasing percentages of ethylene-based synthetic base oils in the blended lubricant product further raises the overall life cycle burden of the finished product. Used oil containing high percentages of high viscosity index and low pour point base oil represents a valuable resource and its proper management should be given the most
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attention. While simple blending of used oils with low quality fuels will recover the energetic value of this material, the latent value of an engineered material containing very low aromatics and waxes is lost. Commercial data is now available for the HyLube re-refining process that allows a more complete life cycle assessment to be made relative to used oil disposal by combustion in a cement kiln. The life cycle comparison of this paper takes into account differences in combustion emissions for avoided fuels, the high quality of the re-refined base oil, byproduct management, published data sets for production of both conventional and synthetic lube base oils, and the current trend toward increasing amounts of synthetic oil in the finished products. The results of this study confirm that in almost every category, re-refining used oil in the Hylube process is more environmentally acceptable than UO combustion in cement kilns. The advantage in CO2 emissions for the UO combustion case where coal is displaced is not likely to be sustainable in the future. Substitution of coal is decreasing because there is a trend for substitution of more expensive fuels or other kinds of waste which have lower CO2 emission than coal (Frankl, 2005).
5. Comments on Sustainability For an oil recycling process to be sustainable, the social and economic aspects surrounding a given project must compliment the environmental benefits. Factors that support re-refining as a more sustainable option for used oil management include contributions to (a) local job creation, (b) regional/national energy security, and (c) competition in the lubrication oil market by replacing foreign production increasingly with local production. However, the key enabler for re-refining is a cost-effective, ecofriendly technology producing consistent high quality, high value products.
Reference Alpha Olefins, 2004, Chem Systems, January 2004 PERP Report, (02/03-04) Energetics Inc., 1998, , Energy and Environmental Profile of the U.S. Petroleum Refining Industry, prepared by Energetics Incorporated for the United States Department of Energy Fehrenbach, H. 2005, Ecological and energetic assessment of re-refining used oils to base oils: Substitution of primarily produced base oils including semi-synthetic and synthetic compounds, Institut für Energie- und Umweltforschung GmbH (IFEU), a study commissioned by GEIR Groupement Européen de l’Industrie de la Régénération. Frankl, P., Fullana, P., Baitz, M., 2005, Europe Life Cycle Considerations on Waste Oils and Implications for Public Policy - Ecobilancio Italia, ESCI Hartmann, C., 2005, Clean, Competitive, and Sustainable: Why the Recycling of Waste Oils Must Remain an EU Policy Priority, GEIR (Groupement Européen de l’Industrie de la Régénération), position paper, Square Marie Lousie 49 – 1000 Brussels High Performance Automotive Lubricants, 1995, Chem Systems, May 1995, 93S4 Taylor Nelson Soffres Consulting, 2001,Critical review of Existing Studies and Life Cycle Analysis of the Regeneration and Incineration of WO, 20 AW 83-5 (December 2005) Winberg, H. 2002, LCA for production of re-refined base oil, Industriellt Miljoskyde, Kungliga Tekniska Hogsloan www.total.com/static/en/medias/topic103/ Total_2003_fs09_Used_lubricant_disposal.pdf
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Effective Process Design Instruction: From Simulation to Plant Design Daniel R. Lewina , Eyal Dassaua and Alon Goldis,b a
PSE Research Group, Chemical Engineering, Technion, Haifa, Israel Nilit, Migdal HaEmek, Israel
b
Abstract This paper describes a new, three-semester sequence of courses, in which students are taught the theory and practice of developing a complete process package. The sequence first teaches the efficient usage of process simulation, followed by a formal process design course, and then rounds off by imparting the skills necessary to perform detailed equipment design. A unique feature is the usage of the same design project, at increasing levels of detail in each of the courses in the sequence. Keywords: Process Design, Process Simulation, Education.
1. Introduction - Status Before The Start Of The New Sequence As in many universities, practical design was traditionally taught to undergraduates at the Technion’s Department of Chemical Engineering by affiliated practicing process engineers. The original organization of the design sequence, shown in Figure 1(a), involved two courses. The first, 054404 Plant Design, consisted of a series of lectures in which the theoretical material was taught in the seventh semester, which prepared the students to take on a modest process design project in the eighth semester (in 054405 Project). The subjects covered in the first of these courses included a review of material and energy balances, how to carry out piping calculations, and the practical design of pumps, compressors, heat exchangers and distillation columns, as well as instruction on the preparation of piping and instrument diagrams and plant layout diagrams. In parallel with the project, students also took the theoretical design course, 054402 Design and Analysis, taught by a full-time faculty member. The required expertise in process simulation, necessary for both of the eighth semester courses, was acquired by the students through simulation exercises introduced into core courses covering heat transfer, multicomponent separations and reactor design. The main problems with the original design sequence were: a. The “academic” and “practical” engineering streams are in parallel with no clear interaction, creating the impression in our students that these two streams are not seriously related. Often, the industrial affiliate reinforced this impression by repeating coverage of several topics in 054404 Plant Design that had already been taught in previous core courses. b. The complexity of the design project that the students could tackle in 054405 Project was limited mainly because in the course of a single semester, they needed to become familiar with the flowsheet, carry out simulations to generate data, and then do the detailed design. This significant time constraint precluded any sophistication in the design as well as limiting the scope of the project. c. Much of the students’ time in the final design project in course 054405 was occupied in filling out specification sheets for each item of equipment and even for instrumentation. This was considered a waste of time by the students, and in our opinion, this impression was justified. Furthermore, students were required to
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tackle two distinct projects in the same semester: a full flowsheet optimization project in course 054402, which combined heat and power integration with optimization, and a detailed plant design of a more limited flowsheet in 054405.
(a)
(b)
Fig. 1. Flowchart of design sequence: (a) before upgrade; (b) after upgrade, with new courses shaded. Shown in brackets is an indication of the course load – a weekly lecture hour is worth 1.0 point, while an exercise hour is worth 0.5 point.
2. Description of New Design Sequence Driven by a desire to streamline the core curriculum, the department decided to upgrade the courses covering engineering design. As shown in Figure 1(b), the proposed new sequence: a. Introduces a new course, 054330 Simulations Laboratory, which gives students the opportunity to learn how to use process simulation efficiently to analyze and synthesize process flowsheets. This course is positioned “just in time” (Lewin et al, 2002) in the sixth semester, in parallel with courses in multiple-stage separation, reactor design, and process control, and one semester ahead of and, together with 054305 Separation Processes, a prerequisite for 054402 Design and Analysis. This course calls for weekly computer-lab sessions, each of two hours, during which students systematically learn how to use the main features of the commercial flowsheet simulator, HYSYS.Plant®, using self-paced multimedia support (Lewin et al, 2003). As part of the course, students are expected to familiarize themselves with the simulation of a generic flowsheet for what will become their project in the subsequent course sequence. b. Moves the course 054402 Design and Analysis to the seventh semester. This course integrates all of the engineering knowledge acquired over the previous six semesters of study, closely following Seider et al (2004). On completing the course, a successful student is expected to be able to: (a) Carry out a detailed simulation of a chemical process using HYSYS and interpret the results; (b) Design a train of separation units, including both ordinary multicomponent and azeotropic towers; (c) Design a heat exchanger network (HEN) for a chemical process such that the maximum energy is recovered or the minimum number of exchangers is used; (d) Suggest reasonable plantwide control configurations using qualitative methods; (e) Carry out a HAZOP analysis using a piping and instrumentation diagram
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(P&ID); (f) Formulate and solve linear optimization problems using linear programming and optimize small-scale processes using HYSYS. As part of this new arrangement, a major class objective involves the optimization of the flowsheet of the same process simulated in the previous semester. This is tackled by groups of up to five students, whose project grade is in proportion to the venture profit (VP) of their design. The course 054401 Economic Considerations is an engineering economics course, whose materials complements 054402 and taught in parallel in the new sequence. c. Merges the old courses 054404 Plant Design and 054405 Project, which together had a 5-point load, into a single course 054410 Plant Design, with a load of only 3.5 points. This new course is positioned after 054402 Design and Analysis, which is now its prerequisite, and combines the critical lecture materials in the old version of the course with a project component. More importantly, with the new positioning of 054402, this opens an opportunity for changing the scope of the design project tackled in 054410, now seeded by the three best designs submitted in 054402. The scale of the project leads to new, never-before considered, but nonetheless valuable, management and leadership challenges, since to get the work done on time, the student groups are now imaginary “companies” of between 20-21 students. The registered students are divided between three companies by the course lecturer to achieve a balance of skilled labor, but the internal organization into design teams by process section, the division of labor between the design-teams, and the selection of team leaders are all decisions made by the students themselves. This is an ambitious game-plan for an undergraduate course, and required two main implementation issues to be addressed: 1. Timetable. Given the serious time-constraints involved in this course, it was decided to present the course lectures (10 lectures of between 2-3 hours each) twice a week instead of the accepted once a week. In this way, all of the theoretical material required by the students to efficiently perform their work was covered in the first five weeks of the semester, leaving the remaining eight weeks for project work. During the semester, there is time for five formal group meetings with each company to obtain updates on progress. 2. Assessment. The assessment of student performance in cases involving large groups is obviously problematic. For the first offering, we decided to grade the students as follows: 30% on a closed-book mid-term proficiency exam, 60% on the project, and 10% personal performance. An addition 5% bonus was added to the grade of selected team leaders, on the basis of merit. Most of the project grade (80%) was for the final report (typically a five-volume set of some 300 pages), with additional credit for the group performance in the final presentations. Note that the project grade is given to the company as a whole and not to each individual section separately; the students were advised at the start of the importance of balancing load and skills between the design teams in each company. The personal grade was for each student’s individual presentations, which was given at one of the weekly meetings during the semester, and for the performance of each student in the final oral defense of the design. In retrospect, the distribution of the grading should be shifted to allow a larger component to individual performance (the split for next year will be 20:50:30 and more reliable methods will be used for individual assessment).
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3. First Implementation The class of 2006 was the first to be exposed to the new approach. Over the threesemester sequence, they worked on the HDA process, in which benzene is manufactured from toluene and hydrogen, producing also methane and biphenyl. In Semester 6, they simulated the process, working in groups of two students. In the following semester, they were given a preliminary design, shown in Figure 2, which they were told was proposed by the engineering office of Emek Projects Ltd (EPL), which: (a) produces light gases (LG) under specification that cannot be sold, with a resulting loss of revenue; (b) requires the treatment of a biphenyl waste stream at an annual cost of $2MM; (c) features relatively little heat recovery and consequently needs a large furnace with huge energy consumption. As a result of all of these deficiencies, the EPL design has a negative VP of about $6MM. The students were told that as a consequence, the services of EPL were discontinued, and a tender was transmitted instead to 20 competing groups of students of 054402 in October 2004. By February 2005, all 20 groups had generated designs with the following common features: (a) On-spec benzene product (99% pure); (b) An additional column to purify biphenyl as a marketable product (99% pure), saving the cost of waste-treatment; (c) Production of on-spec LG (99% pure), for more profit; (d) Usage of pinch analysis for a HEN synthesis to improve overall economics; (e) An average VP of $5MM/year - The least cost-effective design submitted had a VP of $4MM/year, while the best ones had VPs close to $6MM/year.
Fig. 2. EPL’s flowsheet for the HDA process: This design requires the expensive treatment of an organic waste stream (the purged biphenyl byproduct), does not produce LG on spec at additional loss, and has minimal heat integration. The VP for this solution is -$5.93MM.
In Semester 8, the three best designs submitted in Semester 7 were used as seed designs for three competing “companies” of 20-21 students. The organization of each of the three companies was the same - four engineering teams, each responsible for the design of one of the process sectors. Each company had to decide how the work to be done would be divided between the four groups. The students assigned to the three companies were selected by the instructors of the course. However, the internal company organization, from the make-up of four separate engineering groups, the selection of group leaders, to the format and public presentation of the final design, and
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even the design of company logos, was left to the students themselves. It has been observed that the sequence as a whole, and especially the last course, promotes teamwork and leadership skills among the students. 4. Feedback from Students All Technion lecture-based courses are polled once in the last few weeks of the semester. A lecturer is graded on a scale between 1 (poor) – 5 (excellent) on a number of categories such as: his preparedness, the course organization, the clarity of presentation, and the degree to which students questions and concerns are addressed. The course 054402 Design and Analysis has consistently scored above average: the last two polls returned scores of 4.28 and 4.62, where the Technion average is about 4.0. The score for 054410 Plant Design was 4.66, compared to a score of 3.23, which was the last grade obtained by the previous lecturer of the same course, but in the original format. In addition to the Technion polls, a questionnaire was prepared to address specific questions associated with the three individual courses in the sequence as well as to gauge the students’ impressions of the complete sequence. The questionnaire consisted of 15 questions, each of which required a response on a scale of 1 (poor/low) – 5 (very good/high). Table 1 lists the questions, together with the score average and the percentage of responders who responded with a grade of 4 and above (referred to as the percentage approval). The overall impression is a very positive one, both with respect to the individual courses as well as to the overall sequence. Of particular note was the positive responses regarding: (a) the usefulness of exposure to the design project in the sixth semester (Question 2), (b) the effectiveness of the sequence in teaching students to make good design decisions (Questions 4 and 7), and (c) the level of support provided to them (Questions 3, 11 and 15). Less favourable responses were returned on issues relating to individual assessment of students (Questions 9, 10 and 14). Opportunity was provided on the questionnaire for specific comments, and there too, the general opinion of the students was that the experience gained was worth the large amount of effort expected of them. 5. Conclusions and Significance This paper has provided details of the structure and objectives of each of the three courses in the new design sequence, and also described its first implementation. The objective is to have our students experience large-scale process design, and the evidence indicates that the new sequence enables this outcome. A process package on this scale can only be completed in a semester if students arrive at the detailed design stage with an optimized, heat-integrated process design, which itself can only be attained if proficiency in the usage of process simulation is ensured in advance. The paper has includes data on the impact of the sequence on our graduating students. It is clear that the current design sequence is a clear improvement over the previous version: (a) The materials are transmitted to the students in a more efficient and streamlined fashion; (b) The focus and responsibility has shifted to the students – it is their responsibility to learn the materials in time and to perform adequately; (c) Students are and feel better prepared to take on large-scale design and development projects; (d) The sequence promotes teamwork and leadership skills among the students. At the time of writing, we proceed with the second implementation of the sequence, in which students will ultimately produce a complete process package for an LNG plant, motivated by the recent agreement in which Egypt will supply natural gas to Israel, starting in October 2006.
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4. Acknowledgements The sequence upgrade was supported by a grant from the Committee for Planning and Budgeting of the Israeli Council for Higher Education. We also appreciate the contributions of Josh Golbert, Roman Sheinman, Alex Tesler and Eytan Filiba, who provided assistance to students in the Simulations Laboratory, to Profs. Ram Lavie and Ephraim Kehat, who served as consultants to the student companies in the plant design course, and to Prof. Ishi Talmon, the Dean of Chemical Engineering, who both enthusiastically supported the design sequence upgrade and provided funds for beer and pizzas on the Plant Design Presentation Day. References [1] Lewin, D. R., W. D. Seider and J. D. Seader (2002). “Integrated Process Design Instruction,” Comput. Chem. Eng., 26(2), 295-306. [2] Lewin, D. R., W. D. Seider, J. D. Seader, E. Dassau, J. Golbert, D. Goldberg, M. J. Fucci, and R. B. Nathanson (2003). Using Process Simulators in the Chemical Engineering Curriculum – A Multimedia Guide for the Core Curriculum, Version 2.0, Multimedia CD-ROM, John Wiley, New York. [3] Seider, W. D., J. D. Seader, and D. R. Lewin (2004). Product and Process Design Principles: Synthesis, Analysis, and Evaluation, John Wiley and Sons, New York. Table 1. Questionnaire and Results for the Class of 2006. Question 1. In 054330 Simulations Lab, how effective was the multimedia in helping you to learn HYSYS? 2. In 054330 Simulations Lab, how important was it for you to start getting experience with the project before 054402? 3. How well did the lectures and exercises in 054402 Design and Analysis prepare you to take on the course project? 4. In 054402 Design and Analysis, how effective was the project in helping you to learn to make good design decisions? 5. What is your opinion of the way 054410 Plant Design was organized? 6. How well did the sequence 054330 and 054402 prepare you for 054410 Plant Design? 7. In 054410 Plant Design, how effective was the project in helping you to make good design decisions? 8. In 054410 Plant Design, is the size of the design groups (20-21 students) appropriate for the project scale? 9. In 054410 Plant Design, was the requirement to have every student make a presentation helpful to your self-confidence? 10. Did 054410 Plant Design improve your ability to work effectively as a team member? 11. Are you satisfied with the support given by the course instructors of 054410? 12. How effective was the overall sequence in giving you a better idea of real world design? 13. Is continuing the project over three semesters a good idea? 14. Was the continuing competition between groups over the sequence a good idea? 15. How effectively did the websites support the course sequence?
Average
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3.74
67
4.04
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4.08
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4.14
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3.83
64
3.92
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4.12
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3.72
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3.55
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3.54
58
4.12
78
3.79
71
4.02
71
3.71
60
4.40
84
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Developments in the Sequential Framework for Heat Exchanger Network Synthesis of industrial size problems Rahul Anantharaman and Truls Gundersen Dept. of Energy & Process Engineering, Norwegian University of Science and Technology, Kolbjorn Hejes vei 1B, NO-7491, Trondheim
Abstract A Sequential Framework for Heat Exchanger Network Synthesis (HENS) is presented and the philosophy of this iterative methodology is explained. There are two main advantages of the proposed methodology. First, the design procedure is, to a large extent, automated while keeping significant user interaction. Second, the subtasks of the framework (MILP and NLP problems) are much easier to solve numerically than the MINLP models that have been suggested for HENS. The limiting factors of the methodology are the NLP and MILP models where enhanced convex estimators are required to reach global optimum in the former while significant improvements are required to prevent combinatorial explosion in the latter. This paper makes an attempt to address a few of the limiting elements of the framework. Keywords: Heat Exchanger Network Synthesis, Sequential Framework, Optimization
1. Introduction The Heat Exchanger Network Synthesis (HENS) problem involves solving a three-way trade-off between energy (E), heat transfer area (A), and how this total area is distributed into a number of heat transfer units (U). For details about the subject, see exhaustive reviews [1] and [2]. Optimization methods have been routinely applied in an effort to solve the complex and multiple trade-offs that are inherent to the HENS problem. Simultaneous MINLP models (for example [3]) can, in theory, address and solve the trade-offs in the HENS problem. These models have demonstrated severe numerical problems related to the non-linear (non-convex) and discrete (combinatorial) nature of the HENS problem. Even with the rapid advancements in computing power and optimization technology, the size of the problems solved with these methods does not meet industrial needs. The HENS problem has been proved to be NP-hard in the strong sense by [4] and has prompted a renewed interest in synthesis methods for HENS that utilize the strategy of dividing the HENS problem into a series of sub-problems to reduce the computational complexity of obtaining a network design. This paper presents developments in a Sequential Framework that combines and takes maximum benefit from thermodynamic insight, logic, heuristics and efficient optimization techniques.
2. The Sequential Framework As a compromise between Pinch Analysis and simultaneous MINLP models, a sequential and iterative framework has been in development in our group with the main objective of finding near optimal heat exchanger networks for industrial size problems.
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The subtasks of the process are solved sequentially using Mathematical Programming. Briefly, these steps involve: establishing the minimum energy consumption (LP), determining the minimum number of units (MILP), finding sets of matches and corresponding heat load distributions (HLDs) for minimum or a given number of units (MILP), and network generation and optimization (NLP) as shown in Figure 1. The Sequential Framework is based on the recognition that the selection of HLDs impacts both the quantitative (network cost) and the qualitative aspects such as network complexity, operability and controllability. The Vertical MILP model for selection of matches and the subsequent NLP model for generating and optimizing the network form the core engine of the framework. Significant user interaction is built into the framework in the form of iterative loops to enable the designer to explore and evaluate the most promising networks with respect to Total Annual Cost (TAC), network complexity (number of units, splits, etc.), operability and controllability. 2.1. Rationale for the loops in the framework The loops in the framework simulate the three-way trade-off indicated in the introduction. Loops 1 and 2 can be thought of as the area loops, loop 3 as the unit loop and finally loop 4 as the energy loop. 2.1.1. Initialization The level of heat recovery (represented by HRAT, the Heat Recovery Approach Temperature) is initialized by a pre-optimization procedure such as SuperTargeting (ST). The number of units (U) is initialized, for the corresponding HRAT, to be the minimum number of units (Umin) using a simple MILP Transhipment model allowing the Exchanger Minimum Approach Temperature (EMAT) to be zero. Using an EMAT in addition to HRAT, where EMAT = HRAT, allows heat exchange across pinch points and hence more feasible solutions. The EMAT for the Vertical Transportation Model in the core of the framework is initialized to a small value (ex. HRAT/8). 2.1.2. Loop Sequence The logical sequence of actions is indicated in Figure 1 as the following nested loops: 1. Derive networks for the second or the third best HLDs, keeping U, EMAT and HRAT unchanged: Experience has shown that the Vertical MILP Transportation Model identifies an almost perfectly ranked sequence of HLDs that leads to networks with increasing cost. The HLD loop is mainly relevant for the qualitative aspects of the network, as described earlier. 2. Adjust the value of EMAT slightly above the earlier value: Choosing EMAT is not straightforward in the Vertical MILP Transportation model as the value of EMAT is used to create additional enthalpy intervals and has to be balanced. If it is set too low, the non-vertical values of ΔTLM,mn (see Eq 1) become very small and HLDs with such non-vertical heat transfer will face large penalties. On the other hand, if EMAT is set too high, potential HLDs will be excluded from the feasible set of solutions. EMAT can be varied in two or three steps between HRAT/8 and HRAT/2. It is worth noting that EMAT performs a similar function to the +X/-X rule when optimizing networks in the Pinch Design Method. 3. Increase the number of units by one: For a given value of HRAT, the best solution is one where U is close to the corresponding Umin. Hence starting at Umin ensures that the number of loops the designer has to run through to obtain the best solution is minimal. Also, in the first run of the framework, with U = Umin, EMAT does not affect the HLDs obtained - thus loop 2 can be ignored. This loop is terminated when increasing the value of U does not lead to a decrease in the TAC.
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Figure 1: A Sequential Framework for HENS with the Vertical Transportation model
4. Adjust the value of HRAT. From the above discussion it is evident that, though there are a number of loops in the framework, the best solution is arrived at early in the synthesis process.
3. Temperature Intervals (TIs) in the Vertical MILP Transportation model The original Vertical model [5] was developed to include area considerations in the selection of HLDs. This transhipment model was modified [6] to a transportation model to include effects due to the differences in film heat transfer coefficients. This Vertical Transportation model was further enhanced [7] to reduce the size of the model. The Vertical MILP Transportation model minimizes pseudo-area as given below: ⎡ Qim, jn ⎤ min ∑∑∑∑ ⎢ ⎥ i j m n ⎣ ⎢U ij ⋅ ΔTLM ,mn ⎦⎥
(1)
where i and j are the hot and cold streams respectively and, m and n are the hot and cold temperature intervals. The model gives a ranked sequence of increasing network area when the pseudo-area of Eq 1 replicates the actual area of heat exchangers in the network. This is possible when the sizes of the TIs are small (or a large number of TIs). This can be visualized as: creating more intervals allows matching corresponding to the spaghetti structure – and thus minimum area. The transportation model is a polynomial time algorithm [4]. Hence the number of TIs must be limited to reduce computational time while ensuring that the model predicts the accurate ranked sequence. A heuristic approach for creating TIs, described in [8] for an area targeting model, was tested but failed to meet our requirements. The procedure below describes how to generate Temperature Intervals based on Enthalpy Intervals (EIs) of the balanced composite curves – a Vertical model. Step 1. Establish the balanced composite curves, using HRAT, stream and utility data. Step 2. Supply and target temperatures of all streams, including utility streams, are set to be the Primary Hot/Cold Temperatures.
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Figure 2: The Modal Trimming method
Step 3. For all cold supply and target temperatures, find adjacent hot temperatures placed vertically above the kinks of the cold composite curve. These are the Secondary Hot Temperatures. Similarly, for all hot supply and target temperatures, find adjacent cold temperatures placed vertically below the hot composite curve. These are the Secondary Cold Temperatures. Step 4. For all cold supply temperatures, find the corresponding Tertiary Hot Temperatures by adding EMAT. Disregard any hot temperature that is colder than the coldest hot target temperature. Similarly, for all hot supply temperatures, find the corresponding Tertiary Cold Temperature by subtracting EMAT. Disregard any cold temperature that is hotter than the hottest cold target temperature. Step 5. Quaternary Hot/Cold Temperatures are calculated by adding/subtracting EMAT to the Secondary Cold/Hot Temperatures. Step 6. The hot/cold temperatures from Steps 2 to 5 are merged. They are then sorted and duplicate temperatures removed to give the corresponding hot and cold TIs. Note: The number of hot TIs need not equal the number of cold TIs.
4. GLOBAL OPTIMA FOR NETWORK GENERATING NLP MODEL In the Sequential Framework, network generation and optimization is performed by an NLP formulation, where the actual network topologies are extracted from the stream superstructure given in [9]. All possible network structures for a given set of HLDs are included in this superstructure. The non-convexities associated with the formulation and convex estimators to overcome these are discussed in [10]. In this work, global optimization of the NLP formulation by the Modal Trimming method is explored. 4.1. The Modal Trimming Method The Modal Trimming Method [11] is based on the Tunneling Algorithm [12] that has evolved into a robust method for global optimization with many applications. The modal trimming method consists of the following two phases: 1) finding a local optimal solution to obtain a global quasi-optimal one, and 2) finding a feasible solution with the
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value of the objective function equal to that of the global quasi-optimal one to obtain a starting point for finding a better local optimal solution. These steps are repeated until a feasible solution is not found. The current global quasi-optimal is taken to be the global optimal solution. Figure 2 shows the functioning of the modal trimming method. The second step, the search for feasible solutions, is the most important step in this method. An extended Newton Raphson method is applied to search for a feasible solution where the value of the objective function equals the current global quasioptimum. The Moore-Penrose generalized inverse of the Jacobian is utilized in the normal Newton-Raphson method to improve the Newton-Raphson method for this case. GAMS/CONOPT is used to find the local optima (Step 1), while Visual Basic with Matlab-Excel Link is used to search for feasible solutions (Step 2) when implementing the Modal Trimming method. The four stream, six exchanger example presented in [10] was used as a test case. The method proved to be inefficient, taking over two hours to attain the global optimum. The method and its implementation will need further testing to identify cause(s) of the inefficiency. It is also unclear, at this stage, if the method is suitable for this NLP formulation.
5. EXAMPLE In this section, a medium sized problem is solved to illustrate the use of the Sequential Framework. SeqHENS, an Excel Add-in, for the Sequential Framework, was used to generate networks for the given stream data. For comparison purposes, the operating cost of the solution presented in [13], 1,014,323 $yr-1, was unchanged. This corresponds to a HRAT of 20.35 °C. The fourth step of the framework generates the best solution with a TAC of 1,532,148 $yr-1. This compares well with the solution for base case given in [13] with a TAC of 1,530,063 $yr-1. The network generated is shown in Figure 3. It is also worth noting that the Vertical MILP transportation model for selecting HLDs in the Sequential Framework allows only one match between streams. This “simplification” of allowing just one match between streams in [13] gives a TAC of 1,568,745 $yr-1. Yet another example with more details on computations and the search for the ‘best’ network can be found in [7].
6. CONCLUSION A Sequential Framework for Heat Exchanger Network Synthesis and the rationale behind the method and the loops in the framework has been presented. The importance of TIs in the Vertical MILP transportation model was discussed and a procedure for establishing a set of optimum TIs is detailed. Extensive tests were performed to ensure that the Vertical Model gives the correct ranked sequence of HLDs with these TIs. The Modal Trimming method was applied to the NLP formulation to achieve global minimum. This system was found to be inefficient and improvement in orders of magnitude is required. Further testing must also be done to check the suitability of this method to solve the NLP formulation in the Sequential Framework. This framework has been applied successfully to a test problem of 15 streams and gave comparable solutions to the previously published results. Significant improvements are required to solve the minimum number of units MILP model and the Vertical MILP Transportation model (in addition to the solution methods for the NLP model) for industrial cases. These are areas of ongoing research in our group.
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Figure 3: Best network for example problem
References 1. T. Gundersen and L. Naess, 1988, The synthesis of cost optimal heat exchanger networks, Comp. & Chem. Eng., 12, 6, 503-530. 2. K.C. Furman and N.V. Sahinidis, 2002, A critical review and annotated bibliography for heat exchanger network synthesis in the 20th century, Ind. Eng. Chem. Res., 41, 2335-2370. 3. T.F. Yee and I.E. Grossmann, 1990, Simultaneous optimization models for heat integration – II. Heat exchanger network synthesis, Comp. & Chem. Eng., 14, 10, 1165-1184. 4. K.C. Furman and N.V. Sahinidis, 2001, Computational complexity of heat exchanger network synthesis, Comp. & Chem. Eng., 25, 1371-1390. 5. T. Gundersen and I.E. Grossmann, 1990, Improved optimization strategies for automated heat exchanger network synthesis through physical insights, Comp. & Chem. Eng., 14, 9, 924-944. 6. T. Gundersen, P. Traedal and A. Hashemi-Ahmady, 1997, Improved sequential strategy for the synthesis of near-optimal heat exchanger networks, Comp. & Chem. Eng., 21, Suppl., 59-64. 7. R. Anantharaman and T. Gundersen, 2005, Revisiting the sequential framework for nearoptimal heat exchanger network synthesis, In Proceedings from PRES 2005, Vol 1, 67-72. 8. J.M. Jeżowski, H.K. Shethna and F.J.L. Castillo, 2003, Area targets for heat exchanger networks using linear programming, Ind. Eng. Chem. Res., 42, 1723-1730. 9. C.A. Floudas, A.R. Ciric and I.E. Grossmann, 1986, Automatic synthesis of optimum heat exchanger network configurations, AIChE Journal, 32, 2, 276-290. 10. A. Hashemi-Ahmady, J.M. Zamora and T. Gundersen, 1999, A sequential frameowrk for optimal synthesis of industrial size heat exchanger networks, In Proceedings from PRES ’99, 329-334. 11. R. Yokoyama, M. Inui, K. Ito, 2005, Prediction of energy demands using neural networks by a global optimization method, In Proceedings of ECOS ’05, Vol. 2, 609-616. 12. A.V. Levy and A. Montalvo, 1985, The tunneling algorithm for the global optimization of functions, SIAM Journal of Scientific and Statistical Computing, 6, 1, 15-29. 13. K. Björk and R. Nordman, 2005, Solving large-scale retrofit heat exchanger network synthesis problems with mathematical optimization methods, Chem. Eng. Process, 44, 869-876.
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Linking experiments to modeling in biodiesel production Anton A. Kiss, Alexandre C. Dimian, Gadi Rothenberg University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands
1. Abstract The problems associated to current biodiesel manufacturing are outlined. This study shows that linking experiments to modeling leads to feasible solutions for the production of biodiesel by a novel design based on catalytic reactive distillation. The key features of the methodology are discussed and the experimental results using solid catalysts and simulated designs are presented. 2. Introduction Sustainable energy management is one of the key issues of modern society. Interest in biodiesel is growing following tighter legislation on vehicle emissions. Biodiesel is the only alternative fuel currently available that has an overall positive life cycle energy balance.1 It consists of fatty acid methyl esters (FAME), currently manufactured by trans-esterification using liquid NaOH catalyst, or batch esterification of fatty acids using H2SO4 as catalyst. These corrosive catalysts require expensive neutralization and separation, making biodiesel an nice but still costly alternative fuel. Moreover, the H2SO4 catalyst removal is imperative due to EU restrictions on sulfur content in diesel fuels. To solve these problems, we propose the replacement of the homogeneous acid catalyst with a solid acid catalyst (SAC) and develop a sustainable esterification process based on catalytic reactive distillation (RD).2 Solid acids can be easily separated from the product. They need less equipment maintenance and form no CO + light polluting by-products. The No global warming development of the catalyst Biomass CO release to atmosphere should be integrated in the design at an early stage. We aim for a superacidic catalyst that is active, Biodiesel synthesis Use in cars and trucks selective, water-tolerant and stable at high temperatures.3 Figure 1. Life cycle of biodiesel as green fuel. 2
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Novel processes such as RD make use of SAC to produce fatty esters.2 The Minimum kinetic CFD modeling requirements main problem is finding a suitable catalyst that is active, selective, and Catalyst development stable under the process conditions. Catalyst ¾ Fast screening The non-ideal nature of the reactants ¾ Detailed design implementation ¾ Catalyst synthesis mixture leads often to segregation of the aqueous phase that may easily deactivate the solid acid catalyst. Bench experiments Optimal exp. design To solve these problems, we are ¾ Planning ¾ Kinetic modeling developing catalysts based on optimal ¾ VLLE data Final design experiment design (Figure 2). Usually, the chemist prepares the catalyst and the experimental results are used later Figure 2. Linking experiments to simulation. in design by engineers. Often the design fails since the feasible conditions in term of concentrations and temperature are different from those used in the kinetic experiments. By integrating lab research with optimal experiment design guided by process simulation, the scaling and validation steps are minimized. In the method proposed here, the requirements for the catalyst are calculated by simulation. Based on an economic evaluation the simulation sets the minimum kinetic requirements for catalyst in terms of reaction rate and selectivity. These targets are used directly for the fast screening of catalysts, followed by synthesis, characterization of surface activity and internal diffusion. On this basis few catalysts are selected for refined synthesis and characterization. The concentration of the acid source, and the calcination temperature and time were among the optimization parameters that influence the catalytic properties. 4. Results and discussion The wider goal of this study is the production of fatty esters, and in particular FAME, by integrating the reaction and separation into one unit (RD) that can intensify the mass transfer. Fatty acid esterification using solid acids is not yet well established in industry, as it is much more difficult to find a suitable solid acid catalyst to do the job compared to shorter acids such as acetic acid. The catalyst should be very active and selective (as by-products formed in secondary reactions may render the process uneconomical), water-tolerant (water byproduct may deactivate the catalyst) and stable at relatively high temperatures. In addition, it should be an inexpensive material that is readily available on an industrial scale. Our previous study has shown that zeolites, and carbon-based acid catalysts exhibit low-activity in fatty acid esterification.4 Therefore, in this work we tested heteropoly-acids, ion-exchange resins and mixed metal oxides.
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The experimental results are presented on the use of solid catalysts in esterification of dodecanoic acid with 2-ethylhexanol and methanol. Tungstophosphoric acid exhibited high activity, close to H2SO4 used as a benchmark (Figure 3, left). However this is soluble in water hence not usable as SAC. Its cesium salt (Cs2.5) is also super acidic and its mesoporous structure has no limitations on the diffusion of the reactants. However, due to its high molecular weight, the Cs2.5 catalyst exhibits only average activity per weight of catalyst. Amberlyst-15, a styrene-based sulfonic acid, showed high activity but its low thermal stability (up to 150°C) renders this SAC unsuitable for RD applications. Compared to the sulfated zirconia (SZ) catalyst, the other mixed metal oxides prepared and tested (titania and tin oxide) performed slightly better. However, SZ is less expensive and it is readily available at industrial scale. 100
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Surface area [m2/g] 163 118 129 100
Catalyst sample Cs2.5H0.5PW12O40 ZrO2/SO42- / 650°C TiO2/SO42- / 550°C SnO2/SO42- / 650°C
Pore volume [cm3/g] 0.135 0.098 0.134 0.102
Pore diameter max./mean/calc. [nm] 2 / 5.5 / 3 4.8 / 7.8 / 7.5 4.1 / 4.3 / 4.2 3.8 / 4.1 / 4.1
Sulfur content [%] N/A 2.3 2.1 2.6
Moreover, it is very selective and thermally stable. By increasing the amount of catalyst used the conversion can be further increased (Figure 3, right), making this catalyst suitable for RD. In a separate set of experiments, the activity dropped to cca. 90% after five consecutive runs, and remained constant thereafter. Re-calcination of the used catalyst restored its original activity. Considering these promising results, SZ was tested also with methanol (Figure 4). Compared to the previous case the reaction rates are higher, due to the alcohols’ relative sizes. The sulfated metal oxides proved to be active, selective, and stable under the process conditions and integrated easily in developing a sustainable esterification process based on catalytic reactive distillation. Unsurprisingly, the characteristics of these catalysts are quite similar (Table 1). The hydrophobicity of the catalyst surface and the density of the acid sites are of paramount importance in determining the catalyst’s activity and selectivity. This is especially true for fatty acids and long chain alkyl alcohols, that are both very lipophilic compounds. In the ideal case, the fatty acid molecules could adsorb perpendicular to the surface, with the tails forming a local hydrophobic environment. For too many adjacent acid sites, the byproduct water from the esterification would adsorb on the surface, and the catalyst would lose its activity (Figure 5, left). Several secondary reactions are possible, but these can be avoided by using a selective solid catalyst such as sulfated metal oxides. Figure 5 (right) shows the reaction pathways and the possible products. REACTANTS Fatty Acid
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Figure 5. Effect of hydrophobic surface on catalytic activity (left). Reaction pathways (right).
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Dodecanoic acid
0.6
VLLE
0.4
Feed ratio 1:1
0.2
Methanol
Evaporator
0
Alcohol, 65ºC
0
0.2
0.4
0.6
0.8
1
X1 (water+acid)
Water, 100ºC
Bottom: 473 K
Ester ≥ 99.9 %
Figure 6. Generalised CPE diagram (left). FAME production flowsheet (right).
The catalyst development was integrated in the process design at an early stage, by data mining and embedding of reaction kinetics in the process simulation. The analysis of physico-chemical properties shows very high boiling points for dodecanoic acid and esters (Table 2). Hence, the ester will be separated in the bottom of the RD column and water by-product is removed as top product. By removing water by-product the equilibrium is shifted towards ester formation. Table 2. Normal boiling points of chemical species involved in the process.
Chemical name Dodecanoic (lauric) acid Methanol 2 Ehyl hexanol Methyl dodecanoate 2 ethylhexyl dodecanoate Water
Chemical formula Mw (g/mol) C12H24O2 200 CH4O 32 130 C8H18O 214 C13H26O2 312 C20H40O2 18 H2O
Tb (K) 571 338 459 540 607 373
Tb (°C) 298 65 186 267 334 100
METHANOL
Temperature C 160 180
ACID ESTER WATER
140
0.25
0
RDC: Temperature Profile
200
220
RDC: Liquid Composition Profiles
X (mole frac) 0.5 0.75
1
Rigorous simulations were performed in Aspen Plus. The generalised chemical and phase equilibria (CPE) diagram is shown in Figure 6 (left). Each point inside represents the liquid composition at phase and chemical equilibrium. The esterification reaction must take place in the homogeneous region.
3
6
9 Stage
12
15
0
5
10 Stage
Figure 7. Liquid composition profiles (left) and temperature profile (right) in RDC.
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Fatty acid
Total alcohol conversion
Methanol recovery
Triglycerides Fatty acids
Highest reaction rate
RDC Methanol
Alcohol
Water
Distillation column
Total acid conversion
Glycerol
FAME Biodiesel
Figure 8. Reaction rate profile in RDC at optimal reflux ratio (left). Combined RD process (rigt).
The fatty acid feed must be fed in the top of reactive zone to reduce the contamination of the final product. Figure 6 (right) presents the flowsheet of the process. An additional evaporator is used for further ester purification. Hence, high purity final products are feasible. The composition and temperature profiles in the RD column are shown in Figure 7. The RD column uses an acid reflux. For the optimal reflux ratio the maximum reaction rate is located in the centre of RDC, providing complete conversion of reactants at the ends of the column (Figure 8, left). Alternatively, both reactions (trans-esterification & esterification) can be combined with separation, shifting the equilibrium towards products by continuous removal of water and glycerol (Figure 8, right). 5. Conclusions Optimal experiments design using targets set by simulation allow the systematic study of initial reaction rates in screening solid acid catalysts. The novel alternative to make biodiesel by RD replaces the liquid catalysts with solid acids. This can dramatically improve the economics of current biodiesel synthesis and reduce the number of downstream steps. The key benefits are: 1. High unit productivity, up to 6~10 times higher than of the current process. 2. Lower excess alcohol requirements (stoichiometric ratio at reactor inlet). 3. Reduced capital and operating costs, due to less units and lower energy use. 4. Sulfur-free fuel, since solid acids do not leach into the product. 5. No waste streams because no salts are produced (no neutralization step). Acknowledgement. We thank Dutch Technology Foundation STW (NWO/CW Project 700.54.653) and Cognis, Oleon, Sulzer, Uniquema, Engelhard for the financial support.
References 1. 2. 3. 4.
B. Buczek, L. Czepirski, Inform, 2004, 15, 186. A. C. Dimian, F. Omota, A. Bliek, Chem. Eng. Sci., 2003, 58, 3159 & 3175. T. Okuhara, Chemical Reviews, 2002, 102, 3641. A. A. Kiss, A. C. Dimian, G. Rothenberg, Adv. Synth. & Cat., 2006, 348, 75.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Optimization studies in sulfuric acid production Anton A. Kiss,a Costin S. Bildea,b Peter J.T. Verheijenb a University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands b Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
1. Abstract Current legislation imposes tighter restrictions to reduce the impact of process industry on environment. This work presents the dynamic simulation and optimization results for an existing sulfuric acid plant. Operational problems may occur when the process is disturbed due to production rate changes or catalyst deactivation, the non-linear response of the plant leading to sustained oscillations. Since the plant is operated near full capacity, only minor increases in energy production can be achieved. However, the SOx emissions can be significantly reduced by ~40% or more, by optimizing the operating parameters. 2. Introduction Most sulfuric acid plants are rather old and are facing now additional challenges that aim to maximize the amount of energy produced while reducing the environmental impact. Sulfuric acid is the chemical product manufactured in largest quantity in terms of mass, with about 40 million tons produced annually only in USA. It has a wide range of uses and plays an important role in the production of almost all manufactured goods. Approximately 65% of the H2SO4 produced is used in the production of agricultural fertilizers. This study presents the results of the dynamic simulation and optimization of an existing sulfuric acid plant (PFI – Phosphoric Fertilizers Industry, Greece). Due to the partnership with Process Systems Enterprise Ltd. (PSE), and thanks to its powerful features, gPROMS was selected to perform all simulation tasks.1 The dynamic model developed in this study includes also a graphical user interface (GUI) built in MS Excel, that allows scenario evaluation and operator training. The model has been successfully used for dynamic simulations to evaluate the non-steady-state behavior of the plant and detect changes in product quality, as well as to minimize the total amount of sulfur oxides released in atmosphere. For the major units of the flowsheet, the main equations describing the dynamic model are given using the standard notation. Due to space limitations and model complexity, some modeling details were omitted but they are available upon request. Here we limit to present only the most important results of this study.
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3. Process description The main steps in this process consist of burning sulfur (S) in air to form sulfur dioxide (SO2), converting SO2 to sulfur trioxide (SO3) using oxygene (O2) from air, and absorbing the sulfur trioxide in water (H2O) or a diluted solution of sulfuric acid (H2SO4) to form a concentrated solution of acid (>96%). Because the reaction is limited by the chemical equilibrium a multi-bed catalytic adiabatic reactor is used. The catalyst used is vanadium oxide (V2O5) mixed with an alkali metal sulfate. This mixture is supported on small silica beads, and it is a liquid at the high temperature inside the reactor. Several conversion steps, addition of fresh air and inter-stage cooling are necessary as the reaction is reversible and exothermal.2 For heat integration reasons, two feed-effluent heat exchangers (FEHE) are used. SO2 conversion is further improved and tail gas emissions are reduced through an intermediate SO3 absorption step. The absorption of SOx is finalized in the second absorber. The simplified flowsheet of the sulfuric acid production process consists of a sulfur burner, multi-pass converter, heat exchangers and absorbers (Figure 1). The process is designed to give a conversion of sulfur dioxide to sulfuric acid of over 99.7%. This is achievable in practice due to the intermediate absorption (after the fourth bed of catalyst) that changes the gas composition, thus shifting the equilibrium curve to higher conversions (Figure 2). The model parameters were measured or estimated using standard correlations. Air
Steam
Furnace Boiler
SO2
HX1
1
2
S (lq) Air 3
FEHE1
4
FEHE2
5
SO2
Reactor
H2SO4 HX2 Abs1
HX3
HX4
H2SO4 Abs2
H2SO4
Figure 1. Simplified flowsheet of sulfuric acid production plant.
H2SO4
Optimization Studies in Sulfuric Acid Production
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Dynamic model 1. Oxidation reactor (multi-pass adiabatic converter, five beds of catalyst) Mass balance:
dci dc d 2c = −u i + ρb ⋅ν i ⋅ r + Dz 2i dt dz dz
(1)
Energy balance:
(ε ⋅ ρ
f
⋅ c p , f + ρb ⋅ c p ,cat ) ⋅
Pressure drop:
dT dT d 2T = − ρ f ⋅ c p, f ⋅ u + ρb ⋅ r ⋅ ( −ΔH r ) + k z 2 dt dz dz dP u2 = − ff ⋅ρf ⋅ dz Dp
(2) (3)
A kinetic model similar to the one proposed by Froment & Bischoff was used:2 § · pSO3 k1 ⋅ pO2 ⋅ pSO2 ⋅ ¨1 − 1/2 ¨ K p ⋅ pSO ⋅ pO ¸¸ 2 2 ¹ © r= 2 22.414 ⋅ 1 + K 2 ⋅ pSO2 + K 3 ⋅ pSO3
(
)
(kmol/kg⋅cat⋅s)
where K p = exp ( -10.68 + 11300/T ) and k1 = k1,0 ⋅ exp ( − Ea,1 / T )
(4)
(5)
2. Heat exchangers (HEX with constant coolant temperature and FEHE) Energy balance (HEX): Energy balance (FEHE):
dT dT 4 = − ρ f ⋅ c p, f ⋅ u − ⋅ H w ⋅ (T − Tc ) dt dz D Hw dT1 dT 4 = −u1 1 − ⋅ (T1 − T2 ) dt dz D ρ1 ⋅ C p ,1
ρ f ⋅ c p, f ⋅
Hw dT2 dT = u2 2 + Av ⋅ (T − T ) ρ 2 ⋅ C p ,2 1 2 dt dz
(6) (7)
(8)
3. Absorbers (A – SO3 in gas phase; B – H2O in liquid phase) Mass balance gas (A): Mass balance liquid (B): Energy balance (gas):
dcA 4 d =− ( F ⋅ C ) − N A ⋅ Av dt π D2 dz G A dc 4 d ε L ⋅ B = − 2 ( FL ⋅ CB ) − N B ⋅ Av dt π D dz 4 dT1 dT H ⋅ A ε⋅ =− ⋅ FG 1 − w v (T1 − T2 ) 2 dt πD dz ρG ⋅ c p ,G
εG ⋅
Energy balance (liquid): H ⋅A N ⋅A 4 dT dT (1 − ε ) ⋅ 2 = 2 ⋅ FL 2 + w v (T1 − T2 ) + A v ⋅ ( −ΔH R ) dt π D dz ρ L ⋅ c p , L ρ L ⋅ c p,L
(9) (10) (11)
(12)
dP u2 (13) = − ff ⋅ρf ⋅ dz Dp Note: The standard notation was used in describing the model equations (Eq. 1-13). Pressure drop:
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4. Results and discussion
1 0.9 0.8 Conversion SO2 / [-]
The path of the reaction can be conveniently shown in the X–T diagram (Figure 2). The horizontal lines represent gas cooling only, as no conversion occurs when the gas is not in contact with the catalyst. The diagonal lines represent adiabatic temperature rise of the gas within the different converter passes. Their slopes are directly proportional to the specific heat capacity of the gas. Each initial gas concentration of SO2 has its own equilibrium curve. For a given gas composition, the adiabatic temperature-rise can approach the equilibrium curve but never cross it.
0.7 0.6 0.5 0.4 0.3
PFI-Reactor
0.2
EQ curve (initial)
0.1
Simulated reactor
EQ-Reactor
EQ curve (final)
MaxRate Curve
0 675 725 775 825 875 925 975 Temperature / [K]
Figure 2. Temperature-conversion diagram.
Operational problems may occur when the process is disturbed due to production rate changes, catalyst deactivation, or variation of air feed flow rates caused by day-night or summer-winter temperature differences and constant volumetric flow operation. For certain disturbances (–5% feed flow rate), the response of the plant is nonlinear and leads to sustained oscillations (Figure 3). Obviously, these oscillations are not acceptable since they propagate from the reactor to absorbers, rendering the plant unstable and the product off-spec. The non-linear behaviour is explained by the presence of an inverse response3 for the temperature through the reactor beds, combined with the positive energy feedback induced by the two feed-effluent heat exchangers (FEHE). 900 Temp_R1 Temp_R2 Temp_R3 Temp_R4 Temp_R5
850
Temperature / [K]
Temperature / [K]
900
800
750
700
Temp_R1 Temp_R2 Temp_R3 Temp_R4 Temp_R5
850 800 750 700 650
0
2
4 6 Time / [hr]
8
10
0
2
4 6 Time / [hr]
8
10
Figure 3. Reactor temperatures time-profile, for ± 5% change in air feed flow rate (initial case).
Optimization Studies in Sulfuric Acid Production
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Figure 4. Reactor temperatures time-profile, for ± 10% changes in air feed flow rate (final case).
The plant can be stabilized by changing the catalyst distribution among the conversion stages (e.g. shortening the length of several catalyst beds) and employing feedback temperature controllers. By applying these changes to the reactor, the sustained oscillations are avoided and the dynamic response becomes acceptable even for larger disturbances in the feed flow rate (Figure 4). The molar fractions of SO2 and SO3 at the outlet of the final absorption column, are presented for similar disturbances in Figure 5. Note the non-symmetrical response (–50%...+200% for SO2) to these ±10% disturbances. However, the composition reaches a new steady state in relatively short time, less than 1 hour. Multi-variable optimization was performed for several production rates, corresponding to the amount of sulfur fed into the plant (nominal value and ± 510% changes). Five key variables were identified and manipulated accordingly to carry out the optimization: the amount of air fed into the sulfur burner, the flow rates of air fed into converter pass 3 and 4, and split fractions (by-pass) for cold streams entering the gas-gas heat exchangers (FEHE1 and FEHE2). In both optimizations the cold streams should not be split to bypass the heat exchangers. 2.20E-05
Molar fraction SO 3 / [-]
Molar fraction SO 2 / [-]
6.0E-04 5.0E-04 4.0E-04 3.0E-04 F-10% F+10%
2.0E-04 1.0E-04 0.0E+00
2.15E-05 2.10E-05 2.05E-05 F-10% F+10%
2.00E-05 1.95E-05 1.90E-05
0
1
2
Time / [hr]
3
0
1 2 Time / [hr]
3
Figure 5. Molar fractions of SOx after absorption, for ± 10% disturbance in the air feed flow rate.
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500 f_obj initial f_obj final
f_obj / [ppm SO x ]
Conversion / [-]
0.9 0.8 0.7 F_air -10% F_air - 5% F_air_std F_air + 5% F_air +10%
0.6
400 300 200 100 0
0.5 0
1
2 3 Reactor Beds / [-]
4
5
285
300 315 330 345 Sulfur Flowrate / [kmol/h]
360
Figure 6. Conversion profiles in the reactor, for ± 10% changes in the air feed flow rate (left). SOx emissions (lower is better) before and after optimization at various feed flow rates (right).
The optimization targets were to maximize the amount of energy produced, and minimize the total amount of SOx released in atmosphere (i.e. not absorbed in the final absorption column). The conversion profiles in the reactor for changes of ± 10% in the feed flow rate are given in Figure 6 (left). Increasing energy production is equivalent to maximizing the amount of SO2 converted into products. A flat optimum is expected, since the heat generated in the reaction is the one that is recovered. As the conversion of the process is almost 99.85%, any further increase is insignificant. Not surprinsingly, the energy production can be increased by only ~1%. However, the SOx emissions can be drastically reduced by ~40% (the nominal case) or even more (Figure 6, right). Therefore, the plant can be fully exploited while respecting the ecological restrictions. 5. Conclusions The dynamic model developed in this work provides reliable results that are in excellent agreement with the data available from the real plant, the relative error being typically less than 1%. Along with minor benefits in energy production the amount of SOx emissions could be significantly reduced by ~40% just by optimizing operating parameters such as feed flow rates or split fractions. Besides controllability, operability and optimization studies the plant model coded in gPROMS is also useful for operator training and scenario assessments. Acknowledgement. This project (OPT-ABSO, Contract nr. G1RD-CT-2001-00649) was funded by the European Commission. We also thank PFI and PSE for assistance.
References 1. gPROMS Advanced User Guide, Process Systems Enterprise Ltd., 2004. 2. Froment, G.F., Bischoff, K.B., Chemical Reactor Analysis and Design, Wiley, 1979. 3. Morud J.C. and Skogestad S., AIChE Journal, 1998, 44, 888.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Integrating Advanced Thermodynamics and Process and Solvent Design for Gas Separation Emmanuel Keskesa∗ , Claire S. Adjimana , Amparo Galindoa , and George Jacksona a
Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London SW7 2AZ, U.K.
The integrated design of a solvent and process for high-pressure gas separations requires thermodynamic models which can reliably predict high-pressure phase equilibrium as a function of the solvent’s molecular structure. A methodology to tackle such a design problem is developed using the SAFT-VR equation of state to obtain the thermodynamic properties of the materials involved. It is applied to the problem of CO2 capture from methane, given a high-pressure feed of high-CO2 content. Optimal operating conditions and an optimal alkane solvent (C20 H42 ) are identified and found to maximise the recovery of both CO2 and CH4 , thus offering a promising alternative to current processes. 1. INTRODUCTION The increasing importance of natural gas as a source of energy poses difficult gas separation design challenges, as the streams recovered from gas fields are at high pressures (typically about 10 MPa) and can contain a high proportion of CO2 (up to 70%). In addition, as the implementation of the Kyoto protocol would require the capture of large quantities of CO2 , the injection of CO2 in depleted or neardepleted reservoirs for enhanced oil/gas recovery operations will become increasingly frequent. This is likely to result in gas streams which are even richer in CO2 . Yet, conventional CO2 separation techniques are usually restricted to low CO2 content or low-pressure feeds. 1.1. Choice of separation technique The techniques used in the gas separation industry include adsorption on solid substracts, chemical absorption, gas permeation and physical absorption [1]. Adsorption is economical for purification, typically reducing the CO2 content from 3% down to 0.5%. Chemical absorption has been used successfully for low-pressure gas streams containing between 3% and 25% of CO2 , but involves large solvent regeneration costs which hamper its application to higher CO2 contents. Gas permeation techniques are compact and flexible, adapting easily to changes in CO2 content . However, reliability is a concern, especially because natural gas contaminants can lead to the deterioration of the membrane. Physical absorption can also be used successfully, its main advantage is that physical solvents have no absorption limitation (unlike chemical absorption, which is limited by stochiometry). The amount of CO2 absorbed by the solvent is determined by the vapour-liquid equilibrium of the mixture, which is specified by the pressure and temperature. At high CO2 partial pressure, the CO2 loading capacity of the solvent is higher for a physical solvent than for a chemical solvent. Hence, physical absorption processes are particularly appropriate for the treatment of CO2 -rich gas streams. 1.2. Choice of solvent for physical absorption The choice of solvent is one of the key decision variables which impacts on the performance and economics of a physical absorption process. Many solvents have been used for the absorption of CO2 and CH4 including, formulations of, tributyl phosphate, polycarbonate, methylcyanoacetate, and N-formyl morpholine [2]. Unfortunately, there are major drawbacks with all of these for offshore operations: the solvents are not easily disposable and could be involved in side reactions with other natural gas constituents. A more suitable solvent, which does not react and can easily be handled in an oil and gas environment, is n-butane, which has been used in the Ryan-Holmes cryogenic separation process ∗ Financial
support from Schlumberger Cambridge Research is gratefully acknowledged.
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[3]. The process has a satisfactory CO2 /CH4 separation factor, but operation at low temperatures is very energetically demanding. Like n-butane, other alkanes such as n-decane are known to absorb CO2 preferentially to CH4 . Experiments evidence is available [4], and show some K-values ranging from 1.2 to 1.8. The use of higher alkanes or alkane blends may provide a promising route towards adapting the Ryan-Holmes process to the temperatures and pressures typical of gas fields. In addition, the alkane solvent presents the advantage to be cheap, easily available, and tunable (mixture of alkanes). Often, the choice of solvent is made prior to the design of the flowsheet structure and operating conditions. These decisions are however,closely linked, and the best process performance can only be achieved by considering them simultaneously. Several methods for integrated process and solvent design have been proposed [5–7], but they usually rely on thermodynamic models which are only applicable at low pressures. The objective of this work is to show how the incorporation of advanced thermodynamic methods can be used to tackle the design of a solvent and a process for high-pressure separation in an integrated manner. In this work, we focus on identifying the optimal combination of solvent and operating conditions based on separation performance. 2. METHODOLOGY In this section, we give a short description of the process flowsheet and model, the thermodynamic model, and the optimisation problem. 2.1. Process flowsheet The flowsheet which is proposed to carry out the gas separation is shown in Figure 1. A feed of CH4 and CO2 enters the absorber from the bottom. CO2 is transferred to the solvent stream, which enters through the top of the absorber. The gas leaving the absorber is the clean gas (CH4 ) product. The charged solvent leaves the absorber from the bottom and passes through a series (four) of flash units, where the pressure is reduced in stages. The vapour streams from the first two flash units are re-compressed and mixed with the absorber gas feed. The vapour streams from the last two flash units are almost pure CO2 , and form the CO2 product stream, which can be stored or used for enhanced recovery. The clean solvent leaving the last flash unit is re-compressed and re-injected into the absorber, together with a make-up stream of fresh solvent. Two coolers and a heater have been added on various streams to control the recycle temperatures. 2.2. Process model Steady-state mass and energy balances for all the units in the flowsheet have been derived. Thermodynamic equilibrium has been assumed in all units. The model is able to identify whether one or two phases are present in any of the units. This is particularly important in the compressors, where vapour phase conditions should be maintained. Two-phase equilibrium is specified via the equality of pressures and chemical potentials of all components in the two phases. In the first instance, full efficiency has been assumed for every equilibrium tray in the absorber. The process model requires the following physical properties: enthalpy, entropy, heat capacity, chemical potential and pressure, which should be calculated using an equation of state given the high-pressure conditions and non-ideality of the mixture. The SAFTVR equation of state [8,9] has been chosen for its ability to represent phase behaviour accurately over a wide range of conditions, and to treat homologous series of mixtures, in a predictive way [10]. In addition, we would like to compare the alkane solvent with the conventional amine solvent; and SAFT-VR offers a unified platform for using the association for the chemical reaction modelling. The equations have been implemented in gPROMS [11]. The SAFT-VR calculations are implemented in Fortran90 and accessed via a Foreign Object interface [12]. 2.3. SAFT-VR model for CO2 /CH4 /n-alkane mixtures The SAFT-VR equation allows a complete family of solvents to be considered in design. When treating systems which do not associate with the SAFT-VR equation of state, as is the case here, each component i is described by two molecular size parameters, the number of spherical segments in a molecule mi , and the hard-core diameter of the segment σii , and two energy parameters, the strength ii and range λii of the dispersive interactions. Parameters for CO2 and CH4 have previously been refined to describe coexistence data, which leads to an over prediction of the critical pressure and temperature [10,13]. As the region near the critical point of these two compounds is of interest here, the size of and energy
Integrating Advanced Thermodynamics and Process and Solvent Design
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Figure 1. Proposed flowsheet for the CO2 /CH4 separation process. parameters σii and ii have been rescaled to reproduce the critical temperature and pressure (see also [10] for more detail). The resulting parameters are presented in table 1. For the n-alkane solvent to be
Table 1 SAFT-VR parameters for CH4 and CO2 . Component i mi σii (˚ A) CH4 1 4.0576 2 3.1364 CO2 kB is the Boltzmann constant.
ii /kB (K) 156.50 168.89
λii 1.4479 1.5157
used in the process, a correlation that gives the SAFT-VR parameters as a function of the molecular used [13]: weight M Wa , in g.mol−1 , of n-alkane a has been ma = 0.02376 × M Wa /(g.mol−1 ) + 0.6188 (1) 3 A) = 1.53212 × M Wa /(g.mol−1 ) + 30.753 ma (σaa /˚ ma (aa /kB )/K = 5.46587 × M Wa /(g.mol−1 ) + 194.263 ma λaa = 0.04024 × M Wa /(g.mol−1 ) + 0.6570
(2) (3) (4)
Standard mixing rules can be used to model mixtures of different molecules [9]. This requires binary parameters to describe the interactions between species i and j. The following relations have been used: σij =
σii + σjj ; 2
√ ij = (1 − kij ) ii jj ;
λij =
σii λii + σjj λjj , σii + σjj
(5)
where kij is an additional parameter, which captures the deviation of the unlike interaction energy from the geometric mean. In our work, this interaction parameter has been estimated based on isothermal
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vapour-liquid equilibrium data over a wide range of temperatures and pressures for each of the three relevant binary mixtures (CO2 /CH4 , CO2 /n-decane, and CH4 /n-decane). For CO2 /CH4 mixtures, we have used all the experimental data available. N-decane (C10 ) has been used as a representative compound in the n-alkane series as large sets of experimental data are available. For CO2 /C10 and CH4 /C10 , we selected experimental data at pressures below 10 MPa and temperatures below 477 K; this corresponds was estimated by using the maximum likelihood objective to the operating range of the process. Each kij function for the mixture pressure and the vapour mole fraction, as a function of liquid mole fraction and temperature. A constant variance model was used. All the experimental data available in Detherm [15] were used, namely 111 points for CH4 /C10 and 85 points for CO2 /C10 , both over 15 temperature values and 312 points over 24 temperatures for CH4 /CO2 . The error for the pressure and the vapour mole fraction can be found in table 2. Table 2 Errors for the predicted vapour-liquid equilibrium pressure and vapour mole fraction. Mixture AAPE% for the pressure AAD[mol] for the vapour mole fraction CH4 /CO2 0.021 0.0236 0.099 0.0010 CO2 /C10 0.076 0.0027 CH4 /C10
The following interaction parameters were obtained: = −0.053006; kCH 4 ,C10
kCH = +0.036798; 4 ,CO2
kCO = +0.089642. 2 ,C10
(6)
Since the n-alkanes belong to a homologous series, we assume that the interaction parameters between CH4 (or CO2 ) and any n-alkane solvent are the same as those between CH4 (or CO2 ) and n-decane. = 0. To a first Furthermore, n-alkane i/ n-alkane j interactions are described by Eqs. (5) with kij approximation, this allows for n-alkane solvent blends to be represented by treating the blend as a single species with molecular weight equal to that of the blend. 2.4. Formulation of the design problem For a feed of fixed flowrate and composition, and a fixed number of trays in the absorber (N = 5), the vector of design variables, d, for the gas separation process consists of the solvent flowrate (F ), the absorber pressure (P0 ), the flash unit pressures (P1 , P2 , P3 , P4 ), the heating duty (QW ), the cooling duty for the cooler on the solvent recycle (QC), the compressor cooling duty (QCcomp ), and the molecular weight of the alkane solvent (M Wa ). Only n-alkanes with a molecular weight between that of C7 H16 and that of C20 H42 are considered, because smaller n-alkanes are likely to be too volatile, while larger n-alkanes would be too viscous. The general optimisation problem is formulated as follows: max f (d, x) d
s.t.
h(d, x) = 0 P0 ≥ P1 ≥ P2 ≥ P3 ≥ P4 ≥ 0 QC ≤ 0; QCcomp ≤ 0; QW ≥ 0 M WC7 ≤ M Wa ≤ M WC20 F ≥0
(7)
where x is a vector of process state variables and SAFT-VR solvent parameters, f is a performance measure, h are the model equations, which include material and energy balances, thermodynamic relations, and the solvent structure-parameter relations, i.e. Eqs. (1)-(4). Variable M Wa is a continuous variable as it represents the molecular weight of the blend. This problem is therefore a NLP, which can be solved with standard solvers in gPROMS. For an initial investigation of the technical viability of this process, a performance measure based on the purity of the captured CO2 stream and of the clean gas stream is chosen. It is given by f = yCO2 ,CO2 stream + yCH4 ,cleangasstream , where yi,k denotes the mole fraction of component i in stream k.
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3. RESULTS The ability of the thermodynamic model to represent ternary phase behaviour under processing conditions is first studied. The proposed optimisation formulation is then solved to assess process feasibility. 3.1. Prediction of the vapour-liquid equilibrium for CO2 /CH4/n-decane One key assumption of the thermodynamic model in section 2.3 is that ternary phase behaviour can be captured by using binary interaction parameters. To test this hypothesis, we compare predicted tie-lines for mixtures of CO2 , CH4 and n-decane with experimental data [4]. The SAFT-VR thermodynamic model is in very good agreement with the data, even at pressures above 10 MPa, as shown in Figure 2.
C10
C
0 10
0
0.2
0.2
0.8
0.4
0.8 0.4
0.6
0.6
0.4
0.8
C
1
0.2 0.4 0.6 P=4.9MPa
0.8
0.4
0.8
0.2
0
0.6
0.6
0
CO2
0.2
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C1
0.2 0.4 0.6 P=9.8MPa
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0.6 0.4
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2
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C1
0
0.2 0.2 0.4 0.6 P=19.6MPa
0.8
0
CO
2
Figure 2. SAFT-VR predictions (dashed lines) versus ternary experiments (continuous lines) at constant temperature 344.15K and different pressures. 3.2. Process optimisation A feed of 20 kmol.h−1 consisting of 50 mol% CO2 and 50 mol% CH4 , at 10 MPa and 50o C, is considered. The heater is not used in this initial study (QW = 0). A target outlet pressure of P4 = 0.1 MPa is set for the CO2 stream. By solving the optimisation problem, it is found that a clean gas stream of 98.2% CH4 and a captured CO2 stream of 98.8% CO2 can be recovered, with negligible solvent losses. The optimal solvent is found to be C20 H42 (M Wa =281.7). The optimal operating conditions are P0 = 9.7 MPa, P1 = 3.3 MPa, P2 = 1.2 MPa, P3 = 0.3 MPa, F = 19.9 kmol.h−1 , QC=42 kW and QCcomp =13 kW. These preliminary results indicate that the separation of CO2 and CH4 from a high-CO2 content stream using an alkane solvent is technically feasible. The fact that the highest molecular weight solvent that we allowed is optimal can be explained from phase separation arguments (the heaviest n-alkane will
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have the highest CO2 /CH4 selectivity). This result indicates that the effect of solvent viscosity on the process should be included in the model. 4. CONCLUSIONS The integrated design of a solvent and process for high-pressure gas separation requires the use of reliable thermodynamic models which can relate solvent structure to properties. The SAFT-VR equation of state is such a model. We have shown for the first time how it can be incorporated in the overall design problem. The proposed approach has been applied to the increasingly important problem of CO2 capture from CH4 . A thermodynamic model has been developed for CO2 /CH4 /n-alkane mixtures and its use in the solvent/process design problem has shown that physical absorption with an alkane solvent is a promising alternative to standard techniques when the feed is CO2 -rich. Further investigation of CO2 capture using n-alkane solvents will focus on the use of a cost-based performance measure, accounting for capital and operating costs, and on incorporating the effect of solvent viscosity. Further extensions of the integrated solvent and process design methodology using SAFT-VR will focus on integer decisions (e.g. the number of trays in the absorber), and the extension of the capabilities of SAFT-VR to relate solvent structure to SAFT-VR parameters. This type of methodology could be applied to the separation of H2 S/CH4 or N2 /CO2 which are two important applications of capture of greenhouse gases, or also to investigate other solvents like amine. 5. ACKNOWLEDGMENT Emmanuel Keskes would like to thank Schlumberger Cambridge Research for funding a studentship. We also acknowledge the use of the EPSRC’s Chemical Database Service at Daresbury. REFERENCES 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
A.L. Kohl and R.B. Nielsen, Gas purifcation. Gulf, 5th edition, Houston (1997). S.A. Newman, Acid and sour gas treating processes, Gulf, Houston (1985). A.S. Holmes and J.M. Ryan, US Patent No. 4 318 732-A (1979). I.I Dunyushkin, V.G. Skripka and T.L. Nenartovich, Phase Equilibria in the Systems Carbon Dioxide - n-Butane - n-Decane and Carbon Dioxide - Methane - n-Decane, Imperial College London, VINITI issue, 2180-77 (1977). M.R. Eden, S.B. Jorgensen, R. Gani and M.M. El-Hawagi, Chem. Eng. and Proc., 43 (2004) 595-608. E.N. Pistikopoulos and S.K. Stefanis, Comp. Chem. Eng., 22 (1998) 717-733. E.C. Marcoulaki and A.C. Kokossis, Comp. Chem. Eng., 22 (1998) S11-S18. A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J. Mills, G. Jackson, and A.N. Burgess, J. Chem. Phys., 106 (1997) 4168. A. Galindo, L.A. Davies, A. Gil-Villegas and G. Jackson, Mol. Phys., 93 (1998) 241. A. Galindo and F.J. Blas, J. Phys. Chem. B, 106 (2002) 4503. Process Systems Enterprise, www.psenterprise.com. N.M.P. Kakalis, A.I. Kakhu and C.C. Pantelides, Proc. 6th International Conf. on Foundations of Computer Aided Process Design, CACHE Publications (2004) 537. P. Paricaud, A. Galindo and G. Jackson, Ind. Eng. Chem. Res., 43 (2004) 6871. D.A. Fletcher, D.A., R.F. McMeeking and D. Parkin, The United Kingdom Chemical Database Service, J. Chem. Inf. Comput. Sci., 36 (1996) 746-749. DECHEMA, www.dechema.de/detherm-lang-en.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Integrated Approach to Crystallization Process Design for Fine Chemicals and Pharmaceuticals Christianto Wibowo, Ketan D. Samant, and Lionel O’Young ClearWaterBay Technology, Inc., 20311 Valley Blvd., Suite C, Walnut, CA 91789, USA
Abstract Designing a crystallization process that can recover the right product with a reasonably high yield and sufficiently high purity is often a key task in ensuring overall process feasibility in the fine chemical and pharmaceutical industry. An integrated approach for crystallization process design which combines synthesis with modeling and experimental activities is presented. By first modeling the thermodynamic behavior of the system, feasible operating regions and the need for further experimental verification can be identified. Kinetics and mass transfer effects are considered next. Modeling of phase behavior and relevant downstream operations helps to design experiments and organize the results in a meaningful way, and more importantly, points to the right direction as to what should be done next, thereby minimizing the required time, effort, and cost for developing the process. Keywords: Crystallization, Solid-liquid equilibrium, Process synthesis.
1. Introduction Most of the newly discovered fine chemical and pharmaceutical compounds are high molecular weight chemicals, which are commonly recovered as solids via crystallization. Consequently, developing a crystallization process that can recover the right product with a reasonably high yield and sufficiently high purity is often a top priority. In contrast to distillation, for which the use of vapor-liquid equilibrium models and residue curve maps has become a routine practice, conceptual design of crystallization processes are often based on scattered solubility data and inadequate information about the compounds involved in the system. Furthermore, limited availability of the new compound as well as time constraints prevent an extensive experimental study. This problem has been addressed from different angles. For example, systematic procedures for synthesizing crystallization processes using solid-liquid equilibrium (SLE) phase diagrams [1] as well as downstream processes [2] have been developed. Much effort has been devoted into modeling and visualization of SLE phase diagrams for synthesis purposes [3,4]. To deal with the lack of reliable model parameters, an objective-driven scheme for experimental determination of such diagrams has also been developed [5]. In light of the need for combining experimental and computational efforts, regression methods for backing out the necessary parameters from experimental solubility data has been proposed [6]. This paper discusses an integrated approach to crystallization process design, which combines synthesis effort with the right amount of modeling and experimental activities to minimize the required time, effort, and cost for developing the process. Since the approach offers both quantitative results and qualitative insight into the problem, it provides a much better chance of success compared to the traditional trial-and-error approach.
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2. Equilibrium-Based Process Design A development project involving crystallization of fine chemicals or pharmaceuticals typically begins in the laboratory with the synthesis or isolation of a target compound in a small scale. At this early stage of development, the best strategy is to focus on thermodynamics first. A preliminary picture of the SLE phase diagram should be generated based on minimum information. With limited amount of available samples, most of which is often reserved for efficacy or clinical tests, preliminary experiments which require a relatively small amount of material (such as DSC, PXRD, Raman spectroscopy, and so on) are performed. The objective is to obtain basic information on physical properties such as melting point and heat of fusion, as well as the presence of polymorphs, solvates, and hydrates. Preferably supported by some experimental solubility data, one should first proceed with modeling to calculate the phase diagram. While not expected to be highly accurate given the limited amount of data available at this point, the diagram should provide a good starting point for process synthesis. By going through the synthesis exercise, one can then identify important regions on the diagram which needs to be verified further. Additional experiments are then planned accordingly, focusing on these regions. The results are then used to improve the model accuracy. This way, a reliable model can be obtained with minimized computational and experimental effort. 2.1 SLE Modeling The basic model representing the SLE phase behavior can be derived by equating the fugacities of each component in the liquid and solid phases. Assuming the solid phase is pure, the generic equation
fi S = γ i xi fi L
(1)
represents the saturation variety for component i, containing all liquid phase compositions at which component i is saturated, or in other words, is in equilibrium with the solid phase [6]. The SLE phase diagram of the system is basically a collection of such saturation varieties for all components in the system. The left hand side of Eq.(1) is the ratio of fugacities of pure component i in solid and liquid phases at the same conditions, which can be computed from pure component melting point and heat of fusion data. The right hand side represents the activity of component i in the liquid phase at a specified temperature and pressure, which is the product of the mixturedependent activity coefficient γ i and the mole fraction xi. A potential problem at this stage is that the melting point and heat of fusion of some impurities are unknown due to unavailability of pure component samples. In such a case, group contribution-based estimation methods [7,8] can be used despite of their limited accuracy in predicting the properties of compounds with complex structures. Finding the physical properties of the solvent should not be a problem since they are available in various databases and reference books. Standard activity coefficient models such as NRTL, UNIQUAC, etc. can be used with Eq.(1) to solve for the liquid phase composition, provided that the parameters for the particular system is available. When dealing with new compounds for which such parameters are typically unavailable, two approaches are possible. The more preferable one is to perform experiments to obtain solubility data at different temperatures. As most standard models only incorporate binary interaction parameters, it is often good
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enough to measure binary SLE data, such as pure component solubility in the solvent and the melting point of a binary mixture. If such measurements are not possible due to the limited availability of samples, an alternative approach is to calculate the solubility from chemical potentials, which can be computed using quantum chemistry and statistical thermodynamics [9]. The results can then be treated as experimental data. Model parameters can be obtained from the data by solving an optimization problem in the following generic form:
(
min F = ∑ wk γ icalc − γ iexpt k
)
2
k
(2)
where the index k indicates data point and wk represents the weighting factor for each data point. Component i is the one whose solubility is being measured. The term γ icalc can be replaced by the appropriate expression of the activity coefficient model, while the experimental value of the activity coefficient ( γ iexpt ) is calculated from Eq.(1). 2.2 Conceptual Design Using the procedure of Wibowo and Ng [1], one can synthesize feasible process alternatives based on the calculated SLE phase diagram. Besides getting a preliminary image of the process, one should also identify the possible operating regions on the phase diagram and perform rough material balance calculations. As an example, consider a case in which a valuable product A has been successfully synthesized in the laboratory by a coupling reaction which also results in several byproducts, B being the most plentiful. The reaction product consists of 75% A, 19% B, 1% other byproducts, and 5% “lights” (e.g. reactants with much lower molecular weights compared to A). Since A and B are heat-sensistive, purification using distillation alone is expected to be expensive with the need to operate under very low pressure. It is therefore desirable to develop a crystallization-based process for purifying product A at a near-atmospheric operating condition. The target is to obtain 99.5% purity with 80% recovery. Assuming the lights can be removed prior to crystallization and the effect of other byproducts can be neglected, the problem is reduced to separation of a binary mixture containing about 80% A and 20% B. After preliminary screening, two solvents S1 and S2 are identified as suitable candidates to serve as crystallization solvent. Fig.1 shows a sketch of the phase diagram involving A, B, and S1 as the solvent, along with the process flowsheet. The feed (point 1) is mixed with fresh solvent and recycle stream in a dissolver to produce the first crystallizer feed (point 2) which is located in compartment A (that is, the region where pure A can be obtained). Upon evaporating some solvent at T1, solid A crystallizes out and the solution composition moves to point 3, which is close to the boundary between compartments A and B. The mother liquor is then fed to a cooling crystallizer operating at T2, resulting in co-crystallization of A and B. The mother liquor from this crystallizer (stream 4) is then recycled. The theoretical overall recovery of A for this configuration is about 87%. It is obvious that the location of the A/B compartment boundary, especially between T1 and T2, is especially important. The per-pass recovery of A in the first crystallizer can be maximized when point 3 is closer to the boundary (farthest away from A). Furthermore, the location of point 4 determines the composition of the purge (stream 5), which in turn dictates the overall recovery of A. Thus, experimental verification is necessary to ensure that the boundary location is correct. Upon obtaining the additional data, regression should be repeated to obtain updated model parameters and the material balance has to be recalculated with a phase diagram that is consistent with experimental results.
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S1
Isotherm at T2 Isotherm at T1
1 80% A 20% B
4 2
Diss
2
3
Cry T1
3
1' 2'
Important region
Boundary 1
4
Cry T2
A
Compartment A
A
S1
5
35% A 65% B
Compartment B 5
B
Fig. 1. Phase diagram and process flowsheet for purification of product A using solvent S1.
Note that T1 and T2 are actually design variables, which affects material and energy balances of the process. Therefore, once a reliable phase diagram is obtained, the overall recovery and energy consumption can be optimized with respect to the design variables. Furthermore, the same exercise should also be repeated using solvent S2, and if deemed necessary, mixtures of S1 and S2. The results can then be compared to determine which solvent gives the better performance. It should be noted, however, that operational considerations such as solution viscosity at crystallization temperature may also affect the final choice of solvent as well as operating temperatures. In other words, the mathematically optimum solution may not always be the best one in practice.
3. Beyond Solid-Liquid Equilibrium The equilibrium-based design obtained by solely considering the SLE behavior does not represent the final design. Equally important to consider in designing an operable crystallization process are kinetic and mass transfer effects such as impurity incorporation into the solid product via trapping during crystal growth or imperfect removal of mother liquor from the solids after filtration. A related issue is the particle size distribution (PSD) of the crystals, which strongly affect filtration, cake washing, and deliquoring performance. Once the thermodynamically feasible operating region has been identified, the next step is to design the entire crystallization-filtrationwashing-deliquoring train in an integrated manner such that all product specifications can be met [10]. A similar despite somewhat less fundamental approach incorporating modeling and experiment into process synthesis should be taken at this stage. The equilibrium-based process flowsheet is expanded to include downstream operations such as filtration, washing, and deliquoring. Since first-principle modeling of crystallization kinetics and downstream operations is often impractical due to the complexity of the involved phenomena, a better approach is to use semi-empirical models that capture the dependence on key design variables but contain adjustable parameters that can be fitted to experimental data. Calculation results using such models can serve as the basis for preliminary equipment design.
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As an illustration, consider the impurity management for the production of A in the above example. Even though thermodynamics dictates that pure A can be obtained in the first crystallizer, in reality some mother liquor that contains B can be trapped inside the crystals during crystallization, forming inclusion impurities. In addition, it is not possible to completely remove all mother liquor from the solids. This is why the downstream process must be designed properly to achieve the 99.5% purity target. Fig.2 shows a schematic of the downstream process after the first crystallizer along with the amount of impurities at each location. In practice, the three operations typically occur within a single equipment, such as a rotary drum filter or a centrifuge. A large portion of the mother liquor is removed from the slurry (1) during filtration, but some residual liquid remains in the wet cake (2). Most of the impurities goes away with the mother liquor, but the inclusion impurities stay with the solids. Washing can then be employed to replace the “dirty” residual liquid, which has the same composition as the mother liquor, with fresh solvent that contains less or no impurities at all. As a result, the amount of impurities in the residual liquid (3) again decreases. In addition, the amount of inclusion impurities may also decrease due to partial dissolution of the crystals. Finally, deliquoring removes some of the “clean” residual liquid to give a drier cake (4), thus further reducing the amount of impurities. Note that normally the dissolved impurities in the remaining liquid will be incorporated into the solids upon drying, since they do not evaporate with the solvent. By performing material balances with separated solid and liquid streams, the impurity content in all streams can be traced. Besides the SLE model (which is necessary to determine the mother liquor composition and the extent of dissolution during washing), constitutive models are needed to describe the dependence on system-related variables such as cake permeability, liquid viscosity, and so on. Table 1 summarizes the generic form of these models. The key variables on which each function shows strong dependence can be identified from first principles, but obtaining an explicit formula is impossible without considerable simplifications such as representing particle size distribution by mean size and porosity. Therefore, it is much easier in practice to introduce empirical parameters. For example, the washing model can be written as
xi7 − xi6 = aW 3 + bW 2 + cW + d xi2 L − xi6
(3)
Washing liquid
Impurity amount
6 1
Filtr
2
Wash
5 Filtrate
3
Deliq
7
In residual liquid
4
8
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Residual liquid
In solids
1 Slurry (solids + mother liquor)
Solids + dirty residual liquid
Solids + clean residual liquid
Solids + clean residual liquid
2
3
Location
Fig. 2. Schematic of the downstream process after the first crystallizer.
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Generic form
Inclusion
xi(1S ) = f 1 x (1L ) , G , PSD
Filtration
X ( 2 ) = f 2 (t f , k , μ , σ )
Washing
x i( 7 ) − x i( 6)
(
x i( 2 L ) − x i( 6) Deliquoring
Explanation
)
= f 3 (W , D n )
X ( 4 ) = f 4 (t dl , k , μ , σ )
Dn = dispersion number, which depends on cake properties and diffusivity of solute in liquid G = crystal growth rate k = cake permeability tf = filtration time tdl = deliquoring time W = wash ratio (kg of wash liquid per kg of residual liquid at the beginning of washing) x = composition X = cake wetness (kg of liquid per kg of solid) γ = liquid viscosity µ = liquid surface tension Superscripts: stream number; S=solid, L=liquid
with parameters a, b, c, and d depends on Dn. Such a polynomial form is selected because it can fit an S-shaped curve, which is the typical shape for most systems [11]. An experimental study to determine the values of dimensionless wash liquor concentration (the left hand side of Eq.(3)) at different values of W can then be performed. If desired, cakes having different PSD (corresponding to different values of Dn) can be studied in separate runs to obtain the dependence of a, b, c, and d on Dn.
4. Conclusion An integrated approach for crystallization process design which combines synthesis with modeling and experimental activities is presented. Beginning with thermodynamics and considering kinetic and mass transfer effects afterwards, fundamental and semi-empirical models supported by experiments focusing on achieving the process objectives are used to guide the development of process alternatives and provide quantitative measures to evaluate them. Using such an approach, the required time, effort, and cost for developing the crystallization process are minimized.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
C. Wibowo and K. M. Ng, AIChE J., 46 (2000) 1400. W.-C. Chang and K. M. Ng, AIChE J., 44 (1998) 2240. K. D. Samant and K. M. Ng, AIChE J., 47 (2001) 861. K. D. Samant, D. A. Berry, and K. M. Ng, AIChE J., 46 (2000) 2435. K. S. Kwok, H. C. Chan, C. K. Chan, and K. M. Ng, Ind. Eng. Chem. Res., 44 (2005) 3788. K. D. Samant, L. O'Young, M. Kwok, and K. M. Ng, FOCAPD 2004 Proceedings, Princeton, New Jersey, 2004, pp.385-389. K. G. Joback and R. C. Reid, Chem. Eng. Comm., 57 (1987) 233. J. Marrero and R. Gani, Fluid Phase Equilib., 183–184 (2001) 183. A. Klamt and F. Eckert, Fluid Phase Equilib., 172 (2000) 43. C. Wibowo, W.-C. Chang, and K. M. Ng, AIChE J., 47 (2001) 2474. R. J. Wakeman and E. S. Tarleton, Filtration: Equipment Selection, Modelling and Process Simulation, Elsevier, Oxford, 1999.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Improved solutions for zebra mussel (Dreissena polymorpha) control – a chemical product engineering approach R. Costaa, P. M. Saraivab, P. Elliottc, D. C. Aldridgec, G. D. Moggridgea a
Structured Materials Group, Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA,UK b GEPSI-PSE Group, Department of Chemical Engineering, University of Coimbra, Polo II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal a Aquatic Ecology Group, Department of Zoology, University of Cambridge, Downing Street, Cambridge, CB2 3EJ,UK
Abstract Systematising the design of chemical products is a crucial challenge for chemical process-related companies. Recognising the need to guarantee and reinforce the profession’s competitiveness, the concept of chemical product engineering (CPE) has been emerging in the vocabulary of chemical engineering. Process system engineering (PSE) thinking and tools provide a sound support to lead CPE activities. Thus, there is plenty of room for the PSE community to identify and address research opportunities connected with CPE. This paper discusses the scope of CPE, proposing a model to structure the discipline. The development of improved solutions for zebra mussel (Dreissena polymorpha) control is used to illustrate how CPE and PSE thinking can be integrated with mutual benefits. Keywords: product engineering, chemical product design, zebra mussel control, PSE
1. Introduction New product development is a crucial task for the success of modern corporations. This activity must be the result of strategic and organizational actions combined with technical effort. Whilst in some industrial and engineering sectors the technical side of new product development has always been a major issue, in the chemical processrelated industries the systematic and efficient design of new products, seen as a specific branch of chemical engineering, is relatively recent. Process System Engineering (PSE) thinking and tools can be very helpful in conducting chemical product engineering (CPE) activities. It is therefore up to the PSE community to identify and address research opportunities in this field. This paper discusses the scope of CPE, proposing a model to structure the discipline. The development of improved solutions for zebra mussel (Dreissena polymorpha) control is used to illustrate how CPE and PSE thinking can be integrated and assist each other in handling such a multi-faceted problem.
2. Chemical product engineering: an emerging paradigm The concept of CPE has been emerging amongst the chemical engineering community, as an autonomous branch, within the last decade (Fig. 1).
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756 March 1995 The term formulation engineering is used to designate the discipline concerned with the design of the desired properties of a product [1]. November 1996 The conference entitled Putting Structure into Chemical Engineering reinforces the need to include product engineering in the chemical enginnering curricula [11].
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1996
January and July 2000 Chemical product design courses taught to chemical engineering undergraduates are referred to in the literature [5, 12].
September 2003 The first European Symposium on Product Engineering is held as part of the 4th European Congress of Chemical Engineering in Granada (Spain). September 2004 The EFCE Section on Product Design and Engineering is started.
May 2001 Cussler and Moggridge [13] publish the first textbook specifically addressing the field of chemical product design.
November 2004 A special issue of Chemical Engineering Research and Design devoted to chemical product design comes out [14].
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November 2004 One of the first review papers in the field of product technology is published [15]. November 2004 The second European symposium on the field of chemical product engineering is held in Groningen (Netherlands), for the first time as an autonomous event and under a more general title (Second European Symposium on Product Technology).
2003
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Figure 1 – Chronological diagram of important events and landmarks in the emergence of chemical product engineering as a discipline within the scope of chemical engineering. Although some efforts have been made to elucidate the scope of CPE and place it in the context of chemical engineering science and practice [2, 3, 6-8, 10], the field is broad and developing in many directions, and a formal structure for it has not yet been widely accepted [15]. In the next section, a framework for CPE is proposed based upon PSE paradigms.
3. Structure of chemical product engineering The model proposed in this paper to structure CPE is based upon the following facets: chemical product, property function and process function, multi-faceted approach, chemical product design and the role of process engineering (Fig. 2). Chemical product
Role of process engineering
Porperty and process functions
Chemical product engineering
Multifaceted approach
Chemical product design
Figure 2 – Facets structuring chemical product engineering. 3.1. Chemical product In the chemical engineering context, the concept of product engineering has often been taken as synonymous with formulation engineering. However, chemical process-related companies, and hence chemical engineers, have to deal with a wider range of products (e.g. polymers, advanced materials, medical devices, bioproducts). Similar principles and fundamentals can be applied to the analysis and design of all of such chemical products. Although an analogy between these products cannot be established regarding their specific appearance or performance, similarities between them do exist in terms of development and manufacturing. 3.2. Property function and process function One of the important features of chemical products is the fact that customers generally do not judge their value based on technical specifications, but rather according to
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functionality and performance attributes such as smell and handling properties, which are usually referred to as quality factors. These quality factors are often qualitative and subjective, and hence quantitative performance indices must be conceived to measure them. Performance indices are determined by the composition and physico-chemical properties of the materials composing the product and its structure. Such dependences can be systematised based on the concept of property function, initially proposed by Rumpf [16]. The desired product structure can be obtained by selecting the proper product ingredients/materials and recipe, as well as a suitable manufacturing process. The relationship between process conditions and structural attributes of a chemical product can be described by what is termed as process function. Depending on the degree to which the underlying phenomena are understood, one out of three approaches can be adopted to derive property functions and process functions: (i) detailed analysis and rigorous modeling, (ii) order-of-magnitude analysis based upon simplifying assumptions, or (iii) black box analysis based on neural networks or regression analysis techniques. The idea of a materials-product-process triangle is introduced in this paper to systematise the concepts of property function and process function (Fig. 3). Quality Factors
Product
Quantitative Composition
Materials
PhysicoChemical Properties
Chemical Product Engineering
s es oc on Pr cti n fu
Pro p fun erty ctio n
Performance Index Property Function Structural Attributes
Operating Variables
Process
Figure 3 – Materials-process-product triangle. 3.3. Multi-faceted approach Since the quality of a chemical product, marketed at a macro-scale, is defined at the nano and micro-scale of its constituents, as well as at the meso and macro-scale of the manufacturing process, the concept of a multi-faceted approach described by several authors [1-4, 7, 9, 10] is also useful to structure CPE. The ultimate goal of CPE is the translation of phenomenological laws and models, expressed by property functions and process functions, into commercial product technology. 3.4. Chemical product design Recently, Cussler and Moggridge [13] developed the concept of chemical product design (CPD), a holistic approach to the design process comprising four essential steps: (i) identification of needs, (ii) generation of ideas, (iii) selection of ideas, and (iv) product manufacture. Other frameworks, specifically addressing the design of chemical products, have also been proposed [5, 17]. The concepts of CPD and CPE should not be taken as synonymous, just as the traditional concepts of process design and chemical engineering have not been seen as equivalents. CPD corresponds to the systematic procedure or framework of methodologies and tools targeted to provide a more efficient
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and faster design of products able to meet market demands. CPE is, in its turn, the science and art of creating such products. 3.5. Role of process engineering CPE and process engineering should be viewed from an integrated perspective, rather than sequentially. CPE covers the whole process of conversion of product discoveries into marketable products.
4. Development of improved solutions for zebra mussel control The zebra mussel, Dreissena polymorpha, is an invasive species, whose biofouling activity causes serious economic problems in fresh water-dependent industries. The losses due to this pest, including monitoring and control costs, have been estimated to be on the order of US$ 5 billion per year [18]. Methods currently available for the control of this bivalve have a number of significant disadvantages. A project aiming at the development of improved control strategies with enhanced effectiveness and greater environmental acceptance is currently in progress. The development of such a control strategy is a complex problem, which involves technical, economic, ecological and environmental considerations and emerges at the interface of chemical engineering with zoology. In order to deal with this complexity, the project has been structured in terms of a CPD template (Fig. 4). The first phase of the development process has been concluded and has resulted in two promising control concepts. The formulation and testing of potential control strategies based on these concepts has also been initiated. In the following paragraphs, details of these early phases of the development process are presented. Development of control concepts Identify customer needs Establish target specifications Generate ideas of control concepts Select ideas of control concepts
Formulation of the control strategy In vivo toxicity testing Selectivity assessment Process development Final testing and refinement Regulatory registration
Figure 4 – CPD approach to the development of zebra mussel control solutions. 4.1. Control concepts Two concepts emerged from the first phase of the development process: (i) control based on the encapsulation of toxins and (ii) control based on the combination of toxins. The formulation of control strategies based on encapsulated poisons exploits the huge filtration capabilities of zebra mussels [19, 20]. In addition to obvious economic benefits, an enhancement of the toxicity of molluscicides provided by encapsulation should also offer environmental advantages since microcapsules can be engineered to degrade before discharge into waterways. Control strategies based on the combination
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of poisons with distinct toxicity mechanisms take advantage of their cumulative actions as well as synergetic effects. It has been proved that the combination of two poisons, one of which has ciliostatic activity and paralytic effects on mussels’ adductor muscle, outperforms the sum of their independent effects. Control strategies based on these two concepts have the potential to meet the application needs previously identified (Table 1). Table 1 Needs that a control strategy should meet. Highly efficient solution Viable cost Non polluting Selective Minimal bio-accumulation Meet governmental regulations Guarantee total plant protection Ease to apply in existing installations Ease to retrofit
4.2. Formulation and testing stages The formulation of viable control methods based on the concepts above involves several inter-related steps (Table 2). This stage has been integrated with in vivo toxicity tests and complemented by endoscopy and video image analysis. Toxicity bioassays assess the performance of potential control methods (expressed in terms of a median lethal concentration, LC50). Endoscopic examination is particularly relevant for the formulation of methods based on particulate toxins. Table 2 Steps involved in the formulation of potential strategies for zebra mussel control. Strategies based on the encapsulation of toxins Selection of the toxin Selection of the coating Selection of the manufacturing process Microcapsules manufacture Physical characterisation
Strategies based on the combination of toxins Selection of the toxins Definition of the application scheme Mixture design
The experimental work involved in the stages of formulation, testing and process development will be designed to model dependencies between product functionality, composition, structure and process conditions. In particular, experiments aiming at the study of the influence of particle size on the toxicity of encapsulated toxins will be conducted in order to specify a materials-process-product triangle (Fig. 5). The use of coatings that are edible to zebra mussels should maximize the filtration of the particulate toxins. Additionally, their size and retardant power are determinant for their toxicity. Adult zebra mussels can filter particles in the size range 0.7 – 450 μm with increased retention of those whose size is between 5 and 35 μm [19]. The retardant power of the particles should be such that a significant amount of the toxin is not released in the water system before the product reaches the mussels.
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Toxicity
Particulate toxin
LC50 Property Function
Quantitative Composition
Toxin and coating
Nature of the toxin and coating
Development of strategies based on particulate toxins
s es oc ion Pr nct fu
Pro p fun erty ctio n
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Operating Variables (depend on the process)
Encapsulation process
Figure 5 – Materials-process-product triangle in the development of control methods based on particulate toxins. The adoption of PSE principles and tools has been decisive in this CPE research work. At a conceptual level, they were crucial in the development of an overall, systematic and holistic model that has been used to define and apply CPE as described. At a more operational level, they were important in the definition of the different development stages for a new zebra mussel control solution, helping in formulating goals, building proper models to address them and reach sound conclusions. The CPD cycle, which is still in progress, will hopefully result in an improved final product compared to the currently available technologies.
5. Concluding remarks In this paper, a model for CPE was proposed in an attempt to structure the discipline as an autonomous branch of chemical engineering. The development of improved solutions for zebra mussel control was discussed in order to illustrate the practical application of such a structure. It was also shown how PSE principles and tools can be integrated in this approach.
Acknowledgements Financial support from the Portuguese Foundation for Science and Technology (PhD fellowship SFRH/BD/18731/2004 and research project POCI/EQU/59305/2004) is gratefully acknowledged.
References [1] J. Villermaux, 1995, Future challenges in chemical engineering research, Trans IchemE – Part A, 73, 105-109. [2] K. Wintermantel, 1999, Process and product engineering – achievements, present and future challenges, Chem. Eng. Sci., 54, 1601-1620. [3] K. Wintermantel, 1999, Process and product engineering – achievements, present and future challenges, Trans IchemE – Part A, 77, 175-188. [4] J. Charpentier and P. Trambouze, 1998, Process engineering and problems encountered by chemical and related industries in near future, Chem. Eng. Proc., 37, 559-565. [5] A. W. Westerberg and E. Subrahmanian, 2000, Product design, Comp. Chem. Eng., 24, 959966.
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[6] J. A. Wesselingh, 2001, Structuring of products and education of product engineers, Powder Techn., 119, 2-8. [7] J. C. Charpentier, 2002, The triplet “molecular processes – product – process” engineering: the future of chemical engineering?, Chem. Eng. Sci., 57, 4667-4690. [8] E. Favre, L. Marchal-Heusler and M. Kind, 2002, Chemical product engineering: research and educational challenges, Trans IchemE – Part A, 80, 65-74. [9] I. E. Grossmann, 2003, Challenges in the new millennium: product discovery and design, enterprise and supply chain optimization, global life cycle assessment, Proceedings of the 8th International Symposium on Process System Engineering – PSE-2003, 28-47. [10] J. C. Charpentier and T. K. McKenna, 2004, Managing complex systems: some trends for th future of chemical and process engineering, Chem. Eng. Sci., 59, 1617-1640. [11] J. Villadsen, 1997, Putting structure into chemical engineering, Chem. Eng. Sci., 52, 28572864. [12] G.D. Moggridge and E. L. Cussler, 2000, An introduction to chemical product design, Trans IchemE – Part A, 78, 5-11. [13] E. L. Cussler and G. D. Moggridge, 2001, Chemical Product Design, Cambridge University Press, Cambridge. [14] T. Broekhuis, 2004, Special issue – product design and engineering. Chem. Eng. Res. Des., 82, 1409-1410. [15] R. M. Voncken, A. A. Broekhuis, H. J. Heeres and G. H. Jonker, 2004, The many facets of product technology, Chem. Eng. Res. Des., 82, 1411-1424. [16] H. Rumpf, 1967, Über die eigenschaften von nutzstäuben, staub – reinhalt, Luft, 27, 3-13. [17] C. Wibowo and K. M. Ng, 2002, Product-centered processing: manufacture of chemicalbased consumer products, AIChE Journal, 48, 1212-1230. [18] M. Khalanski, 1997, Industrial and ecological consequences of the introduction of new species in continental aquatic ecosystems: the zebra mussel and other invasive species, Bulletin Français de la Peche et de la Pisciculture, 345, 385-404. [19] P. Elliott, 2005, The zebra mussel in England: biology, impacts, and control using microencapsulated toxins, Doctor of Philosophy Thesis, University of Cambridge. [20] P. Elliot, D. C. Aldridge and G. D. Moggridge, 2005, Proceedings of the 7th World Congress of Chemical Engineering.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Success Factors for CAPE in the Engineering Practice of a Process Plant Contractor Gabriele Engl, Andreas Kröner Linde AG, Linde Engineering Division, Dr.-Carl-von-Linde-Str. 6-14, 82049 Hoellriegelskreuth, Germany
Abstract The CAPE strategy and key success factors of a process plant contractor are presented from a business and management perspective. As a company with a long tradition and experience in developing own CAPE tools, Linde Engineering is not following "mainstream" strategies observed in process industry. It is shown how this position is justified by key business drivers, and which challenges typically arise. Rather than presenting technical details of new CAPE developments, the contribution gives an overview of CAPE topics in an industrial environment from a practical, rather than academic point of view. Even though many aspects seem to be well-known, they are outlined since they are crucial for the business success and thus demand high attention. Keywords: Engineering company, business drivers, CAPE strategy, process simulation, success factors.
1. Introduction Computer Aided Process Engineering (CAPE) plays a key role in the industrial practice of the Linde AG, Linde Engineering Division. As a leading international engineering and contracting company, Linde Engineering designs and builds turnkey process plants for a wide variety of industrial users and applications: The chemical industries, air
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separation, manufacturers of hydrogen and synthesis gases, natural gas processing and more. From this practical perspective, key business drivers and their impact on the CAPE strategy are presented. Resulting objectives and challenges for CAPE are discussed, where the practical relevance and significance of work is shown by typical examples. As a conclusion, success factors for CAPE are derived.
2. Business Alignment and CAPE Tools In a highly competitive industrial environment, the alignment of the CAPE strategy with the business strategy is essential. Key elements of the business strategy, such as the know-how of own processes and technologies as well as new market developments, are reflected in the CAPE strategy for process simulation, equipment simulation and physical properties: High quality methods and tools, which are available at universities or by commercial suppliers, are combined with internal developments to achieve optimal solutions. Furthermore, each single CAPE project has to be based on a promising business case, i.e. an approved analysis of costs and benefit. 2.1. Process design For process design, Linde Engineering is using commercial process simulation tools as well as its in-house process simulation program OPTISIM®. The latter has been developed for more than 20 years in order to fill a gap in the market and thus to achieve a competitive advantage, e.g. by using equation-oriented technologies to efficiently optimize processes with a large number of parameters including integer variables such as distillation column tray positions, see [1,2]. Even though progress is observed in the commercial market with regard to former gaps, Linde Engineering is still pursuing its strategy of own development, mainly for the following reasons: While commercial developments of equation-oriented technology are focused on plant operation applications, specific requirements for plant design are still not covered. For example, heat exchanger design models pose challenging constraint problems within equation-oriented optimization. Sophisticated algorithms were developed, see [3], which enabled OPTISIM® for the efficient design optimization of large and complex processes such as natural gas liquefaction (LNG plants). In addition, any migration to a commercial tool involves large investments which could not be justified so far. 2.2. Physical properties For physical property calculation, Linde Engineering has made significant investments into its GMPS package (General Multiphase Property System) and corresponding know-how. The system is integrated with the standard process and equipment design tools as well as engineering systems in order to support a consistent engineering workflow. Similar to process simulation, there is a more than 20-year-old history of developing own physical property methods. With regard to technology, specific multiphase
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calculation capabilities are still outstanding compared to commercial offerings. Further advantages are given by the flexibility to rapidly react to new requirements, and by the internal expert know-how which allows highly reliable guarantees, e.g. for product purities. 2.3. Equipment design For equipment design, in-house tools are essential for the simulation of own technologies such as cracking furnaces of olefin plants, reactors, spiral wound or platefin heat exchangers. The following figure shows the interconnection of internal programs and methods with commercial programs and external methods. Physical Properties GMPS
Equipment Design Process Design
External methods
OPTISIM Commercial Programs
Inhouse methods
Inhouse Programs Commercial Programs
Downstream Engineering Systems
3. Challenges Based on the business strategy, the following challenges for CAPE arise, which will be demonstrated by practical examples and some further specific presentations during the conference: 3.1. Adaptation and/or application of available methods to practical problems While universities and commercial suppliers offer a wide variety of high quality methods and tools for the solution of mathematical problems arising in practice, the successful application often exhibits a major challenge. As an example, the optimal design of separation processes poses a Mixed Integer NonLinear Problem (MINLP). When employing state-of-the-art MINLP algorithms in Linde Engineering's proprietary process simulation tool OPTISIM®, internal strategies were successfully implemented to cope with challenges such as instabilities of the underlying distillation column models and the large scale of complete process plant models, see [4].
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3.2. Interfacing external and internal methods and tools In order to support the integration of CAPE tools, a major effort has been invested for the development of interfaces and standards by universities, commercial suppliers and industry. The successful use of the interfaces in practice, however, can be a major challenge for the involved parties. As an example, Linde Engineering is integrating its proprietary physical property system GMPS with commercial process simulation systems. From a strategic point of view, the conflict is faced that standard interfaces like CAPE-OPEN are generally supported, but that, however, additional supplier-specific interfaces might be recommended due to significant advantages with regard to performance. From a technical point of view, a big effort and very tight cooperation of the supplier and Linde Engineering are necessary in order to support special features of the in-house methods, like the multiphase capabilities of GMPS as one of its competitive advantages. As a consequence, this involves an additional conflict with regard to know-how protection. 3.3. Innovation exceeding available methods and tools, e.g. modeling and simulation of own equipment The know-how of own processes and technologies, as a key element of the business strategy, also involves the need of excellent internal know-how to develop CAPE tools for the modeling and simulation of these technologies, and practical solutions are a big challenge. Plate-fin heat exchangers are an example for key plant components where Linde Engineering has its own technology including manufacturing facilities. Commercial tools are available for the detailed design of this equipment, allowing to incorporate proprietary geometry features. In order to achieve a competitive advantage, however, an innovative optimization strategy was successfully implemented and integrated into the existing design workflow. The solution is based on the equation-oriented framework of OPTISIM® and allows to efficiently compute an optimal geometry for given boundary conditions (such as heat transfer and process conditions). From a technical point of view, major challenges have been to find an optimal trade-off between the level of model detail and efficiency, and to implement an equation-based model of an equipment where discontinuities occur due to phase changes in process streams. 3.4. Filling the “gap” between CAPE developing experts and engineering practitioners Even though working in the same company, the CAPE developing experts and the engineering practitioners seem to live in "different worlds". While the first are deeply involved in the theory of modeling, solution methods, programming technology and IT issues, the latter have to cope with the every-day pressure of project work while applying CAPE tools to practical problems. Often it is a challenge for both sides to somehow understand the "language" of the other party, i.e. to understand the needs and requirements of the practitioner on one side, and to understand possible CAPE potentials on the other side. A successful communication is crucial to deliver CAPE tools with maximum benefit.
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3.5. Successful change management by achieving acceptance and usage of tools in practice After the development of CAPE tools it might be an even bigger challenge to achieve the acceptance and the usage of tools by the broad user base. A successful change management includes an early involvement of pilot users, careful testing, presentation of benefits, training, intensive support during the implementation phase, careful review of the user feedback and further improvements based on the experience. As an example, high acceptance of the new optimization framework for plate-fin heat exchangers (see above) could be achieved by supplying a new user interface, which completely hides the complex underlying OPTISIM® program. The new program PLATO was designed to allow a maximum degree of workflow automation for the equipment design specialist, to communicate only the essential ("practical") input and output parameters by the user interface, and to hide any additional (more "theoretical") parameters like optimization tolerances. Only the close cooperation between developers and users especially throughout the first phase of usage finally lead to a well-fit and well-accepted tool, and thus to the practical benefit of significantly reducing the costs of plate-fin heat exchangers. 3.6. Set-up of successful business cases to achieve investments in spite of cost reduction initiatives In the highly competitive market for process plants it is a challenge to achieve major investments into innovative CAPE developments, or strategies to use innovative CAPE tools, since cost reduction initiatives are highly prioritized. Often the costs and benefits are not clear in advance, but promising business cases are required for each single CAPE project. As an example, the use of dynamic simulation and optimization methods for the process and operation design is still far from being established as a standard in the engineering workflow. Linde Engineering successfully implemented dynamic optimization methods in OPTISIM® to compute optimal load change strategies for air separation plants, see [5]. Since the set-up of dynamic process models in the engineering work-flow will significantly increase the engineering man hours and thus the cost of the plant, it will not become standard before the practical benefit has been shown in detail. This is an ongoing project.
4. Success Factors The following key success factors for CAPE can be derived from these challenges: •
Understanding of the underlying business processes and high commitment with regard to the business strategy, set-up of promising business cases
•
Excellent technical know-how of modeling technology, mathematical solution methods, programming technology and information technology
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Capability, combined with creativity and patience, to implement (often empirical) strategies to overcome practical barriers such as model instabilities or discontinuities
•
High degree of support and interest by commercial suppliers in order to adapt commercial tools and interfaces for the successful integration with sophisticated in-house methods
•
Capability and willing to find an optimal trade-off between the best technical solution and an efficient practical solution
•
Excellent "social skills" for a beneficial communication between CAPE developers and users, for the presentation of new CAPE features and potentials, user training, support during the roll-out phase etc.
References [1] E. Eich-Söllner, P. Lory, P. Burr, A. Kröner: Stationary and dynamic flowsheeting in the chemical engineering industry. Surveys on Mathematics for Industry 7 (1997), pp. 1-28. [2] A. Kröner: An Engineering Company’s Approach to Filling "CAPE Gaps" in Process Simulation. Submitted to ESCAPE-16 and PSE'2006, Garmisch-Partenkirchen, Germany, July 2006. [3] G. Engl, H. Schmidt: The optimization of natural gas liquefaction processes. In: Progress in Industrial Mathematics at ECMI 96. M. Brøns, M. P. Bendsøe, M. P. Sørensen (editors), B.G. Teubner, Stuttgart, 1997, pp. 356-363. [4] I. Thomas, A. Kröner: Mixed-Integer Optimization of Distillation Column Tray Positions in Industrial Practice. Submitted to ESCAPE-16 and PSE'2006, Garmisch-Partenkirchen, Germany, July 2006. [5] A. Kröner, T. Kronseder, G. Engl, O. von Stryk: Dynamic Optimization for Air Separation Plants. Proceedings of ESCAPE-11, Kolding, Denmark, 2001.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Polyurethane design using stochastic optimization John Eslicka, Kyle Camardaa a
The University of Kansas, Department of Chemical and Petroleum Engineering, 1530 W. 15th Street, Lawrence, Kansas 66045-7609
Abstract The objective of this work is to develop a method for predicting polyurethane structures having physical and chemical properties matching a given set of properties. Preliminary quantitative structure property relations (QSPRs) are developed for mechanical properties of polyurethane elastomers. A mixed integer nonlinear program (MINLP) optimization problem is solved using Tabu search, a stochastic optimization method, to find a set of polymers with properties matching the target set. Results are obtained for the design of encapsulant polymers for electronics applications. Keywords: Product Design, Optimization, Polyurethane Polymers
1. Introduction Polyurethanes are used in a variety of applications such as foams, adhesives, coatings and plastic parts. Computational molecular design can be used to find candidate polymers that provide better properties for a particular application or to find more environmentally friendly alternatives to existing polymers. Some chemicals used in the production of polyurethanes are hazardous, and new polymers can be designed which are safer to make and still have similar properties to the polymers they replace. There are several techniques commonly used to predict properties of simple polymers bases on topological descriptors. The most common descriptors used are group contributions (Van Krevelen 1990; Venkatasubramaniam, et al. 1994; Maranas 1996; Gani et al. 1989) and connectivity indices (Kier and Hall 1976; Bicerano 1996, Camarda and Maranas 1999; Chavali et al. 2004). This work extends the use of connectivity indices for property prediction to the computational molecular design of cross-linked polymers. Polyurethanes generally consist of three segments, a diisocyanate (hard segment), a relatively long segment made of polyester or polyether (soft segment), and a short polyol chain extender. Polyurethanes tend to form micro-phases, with hard and soft segments forming separate phases. Covalent cross-linking is also possible; this is often accomplished using a chain extender with more than two functional groups. The goal of this work is to extend existing topological descriptors to provide a better description of the structure of polyurethanes, and develop new structure property relationships. The relationships are then used as the basis for an optimization problem to find a set of candidate polymers with a given set of properties. The optimization problem here is solved with Tabu search. Tabu search is a stochastic algorithm which has been shown to find local solutions to computational molecular design problems efficiently (Zhao et al. 2004). Use of Tabu search is relatively new in molecular design (Lin et al. 2005).
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2. Property Prediction To determine how topological descriptors should be adapted for this application, experimental data on polyurethane elastomers was collected and regressed. Mechanical property data are available in the literature for many elastomers. For the preliminary investigation, a set of 35 polyurethane elastomers was used. Data for three mechanical properties (tensile strength, elongation at break, and 300% modulus) was collected. A combination of connectivity indices for polymers as defined by Bicerano (1996) and some qualitative knowledge of polyurethane properties was used to produce a first attempt at new topological descriptors. Polyurethanes usually contain three segments. Each of the segments has a different effect on the overall polymer properties, and it is known that the relative length of the soft segments to the rest of the polymer has a significant effect on the properties. The ratio of the number of chain extender segments to soft segments also effects the properties. The set of descriptors used as property predictors starts with the connectivity indices for the whole polymer as described by Bicerano (1996). In addition, a separate set of connectivity indices for each type of segment is calculated. For connectivity indices related to groups of atoms (second order and higher), some groups of atoms will span segments. In such cases, the contributions of the groups are split between segments using the fraction of atoms in the group that are contained in a segment. The fractions of the total number of non-hydrogen atoms in each segment type are also used as descriptors. Multiple linear regression is used to develop property models. The predictors are the connectivity indices for the hard segment, soft segment, and the whole polymer, and the atom fractions for the hard and soft segments. Simple transformations of the descriptors are included. The connectivity indices and atom fractions form a closed set. The fractions of atoms in each segment add up to one, and the given the connectivity indices for each segment, the connectivity indices for the whole polymer can be found, so the descriptors for the chain extender segment do not need to be included. There are 56 possible predictors when the transformed predictors are included. To find models with a reasonable number of predictors, statistical software was used to analyze all possible combinations of up to five descriptors. The best models obtained are given by equations 1, 2, and 3.
1 = −0.98743 − 0.20943 f 0 Pts − 0.13440 ln( f1 ) + 0.44338ξ
2 1v ,t
+
0.46792
ξ1,t
(1)
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1 0.0025042 = 0.1143933 + − 0.0363393ξ 02v ,t Pel f1
+ 0.1317809 ln(ξ1v ,1 ) −
0.0611842
ξ 0 ,t
+
(2)
0.0414401
ξ1v ,1
1 = 23056.40 − 0.0875589 ln(1.0 − f 0 − f1 ) Pmd
− 4029.83ξ 0 v , 0 + 18610.08ξ
2 0v,0
+ 10896.00 ln(ξ 0, 0 ) −
7.094436
(3)
ξ 0, 0
Where fi is the fraction of atoms in segment i, ξj,i is the jth order intensive simple connectivity index for segment i, and ξjv,i is the jth order intensive valence connectivity index for segment i. The t subscript indicates the connectivity index is for the whole polymer. Pts is the tensile strength in MPa, Pel is the elongation at break, and Pmd is the 300% modulus in MPa. These QSAR expressions allow for a preliminary design of polyurethane encapsulants. Other important factors to be considered include cross-linking density, processing methods. Additional data will allow development of property models accounting for a wider variety of molecular structures, and refinement of the structure property relations.
3. Optimization Once structure property expressions are obtained, they can then be embedded in an optimization problem to find a list of candidate molecules with properties closely matching the desired properties. 3.1. Problem Formulation The objective function used here is the total scaled deviation of the properties from their targets, as shown in equation 4. Wi is the weight assigned to each property; it can be used to scale the objective function and adjust the relative importance of each property.
min ∑ Wi
Pi − Pi , target Pi , target
(4)
In general, the large size and nonconvexities of the MINLP problems which arise in molecular design make deterministic approaches impractical. Furthermore, the inherent inaccuracies in the property correlations make the search for a true global optimum less important than the generation of feasible solutions which are locally optimal and close to the lower bound of zero on the scaled property deviation. For these reasons, stochastic algorithms such as Tabu search and genetic algorithms have been employed in previous computational molecular design studies (Venkatasubramaniam, et al. 1994;
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Zhao et al. 2004). Tabu search is used here based on the success of previous studies applying this method to computational molecular design problems (Lin et al. 2005). The data structure used to represent the polymers segments is a vector of integers. Each integer represents a predefined group used to build the segment chains. The groups used to build the soft segment chains are given in Table 1, and the hard segment groups are given in Table 2. The hard segment is constrained to start and end with an isocyante group cap group. Table 1. Soft segment groups 1
2
3
4
5
Table 2. Hard segment groups 1
2
3
4
5
6
Cap
The soft segment of a polyurethane is usually a polymer itself. The soft segment vector represents a repeat unit and another integer is used to represent the number of times it repeats. The vectors for each segment do not have a fixed length. The size may change as groups are added or deleted. The polymers considered here are subject to some size constraints. The hard segment vector has a minimum length of three and a maximum length of ten including the two cap groups. The soft segment vector has a minimum length of three and a maximum length of forty. The soft segment repeat unit can repeat up to fifty times. For the work presented here, the chain extender was fixed, but the ratio of chain extender segments to soft segments is allowed to vary. Some structural constraints are also considered to provide realistic molecular structures. For example, two –O– groups cannot appear next to each other. The soft segment repeat unit is required to have an –O– group at exactly one end, and the entire soft segment has an –O– group at both ends. 3.2. Tabu Search Tabu search is an stochastic optimization technique similar to a random walk, but with a memory list that prevents it from revisiting solutions close to ones it has recently visited. This adaptive memory helps encourage the search to move out of local optima and find better solutions.
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Tabu search starts with an initial feasible solution or set of solutions, which can be generated at random or specified. The initial solution is added to a Tabu list. A set of neighboring solutions is generated using a set of moves. The moves listed below are allowed. 1. 2. 3. 4. 5.
A group in a segment can be replaced by a new random group A group can be added to a segment A group can be deleted form a segment The number of repeat units in the soft segment can be changed The number of chain extender segments can be changed
The best non-tabu solution from the set of neighbor solutions is chosen as the new solution. Tabu rules are applied to compare new solutions to each solution in the tabu list to determine weather a new solution is tabu. The tabu rules are very flexible, and can be anything from being tabu if the solutions are exactly the same to being tabu if the same type of move was used to generate the solution. Once a new solution is selected, it is added to the tabu list, and the oldest tabu solution is removed if the list is full. The process is repeated until a set number of non-improving iterations is reached. The tabu search can then be repeated with relaxed tabu rules to refine the solution further. The entire search may be repeated with a new randomly generated starting point to find more candidate solutions.
4. Example The structure property relations given by Equations 1 through 3 are used to solve an example polymer design problem with Tabu search. The optimization problem and constraints are given in the previous section. The target properties are given in Table 3, and represent reasonable values for a polyurethane encapsulant for electronics applications. For this example, each property was equally weighted in the objective function. Table 3. Best Solution Property
Tensile strength (MPa) Elongation (%) 300% Modulus (MPa)
Pi,target
Pi
error
%error
35 670 9
35.0077 670.6664 9.4745
0.0077 0.6664 0.4745
0.022 0.099 5.272
Tabu search was run until 150 consecutive non-improving iterations were completed. The tabu restrictions were then removed and the search was run again to refine the solution. The best solution found is shown in Figures 1 and 2. A set of 5 structures which are other candidates for further investigation were also determined. All of these structures were found in a time of around 5 seconds on a Pentium-M 1.7 GHz computer. The best objective function value was 0.056. The Tabu search routine found a solution less than 0.1 about 20% of the time.
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Figure 1: Hard Segment
Figure 2 :Soft Segment
5. Conclusions This paper describes an extension of the use of connectivity indices within a computational molecular design framework to the design of novel polyurethane structures. Results show the effectiveness of Tabu search in solving the resulting MINLP to local optimality, and in generating a list of potential solutions which greatly narrows the number of polyurethane polymer structures to be synthesized and tested.
References J. Bicerano, (1996), Prediction of Polymer Properties. Marcel Dekker, New York. K. Camarda, and C. Maranas, (1999), Optimization in Polymer Design using Connectivity Indices. Ind. Eng. Chem. Res., 38, 1884-1892. S. Chavali, B. Lin, D. C. Miller and K V. Camarda, (2004), “Environmentally-benign Transitionmetal Catalyst Design Using Optimization Techniques,” Comp. Chem. Eng., 28, 605-61. R. Gani, N. Tzouvars, P. Rasmussen, and A. Fredenslund, (1989). Prediction of Gas Solubility and Vapor-Liquid Equilibria by Group Contribution. Fluid Phase Equilib., 47, 133. L. Kier and Hall, (1976). Molecular Connectivity in Chemistry and Drug Research. Academic Press, New York. B. Lin, S. Chavali, K. Camarda, D. Miller, (2005). Computer-aided Molecular Design using Tabu Search. Comp. Chem. Eng., 29, 337-347 C. Maranas, (1996). Optimal Computer-aided Molecular Design: A Polymer Design Case Study. Ind. Eng. Chem. Res., 35, 3403. D. Van Krevelen, (1990). Properties of polymers : their correlation with chemical structure, their numerical estimation and prediction from additive group contributions. Elsevier, Amsterdam. V. Venkatasubramanian, K. Chan, and J. Caruthers, (1994). Computer-Aided Molecular Design using Genetic Algorithms. Comp. Chem. Eng., 18, 833-844. H. Zhao, J. P. Ralston, R. C. Middaugh and K. V. Camarda (2004), “Application of Computational Molecular Design to Gene Delivery Polymers,” Proceedings of Foundations of Computer-Aided Process Design – 2004, 415-418, Computer Aids for Chemical Engineering Education, Austin, Texas.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Real-time Imaging and Product Quality Characterization for Control of Particulate Processes Ying Zhou a,b, Xuan-Tien Doana, Rajagopalan Srinivasan,a,b,* a
Institute of Chemical and Engineering Sciences, 1 Pesek Road, Jurong Island, Singapore 627833 b Department of Chemical and Biomolecular Engineering National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Abstract It is important to real-time in-situ measure the product quality, such as the shape and size in the particulate process for the purpose of process monitor and control. A methodology combing image processing techniques and statistical multivariate image analysis is developed in this paper to measure particle shape and size distributions from in-process video sensors and digitize them for closed-loop control. Monosodium glutamate (MSG) and sodium chloride (salt) crystallization processes are selected as case study and both the methods can correctly classify the two kinds of particles. Keywords: Particle shape and size, In-Process video measurement (PVM), Image analysis technology, Statistical multivariate image analysis
1. Introduction Traditionally, manufacture of pharmaceuticals and fine chemicals have relied extensively on empirical experience. The lack of understanding of the process and the underlying phenomena arise from complex chemistries, non-availability of detailed models, and the lack of in-situ sensors to directly measure product quality. The last issue comes to the fore in particulates processes, for example crystallization, filtration, drying, granulation, etc. While technologies for offline particle size and shape measurements such as microscopy have been available and widely used, it is but recently that inline measurements are becoming possible. With the advances in real-time imaging hardware, such as video cameras and fiber optics (exemplified by Focused Beam Reflectance Measurement, FBRM and Particle Vision and Measurement, PVM, both from Lasentec), and the concomitant developments in image analysis technology, there is an opportunity to control these processes based on direct observation of the product. Initial successes in gaining insights into crystallization process through these means have been reported by Yu et al. [1]; Birch et al. [2]; and Barrett et al. [3]. With regulatory initiatives such as the US Food and Drug Administration’s (FDA) Process Analytical Technology (PAT) program for the pharmaceutical industry, there is a clear need to develop general-purpose approaches which can incorporate image-based advanced sensors for closed-loop control. In this paper, we describe an approach to measure particle shape and size distributions from in-process video sensors and digitize them for closed-loop control. The proposed method combines image processing techniques from artificial intelligence with statistical multivariate image analysis (MIA). Image processing approaches for
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similar problems have been reported by Patience [4] and Calderon De Anda et al. [5]. MIA has been employed by MacGregor and coworkers for process monitoring and control [6-7]. In-process video offers 2-D images of 3-D objects, further these images suffer from various aberrations including background noise, bubbles, out-of-focus objects, and particles in motion. In the proposed approach, image processing is therefore used to extract the maximum information possible from the digital image. This involves image enhancement to remove noise, separating the object from the background by filtering and histogram equalization and combining information from multiple images to eliminate the effect of particulate motion. Feature extraction is then used to identify morphological descriptors (e.g.: shape factors) of the objects of interest through edge detection and other transformations. In parallel, multiway principal component analysis is also performed on the image and statistical descriptors derived from the scores plot. The morphological and statistical descriptors are jointly regressed to product shape and size distributions to develop a data-driven model. This model is used in real-time along with descriptors of inline images to infer the product quality in real-time. We describe this hybrid image analysis methodology in this paper and illustrate its efficacy using video images from an industrially motivated crystallization process.
2. Methodology 2.1. Overview Lasentec Particle Vision and Measurement (PVM) is an in-situ instrument which can be inserted into the liquid solution to take real-time digital images of the process. The PVM used in this research can take up to 10 images per second with a field of view of 860x645μm2, each image f(x,y) is represented by 640x480 pixels, and the value at each pixel is ranged from 0 to 255. The original PVM images are quite blur and hard to directly extract information from it. A new methodology that combine the image processing technology and multivariate image analysis technique is developed to extract maximum useful information from the PVM images, as shown in Figure 1, the acquired PVM images first go through the operation of image enhancement such as background subtraction and contrast-limited adaptive histogram equalization to improve the quality of images, then the operation of edge detection is followed to find the edges of interested objects from images, after that morphology operation, such as image closing, filling and opening, is applied to extract the area of the interested objects for further processing. Based on the extracted objects, both the imaging analysis technology and multivariate image analysis technique are applied to extract more information of each object, such as particle shape and size of crystallization process presented in this paper as a case study, where image analysis technology fits the object into a minimum area rectangle with the rotating clipper method [8-9] to classify two kinds of particle shape and quantify particle size, while multivariate image analysis technique classifies the particles with principal component analysis (PCA) method. 2.2. Image processing Techniques Image Enhancement As shown in Fig. 2(a), it is common in the original PVM images that the center part of the background is much brighter than the edge part due to the illumination of light, especially the four corners are quite dark. This generates a problem when a threshold is selected to transform the gray-level images into binary images, for example, if a high pixel value is used as threshold, the four corners will be considered as objects in the binary images, while a low threshold value will blank out the particle objects as well. In our methodology, a reference image is taken before any particles coming out at the beginning of each experiment, and all the following images
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are processed to subtract this reference image, this subtraction uniforms the image background and highlights the objects. Further image enhancement of contrast-limited adaptive histogram equalization is applied to improve the contrast of objects to background without amplifying the noise in the homogeneous background area. PVM images
Image Enhancement
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Fig. 1. Proposed image analysis method
Edge Detection The operation of edge detection could find the edges of objects from image, and among all the available edge techniques, such as Sobel, Prewitt and Roberts etc, Canny edge detection [10] is selected for this paper. Canny edge detection uses two thresholds to detect both the strong and weak edges by looking for the local maximum modulus of the first derivative of the image that is smoothed by a Gaussian filter, and it is sensitive to the two threshold values and the standard deviation of the Gaussian filter. Morphology Operation The objective of morphology operation is to extract the interested objects from the image. Image closing is first operated to merge the closely adjacent objects, image filling is followed to fill holes of the closed objects’ area and make the objects smoother, after that image opening is performed to remove small isolated pixels, at last only the objects that are big enough are selected for further processing. Feature Extraction To classify the particle shape and quantify particle size, each extracted object is fit into a minimum area rectangle. Rotating clipper method gives an applicable algorithm, where the convex polygon for each of the extracted object is firstly obtained, and it is found that one edge of the searched rectangle should coincide with one edge of the convex polygon. If an edge of the convex polygon is selected as one edge of a rectangle, a parallel line through the farthest point to this polygon edge is another edge of the rectangle, the other two edges of the rectangle are perpendicular to this polygon edge and pass through the two most separated points, such the object is fit
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into a rectangle. Based on each edge of the convex polygon, a rectangle is obtained, and the one with minimum area is selected to contain the object. The length and the width of the rectangle can be used to quantify the size of object, and the ratio of the length to the width could be used to classify the shape of object. 2.3. Multivariate Image Analysis Technique In this study, we apply a modification to the well-known Multiway Principal Component Analysis (MPCA) for particle shape classification and name it Image MPCA (IMPCA). As shown in Fig. 1, the technique follows image pre-processing which has identified both particle boundary as well as the number of such boundary. A square is fitted to each of the boundaries in such a way that it captures entirely the particle. A re-sampling procedure is carried out on those pixels that are within the fitting square to obtain an intensity-averaging matrix X (nxn, where n=8 in this study). As the matrix X captures entirely the particle as well as the boundary where there is transition between image background and particle image, it will carry the characteristics that enable particle classification. The well-known Multiway Principal Component Analysis (MPCA) [11] can be used to build a model for particles as represented by X. However, MPCA considers X as a batch data set and thereby takes different rows in X as independent measurement vectors, which inadvertently ignores the dynamics of image intensity across X rows (ie. along X columns). To overcome the limitation, we propose the following modification: Denote X1 to be the augmented version of X (with time lag d) ⎡ x(d + 1) T ⎢ x(d + 2) T X1 = ⎢ ⎢ ⎢ T ⎢⎣ x( K )
x( d ) T x(d + 1)
T
x( K − 1) T
⎤ ⎥ ⎥ ⎥ ⎥ … x( K − d ) T ⎥⎦ …
x(1) T
…
x(2) T
where K is the number of rows and x is the column vector in original matrix X. Similarly, denote X2 to be the augmented version of XT (with time lag d). The matrix Y=[X1 X2] preserves “image dynamics” in both directions: vertically and horizontally and hence is used for MPCA analysis in which Hotelling s T2 statistic [12] is employed as a classification index.
3. Case Study In this paper, the PVM images of the monosodium glutamate (MSG) and sodium chloride (salt) particles in the crystallization process are used as a case study. Fig. 2 demonstrates the procedure of image processing technology, both the needle-shaped MSG particle and cubic-shaped salt particle can be fit into a rectangle. If a ratio is defined as the rectangle length over rectangle width, the ratio of the MSG particle is much bigger than that of the salt particle, and the ratio is usually above 1.5 for MSG and between 1 to 1.3 for salt obtained from a training set of 34 MSG particles and 25 salt Particles. The ratio value of 1.4 is selected as a threshold to classify the two kinds of particles, which means, the particle is recognized as a MSG if the ratio is greater than 1.4, otherwise it is recognized as a salt. Fig. 3 shows online classification results from IMPCA method. The PCA model was built using 32 MSG particle images with time lag d=2 and 4 principal components retained. Both the image processing techniques and IMPCA methods are applied to a set of test images, which is consisted of 12 images containing 16 MSG particles and another 10 images containing 12 salt particles. Table 1 summarizes the result. By IMPCA method, 12 out of 16 MSG particles were correctly identified, one MSG crystal was incorrectly
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classified as salt and another 3 MSG particles were missed completely (i.e. not shown up in Fig. 3). Further investigation of the actual images revealed that the other two MSG blue circles below the classification boundary were actually corresponding to small part of the particles that had already been identified. The reason might be that in some cases, the constructed boundary was not able to capture the whole particle and hence results in two boundaries for only 1 particle. For salt crystals, 11 out 12 were identified correctly. The false classification was due to the fact that there were two cases where a dark region in certain images was incorrectly identified as a crystal (one red star above and one red star below the classification boundary). While the image processing technology can correctly classify the 16 MSG and 11 salt particles, the 1 misclassification of salt and 5 fault (3 MSG and 2 salt) are due to the dark areas in the image are also detected as particles and classified.
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Fig. 2. The MSG and Salt Particles. (a) Original PVM image, (b) Image quality enhancement, (c) Detection of all edges in image, (d) Extraction of large objects through morphology operation, (e) Particle isolation (f) Identification of particle bounding rectangle.
4. Conclusion A methodology combing image processing techniques and statistical multivariate image analysis is presented in this paper, and it is applied on PVM images to classify particles with different shape. From the experimental result, the two methods are consistent with
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each other and can correctly classify the particles. The future work will quantify the particle size distribution, such as the fitted rectangle of each particle already give some clue of particle size, and quantify the particle shape and size distribution along with time is another import thing to work on.
Fig. 3. Classification of salt and MSG particles by T2 statistic Table 1 Experimental Result
IMPCA Image Num
Particle Number
MSG 12 16 Salt 10 12 Percentage (%)
Correct Classify
MissedClassify
12 11 82
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Fault
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16 11 96
0 1 4
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References [1] Yu L. X., R. A. Lionberger, A. S. Raw, R. D’Costa, H. Wu, and A. S. Hussain, Advanced Drug Delivery Reviews, 56 (2004) 349. [2] Birch M., S. J. Fussell, P. D. Higginson, N. McDowall, and I. Marziano, Org. Process Res. Dev., 9 (2005) 360. [3] Barrett P., B. Smith, J. Worlitschek, V. Bracken, B. O’Sullivan, and D. O’Grady, Org. Process Res. Dev., 9 (2005) 348. [4] Daniel Bruce Patience, PhD. Thesis (2002), University of Wisconsin-Madison [5] Calderon De Anda J., X. Z. Wang, and K. J. Roberts, Chem. Eng. Sci., 60 (2005) 1053. [6] Yu H. and J. F. MacGregor, AIChE J., 50 (2004) 1474. [7] Yu H., J. F. MacGregor, G. Haarsma, and W. Bourg, Industrial Engineering and Chemistry Research, 42 (2003) 3036. [8] Godfriend T. Toussaint, 1998, http://www-cgrl.cs.mcgill.ca/~godfried/research/calipers.html. [9] Hormoz Pirzadeh, 1998, http://cgm.cs.mcgill.ca/~orm/rotcal.html. [10] Canny, John, IEEE Transactions on Pattern Analysis and Machine Intelligence,Vol. PAMI-8, No. 6 (1986) 679. [11] Nomikos P. and J. F. MacGregor, AIChE J., 40 (1994) 1361. [12] Qin S. J., journal of Chemometrics, 17 (2003) 480.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
An Engineering Company's Approach to Filling "CAPE Gaps" in Process Simulation Andreas Kröner Linde AG, Linde Engineering Division Dr. Carl-von-Linde-Str. 6-14, D-82049 Hoellriegelskreuth, Germany
Abstract Process simulators are important CAPE tools applied in the process engineering work flow for various purposes. Commercial process simulation software does often not match the user's actual needs. These gaps between available and required features on a technological, logistic and software engineering level are identified from an engineering company and process plant contractor's perspective. As these gaps may lead to several ® drawbacks, Linde Engineering is still developing its own process simulator OPTISIM and is thus no fully adopting actual mainstream strategies in process industry. An ® outline of the simulator's key features is presented. OPTISIM has been selected as the process simulation tool for a large scale engineering project where it proved its strength in process design and rating as well as in dynamic simulation. Keywords: CAPE, process simulation, optimization, dynamic simulation
1. Introduction The Linde Engineering Division of the Linde AG (Linde LE) is a world leader in process engineering and contracting, designing and building process plants for the chemical and petrochemical industry, natural gas producers, air separation companies, manufacturers of hydrogen and synthesis gas, and others. As a technology driven company Linde Engineering supplies the best suited processes and equipment to its customers. This in return requires the application of highly sophisticated Computer Aided Process Engineering (CAPE) tools and methods at every step of the engineering design process adapted to the company's work flow. Even with the most appropriate commercial software already in use for many years, everyday work is still hindered by gaps between an engineer's needs and available commercial solutions. As an alternative Linde Engineering has been developing the in-house process simulation program ® OPTISIM with the aim to fill and to circumvent various gaps encountered with external process simulation tools. Key features, specific solution approaches and an example are presented.
2. "CAPE Gaps" in Process Simulation 2.1. Process Simulation in the Engineering Work Flow An essential contribution to an outstanding performance in the engineering business is the provision of the best suited integrated CAPE tools for process design, process rating, and optimization while supporting and linking important steps of Linde LE's process engineering work flow as shown in figure 1.
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Figure 1: Process simulation tasks evolving along the engineering work flow
• In the work flow's primary step, the process development, CAPE tools support the design and execution of experiments on a phenomenological and process topological scale. • With a process simulation tool for global heat and material balances around process groups and the entire process sales staff is able to predict investment and operation costs based on proprietary cost estimation models. • Given a process concept and a set of stationary operation cases, the process is designed in the basic engineering step. All relevant process units are defined in the process flow diagram (PFD) together with interconnecting material, heat, and information streams. As always the best process design is searched for, process design calculation and design optimization are key CAPE features [1]. • Detail engineering refines the process design further fixing equipment design parameters based on the chemical and physical phenomena in each process step. Both, the process and the equipment design are evaluated together for their safety margins by steady state and dynamic rating calculations of off-design cases. • In the start-up phase dynamic process simulators predict the transient process response when linked to control system design tools or to operator training systems. • Once a plant is in operation the owner's major interest is the plant producing at optimal steady states and transient phases while satisfying physical, economical and ecological constraints. Plant optimizers and operation management tools heavily rely on steady state process simulation models. It is apparent that each step of the engineering work flow calls for distinguished process simulation applications: over-all or detailed plant models, design calculations and optimizations, as well as steady state or dynamic plant rating. Although all applications refer to the same process, different sets of data must be provided and results need to be archived. At Linde LE data among process simulation applications and among different CAPE tools are interchanged across a process equipment workbench [2]. 2.2. Identifying "CAPE Gaps" in Process Simulation Practice There are commercial process simulation tools available that support many steps of the work flow. More over, research in process and software engineering as well as mathematics continuously provides novel modeling approaches and enhanced numerical algorithms. But the successful application of many tools together with the latest research results in everyday work suffers from various gaps. 2.2.1. The Technology Gap Always having the most efficient software tools available on the engineer's desk top is a vital competitive advantage. The latest developments and research results need to be
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transferred to engineering programs with minimum delay. Continuous experience from using commercial process simulation programs however shows that software vendors will incorporate new features into their programs only if they can achieve a monetary benefit within reasonable payout time. To shortcut this process software tools may be available as user specific versions or allow user supplied extensions. While the first option will increase license fees, the latter may cause considerable coding and maintenance effort for the user. Most process simulators allow the implementation of user defined unit models. But the flexibility to extend or add numerical solution algorithms is quite rare. What lack many process simulators is an equation oriented solution strategy for efficiently computing solution and constraint sensitivities required in robust and high performance optimization. Here, significant progress can be observed in the last years. Migration to commercial process simulators however, involves considerable effort which needs to be justified. 2.2.2. The Logistic Gap Bug fixes and newly added features of commercial software are available depending on the vendor's release policy. Once a new version is issued it will not roll out immediately to the end user. Testing of new versions with reference calculations and the compatibility check of user supplied extensions may take time – even longer than the vendor's release period. If no feasible workarounds are at hand such delays impair the engineering work and cause many severe competition disadvantages. Support is decisive to the acceptance of any program. Due to secrecy regulations simulation problems are allowed to be passed on to the software vendor's helpdesk only after a generalization. This combined with transmission delays and language barriers easily prevent a user to ask for support. 2.2.3. The Software Engineering Gap External process simulation software applied at Linde LE needs customization at various levels: proprietary unit models need to be integrated into the simulator's model library, physical property software must be interfaced and data need to be interchanged with other programs. Software vendors usually commit to standards, such as CAPEOPEN or STEP, but when it comes to establish a reliable and fast communication proprietary interfaces are often necessary resulting in an unexpected increase in the migration and maintenance budget. The adaptation of commercial software to a company's workflow significantly contributes to the total investments necessary.
3. A Successful Solution to Fill the Gap With the intention to facilitate the filling of many CAPE gaps encountered with various commerical process simulation tools or to even avoid some of them Linde LE has been ® developing its in-house simulator OPTISIM for more than 15 years. As an on going project it is continuously customized to the users' needs. Highly integrated cryogenic processes, e.g. air separation and natural gas liquefaction, are the main areas of ® application. Key features of OPTISIM are: • an equation oriented solution strategy suited for steady state and dynamic simulation. • a library of specialized steady state and dynamic process unit models with nonstandard modeling approaches that allow an efficient construction of process models for design and rating. Some outstanding unit operations are multi phase flash, threephase distillation column, multi stream plate fin and spiral wound heat exchangers
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with detailed phenomenology for equipment rating, equation based heat sum curve calculation, and turbo machinery. efficient heuristic starting value algorithms to allow convergence even from starting points far from a solution. tailored numerical software for linear and non-linear equation solving accounting for discontinuous model switching. efficient optimization methods for solving non-linear programs (NLP) and mixed integer non-linear programs (MINLP) based on customized sequential quadratic program solvers. An outline of the MINLP approach and its application in optimal tray positioning in distillation columns is presented in [1]. an initial value solver for differential algebraic model equations with a maximum differentiation index of 2 fully exploiting the model structure for discontinuity handling and consistent initialization. dynamic optimization algorithms for optimal control and dynamic data identification. full integration with Linde LE's multiphase physical property package for any process unit model and any type of application from steady state design to optimal control. tight work flow integration with interfaces to the central process and equipment work bench [2], machine design programs and control design software. ®
Having full control of OPTISIM 's entire source code means a valuable competitive advantage to Linde LE. It is a prerequisite to continuous improvements in process unit models and to flexible incorporation of highly efficient mathematical solution methods thus minimizing any technological gaps. As software development and user support is done by the same staff that is also familiar with user processes there arises hardly any logistic gap. Furthermore, by internally designed software the interface complexity is minimized. Together with an ISO 9001 certified software management most software engineering gaps can be reduced.
4. Example ®
OPTISIM has been chosen as the simulation tool for process design and rating of the first baseload natural gas liquefaction plant based on the novel mixed fluid cascade ® (MFC ) process developed jointly by Statoil and Linde LE [3]. The core of the process are three combined mixed refrigerant cycles for precooling, liquefaction and subcooling of a natural gas stream (NG) to give liquefied natural gas (LNG), cf. figure 2. Views on the construction site and on the preassembled process barge with the core process groups being shipped to the site are given in figure 3.
Figure 2: Mixed fluid cascade process
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The OPTISIM steady state process design model comprises the MFC process and numerous auxiliary process groups. The objective of the design optimization is to minimize the total compressor shaft power while adjusting the refrigerants' compositions, flow rates and pressures. Constraints to the optimization problem arise from plate fin and spiral wound heat exchangers' temperature differences not exceeding prescribed limits. A process simulation tool that is capable of solving steady state design optimization models of the size listed on the left column of table 1 under real life engineering project conditions is - to author's knowledge – unique and hence a key competitive advantage for Linde LE. For safety analysis, e.g. cooling water pump failures or compressor trips, and evaluation of control strategies, e.g. refrigerant composition adjustment, dynamic simulation ® models of the relevant process sections were also implemented in OPTISIM . The right column of table 1 shows the dimensions of a typical dynamic simulation model ® consisting of the MFC process and one auxiliary process group. For a fast and comprehensive set up of steady state and dynamic plant simulation ® models process engineers and OPTISIM development specialists closely cooperated. Innovations in process unit models e.g. augmented rating models for spiral wound heat exchangers and turbo machinery, and in numerical solution algorithms were implemented straightforward with minimum time delay.
Figure 3: The LNG plant under construction…
and the process barge being shipped to the plant
Table 1 Characteristics of LNG plant models
steady state design model process unit models streams components state variables objective function optimization parameters optimization constraints calculation time for one design optimization
1700 1500 34 52000 compressor shaft power 31 1300 60 min
dynamic simulation model process unit models streams components state variables modeled control loops
calculation time for a typical transient
500 490 20 32000 65
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5. Summary It has been shown that process simulation is an important tool in all phases of the process engineering work flow. For various reasons commercial process simulation software does often not match right the engineers' needs. Discrepancies, so called gaps, were identified that appear on a technological, logistic, and software engineering level. To avoid and fill such gaps Linde LE has been developing its own process simulator ® OPTISIM for steady state design and rating, optimization, and dynamic simulation. ® With the powerful process simulation software OPTISIM a new base load LNG plant has been engineered.
References [1] I. Thomas and A. Kröner, Mixed-Integer Optimization of Distillation Column Tray Positions in Industrial Practice, accepted for ESCAPE-16 and PSE 2006, Garmisch-Partenkirchen, Germany, 2006. [2] F. Malcher, S. Tessendorf, R. Zeppenfeld, Integrated Process and Equipment Engineering ® Supported by a Workbench Based on COMOS PT , DARATECH Plant 2002, Houston, TX, January 2002. [3] H. Bauer, A Novel Concept, Hydrocarbon Engineering, May 2002, 59.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A computer-aided methodology with robust design criteria for selection of solvents for reactions Milica Folić*, Claire S. Adjiman† and Efstratios. N. Pistikopoulos Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Abstract Our previous work [1,2] was based on the use of a few reaction rate measurements to build a reaction model, followed by the formulation and solution of an optimal computer-aided molecular design problem (CAMD). Because of the small number of experimental data used, we investigate the impact of uncertainty in the reaction model parameters and formulate and solve a stochastic optimisation problem to arrive at the solvent that gives the best expected performance. These results are compared against the solvents obtained by deterministic optimisation. This methodology is illustrated through application to a solvolysis reaction. Keywords: solvent design, solvatochromic equation, sensitivity analysis, stochastic optimisation, reaction rate.
1. Introduction When used as reaction medium, a solvent can have a great impact on the reactor and process performance. It is widely acknowledged that a solvent can be used to control reaction temperature, to bring reactants together in suitable concentrations and to affect reaction rates dramatically so as to demote or promote particular reactions [3]. Reichardt [4] reports that the solvolysis of 2-chloro-2methylpropane is 335,000 times faster in water than in ethanol, while the reaction between trimethylamine and trimethylsulfonium ion is 119 times faster in nitromethane than in water. In spite of this, the industry relies mostly on experience and intuition when selecting a solvent for a reaction process. There has been little work on systematic approaches to the selection of solvents for reactions. This situation is in striking contrast with the selection of solvents for separation, extraction and distillation processes, where several computer-aided molecular design (CAMD) approaches have been proposed in the last two decades. CAMD is a synthesis activity, which aims to identify a list of candidate molecules that perform a task (or a set of tasks) most effectively [5]. Molecular Design methods are based on the idea that from a small set of atom groups such as CH2 and OH, following certain combination rules, a large number of molecules can be generated and evaluated with respect to a certain performance index. These methods have been successfully applied to a variety of solvent-based problems, allowing a much larger number of solvent molecules to be considered during process design than is possible by experimentation alone. Several of these methods are described in [6]. To develop CAMD methods for solvent design for reactions, we must be able to predict the effect of solvent on reaction rates. One such method is the `reaction fingerprint' developed by Modi et al. [7]. However, this approach applies only to predetermined solvent molecules and therefore would be most suitable for the verification of potential optimal solvents. Recently, Gani et al. [3] have proposed a method for solvent selection for promotion of organic reactions, that combines knowledge from industrial practice and physical insights. The solvent selection involves allocating a score to each * †
Financial support from the ORS scheme and CPSE/EU is gratefully acknowledged. Corresponding author. Tel: +44 (0)20 7594 6638; Fax: +44 (0)20 7594 6606; Email: [email protected]
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solvent from a database of 75 most common solvents. CAMD can also be used to generate a list of solvent candidates, ranked according to their score. Though this method has proved very effective for some application studies, it requires often sensitive information from industrial practice, in order to determine the required scores. Based on these considerations, our goal is to develop a systematic generic approach to solvent design for reactions. The basic premise is that, in the context of reactions, where there is a lack of generic models of solvent effects, an iterative strategy based on targeted experiments, model development, candidate generation by CAMD, and experimental verification must be adopted. The methodology is described in section 2, and is applied to a solvolysis reaction in section 3.
2. Methodology The overall methodology we propose is shown as an algorithm in Figure 1. For a given reaction, the starting step is to perform targeted experiments and gather the kinetic data necessary to build the reaction model. The experiments yield reaction rate constant data for the specified reaction, in different solvents. We have found that eight solvents usually give sufficient data provided that they are diverse in terms of the types of interactions they can have with the species involved in the reaction. As a measure of this, we use the ETN solvent polarity scale [8] and we choose solvents that have ETN values distributed over the entire physical range. In addition, solvents with different functional groups are chosen. Experimental data that have already been reported in the literature should be used at this stage to minimise experimental costs. Choose 8 diverse solvents Obtain rate constant data Build reaction model
Robust design problem formulation
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Figure 1. Overview of the solvent design methodology Once the reaction rate constant data in these eight solvent have been obtained, this information is used to build a reaction model that predicts the reaction rate constant in other solvents based solely on their molecular structure. Next, an optimisation-based computer-aided solvent design problem is formulated and solved. The objective we consider is to find the candidate solvent which gives the highest value of the reaction rate constant. Two formulations of the optimisation problem have been developed: deterministic and stochastic. For the stochastic formulation, we first perform a sensitivity analysis to
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quantify the uncertainty associated with the model. Then, we use these results in order to identify representative scenarios for formulating the stochastic (scenario-based) optimisation problem. By solving this problem, we obtain a solution which gives the best average performance. In the verification step, the predicted rate constant for the best candidate solvent identified is compared to the measured rate constant, to determine whether the reaction model needs improvement. If so, the experimental rate constant for the candidate solvent is added to the set of initial solvents to build an updated reaction model. This procedure is repeated until the model reliability is sufficient. The model building and CAMD step (deterministic and stochastic) are briefly discussed in the next sections. 2.1. Reaction model The reaction model consists of a set of property estimation methods which relate solvent molecular structure to solvent properties, and a solvatochromic equation [9] which relates solvent properties to reaction rate constant for a given reaction. The solvent properties needed in the solvatochromic equation are the so-called solvatochromic parameters, A, B and S, a polarisability correction term, δ, and the cohesive energy density, which is the square of the Hildebrand solubility parameter, δH. The polarisability correction term can be calculated exactly based on molecular structure. The cohesive energy density is estimated through its relation with the molar volume, Vm, and the heat of vaporisation, Hv, as discussed in [10]. Vm and Hv are estimated using the first-order versions of the group contribution techniques of Constantinou and Gani [11] and Marrero and Gani [12], respectively. Group contribution techniques for the hydrogen-bond acidity A, the hydrogen-bond basicity B, and the dipolarity/polarisability S have been presented in [9]. The predicted solvent properties are used in the solvatochromic equation: log k = log k 0 + s ( S + dδ ) + aiA + bB + h i
δ H2 100
(1)
where k is the reaction rate constant, and ko, s, d, a, b and h are reaction-specific parameters. The values of these reaction parameters are obtained via linear regression, based on measurements of the rate constant in eight predetermined solvents. 2.2. CAMD deterministic optimisation Once the reaction model has been developed, it is embedded within a CAMD optimisation problem. This is based on an MILP formulation of the following form: max log k n,y
s.t. structure-property relations property constraints (2) molecular complexity constraints molecular feasibility constraints where n is a q-dimensional vector denoting the number of atom groups present in the molecule; y is a vector of binary variables used to define n. The molecular feasibility constraints consist of the octet rule [13] and the bonding rule (as modified by Buxton et al. [14]). The molecular complexity constraints are limits on the combinations of functional groups and on the total number of groups in the molecule, and relations between the continuous variables n and the binary variables y. The structure-property relations consist of the reaction model described above, and the only property constraint is an upper bound constraint on the melting point Tm to ensure the solvent designed is liquid at room temperature. The group contribution technique of Constantinou and Gani [11] is used to estimate Tm. 2.3. Stochastic design problem formulation By solving the deterministic optimisation problem (an MILP), the optimal solvent candidate for a specific reaction can be identified. However, since a very small amount of data was used to build the reaction model (only 8 solvents compared to the search space of 8500 feasible molecules) we expect there to be uncertainty associated with the reaction parameters. This is confirmed by the very wide 95%
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confidence interval calculated for each of the parameters (for example, see Table 1). Therefore, we propose a strategy to investigate the impact of uncertainty on the model reliability and also to determine what is the optimal solvent candidate given this uncertainty. The first step is to perform a sensitivity analysis in order to identify the key reaction parameters and the most frequently identified “optimal” solvents. We carry out a global sensitivity analysis covering the 95% confidence intervals of the parameters. We analyse the parameters one at a time, and then combinations of two and five parameters at a time. For the one-dimensional case, we use the uniform distribution to sample the uncertainty space, whilst for the two and five-dimensional cases, we use a low-discrepancy sampling technique proposed by Sobol’ [15] that covers the uncertainty space very effectively. Once the parameter space has been sampled, we solve the CAMD problem for each of the sample points and identify the best solvent candidate. We use the number of different designs generated for each parameter as a measure of sensitivity. Thus, the most important parameters are the ones which result in the largest number of different solvent candidates, and further analysis can be focussed on those parameters only. Solving the molecular design problem for combinations of the important parameters provides a map of solutions in the uncertainty space. Then, we can proceed to the second step which is to formulate and solve a stochastic (scenario-based) optimisation problem for a small number of representative scenarios distributed throughout the solution map (i.e. the uncertainty space). The procedure for identifying the scenarios is based on the fact, because of the linear nature of the problem, the candidate solvents found on the map of solutions are clustered. The convex hull for each solvent candidate is first computed. Each convex hull is then divided into three equally sized sub-areas. The centre of mass of each of the sub-areas is then found and added to the list of scenarios. The number of scenarios is therefore three times the number of distinct solvent candidates in the uncertainty space. Each scenario is attributed a weight factor proportional to the size of the corresponding sub-area. Once key scenarios have been determined, we formulate and solve a stochastic optimisation problem. The objective is to find the molecule that has the highest expected value of the reaction rate constant: 1 M max M ∑ wfi log ki n, y i =1 s.t.
log ki = log k0 + si ( S + d iδ ) + ai A + bi B + hi
δ H2
100 other constraints as in problem (2) where M is the number of scenarios and wfi is the weight factor for scenario i.
(3)
3. Case study The case study reported here is for the solvolysis of t-butyl chloride (CH3)CCl → (CH3)3C+Cl- → (CH3)3C+|Solv|Cl- → Products). Solvolysis is a reaction that is induced by the solvent and is known to be susceptible to change in reaction media. We have gathered the experimental kinetic data from the literature for 41 solvents [9, 16-19]. The reaction rate constants reported span over 11 orders of magnitude and the best experimental solvent is glycerol. In our previous work [2], we show how to build the reaction model from data in 8 solvents and solve the CAMD MILP problem. We report satisfactory statistics for the reaction model over the set of 41 solvents. By solving the deterministic optimisation problem, we identified glycerol as the best solvent candidate: this solvent has the highest rate constant measured to date [19]. 3.1. Robust design results To test the proposed approach, we have performed a sensitivity analysis and we report in Table 1 the number of designs generated for several one and two-dimensional cases, and for the five-dimensional case. We have used 1024 sample points in each case. Based on one and two-parameter uncertainty results, we can conclude the design is most sensitive to parameters s and h. The solution map for these
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A Computer-Aided Methodology for Selection of Solvents for Reactions
two parameters is shown in Figure 2. We have obtained the representative scenarios for this space. By solving the stochastic optimisation problem, we have identified glycerol as the solvent with the maximum average rate constant in the whole space of s and h parameter values. The objective function is 0.03. This indicates the robustness of the top solvent identified by the deterministic optimisation in this case. Implementation of integer cut constraints in the problem formulation, allowed the generation of successive candidate solvents. The next best solution generated was 1,1,3-propanetriol with an objective value of 0.022. Table 1. Results of one-dimensional and multi-dimensional sensitivity analysis Reaction Nominal 95% confidence No. of designs parameters varied value Interval generated s 2.99 [-22.8, 28.8] 6 d
0.96
a b h
[-11.7, 13.7]
2
5.35
[-29, 39.7]
2
2.08
[-21.5, 25.7]
3
1.44
[-11.3, 14.2]
6
s, a
15
s, b
10
s, d
4
s, h
28
a, b
11
s, d, a, b, h
40
1 3
4
2
Figure 2. Solution space and convex hulls for the case study. All 28 molecules generated are presented. The largest areas correspond to the following molecules: 1:glycerol, 2:8-nitro-2-(nitromethyl)oct-1-ene,
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3: 4-nitro-2-(nitromethyl)but-1-ene, 4: 2-methoxy-3,4,5-trimethylhexane
4. Concluding remarks A computer-aided methodology with a stochastic design formulation has been proposed for the systematic selection of solvents for reactions. At the core of the CAMD problem, the reaction model is based on the empirical solvatochromic equation, in which the solvent properties are obtained by group contribution techniques and the reaction parameters are regressed from experimental data. We have shown how a sensitivity analysis can be performed to study the impact of uncertainty in the model and we have used the results obtained to formulate the stochastic design problem. By solving this problem, the robustness of solution obtained by solving the determined CAMD problem can be assessed. This algorithm provides a guide for experimentation and can easily be integrated with other CAMD algorithms. We plan to test the methodology for multi-step and parallel reactions, and then proceed to the integration of this solvent design approach in a plant-wide design strategy.
References [1] M. Folić, C.S. Adjiman and E.N. Pistikopoulos, Proceedings of ESCAPE-14, Elsevier (2004) 175. [2] M. Folić, C.S. Adjiman and E.N. Pistikopoulos, Proceedings of ESCAPE-15, Elsevier (2005) 1651. [3] R. Gani, C. Jiménez-González and D. J. C. Constable, Computers chem. Engng. 194-197 (2005) 1661. [4] C. Reichardt, Solvents and Solvent Effects in Organic Chemistry, VCM Publishers, UK (1988). [5] R. Gani and E.A. Brignole, Fluid Phase Eq. 13 (1983) 331. [6] L. E. K. Achenie, R. Gani and V. Venkatasubramanian (eds.), Computer Aided Molecular Design: Theory and Practice, Elsevier, Amsterdam (2003). [7] A. Modi, J. P. Aumond, M. L. Mavrovouniotis and G. Stephanopoulos, Computers chem. Engng. 20 (1996) S375. [8] C. Reichardt and E. Harbusch-Görnert, Liebigs. Ann. Chem. (1983) 721. [9] M. H. Abraham, R.M. Doherty, M.J. Kamlet., J.M. Harris and R.W. Taft, J. Chem. Soc., Perkin Trans. 2 (1987) 913. [10] T. J. Sheldon, C.S. Adjiman and J.L. Cordiner, Fluid Phase Eq. 231 (2004) 27. [11] L. Constantinou and R. Gani, AIChE J. 40 (1994) 1697. [12] J. Marrero and R. Gani, Fluid Phase Eq. 183-184 (2001) 183. [13] O. Odele and S. Macchietto, Fluid Phase Eq. 82 (1993) 47. [14] A. Buxton, A.G. Livingston and E.N. Pistikopoulos, AIChE J. 45 (1999) 817. [15] I. M. Sobol’ Comput. Maths. Math. Phys. 7 (1967) 86. [16] M. H. Abraham, J. Chem. Soc. – Perkin Trans. 2 (1972) 1343. [17] M. H. Abraham, R.W. Taft and M.J. Kamlet, J. Org. Chem. 46 (1981) 3053. [18] G. F. Dvorko, V.V. Zaliznyi and N.E. Ponomarev, Russian J. General Chemistry 72 (2002) 1549. [19] R. M. C. Gonçalves, A.N.M. Simões, R.A.S.E Leitão and L.M.P.C. Albuquerque, J. Chem. Research (1992) S330.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Separation of azeotropes in batch extractive stripper with intermediate entrainer Viktoria Vargaa,c, Erika R. Fritsa,b, Vincent Gerbaudc, Zsolt Fonyob, Xavier Jouliac, Zoltan Lelkesa, Endre Reva,b a
Budapest Univ. Techn. Econ., Dept. Chem. Eng. ,and bHung. Acad. Sci., Res. Group Techn. Chem.; Muegyetem rkp3, Budapest H-1111, Hungary c INPT, Lab Genie Chim, 118 route de Narbonne, Toulouse F-31062, France
Abstract Batch extractive distillation is an attractive technique for separating azeotropic mixtures (Steger et al 2005). These processes can be carried out in a batch rectifier, and in a stripping column, as well; but, unfortunately, the necessary feasibility studies of the latter one are still missing from the scientific communications. Here we present the first preliminary results on the feasibility of separating azeotropes with intermediate boiling entrainer in batch extractive stripper. The following mixtures are studied here: 1: methanol (MeOH) / toluene (TOL) (minimum boiling azeotrope) with triethylamine (Et3N) as intermediate boiling entrainer; 2: chloroform (CHCl3) / ethyl acetate (EtOAc) (maximum boiling azeotrope) with 2-chlorobutane (2ClBu) as intermediate boiling entrainer. Separation of the same mixtures with the same entrainers in rectifier have already been studied (Lelkes et al., 2002, Varga, 2003); thus, the results of the present feasibility study allow comparing the separation processes in a stripper or in a rectifier for the selected specified mixtures. Similar mixtures occure in Rev et al., 2003. The mixture to be separated is charged to the reflux drum (the so-called vessel, or still vessel), of great hold-up capacity. Liquid entrainer is added to the stripping unit for enhancing the relative volatility of the mixture. The boiler at the bottom of the column serves as to evaporate the liquid. The product is removed from the bottom. The entrainer is either premixed to the charge (Solvent-enhanced Batch Stripping, SBS), or it can be continuously fed to the system (Batch Extractive Stripping, BES). The place of the continuous feeding can be the top vessel (BES-T), or some intermediate point in the column (BES-I). Keywords: Extractive distillation, batch stripper, profiles maps
1. Methodology A feasibility method for studying batch extractive stripping is developed on the basis of the profiles maps analysis (Lelkes et al, 1998). The column concentration profile can be modelled with the differential equation
(
dx V = y − y* dh L
)
(1)
where h is dimensionless column height; V and L are vapor and liquid flow rates, respectively; y* is vapor composition in equilibrium with the liquid composition x; and y is the actual vapor composition according to the component balance. The extractive profile is computed top down; the stripping profile is computed bottom up.
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During the study, the phase equilibria were approximated by a modified Raoult-Dalton equation with ideal vapour phase; the liquid non-ideality is taken into account with NRTL model with parameters shown in Table 1. Excel add-in tool BibPhyAddIn© (ProSim SA, 2001) was used for computing equilibria. Table 1. NRTL parameters MeOH (A) – TOL (B) - Et3N (C) MIN. boiling azeotrope – Interm. entrainer Aij Aji pair αij [cal/mol] [cal/mol]
CHCl3 (L) – EtOAc (M) – 2ClBu (N) MAX. boiling azeotrope – Interm. entrainer Aij Aji pair αij [cal/mol] [cal/mol]
A-B
907.825
1025.08
0.4315
L-M
375.569
-619.982
0.8704
A-C
-642.811
1272.826
0.2793
L–N
857.97
-595.47
0.2216
B-C
348.946
-255.684
0.2876
M-N
16.088
118.613
0.3007
Interval arithmetic based computing method, the same presented at ESCAPE-15 (Frits et al, 2005), is applied to determine the exact loci of map singularities. The process is feasible if there is an instantaneous vessel composition from where the appropriate bottom composition can be obtained with the help of the extractive and/or stripping profiles, and if this vessel composition is reachable from the charge either by pre-mixing entrainer or by distillation. The set of these possible vessel compositions constitute a so-called feasible region.
2. Feasibility study of MeOH / TOL / Et3N MeOH/TOL mixture forms a minimum boiling azeotrope as is shown in Figure 1a with RCM. Separation with SBS or BES-T, where only stripping profile is present in the column, is not possible if the bottom product is specified as xW=[10-5+...; 0.95; …], since the feasible region is too small. With changing the purity criteria to xW=[0.001; …; …] (to produce binary TOL / Et3N first product), the process becomes feasible. The feasible region can be reached from the azeotrope with entrainer pre-mixing, as is shown in Figure 1b. TOL 1 0.8
TOL s=infinite Stripping profiles
1 0.8
0.6
0.6
0.4
0.4
0.2 0 Et3 N 0
Az 0.2 0.4 0.6 0.8 /a
MeOH 1
Figure 1. a) Residue Curve Map (RCM)
0.2 0 Et3 N 0
Feasible region borders s=1
premix line s=5 s=10 s=50 Az
0.2 0.4 0.6 0.8 /b
MeOH 1
b) Feasible regions with different reboil ratios
With decreasing reboil ratio, the feasible region is also decreased, and located farther from the azeotrope. As a result, a greater amount of entrainer pre-mix is needed at the beginning of the process to access the feasible region. There is minimum reboil ratio
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(sMin=0.5, in the studied case) below which the feasible region cannot be reached with entrainer pre-mix. As a common effect of simultaneous product removal and entrainer feeding, the vessel composition moves toward the MeOH / Et3N binary side. The first product (TOL / Et3N mixture) is rather dilute as a consequence of the premix line location and the profile curvatures. When reached it, no TOL is remained in the vessel. Entrainer feeding is stopped; and the MeOH / Et3N mixture is separated with conventional batch stripping. The binary TOL / Et3N first product can also be separated in the column with conventional batch stripping. In principle, the separation may be improved by applying extractive section. In case of feeding the entrainer to some intermediate point in the column (BES-I), both extractive and stripping profiles must be considered. Figure 2 shows that high purity product cannot be produced from azeotrope composition since the extractive profiles do not cross the stripping profile from the direction of the MeOH / Et3N edge. TOL diluted with much entrainer (binary first product) can be obtained either without extractive profile (as above), or if the extractive profiles cross the stripping profile from the MeOH / Et3N edge. However, this is not the case, as is shown in Figure 2b. TOL TOL s=infini s=10 1 1 F/L'=0.5 F/L'=1 0.8 0.8 Extractive profiles 0.6 SN 0.4 0.2 0
0.6 Stripping profile Az MeOH
Feasible region
0.4
Still path
0.2
Az
0
MeOH
0.2 0.4 0.6 0.8 1 Et3 N 0 0.2 0.4 0.6 0.8 1 /b /a Figure 2. a) xW=[0.001; 0.5; …] b) xW=[0.001; 0.1; …] Thus, separation of minimum boiling azeotropes in batch extractive stripping with intermediate entrainer is feasible, but practically not promising. On the other hand, high purity product can be obtained in batch rectifier (Rev et al. 2003). Therefore, the separation is better carried out in a rectifier. Et3 N 0
3. Feasibility study of CHCl3 / EtOAc / 2ClBu CHCl3/EtOAc mixture forms a maximum boiling azeotrope, as is shown in Figure 3a. Separation with SBS or BES-T (where only stripping profile is presented in the column) is, in principle, possible with high purity bottom product xW=[…; 0.95; …], as is shown in Figure 3b. The stripping profiles are shortened with decreasing reboil ratio, and the smaller feasible region is located farther from the CHCl3 vertex. Entrainer is needed to pre-mix in all the cases. There is a minimum reboil ratio (sMin=7, in this case) below which the feasible region cannot be reached from the azeotropic composition even with entrainer premix. The feasible region is quite narrow, see Figure 3b showing several such regions with different reboil ratios. Thus, it is difficult to keep the vessel composition in that area, and this configuration may be disregarded in practice.
V. Varga et al.
796 EtOAc 1 Az 0.8
0.4
EtOAc 1 Feasible regions Az 0.8 s=6 0.6 s=7 s=10 0.4
0.2
0.2
s=infinite Stripping profiles
0.6
CHCl3
0 2ClBu 0 Figure 3.
0.2 0.4 0.6 0.8 1 /a
a) Residue curve map
s=50 s=infinite
0
CHCl3
2ClBu 0
0.2 0.4 0.6 0.8 /b
1
b) Feasible regions with different reboil ratios
F/L' increases
The separation becomes feasible with applying an extractive section. The extractive profiles cross the stripping profile started from high purity product composition, as is shown in Figure 4a with total reboil approximation. The stable node of the extractive profiles moves from the azeotrope to the entrainer vertex with increasing the feed ratio. The separation is feasible with such a great feed ratio that the stable node is shifted to the left of the stripping profile. The minimum value of the feed ratio is F/L min=0.05, at s=∞. When finite reboil ratio is applied, an additional separatrix appears on the extractive profile map, close to the CHCl3/2ClBu edge (Figure 4b). The separatrix is indifferent to the feasibility of the process because the extractive profiles cross the appropriate stripping profile from both sides of the separatrix. The feasible region is bordered by the stripping profile itself. EtOAc EtOAc s=50 1 s=infinite 1 F/L'=0.07 F/L'=0.2 Az 0.8 0.8 Still path 0.6 0.6 Stripping profile Stripping profile 0.4 Extractive 0.4 Extractive 0.2 profiles profiles SN 0.2 CHCl3 0 CHCl3 0 0 0.2 0.4 0.6 0.8 1 2ClBu 2ClBu 0 0.2 0.4 0.6 0.8 1 /b /a Figure 4. a) SN moves down with increasing F/L’
b) Feasible separation sequence
Entrainer is not needed to pre-mix, in this case. After filling the azeotrope to the top vessel, the column is heated-up, then operated with continuous entrainer feeding with infinite reboil ratio, until the appropriate bottom composition is reached. Production is then started with finite reboil ratio and continuous entrainer feeding. Entrainer feeding is applied during the production; in order to maintain the product purity, and to control the vessel path. The feeding is stopped when the vessel composition arrives to the stripping profile, and an off-cut is taken away until the vessel composition reaches the binary side. The entrainer is then produced at the bottom with conventional batch stripping, and pure component remains in the top vessel.
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Thus, separation of maximum boiling azeotropes in batch extractive stripping with intermediate entrainer is do feasible, and high quality product can be removed. Since entrainer-diluted, but not pure, product can be produced in batch rectifier (Lelkes et al. 2002), the separation is better planned to be performed in a stripper.
4. Conclusions and recommendations The feasibility methodology originally developed for homogeneous extractive distillation is successfully extended to batch extractive stripping. Separation of minimum and maximum boiling azeotropes with intermediate boiling entrainer has been studied. In case of separating minimum boiling azeotropes, binary product is expected. The process is feasible with a single stripping section, but a first product highly diluted with entrainer can only be produced. This situation cannot be improved with applying extractive section. The process does not seem be attractive because the same mixture separated with batch extractive distillation process in rectifier gives high product purity already in its first product. When maximum boiling point azeotropes are separated with intermediate entrainer, high purity first product can be produced. The feasible region is very narrow when a single stripping profile is applied. There is a minimum reboil ratio below which the premix line does not reach the feasible region. The feasible region can be extended with extractive section, so much that premix of entrainer is not needed. There is a minimum feed ratio for the extractive profiles’ stable node to reach the stripping profile. Minimum reboil ratio belongs to given feed ratio. The separation method is suitable to produce high product, whereas it was impossible with the same method in a rectifier.
Acknowledgements Hungarian OTKA F046282, OTKA T037191, and French Government Grant.
References Z. Lelkes, E. Rév, C. Stéger, Z. Fonyó, Batch extractive distillation of maximal azeotrope with middle boiling entrainer, AIChE Journal 48 (2002) 25242536. V. Varga, Diplomawork, in Hungarian, Budapest Univ. Techn. & Econ., Dept. Chem. Eng., 2003. E. Rev, Z. Lelkes, V. Varga, C. Steger, Z. Fonyo, Separation of a minimumboiling azeotrope in a batch extractive rectifier with an intermediate-boiling entrainer, Ind. Eng. Chem. Res. 42 (2003) 162-174. C. Stéger, V. Varga, L. Horváth, E. Rév, Z. Fonyó, M. Meyer, Z. Lelkes, Feasibility of extractive distillation process variants in batch rectifier column, Chem. Eng. Proc., 44, (2005), 1237-1256 Z. Lelkes, P. Láng, B. Benadda, P. Moszkowicz, Feasibility of extractive distillation in a batch rectifier. AIChE Journal, 44 (1998):810-822. ProSim SA, 2001, www.prosim.fr E. R. Frits, M. Cs. Markót, T. Csendes, Z. Lelkes, Z. Fonyó, E. Rév, Use of Interval Optimization for finding Limiting Flows of Batch Extractive Distillation, p. 661-666. in: European Symposium On Computer Aided Process Engineering 15 , Ed: Puigjaner, L.& Espuña, A., Elsevier, 2005.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Conceptual steady state process design in times of value based management Alexander Wiesel, Axel Polt BASF Aktiengesellschaft, Corporate Engineering, 67056 Ludwigshafen, Germany
Abstract Modern Engineering faces numerous, often antagonistic challenges: the fastest process design at manageable risks and minimum costs. BASF Corporate Engineering’s answer is the development of the i-TCM@GI™-method to minimise total costs. It integrates shortcut equipment design and cost estimation directly into the process simulation environment. For the first time in commercial simulation software, all costs available at this project stage– variable costs as well as investment capital - are considered within the objective function for process optimisations. The result is an accelerated process design with improved starting points for an effective equipment design and efficient processes at minimised total costs. Keywords: process simulation, engineering, optimisation, total costs, i-TCM@GI™
1. Engineering Challenges Global competition, energy prizes, rapidly changing markets and many other factors have added significant pressure on chemical process engineers [Laudicina, 2004]. Results of this change are strict time constraints and tight budgets with probably changing scopes. The objectives are clear but unfortunately antagonistic: the fastest process design at minimum risk and minimum costs. One answer to this is true simultaneous engineering, which helps to reduce development times. However, the unavoidable iteration loops between different engineering functions directly influence engineering and therewith project costs. A sequential strategy reduces engineering costs but will never be competitive in terms of project time. In addition to that monitoring aspects gain attention as expressed by the reflection of processes using value driver trees. These are used to measure the contribution of individuals, business units, function centers or even teams to the overall success. Furthermore the value driver trees offer means to set goals for the future. For example the engineers’ productivity might be measured in terms of the hours spent per item. In the field of conceptual process design the two key performance indicators involved are the effectiveness and the efficiency corresponding with the quality of the design and the time spent for development. However, direct measures are hard to define as the process design is typically unique because of different conditions such as of utilities, technological improvements etc. Nevertheless, the engineers’ focus shifts more and more to economic aspects. Of course the constraints remain - technical feasibility is taken for granted and technical innovations still remain the expectation, not the exception. So the question is how to prepare for this situation?
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2. Current process design limitations 2.1. Engineering workflow The typical process design workflow consists of three steps. Following the task definition a conceptual process design team carries out a first design. A Steady state process simulation delivers mass, atom and energy balances based on thermodynamics. An equipment design team proceeds with the results and delivers dimensions of key equipment. Finally a cost estimation based on data banks or supplier information results in an economic evaluation of the plant concept. Under ideal conditions this sequential workflow offers the opportunity of minimum engineering costs as each function works only once. But obviously it does not result in minimum project costs. In reality the equipment design phase reveals technical drawbacks and iterates with process simulation engineers. Furthermore, the cost estimation might end up with figures that exceed budgets so that typically there are several major alternatives for the process design or at least minor changes for specific tasks respectively units. 2.2. Software integration Another aspect concerning process design efficiency is the degree of software integration. Though computational power has developed astonishingly in the past the engineering tools are still more or less isolated. For example designed equipment is not fully supported in the process simulation software for recalculations and process simulation results are not yet plug & play applicable for following engineering purposes. So there is still a lot of manual data transfer needed for the development and evaluation of process designs.
3. Integration of engineering elements in the process simulation BASF Corporate Engineering’s answer to these challenges is the development of the iTCM@GI™-method* to minimise total costs. The method overcomes the limitations mentioned above at least partially by integrating engineering modules - shortcut equipment design and cost estimation - into the Aspen Plus® process simulation environment. For the first time in commercially available simulation software, all costs available at this stage of the project – variable costs such as utilities, raw materials and waste treatment as well as estimated project costs incl. investment capital – may be considered within the objective function for process optimisations. Whereas process simulations typically delivered stream data, i-TCM@GI™ directly fills the ISBL part with information about equipment sizes and costs as part of the simulation results, see fig. 1. Feeds [€/a]
Utilities [€/a]
ISBL calculated simultaneously - Equipment sizes [d,l,A] - Equipment costs [€]
Waste gas [€/a]
Wastes[€/a]
Products (fixed)
Side products[€/a]
Figure 1: Implementation of ISBL part in process simulation environment
*
intelligent total cost minimization at corporate engineering (company code GI)
Conceptual Steady State Process Design in Times of Value Based Management
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3.1. From process simulation to plant simulation The method offers the possibility to retrieve more information from process simulation than just mass, atom and energy balances. So the integrated short cut equipment design means a step from process simulation to plant simulation and new insights in the process for engineers. The additional results on sizes and costs allow the evaluation of designs and alternatives directly in terms of technical feasibility and costs without multiple iterative and therefore time and cost consuming iteration loops with other engineering disciplines. Furthermore the direct feedback on a given set of process parameters helps to diagnose these and therefore shifts the process simulation quality to a higher level. Furthermore, all ISBL (inside battery limits) related information is available within the simulation framework and not just added for post-solve purposes. This makes these data accessible for optimisation purposes. In i-TCM@GI™ the depreciation of the estimated project based on short cut equipment design is used within the objective function. Together with variable costs such as raw material, utilities, waste treatment etc. this results in total costs, the most objective criteria for effective process optimisation. 3.2. Paradigm shift in advanced process simulation Another major aspect of i-TCM@GI™ is the transparency of costs or better money flows within a plant. The switch from engineering terms to money flows is illustrated in figure 2. The new money flow diagram shows the costs respectively depreciation for each unit as well as utility and feed streams. In the context of Value Based Management this transparency visualizes cost drivers as well as those pieces that add value to the process. The transparency effectively helps direct the limited process design resources to major cost causers and obtain minimal total life cycle costs. 48 4.55
Temperature (C)
48
Pressure (bar) Mass Flow Rate (kg/hr) Vapor Fraction Q
30
4.55
Duty (kW)
1.00
6423
W642
W642-IN
142 k€/a
48 4.55
C642
30370 48
80
4.55
8.00
4453
0.00 6422 160
0.00
46526 0.00
380 k€/a
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34923
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C642-RFX
1350 k€/a
53114
6420
0.00
73 k€/a
PS-642 162 4.63 16125
W641
A-6421
0.00 6421
162
86 k€/a
4.63 374295 W641-IN
0.10
QC=-3454 QR=3604
89 k€/a
Figure 2: Transfer from process flow diagram to money flows
This change from engineering key performance indicators to total costs as focus of process optimisation means a paradigm shift in process simulation that takes up the idea of value based management, technical issues become constraints. 3.3. Impact on engineering efforts As result of the added sizing and cost estimation routines i-TCM@GI™ obviously means higher effort for the steady state process simulation part of conceptual process design. However, instead of working out several designs in detail at high effort and
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costs i-TCM@GI™ allows an effective screening of different designs at minimum total effort. The fast evaluation of several alternatives within the process simulation limits the more detailed engineering effort to the most promising designs. The result is an accelerated process design with improved starting points for an efficient equipment design and effective processes at minimised costs. 3.4. Workflow 3.4.1. Model libraries The implementation effort is significantly reduced by the introduction of model libraries for equipment design and cost estimation. This also means a common basis for objective decision-making and less dependence on individual preferences. The used short cuts are based on in-house design routines and literature sources [VDI, 1997; Weiss, 1996; Kister, 1992; Marr and Moser, 1995]. 3.4.2. Mathematics Prerequisites for i-TCM@GI™ and efficient optimisation runs are a robust simulation in equation oriented formulation and excellent convergence attributes. The optimisation is carried out using the Aspen Plus Optimizer™ based on a DMO solver algorithm. 3.4.3. Objective functions The aim of the optimisation is the minimisation of total costs. For that reason the objective function is the sum of fixed and variable costs respectively the depreciation of capital investment and production specific costs as utilities, raw materials etc., see figure 1. The cost estimation for capital investment reveals the inside battery limits project costs including machines and equipment, buildings, insulations, piping, control systems, construction as well as engineering costs, incentives etc.
4. Applications of i-TCM@GI™ 4.1. Design of green field plants 4.1.1. Effectiveness related results The impact of cost minimisations using i-TCM@GI™ depends on the process history. In case of processes which have been designed and manually optimized, e.g. in case studies, several times before first experiences of rigorous optimisation reveal additional reductions in the lower one digit range. For new processes the saving potentials are significantly higher such as a 15 % reduction in utility costs for a world scale intermediate plant. Besides total costs the engineering features of i-TCM@GI™ add significant value to the process design quality. Without the direct feedback in terms of costs numerous parameters have not been questioned in detail and mainly set up on basis of rules of thumb. However, these do not always consider given constraints, actual prizes etc. Examples are temperature approaches in heat integrators, product recoveries in distillation columns, higher yields versus recycle streams and many more. Now iTCM@GI™ offers an efficient method to set up these parameters based on objective criteria and therefore increases the quality of process simulation and process design. This opportunity is particularly beneficial in studies with different possible sites or other changing boundary conditions 4.1.2. Efficiency related results The integrated shortcut equipment design routines are comparable with feasibility studies, result in good starting values for detailed equipment design studies and therewith reduce iteration loops and process design time. For example technically
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infeasible solutions, e.g. column diameters, can be directly evaluated as they are part of the process simulation results and are not passed to equipment design staff. First experiences with application of i-TCM@GI™ showed that the fast evaluation of process alternatives reduces engineering efforts. For example the question if it would be beneficial to energetically couple two distillation columns in terms of total costs was answered within one day on basis of individually optimized designs with savings of about 0.6 Mio. € per year. 4.2. Revamping existing plants using a modified i-TCM@GI approach Compared with the process design for a green field plant revamping of a given plant requires some additional effort. The aim is to maximize capacity or minimise investment for a given capacity. Installed assets should be used if possible. During this process integrated sizing routines are beneficial to evaluate the utilization of maximum load. 4.2.1. Model library for design and rating Different from the design of new equipment a check of installed equipment requires extended routines, e.g. the calculation of heat exchangers hydraulics. For that reason a second model library was set up which includes a rating as well as a design mode. In the end of each simulation run the technical applicability of the installed equipment is compared with the demands. In case a replacement becomes necessary the design mode gives equipment sizes and cost estimations. Furthermore, the equation oriented solver approach allows the efficient adjustment of operating parameters to fully utilize assets based on the rating results. This approach minimises the number of actions to achieve a given capacity and reduces investment for new equipment to a minimum. 4.2.2. Results of the application of i-TCM@GI to revamps The first experiences with the application of i-TCM@GI™ for two revamping projects led to results of several different aspects. Concerning efficiency the implementation of hydraulics and specific adjustments to the set of operating parameters led to good initialization results for more detailed equipment verifications. A final set of corresponding common values was now obtained after about two iteration loops, which is a significant reduction, compared with previous experiences. Furthermore, possible operating point options are a result. For example a heat exchanger cools down a stream to 40 °C. In case of the given higher capacity this temperature would be 44 °C now. This option can be discussed with operating units. In this manner options and flexibility lead to a significant reduction of equipment to be replace instead of binary yes or no decisions for constant sets of operating parameters. In one of the first projects 3 out of 10 heat exchangers which were found to be replaced in case of identical sets or operating parameters could be reused at the higher capacity with only minor adjustments the process.
5. Summary The application of i-TCM@GI™ to conceptual process design has proven to add significant value to projects in terms of costs. First experiences for older chemical process still reveal opportunities in the lower on digit percentage range. For new concepts, which have not been manually trimmed before the opportunities are significantly higher. However, at the moment this is only a trend on basis of few projects. Further applications are needed for exact figures. A major advantage of i-TCM@GI™ is the fast evaluation of project alternatives on basis on costs. This gain in efficiency enables engineers to investigate more alternatives than before with the aim of an even better design.
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In revamping projects the significant number of iterative working steps could be reduced to about two iterations. Furthermore, instead of copying one set of operating data to another capacity, the implementation of equipment rating allows the utilization of more assets by minor adjustments to lower project costs. Finally, i-TCM@GI™ means a step from process simulation in terms of mass, atom and energy balances to real plant optimization in terms of total costs with underlying detailed information about equipment. Now one can specify the assets and calculate the opportunities resulting in a possible reversal of the engineering workflow.
References P.A. Laudicina, 2004, World out of balance, McGraw-Hill Education R. Marr, F. Moser, 1995, Die Auslegung von stehenden Gas-flüssig-Abscheidern - Schwerkraftund Gestrickabscheider, Verfahrenstechnik, Volume 9, Issue 8, pages 379-382 H.Z. Kister, 1992, Distillation Design, McGraw-Hill, Boston Verein Deutscher Ingenieure; GVC VDI-Gesellschaft, 1997, VDI-Wärmeatlas, 8th ed., SpringerVerlag, Berlin Heidelberg S. Weiss, 1996, Thermisches Trennen, Verfahrenstechnische Berechnungsmethoden, 2nd ed., Dt. Verlag der Grundstoffindustrie, Stuttgart
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Computer Aided Methods & Tools for Separation & Purification of Fine Chemical & Pharmaceutical Products Maria B. C. Afonso2, Vipasha Soni1, Piotr T. Mitkowski1, Loic d’Anterroches1, Rafiqul Gani1, Henrique Matos2 1
CAPEC-Dep of Chem Eng, Tech Univ of Denmark, DK-2800 Lyngby, Denmark Dep of Chem Eng-Instituto Superior Técnico , Av Rovisco Pais, 1049-001 Lisboa, Portugal
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Abstract An integrated approach that is particularly suitable for solving problems related to product-process design from the fine chemicals, agrochemicals, food and pharmaceutical industries is presented together with the corresponding methods and tools, which forms the basis for an integrated computer aided system. The methods and tools are linked through the problems they are able to solve and the associated dataflow. The integrated computer aided system has been used to solve a number of industrial problems and summarized results from a selection, involving separation and purification issues, are presented. Keywords: Fine chemicals, agrochemicals, food, pharmaceutical products, separation, purification, methods and tools
1. Introduction The fine chemicals, agrochemicals, food and pharmaceutical industries need a different set of processes and have different operational constraints than processes producing bulk chemicals. For example, they usually involve batch operations (low production rates) and usually handle chemicals that are temperature sensitive, difficult to separate (because of isomers), and have high purity requirements. Also, at the initial discovery step, the required processing steps need to be configured and tested very rapidly and at the final (clinical) trials, they need to be reliable and efficient. Computeraided tools can provide significant savings in time and resources if reliable models for product-process evaluation were available and could be used in an integrated manner. The objective of this paper is a) to present a set of integrated computer-aided methods and tools that are particularly suitable for application in the synthesis, design and analysis of the separation-purification steps related to the production of high-value chemical products; b) to highlight the application of the developed methods and tools through a set of industrial case studies. The methods and tools include a large database containing pure component data of chemicals, solubility data of typical chemical products, azeotropic data of chemicals and many more. If the data for a chemical is incomplete, a property model program package is available to generate reliable data to fill out the needed missing properties. A unique feature of the property prediction method is its ability to predict missing model parameters from molecular structural information, without the need for additional experimental data [1] and is therefore able to handle a very wide range of compounds and isomers. Also, the azeotrope database
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includes an analysis tool that helps to identify azeotropic systems for which solventbased separation is necessary and azeotropic systems for which it is not necessary. A solvent selection tool [2], integrated to the system, then finds the most appropriate solvents for the desired separation. The same is true for solid-liquid separations involving solution crystallization [3] and liquid-liquid extraction. In all these cases, the sequence of operations needed to achieve the desired separation/purification is identified through generic model-based tools (solubility model creation, generation of saturation curves, sequencing of operations, etc.) that can handle the complex mixture behavior of fine chemicals and pharmaceutical chemicals. The generated synthesis/design alternatives for the separation/purification steps are verified through an integrated modelling system [4] that can be configured to simulate various types of batch/continuous operations. Finally, a chemical system pre-analysis tool has been developed to identify the types of operation needed to achieve the desired separation or purification [5]. For example, should the separation/purification be achieved through batch distillation or short-path evaporation or pervaporation, if a vapor-liquid separation is feasible? Also, when should crystallization be used and under what conditions? Should solvents (and/or anti-solvents) be introduced?
2. Integrated Computer Aided System - ICAS For ICAS, a model-based framework (see Fig. 1) for product-process design has been adopted. A detailed description of ICAS can be found elsewhere [6].
Figure 1: Framework for integration of product-process design For the integrated system (as shown in Fig. 1) to work, it is very important to identify the various tasks (problem types) and the methods and tools that need to be used to solve them. Also, the data-flow for each problem type needs to be identified and matched with the corresponding methods and tools. Table 1 provides a partial list of identified data-flow relationships with respect to problem types while Table 2 provides relationships between methods and tools to the problem types of Table 1. What is not shown in this paper, but is equally important, is the work-flow (or the sequence) in which the various methods and tools need to be used to solve the total problem (broken down into a collection of smaller sub-problems). This is briefly discussed in the next section where the solution of a selection of industrial problems are presented.
Separation and Purification of Fine Chemical and Pharmaceutical Products Table 1. Data flow for a selection of design problems Input Data Problem Type Building blocks for molecules; target Molecular properties and their upper/lower Design (CAMD) bounds and/or goal values List of candidate compounds to be Mixture Design used in the mixture; target properties (CAMbD) and their upper/lower bounds and/or goal values at specified conditions of temperature and/or pressure Desired process specifications (input Process streams, product quality specificaDesign/Synthesis tions, process constraints, etc.) (PD) Desired separation process specifications (input streams, product quality specifications, process constraints, etc.) and desired (target) solvent properties Details of the molecular or formulated product (molecular structure or a list of molecules plus their composition and state) and their expected function Details of the process flowsheet and the process (design) specifications
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Process flowsheet (list of operations, equipments, their sequence and their design parameters) Process flowsheet (list of operations, equipments, their sequence and their design parameters) plus list of candidate solvents Variable values defining the performance criteria
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Table 2. List of methods, algorithms and software tools useful in process-product design Problem Type Method/Algorithm Tools/Software Molecular & Mixture Molecular structure generation; ProCAMD Design (CAMD) Property preciction and database; Screening and/or optimization Process Design/Synthesis Process synthesis/design; Process ICAS (PDS, ICAS(PD) simulation/optimization; process sim, PA) analysis Process-Solvent Design CAMD-methods/tools; Process ICAS (ProPred, Synthesis/Design; Process ProCAMD, PDS, simulation/optimization; Process ICAS-sim, PA) analysis Process Evaluation Property prediction & databse; ICAS (ProPred, Product performance evaluation ICAS-utility, MoT) model; Model equation solver Process Evaluation Process synthesis/design; Process ICAS (ICAS-sim, simulation/optimization; Process ICAS-utility, MoT) analysis The method and tools mentioned above have been integrated and fine-tuned to the needs of the fine chemicals, agrochemicals, food and pharmaceutical industries with respect to their product-process design problems. They have been successfully applied to interesting industrial problems (solvent-based separation of reaction products;
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purification of the active ingredient (pharmaceutical product by short-path evaporation); sequence of crystallization steps to extract the solid product; vacuum batch (or membrane) distillation to purify aroma/food related compounds and many more.
3. Separation & Purification Problems & Solutions Separation processes employed for product development in the high-value chemical product sector are usually batch distillation (also including solvent-based and reacting systems), short-path evaporation, membrane-based separations (nano-filtration, micro-filtration, etc.), crystallization, chromatography, to name a few. As the chemicals involved are complex and have isomers, the separation tasks can be quite difficult. In the text below, a selection of typical separation-purification related industrial problems are discussed. More specifically, results from case studies involving product purification operations and solvent-based product recovery operations, are presented. Note, however, that these methods and tools have also been used to solve problems to improve product yields in specific reaction paths (through solvents and/or hybrid membrane contacting devices), for improved pesticide product performance (through formulation design for higher pesticide uptake through plant leafs), for increased recovery of fruit juice (through membrane distillation), and many more. Because of restrictions related to confidentiality and paper-size, specific details such as chemical names are not given. Instead, we will concentrate mainly on the problem definition, important solution steps and some of the important results. 3.1. Product purification problem A small amount of water needs to be removed from a solution containing an active pharmaceutical product (API), which is sensitive to temperatures above 350 K. Because of the nature of the reaction step, chemicals such as alcohols, acids, aldehydes and ketones cannot be used. At the operating pressure, the solvent must boil at lower than 350 K but should not be too volatile, as this will cause a VOC release. Two options could be considered. Option-1 could be to find a low boiling solvent that is miscible with the API but forms a low-boiling azeotrope with water. In this case, the water could be distilled off with the solvent by operating from the solventrich side. Option-2 could be to find a high-boiling solvent that does not form azeotrope with water and is only partly miscible (or immiscible) with water. In this case also, the water can be distilled off. The problem solution needs to predict the properties of the solvent, the solute and the mixture, needs data on azeotropes with water, needs simulation models for vapor-liquid separation operations (flash or batch distillation). Clearly, option-1 provides a better design alternative (w.r.t. ease of operation, reliability and economic criteria) as it means operation at lower temperatures and guaranteed removal of water. First a search of the database is made to find the chemicals forming azeotropes with water (excluding the chemical types listed above). For purposes of illustration only, Fig. 2 is included to highlight how the chemical type and carbon number of the candidate solvents are identified from plots of binary azeotropes. As shown in Fig. 2 (for alkanes versus carboxylic acids as an example), for a chemical type, an azeotrope may exist only between an upper and lower bound of carbon numbers for that chemical type. Also, azeotropes involving lower carbon numbers are likely to vary with pressure (meaning solvents may not be necessary for a separation task) while those involving higher carbon numbers are less likely to vary with pressure (meaning that solvents would be necessary). Using this information and generating pure component information on the solubility parameter, normal boiling
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point and normal melting point of the API, it is possible to use the ProCAMD tool within ICAS [6] to identify solvents that are partially miscible in water, forms low boiling azeotropes with water, dissolves the API and has boiling points between 350 – 420 K and melting point lower than 250 K. Using ICAS-utility and ICAS-sim, the removal of water from the API solution has been verified. 3.2. Recovery of API from reactor effluent through crystallization operations In this problem, a solvent (and/or an anti-solvent) that will promote the crystallization of the API at a specified temperature with the desired shape of crystals needs to be selected and its performance verified. A secondary objective is to add an anti-solvent so that an additional amount of the API can be crystallized out without further cooling at a reduced temperature. Two sub-problems are highlighted here. The first involves the creation of a customized solubility model, when the available model is not applicable. This is illustrated through a case study involving an API called Cemetidine [7], which is prescribed as a treatment for excessive stomach acid in conditions such as peptic ulcers. Solubility data in selected solvents (not necessarily optimal solvents) are available. First the necessary pure component data is retrieved from the database (Melting point= 412.4 K, Enthalpy of melting=44033 J/mol). The Hildebrand solubility parameter is predicted to be 29.2 MPa1/2. This means that polar hydrogen bonding solvents would be more appropriate for Cemetidine. Using the available solubility data and creating a customized UNIFAC model, this result is verified through Fig. 3, where the solubility of Cemetidine in different solvents are plotted against the solubility parameter of the solvent. By definition, the solvent showing the highest solubility is the best solvent (as usually there is a clear maximum) and the corresponding solubility parameter value of the solvent also indicates the API value for this property (verifying, therefore, the earlier prediction). Primary and secondary alcohols have been found to be most suitable. The second sub-problem involves the selection of solvent-antisolvent mixtures for a sequence of crystallization operations. Using the decomposition method for CAMD proposed earlier [3], solvent mixtures have been identified for drugs like Ibuprofen and Paracetamol. In the case of Ibuprofen, the performance of the solvent and the formation of crystals have been validated through experiments [3]. 3.3. Separation of impurities from heat sensitive product An effluent from a multi-step reaction process consists of the API plus additives and impurities, which have lower boiling points than the API and additives. Vapor-liquid separation is feasible but not with distillation. One option is to use shortpath evaporation, for which a new simulation model needs to be developed and validated through pilot plant data. Also, as the impurities, the API and additives are not present in the properties database, the needed properties (vapor pressure as a function of temperature, specific enthalpies, heats of vaporization, etc.) need to be predicted. The solution steps are as follows: generate a steady state simulation model, introduce new compounds into the database with their predicted properties needed for the simulation, validate model and adjust the model parameters to fit the plant data, and finally, use the validated model to identify the best sequence of operations that can achieve the desired separation (purity of product). Provide the developed model as a model-object that can be run from external software. Using the modelling toolbox (MoT) in ICAS, a model has been developed, model parameters identified and then a
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sequence of purification steps designed to match the desired product purity. More details of the model can be obtained from the authors or from the ICAS reference [6].
Figure 2: Plot of binary azeotropes of different carboxylic acids (different curves indicate different acids) as a function of the carbon number of alkanes.
Figure 3: Plot of solubility of Cemetidine versus solubility parameter of different solvents.
4. Conclusions Because of the nature of the problems in the high-value chemical products sector (new chemicals, process reliability, reliable product performance and complex phenomena), models play a very important role and provide the basis for a wide range of computer aided tools. In this respect, the paper highlights the important issues with respect to the integrated approach as well as the need for a flexible model-based framework, the appropriate modelling tools, the importance and need for property models, and the need for performance models for evaluation of processes and products. For the integration of methods and tools to work, relationships between the data-flow, problem types and methods/tools need to be properly established. Current and future work is developing a collection of case studies of wider scope and significance.
References [1] R. Gani, P. M. Harper, M. Hostrup, I & EC Research, 44 (2005) 7262-7269. [2] L. E. K. Achenie, R. Gani, V. Venkatasubramanian, Computer Aided Molecular Design: Theory & Practice, CACE-12, Elsevier Science b.v., The Netherlands, 2002. [3] A. Karunanithi, L. E. K. Achenie, R. Gani, Chem Eng Science, 61 (2006) 1243-1256. [4] M.Sales-Cruz, R. Gani, in Dynamic Model Development, Eds. S.P. Asprey and S. Macchietto, CACE, 16 (2003), Elsevier Science b. v., The Netherlands. [5] L. d’Anterroches, Process Flowsheet Generation, & Design through a Group Contribution Approach, PhD-Thesis, technical University of Denmark, Lyngby, Denmark, 2005.
[6] ICAS web-address (http://www.capec.kt.dtu.dk/Software/ICAS-and-its-Tools/). [7] P. Crafts, The role of solubility modelling and crystallization in the design of active pharmaceutical ingredients, in Case Studies in Chemical Product Design, Eds K. M. Ng, R. Gani & K. Dam-Johansen, CACE (2005), Elsevier Scieince b. v., The Netherlands (in press).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Integration along the Lifecycle of Calcium Fluoride in the Fluorine Industry Aurora Garea, Rubén Aldaco, Ignacio Fernández, and Angel Irabien Departamento de Ingeniería Química y Química Inorgánica, ETSII y T, Universidad de Cantabria, Avda. Los Castros, s/n, Santander 39005, Spain
Abstract The recovery of fluoride from industrial wastewaters as a product to be reused and its integration along the lifecycle is presented as priority objective of the fluorine industry in order to contribute to the development of sustainable processes. In this work, the computer aided integration of processes is applied to the dry fluorspar process of the Aluminum Trifluoride manufacture from HF, with a fluoride recovery/recycling process based on a fluidized bed crystallization step leading to a significant saving of raw material CaF2. A 5% saving of CaF2 was obtained with a fluoride recovery yield in the crystallization system of 65% as calcium fluoride referred to the total fluoride losses. Keywords: calcium fluoride, computer aided integration, fluidized crystallization
1. Introduction The Computer Aided Process Engineering is currently applied in industry in different phases of the lifecycle of a process and product, focused on the integration of existing sotware tools and support systems that are currently used for an effective support of the design lifecycle in Chemical Engineering (Bayer et al., 2000; Mayer and Schoenmakers, 1998). Sustainable development requires not only to assess multi-dimensional items such as environmental impacts, process safety, product safety, and so son, required from human society, environment, and markets, but also to feedback asssesment results to product and process design/development, with relevance on product lifecycle studies (Naka et al., 2000). A review of lifecycle assessment applications was reported by Azapagic (1999) covering applications in (i) strategic planning or environmental strategy development, (ii) product and process optimization, design and innovation, (iii) identification of environmental improvements opportunities, (iv) environmental reporting and marketing, (v) creating a framework for environmental audits. The application of lifecycle assessment in process selection, design and optimization was remarked as a tool for identifying clean technologies. The importance of lifecycle assessment for process selection has also been recognised by the EU Directive on IPPC (Council Directive 91/61/EC, 1996) which requires that the Best Available Technique must be chosen by considering the environment as a whole, including indirect releases, consumption of raw materials and waste disposal. Recent progress has promising implications on the use of intelligent systems for product lifecycle managements applications in the chemical, petrochemical, pharmaceutical and discrete parts manufacturing industries for better product quality, inherently safer design, operator training, abnormal events management and optimal process operations (Venkatasubramanian, 2005; Palaniappan et al. 2002). The application of inherent
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safety and waste minimization principles can be considered throughout the lifecycle of a process. The concept of waste minimization incorporates any technique, process or activity, which avoids, eliminates or reduces a waste at its source, or allows reuse or recycling of the waste. A systematic methodology for the integration of safety and enviromental impact is reported by Palaniappan et al. (2002). Calcium fluoride as fluorspar is a raw material for the Inorganic Fluorine Industry, with a wide use as reactant in the production of HF and its derived chemicals. The limited natural resources of fluorspar and the world consumption of fluorspar for chemicals production, stabilised around 4 million of tons in 2004 (World Mineral Statistics, 2004) require to make an effort in the integration along its lifecycle which is the aim of this work. The computer aided integration of processes is applied to the dry fluorspar process of the Aluminum Trifluoride manufacture from HF, with a fluoride recovery/recycling process based on fluidized bed crystallization (Irabien et al., 2005; Aldaco et al., 2005a-c) that accomplish with the sustainability criteria of design approach, minimization of materials and energy consumption and integration along lifecycle.
2. Simulation of the integrated process 2.1. Description of the Dry fluorspar process of AlF3 manufacture with fluoride recovery Aluminum Trifluoride (AlF3) is a wide used chemical in the electrolysis of aluminum and the glass and enamel industries. Approximately 65% of total AlF3 production is manufactured by the dry fluorspar process, in two steps as it is shown in Figure 1: (i) generation of gaseous HF from fluorspar (CaF2) and sulphuric acid (H2SO4), and (ii) production of AlF3 from gaseous HF and aluminum hydroxide (Al(OH)3, activated to Al2O3). In this work, the process line from the raw fluorspar to the product AlF3 is integrated with a fluoride recovery/recycling process based on fluidized bed crystallization with the calcium reactant provided by Ca(OH)2 (Figure 1). CaSO4l
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The fluoride recovery process to obtain synthetic calcium fluoride which is recycled to the feed line, consists on a crystallization process in a fluidized bed reactor which is proposed as an alternative to the conventional precipitation in the BREF document on Best Available Technology in common waste water and waste gas treatment and management systems in the Chemical Sector (European Commission, European IPPC Bureau, Draft reference document on Best Available Techniques in the Large Volume Inorganic Chemical, Ammonia, Acids and Fertilisers Industries: LVIC-AAF, draft march 2004). In compliance with the IPPC Directive requirements, Council Directive 96/61/EC, that recognises the importance of lifecycle assessment for process selection, the major advantages of the crystallization process are the elimination of sludge formation, the possibility of material recovery and the reduction of solid waste. The recovery of fluoride by crystallization is an example of process intensification, using compact and flexible units that enable modular set-up and tailor-made materials selection. The idea behind process intensification is the optimal integration of energy, materials, and processing tasks with the goal of minimizing the amount of energy and materials needed and size of equipment required to produce a given quantity of product per unit time (Korevaar, 2004). 2.2. Simulation of the fluoride recovery process Previous works were focused on the study at a laboratory scale in a pilot plant of the fluoride removal and recovery as synthetic CaF2 after crystallization in a fluidized bed reactor working with standard fluoride solutions (Irabien et al., 2005, Aldaco et al., 2005a-c). The analysis of the main variables was performed and the growth rate of product grains was modeled by a function of the supersaturation and the superficial velocity in the reactor (Aldaco, 2005). The growth rate of the fluoride-covered grains accounts to the nucleated precipitation: molecular growth and aggregation on the solid grains, in competition with the discrete precipitation in the liquid phase and mineral abrasion leading to small particles (fines) which leave the reactor in the outlet stream. Taking into account these studies on the crystallization process for the fluoride recovery as solid product CaF2, the aim of the simulation is the modelling of the fluidized bed reactor and scale up of the process for its integration into the AlF3 production, with the experimental facility of a pilot plant operated with industrial fluoride wastewaters proceeding from a process line of the AlF3 production in an industrial installation. The pilot plant was designed for a reactor capacity: 5 m3 h-1 total flowrate, superficial velocities up to 80 m h-1, height/diameter ratio of 10, fluoride inlet concentration 150 mg l-1 or higher in the case of the distributed feeding of the calcium reagent solution along the reactor length, and 10% calcium excess over the stoichiometric ratio. A simplified scheme of the pilot plant installation is shown in Figure 2. The recycling of the treated water at the reactor exit leads to zero consumption of fresh water for adjusting the concentration of fluoride and calcium streams at the reactor inlet. The evolution of solid particles in the reactor (mass and diameter) during time is given by the growth rate expression of the fluoride-covered grains and the fines formation rate (Aldaco, 2005), which are introduced in the set of equations of mass balances related to the tubular reactor, considering the Zaki equation for bed porosity under fluidized conditions. The variables related to the fluoride conversion are used in the forms of fluoride in solution, solid product grains of CaF2 recovered in the reactor, and fluoride as small particles or fines that leave the reactor.
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Figure 2. Scheme of the fluoride recovery installation with the fluidized bed reactor.
The simulation of the crystallization process leads to the description of the pellets growth with the operation time as well as the profiles of the bed porosity, pressure drop, solids load and height occupied by pellets at any time. Aspentech software was used for modelling and simulation. Figures 3-5 show the simulation results corresponding to the pellets growth in the fluidized bed reactor with 325 μm seeds at t=0, for 65% fluoride conversion to solid product grains recovered in the reactor. These results allow to predict the cycle operation time in the reactor and the characteristics of the solid product to be discharged and recycled to the production process of HF. The monitoring of the pressure drop during the operation (Figure 4) is used for the regulation of the pellets drainage valves according to a specified maximum value at which point the drainage cycle would start automatically. 8.E-04
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Figure 5. Profile of pellets growth in a fluidized bed reactor, 325μm seeds at t=0.
This crystallization process is focused to the minimization of raw fluorspar needs with the reuse of the synthetic CaF2 obtained to the HF gaseous generation process as step 1 in the industrial process of AlF3 manufacture (Figure 1). The fluoride content in the wastewater stream at the exit of the AlF3 production in the fluorine industry is approximately 58 kg/ton AlF3 (BREF document LVIC-AAF, draft march 2004). The wastewater stream is introduced to the crystallization process leading to the fluoride recovered in terms of synthetic calcium fluoride, 75 kg CaF2 /ton AlF3 (from a basis of 65% recovery yield as pellets in the crystallization process). Taking into account the total raw material CaF2 required for the AlF3 manufacture, 1.5 ton/ton AlF3, the integration of the crystallization process leads to a 5% saving as raw material, with an important contribution to the product lifecycle.
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3. Conclusions Taking into account the limited natural resources of fluorspar and the increasing consumption of fluorspar for chemicals production, an effort in the integration along its lifecycle is considered to be an important task of computer aided integration of processes in fluorine industry. In this work, the integration of a fluoride recovery unit in the industrial manufacture of AlF3 has been performed by a fluidized crystallization process that allows to reuse the fluoride recovered as synthetic calcium fluoride (CaF2) in the raw material line, with a 5% saving of fluorspar. Acknowledgements This research is financially supported by the Spanish Ministry of Science and Technology (Projects PPQ2003-0546, PTR1995-0799-OP).
References R. Aldaco, P.de Luis, A. Irabien,2005a, Fluidized bed reactor for fluoride removal, Chem. Eng. J., 107, 113-117. R. Aldaco, A. Garea, A. Alcaraz, A. Irabien, 2005b, Comparative study of fluoride recovery from industrial wastewater, 7th World Congress of Chemical Engineering, Glasgow, Scotland. R. Aldaco, A. Garea, A. Irabien, 2005c, Fluoride recovery in a fluidized bed: crystallization of calcium fluoride on silica sand, Ind. End. Chem. Res., in press. R. Aldaco, 2005, Control de fluoruros con recuperación de producto, PhD Thesis Universidad de Cantabria, Spain. A. Azapagic, 1999, Life cycle Assessment and its application to process selection, design and optimisation, Chem. Eng. J., 73, 1-21. B. Bayer, R. Schneider, W. Marquardt, 2000, Integration of data models for process design - first steps and experiences, Comput. & Chem. Eng., 24, 2-7, 599-605. Council Directive 91/61/EC 1996, Concerning Integrated Pollution Prevention and Control, Official Journal of the European Communities, No. L 257, 10 October, HMSO, London, 1996. A. Irabien, A. Garea, R. Aldaco, 2005, Tecnología más limpia para el tratamiento de aguas residuales con fluoruros mediante obtención de fluoruro cálcico sintético, Spanish Patent number P200501234. G. Korevaar, 2004, Sustainable chemical processes and products, PhD Thesis, Technische Universiteit Delft, The Netherlands. H.H. Mayer, H. Cshoenmakers, 1998, Application of CAPE in industry - status and outlook, Comput. & Chem. Eng., 22, suppl., S1061-S1069. Y. Naka, M. Hirao, Y. Shimizu, M. Muraki, Y. Kondo, 2000, Technological information infraestructure for product lifecycle engineering, Comput. & Chem. Eng., 24, 2-7, 665-670. C. Palaniappan, R. Srinivasan, I. Halim, 2002, A material-centric methodology for developing inherently safer environmentally benign processes, Comput. & Chem. Eng., 26, 4-5, 757-774. V. Venkatasubramanian, 2005, Pronostic and diagnostic monitoring of complex systems for product lifecycle management: Challenges and opportunities, Comput. & Chem. Eng., 29, 6, 1253-1263.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Design of sustainable processes: Systematic generation & evaluation of alternatives Ana Carvalho1,2, Rafiqul Gani1, Henrique Matos2 1
CAPEC-Dep of Chem Eng, Tech Univ of Denmark, DK-2800 Lyngby, Denmark Dep of Chem Eng-Instituto Superior TØcnico, Av Rovisco Pais, 1049-001 Lisboa, Portugal
2
Abstract The objective of this paper is to present a new generic and systematic methodology for identifying the feasible design (retrofit) potential of any chemical process. The methodology determines a series of mass and energy indicators (through simple calculations using steady state process data), establishes the operational and design targets, and through sensitivity based analysis, identifies the design alternatives that can match the established design targets. The significance of this indicator-based method is the possibility to direct all the contradicting criteria (factors) to move in the same direction, thereby eliminating the need to identify trade-off based solutions. Finally, the indicators also reduce (where feasible) the safety indicators because the conditions of operation become less extreme. A new indicator sensitivity analysis algorithm has been added to the methodology to define design targets and to generate sustainable process alternatives. Through the process flowsheet for the production of Methyl Tertiary Butyl Ether, the application of the indicator based methodology is highlighted. Keywords: Process design, Sustainability metrics, Mass and energy indicators, Safety index, indicator sensitivity algorithm
1. Overview of the methodology Increasing productivity, reduction of waste and energy consumption, reduction of valuable raw materials, recovery of products, and the last but not the least, the requirements related to process safety and process controllability, all represent conditions (or constraints), which together, formulate the conditions for a more sustainable process. In order to achieve a sustainable development, much progress is needed in the application of science for the identification, design and development of appropriate products and processes that will produce them, which is governed by the way we manage our valuable natural resources, design industrial products and processes, safeguard human health. Indeed grow our food is undoubtedly influenced by how we use our material resources. The aim of this paper is to present a new generic and systematic methodology for identifying and screening potential improvements in any chemical process, with respect to making them more sustainable and consequently more adaptable to the environmental restrictions that may be present at specific periods of time or location. The first step of this method is to obtain the necessary process data so that the indicators can be calculated. The indicators, first introduced by Uerdingen [1], give information about costs, benefits and accumulations in the process. They are related with mass and energy paths present in the overall process, thereby making it possible to identify the critical points in the process, and to precisely define the design targets that will lead to sustainable process improvements (without significant trade-offs).
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2. Transformthe flowsheet to a process graph 3. Calculate:
3.1 Mass and Energy Indicators
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3.3 Sustainability Metrics
4. Set indicator targets through an algorithm(ISA) 5. Sensitivity Analysis to determine the parameters which have more effect in the targets No 6. Design alternatives that match design targets Does the new alternative improve the indicators and improve or maintain the sustainability metrics and the safety index? Yes Through the indices and metrics choose the best alternative
Consequently with the indicator information it is possible to generate new process alternatives that can make the process more sustainable. The variables present in the indicators model equations are usually those that are either measured or manipulated in the actual process-plant or can be generated through process simulations. It is also important to mention that theses equations are very simple and consequently easy to calculate. The second step is to represent the process flowsheet in terms of mass and energy graphs that list the set of open and cycle paths present in the process. The open paths trace the mass of each component (and the stream energy) from its introduction in the process (for example, as a feed stream or as a product of any reaction) to its exit (as part of a demand stream or as a reactant in a reaction), while the cycle paths trace the mass of each component (and the energy) that remains in the process and does not go out. The mass and energy indicators
Figure 1: Flow-diagram of the method are calculated for each open and cycle path according to the models defined by Uerdingen [1] in the third step, where also the sustainability metrics [2] as well as the safety index [3] are calculated. These last two sets of parameters are employed as performance criteria to analyse the new process (design) alternatives and the likely process improvements. The 49 sustainability metrics previously proposed [2], which cover aspects related to social, economical and environmental area, follow the rule that lower values make a lower impact and therefore, the corresponding process design becomes more sustainable In this work, those related to the environmental metrics, have been replaced by the parameters calculated through the WAR algorithm [4], as they are similar but the WAR algorithm parameters are supported by models which are available as a tool in ICAS (Integrated Computer Aided System). This package allows the study of integrated process synthesis, design and analysis. Note that the sustainability metrics related to mass and energy are also related to profit and hence, if the impact is reduced, the process becomes more profitable, then reducing the release of products or raw materials will reduce the waste and subsequently the environmental impact is improved. The safety of the process is another issue that is included in the analysis of the overall process. Therefore, the final target is to create new process alternatives that may be sustainable (based on the sustainabilty metrics) but also more safe design. According to
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the method developed by Heikkilä [4], two types of safety index are calculated: an index and its associated sub-indices related to inherent chemical safety (related on the properties of the chemicals present in the process) and another related to inherent process safety (related to the conditions of operation of the process). The sum of the two types of safety indices gives the total safety index, which must be as low as possible because the sub-indices are calculated on a scale of 0-5 where lower numbers indicate safer options. Once the indicators together with the sustainability metrics and the safety indices have been established for a base case (or existing) process design, the next step is to analyse them to identify opportunities for sustainable and safer process improvements (steps 4-6 in Fig. 1). A new algorithm called the Indicator Sensitivity Algorithm (ISA), which generates process alternatives matching specified design targets has been developed and incorporated in the framework of the overall systematic indicator-based methodology. Here, the indicators that have the potential to make the greatest impact are identified and targets (design) are set for their new values that any new process alternative must match. Since the same process and manipulated variables that define the indicators also define the process alternatives, it is also possible to simultaneously ensure (without additional simulation or optimisation) that the desired process improvements will be achieved. The main steps of ISA are given below: a. Select the indicators which show the potential for the biggest impact (based on a set of guidelines [1] ) and consequently, the locations in the process flowsheet with potential design-operational deficiencies; b. Define the process improvement objective with respect to process-manipulated variables together with parameters such as prices of materials and utilities; c. Identify the variables which influence the selected indicators through an incidence matrix where each row represents an indicator model equation and each column represents the corresponding variables that are present in it; d. From the analysis of the incidence matrix of step c, identify the common set of variables that belong to the objective expression as well as the selected indicators. These variables will define the design targets; e. Analyze the selected variables following the three steps below: 1. Calculate for all variables the percentage of their global value that will be affected by a given path; 2. Calculate the percentage of each term present in the objective expression in the total value of the variables; 3. Identify the most sensitive variables by determining of the constant parameters which are multiplied by the variables, and then calculate their percentage in the sum of all coefficients; f. Establish limiting values of the selected variables based on a relation between percentage changes in the variables against improvements achieved in a scale of 0-5; g. Analyze the percentages defined in steps e-2 and e-3 and transform them to scores through the relations set in step f; h. Determine the score for all indicators by the sum of the points given by the variables which influence them; i. If the investment costs were not considered in the profit expression give extra points to the paths that makes significant changes in the total investment cost; j. Set the highest scored indicator as the target of the improvements.
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In the above algorithm, step a. aims to analyze the indicators and select the ones that comparatively have the largest potential to change the process (i.e., the sustainability metrics and safety indices). Step b. establishes the process improvement objectives. For example, a process which produce large quantities of the desired product the objective for improvement could be to improve the profit margin while simultaneously improving waste reduction potential (and therefore, also the environmental impact) and keeping all other metrics and indices neutral or within insignificant increases. In other cases, such as a pharmaceutical process where the process operational cost may be less important than, for example, the safety of the process or the process operation reliability, the performance could be measured in terms of process sensitivity rather than cost of operation. Therefore, the analysis of the indicators is related to aim of the process improvements to be carried out and consequently the target indicators will vary for each case since all the other following steps are dependent on this one. A sensitivity analysis is done for the selected variables defining the design targets in order to determine which of them represent the greatest percentage change to the total process improvement objective and to determine which of them are the most sensitive (i.e., make the biggest impact on the indicators). Then a scale (0-5) is defined with the values obtained from the sensitivity analyses and the percentages obtained in step e. are translated into scores. Each indicator is dependent on some variable and each target variables has a score. Summing the scores of all the variables present in an indicator identifies the indicator targets. Finally, with the target indicators identified, step 5 (see Fig. 1) determines which of the process-operational variables can affect the biggest changes in the target indicators for smallest changes in their values. This analysis is done for step increments in all the variables and their consequent effect on the target indicators. In this analysis, it is possible to define exactly what must be done, so step 6 (see Fig. 1) becomes easy to execute. Note that at the end of step 6, new sustainable process alternatives will be available, and using the performance criteria (optimization objective), the best alternative can be determined.
2. MTBE Case Study 2.1. Process flowsheet Methyl ter-buthyl ether (MTBE) is manufactured by catalytically reacting isobutylene and methanol. This process involves 11 components such as nButane, Isobutane, 1-Butene, cis/trans 2-Butene, tertButanol, Di-isobutylene, water plus isobutylene, water Figure 2: Process flowsheet for MTBE process and methanol. The first 5 components are fed with isobutylene in the process. The tertbutanol and the Di-isobutylene are secondary products which are produced by the reaction of water with isobutylene and the dimerization of isobutylene, respectively. Water enters the system as the solvent used for the recovery of methanol.
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The process starts with the reaction of isobutylene and methanol in the reactor (RX-1), followed by reaction-separation in a reactive distillation column. The next section in the process is the recovery of methanol with a wash column, where water is used to extract the methanol from the other impurities (fed with isobutylene). The section of the process involves a distillation column to separate methanol and water for recycle. 2.2. Methodology results The first step was the decomposition of the flowsheet into mass and energy open- and cycle-paths. For the MTBE process flowsheet, a total of 26 open paths and 20 cycle paths were found. For all of these paths, the corresponding set of indicators was calculated and only the most sensitive indicators are listed in Table 1 (for open paths) and Table 2 (for cycle paths). Table 1: Calculated MVA and EWC indicators values of the most sensitive open paths in the MTBE process flowsheet Open Path Compound Flow-rate kg/h MVA (103$/y) EWC (103$/y) O7 n-butane 4446,28 -376.41 9.07 O9 Isobutane 20255,53 -1714.78 260.70 O11 1 Butene 3338,42 -282.62 43.41 O14 BTC2/BTT2 4769,13 -403.74 63.91 Table 2: Calculated MVA and EWC indicators values of the most sensitive cycle paths in the MTBE process flowsheet Cycle Path Compound Flow-rate kg/h AF EWC (103$/y) C19 Water 6744,86 626.51 313.35 The sustainability metrics as well as the safety index were also calculated. In this work, only 23 sustainability metrics were calculated, because it was assumed that the corresponding parameters would remain unchanged (for example, new investment, taxbenefits, etc.) since data related to the social metrics was not available. The value of the Chemical inherent safety index was 17 and the value of the Process inherent safety index was 13, finally the value of the total safety index was found to be 30, which indicates that the process is on the safe side. From Table 1, it can be noted that the most sensitive open paths correspond to the mass of inert components present in the process, which enter the process as impurities of the reactant (isobutylene). These open paths show a very negative value of MVA (Material Value Added), indicating that the impurities loose their value through this path. The components are bought at a high price, but their value added is lost across the path. It is also possible to explain the high values of EWC (Energy and Waste Costs). This means high energy consumption due to the high mass flow-rate of these components. These indicators point out the need for a reduction of the flow rates for these paths. In the cycle paths, the only cycle that makes a significant impact is the cycle path C19 for water. In this cycle, high values of AF (Accumulation Factor) and EWC can be observed. The high values of AF indicate the high accumulative behaviour of water in the recycle, and consequently, high energy consumption. These indicators point to a high potential for a process improvement. At this point, five paths are available as potential target indicators. To set target values, it is necessary to define the improvement objective in order to apply ISA. In this case study the main objective was the profit subject to corresponding improvement in environmental impact and insignificant effect on all other metrics and indices. According to ISA, the most
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sensitive indicators were identified to be the EWC and AF indicators in cycle path C19. Finally, a sensitivity analysis was done in order to determine the corresponding target variables, which would produce the best improvements in the target indicators. The flow-rate of cycle C19 was found to the most sensitive variable. Therefore, new process alternatives reducing the water flow rate in cycle path C19, would definitely satisfy the specified process improvement. The feasibility of reduction of water flow rate can be verified through an analysis of the solubility of methanol in water (the high solubility indicates that the water flow rate can easily be reduced). To validate the benefits of the generated alternative, the process was simulated once again, but this time with 20% less of water in the washed system. This new alternative shows very good results. First with this reduction the efficiency of the wash column becomes constant at 99.9% recovery of methanol and water, as well as the efficiency of separation of the impurities that also become constant. The target indicators were now found to be improved by 14.4% and 20%, respectively, for AF and EWC, while the performance criteria was found to be improved by 3% (energy sustainability metric) and by 4% (water sustainability metric). This, therefore, also improved the environmental indices. Therefore, the generated process alternative is acceptable because it makes the process more sustainable and reduces the environmental impact. None of the other indicators or sustainability metrics or safety indices were found to be significantly increased. Finally, a study was made to evaluate the influence of the improvements with respect to profit and it was found that overall, an increase of 1.6% on the gross margin is possible (for an existing process where the equipment design is fixed) but a significant savings on investment cost can be achieved for a new process (as now the equipment sizes will be reduced).
3. Conclusions & Future work The further development and application of a systematic indicator-based methodology for process analysis and improvement have been highlighted through a case study. An useful feature of this methodology is that the initial indicator-based analysis needs to be made only once for any process with respect to a base-case or existing design. Potential (feasible) process improvements can be identified whenever necessary by only following the next steps (the Indicator Sensitivity Algorithm). This methodology has been applied to a number of case studies involving well known processes such as the HDA-process, natural gas purification plant and many more (details of these case studies can be obtained from the authors). In all cases, it was possible to determine sustainable process alternatives. Current and future work is looking at generating operation alternatives for high-value batch-operation based chemical products and more complex continuous chemical processes.
References [1] Uerdingen, E., Fischer, U., Hungerbuhler, K., Gani, R., 2003, AIChE J, 49(9), 24002418. [2] Tallis, B., 2002, Sustainable Development Progress Metrics, IChemE Sustainable Development Working Group. [3] Cabezas, H., Bare, J., Mallick, S., 1999, Computers Chemical Engineering, 23 (4-5), 623-634. [4] Heikkilä, A.-M, 1999, Inherent Safety in Process Plant Design – An Index-based Approach, Ph.D-Thesis, 62-89.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Systematic Procedure for Designing a Microreactor with Slit-type Mixing Structure Osamu Tonomura, Tatsuya Takase, Manabu Kano, Shinji Hasebe Dept. of Chemical Engineering, Kyoto University Nishikyo-ku, Kyoto 615-8510, Japan E-mail: [email protected] (O.Tonomura)
Abstract This paper proposes a systematic procedure for designing a microreactor with slit-type mixing structure. The proposed approach enables us to effortlessly determine the design parameters of the microreactor that can achieve the perfect mixing in the desired residence time. The Fourier number is built into the proposed method to avoid repeating CFD simulation. Furthermore, an index that estimates the difference between the reaction yield calculated by using PFR model and that provided by CFD simulation is proposed. The dependence on CFD simulation can be reduced through the introduction of this index. The combination of the above-mentioned approaches promotes the efficiency in microreactor design. The validity of the proposed design procedure is assessed through a case study. Keywords: Microreactors; Slit-type mixing structure; Systematic design; Fourier number; Computational fluid dynamics; Robustness
1. INTRODUCTION Micro chemical plants potentially enable us to handle materials which cannot be produced in conventional chemical plants and to improve the present production efficiency drastically. Extensive studies have been conducted on micro chemical process technologies [1-7]. In particular, chemical reactions, mixing, separation, and transport phenomena in micro-space have been energetically analyzed, and many types of microdevices have been proposed for each micro unit operation. The cross-sectional dimensions of microdevices are of the order of micrometers to millimeters. The low Reynolds number flow (typically, Re<100) in a microchannel insures laminar flow and the absence of efficient turbulent mixing. To shorten mixing time, the interfacial area between the fluids should be increased and the diffusion length should be decreased. A variety of different micromixers have been described in the literature [2]. These mixing strategies can be broken into two main categories: active and passive. Active mixers induce transverse flow within the microchannel by using an additional power. For example, Oddy et al. [8] developed a micromixer which achieves increased mixing by applying a sinusoidally oscillating potential across a mixing region to create flow instabilities. Passive mixers, on the other hand, use cross-stream diffusion and in some case introduce chaotic advection. The most basic passive mixer is the T- or Y- mixer, in which two adjacent fluids flow and mix via radial diffusion. Recently, Computational Fluid Dynamics (CFD) is a powerful tool for investigating the effect of design parameters of microdevices on the mixing performance. CFD simulation gives us the detailed information about velocity,
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temperature, and concentration profiles. However, it requires considerable computational time and high skills of generating models and meshes. At the beginning of this research, the robustness of the microdevices with two different mixing structures against a change of operating conditions is analyzed by using CFD simulation. Subsequently, we develop a systematic design method for a microdevice with slit-type mixing structure. In addition, an index that estimates the difference between the reaction yield calculated by using PFR model and that provided by CFD simulation is also proposed. The validity of the proposed approach is assessed through a case study.
2. EFFECT OF DIFFERENT CHANNEL STRUCTERS ON MIXING PERFORMANCE As the first step of the investigation, we calculate the mixing efficiency of a microreactor shown in Fig. 1. Figure 1 (a) is the microreactor with multi-hole-type mixing structure. Figure 1 (b) is the microreactor with slit-type mixing structure. The robustness of the microreactors with two different mixing structures against a change of operating conditions is analyzed by using CFD software (Fluent®). The finite-volume method with all variables defined at the center of the control volume is used in Fluent® to solve conservation equations for mass, momentum, and energy. CFD simulation conditions are summarized in Fig. 2. We examine 27 different ternary combinations with flow rate (5.63x10-3, 3.13x10-3, and 6.25x10-4 mm3/s), viscosity (8x10-3, 8x10-4, and 8x10-5 Pa•s), and density (2000, 1000, and 500 kg/m3). In order to assess the micromixer design performance, mixing length and/or time is defined as the length and/or time in flow direction after which the liquid concentration over all positions of a channel cross-section deviates by no more than 1% of the fully mixed concentration. In case of the multi-hole-type microreactor, the average and standard deviation for mixing time is 1.16 s and 0.154 s respectively. In case of the slit-type microreactor, the average and standard deviation for mixing time is 0.884 s and 2.39x10-2 s respectively. Fluid B 流体B
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Fig.1. Multi-hole-type mixer (a) and slit-type mixer (b).
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z Fluent® 5/6 code z Laminar flow mode z Non-slip at wall z Negligible gravity z Uniform inlet velocity (water (298 K))
Fig. 2. CFD simulation conditions.
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Fig. 3. Concentration distribution in a Y-Z plane immediately after Fluid A and B mix.
Therefore, it is clarified that the microreactor with slit-type mixing structure is superior to that with multi-hole-type structure in mixing time as well as robustness. Figure 3 shows the concentration distribution in a Y-Z plane immediately after Fluid A and B mix. The cross-sectional area occupied by the two fluids. Pure white and black indicate Fluid A and B respectively. The concentration distributions shown in the top of Fig. 3 are ideal for realizing the rapid mixing. The simulation results are as follows: For any operating condition, the slit-type mixer can achieve the ideal concentration distribution, that is, the constant diffusion length. Therefore, the slit-type mixer is robust against the change of operating conditions. On the other hand, the concentration distributions in the multi-hole-type mixers differ as to the magnitude of fluid density. The diffusion length of the multi-hole-type mixers is longer than that of slit-type ones. Other parameters such as flow rate and viscosity hardly influence the mixing time. According to the above-mentioned results, the design of slit-type micromixers is focused on in the following sections.
3. SLIT-TYPE MICROMIXER DESIGN 3.1. A Proposed Design Approach In this section, a systematic procedure for designing a microreactor with slit-type mixing structure is proposed. The proposed approach enables us to effortlessly determine the design parameters of the microdevice that can achieve the perfect mixing in the desired residence time. The Fourier number is built into the proposed method to avoid repeating CFD simulation. The Fourier number relates a residence time in the mixing channel to the binary diffusion coefficient and a characteristic length scale, which is the width of the channel. The design procedure is as follows:
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i ) Define design specifications such as flow rate, diffusion coefficient, density, viscosity, desired mixing time, maximum pressure drop, and the upper and lower limits of channel dimensions. ii) Run a two-dimensional CFD simulation for a certain slit-type micromixer with a design and operating conditions, derive the mixing time ( t [s] ), and calculate Fourier number (Fo). Dt Fo = 2 (1) H where D is diffusion coefficient, t is mixing time, and H is the height of channel. iii) Determine H, in which the design specifications are fulfilled, by using Eq. (1) and Fo derived from ii). iv) Determine the other design parameter, i.e. channel width and length. v) Validate the design result through three-dimensional CFD simulation. 3.2. Case Study The proposed approach is applied to a design problem of the slit-type micromixer that can achieve the perfect mixing in the desired residence time. Table 1 shows the main design conditions and assumptions in this case study. As a result of optimal design, Fo is 0.358, H is 37.4 µm, W is 5 mm, L is 21.4 mm, and pressure drop is 6.31 kPa. In addition, the mixing time and pressure drop calculated by three-dimensional CFD simulation are 0.502 s and 6.32 kPa respectively. The effectiveness of the design result is assessed through a case study. Table 1. Design conditions and assumptions Objective function · Minimization of Pressure drop Optimization variables · Channel height H, width W, and length L Constraints · H ≥ 10µm, W ≤ 5mm, 10H ≤ 2W ≤ L · Each flow rate of two fluids : 4 mm3/s · Desired mixing time : 0.5 s · Physical properties of fluids are the same as water (298 K).
4. MICROREACTOR DESIGN 4.1. PFR Model and CFD model CFD simulations can support the precise design and analysis but require the skills of modeling and generating meshes as well as the long computational time. Due to a large number of degrees of freedom to design microreactors, it is not practical to apply CFD simulation directly to the optimal design problem. Plug Flow Reactor (PFR) model is often utilized for designing the conventional tubular reactors. However, PFR
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model doesn’t always produce the good result of microreactor design due to the radial velocity and concentration distributions. In this section, both microfluidic mixing and chemical reaction are treated simultaneously. And then, an index ( ε ) that estimates the difference between the reaction yield calculated by using PFR model and that provided by CFD simulation is proposed. The proposed index is expressed in Eq. (2), provided that channel height is within the range of Eq. (3). H2 (2) D kC n −1 H 2 φ = A0 ≤ 30 (3) xD where H is channel height, D is diffusion coefficient, α is proportion coefficient, k is reaction rate constant, CA0 is inlet concentration of reactant A, n is the order of reaction, and x is reaction conversion. The dependence on CFD simulation can be reduced through the introduction of this index. The proposed design procedure is as follows:
ε =α
a ) Define design specifications such as product yield/selectivity, flow rate, diffusion coefficient, density, viscosity, maximum pressure drop, and the upper and lower limits of channel dimensions. b ) Calculate the upper limit of channel height by using Eq. (3). c ) Apply PFR model to the microreactor design problem and derive the average residence time of fluid, which can achieve the desired product yield. d ) Run a two-dimensional CFD simulation for a certain slit-type microreactor with a design and operating conditions, derive ε , and calculate α in Eq. (2). e ) Determine the design parameter, i.e., channel width, depth, and length under design constraints. f ) Validate the design result through three-dimensional CFD simulation. Table 2. Microreactor design conditions and assumptions Consecutive parallel reactions A (reactant) + B (reactant) → C (intermediate)
r1 = k1CACB
A + C → D (by-product)
r2 = k2CACC
C + E (reactant) → P (product)
r3 = k3CCCE
Objective function · Minimization of pressure drop Optimization variables · Channel height H, width W, and length L Constraints and assumptions · H ≥ 10µm, W ≤ 5mm, 10H ≤ 2W ≤ L · Each flow rate of three reactants : 1 mm3/s · Reaction rate constants k1, k2, and k3: 0.1, 0.03, and 0.3 m3•kmol-1•s-1 · Desired yield of product P : 60 % · Physical properties of fluids are the same as water (298 K).
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4.2. Case Study The proposed approach is applied to a design problem of the slit-type microreactor having consecutive parallel reaction systems. Table 2 shows the consecutive parallel reactions, main design condition and assumptions in this case study. As a result of optimal design, ε is 0.112, α = 0.0172 s-1, H is 80.5 μm, W is 5 mm, L is 15.4 mm (reactant E is mixed at L = 0.27 mm), and pressure drop is 169 Pa. The yield of product P and pressure drop calculated by three-dimensional CFD simulation are 60.9 % and 169 Pa respectively. The effectiveness of the design result is confirmed through a case study.
5. CONCLUSIONS The robustness of the microdevices with two different mixing structures against a change of operating conditions is analyzed by using CFD simulation. As a result, it is clarified that the microdevice with slit-type mixing structure is superior to that with multi-hole-type structure in mixing time as well as robustness. In addition, a systematic procedure for designing a slit-type microreactor is proposed. Furthermore, an index that estimates the difference between the reaction yield calculated by using PFR model and that provided by CFD simulation is also proposed. Both proposed approaches promote the efficiency in microreactor design.
ACKNOWLEDGMETS This research was partially supported by the New Energy and Industrial Technology Development Organization (NEDO), and Project of Micro-Chemical Technology for Production, Analysis and Measurement Systems.
REFERENCES [1] A.J. van der Padt Abrahamse, R.M. Boom, and W.B.C. de Heij, 2001, Process fundamental of membrane emulsification: simulation with CFD, AIChE J., 47(6), 1285-1291. [2] W. Ehrfeld, V. Hessel, V. Löwe (eds.), 2000, Microreactors, Wiley-VCH, Weinheim. [3] W. Ehrfeld, V. Hessel, and H. Löwe, 2000, Extending the knowledge base in microfabrication towards chemical engineering and fluid dynamics simulation, 4th International Conference on Microreaction Technology (IMRET), 3-20. [4] P.D.I. Fletcher, S.J. Haswell, E. Pombo-Villar, B.H. Warrington, P. Watts, S.Y.F. Wong, and X. Zhang, 2002, Microreactors: principles and applications in organic synthesis, Tetrahedron, 58, 4735-4757. [5] V. Hessel, S. Hardt, and H. Löwe (eds.), 2003, Chemical micro process engineering. WileyVCH. [6] K.F. Jensen, 2001, Microreaction engineering – Is small better?, Chem. Eng. Sci., 56, 293-303. [7] H.A. Stone and S. Kim, 2001, Microfluidics: Basic issues, applications, and challenges, AIChE J., 47(6), 1250-1254. [8] M.H. Oddy, J.G. Santiago, and J.C. Mikkelsen, 2001, Electrokinetic instability micromixing, Anal. Chem., 73, 5822-5832.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Model-based Optimal Design of Pharmaceutical Formulations Fernando P. Bernardoa, Pedro M. Saraivaa and Sérgio Simõesb a
GEPSI-PSE Group, Department of Chemical Engineering, University of Coimbra, P lo II - Pinhal de Marrocos, 3030-290 Coimbra, Portugal. e-mail: [email protected] b School of Pharmacy, University of Coimbra / Bluepharma, S.A.
Abstract This paper presents a product/process model for a pharmaceutical formulation, describing the main physico-chemical phenomena responsible for three product quality factors: active ingredient release, active ingredient homogeneity in the formulation and physical stability. Our model consistently quantifies these factors as a function of product composition and process variables, and thus is suitable to support optimal product/process design. Simulation results for the particular ointment under study reveal interesting relationships between product composition and performance. Keywords: product design, pharmaceutical formulations, drug release models
1. Introduction The design of formulated products, such as pharmaceuticals, cosmetics and foods, comprises the specification of proper ingredients, their amounts and also the way they should be structured, in order to meet a set of product functionalities valued by customers (Cussler and Moggridge, 2001; Wibowo and Ng, 2002; Hill, 2004; Schubert and Engel, 2004). Such products usually are complex systems (e.g. emulsions, foams, gels and composite solids), whose properties and performance are determined not only by composition but also by structure at a nano or microscale. Therefore, a rational approach to design and develop them requires a reasonable understanding and quantification of the physico-chemical processes taking place at these scales. Furthermore, since product structure is to a great extent the result of manufacturing process conditions, product and process design should be conducted in an integrated framework, in order to achieve better overall solutions. In previous work (Bernardo and Saraiva, 2005), we have presented a generic optimal design framework, integrating both product and process design issues. Given the relevance of product design in the pharmaceutical sector, in this paper we apply such a framework to a pharmaceutical ointment for skin application, in close collaboration with a national company (Bluepharma). A set of four product quality factors were identified: 1) active ingredient (AI) release rate; 2) AI homogeneity in the formulation; 3) ease of application; and 4) physical stability. Our final objective is to determine an ointment formulation that optimizes the above set of quality factors, also taking into account process related issues. In this paper, we will focus on the development of the physico-chemical models required to quantify the above factors as a function of product composition, product structure and also some process variables. Based on these models and a product formulation made out of six components, we then propose a set of design solutions that illustrate trade-offs between different product quality factors and manufacturing costs. As a forthcoming step (not covered in this paper), we will apply optimization algorithms, using an overall
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integrated product/process performance criterion as objective function, as well as the inclusion of model uncertainties, to come up with the best product/process design solutions.
2. Base Formulation The ointment under study is an emulsion of a solution, S (active ingredient C1 + solvent C2), dispersed in a highly viscous lipid phase, L. In a first product development stage, the lipid phase is considered to be composed of four excipients, with the following main functions: C3 – oily-aqueous excipient (oily excipient with some degree of hydrophilicity); C4 – oily excipient; C5 – thickener and also oily excipient; C6 – surfactant. Our AI is only slightly soluble in the continuous lipid phase, where its solubility increases as that phase becomes more and more enriched in C3. Ointment consistency is mainly controlled by excipient C5. The emulsion formation and its physical stability are promoted by surfactant C6, which adsorbs at the surface of solution S droplets. For the time being, this formulation with six components is considered to be fixed, while in future development stages some ingredients may be substituted or new ingredients added, such as penetration enhancers or bioadhesive agents. The following typical composition is used as a nominal design: wN = {2, 18, 14, 59, 5, 2} (vector of mass percentages for components C1 to C6). Since ointment consistency will not be addressed in this paper, thickener concentration is kept fixed at its standard nominal value of 5%. The ratio w1/(w1+w2) is also considered constant and equal to 0.1.
C1d
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4. blood circulation system
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Figure 1. Representation of transdermal drug release model.
3. Active Ingredient Release The release of the active ingredient from the ointment vehicle is the most important product functionality, upon which depends product approval or rejection. In order to predict this functionality, we have constructed a model describing drug transfer from the ointment to the blood circulation system, comprising four compartments (Figure 1). Drug transfer through these compartments is modelled as a series of interphase equilibria and diffusion processes. Diffusion through the stratum corneum (SC, epidermis outermost layer) is often the rate-controlling process, while viable epidermis is often the most permeable layer, due to its higher level of hydration. The vehicle may offer a significant resistance, especially in the case of highly viscous formulations, like the one we are studying. Besides diffusion, equilibrium phenomena in the interphases
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also control drug release rates and depend on the lipophilic/hydrophilic balance between drug molecule and different phases present. Our complete mathematical model comprises a set of differential and algebraic equations, capable of predicting the spatial-temporal drug concentration profiles in compartments 1, 2 and 3 and the temporal profile of plasma concentration. Model parameters include the applied ointment volume (V), area of application (A), dispersed phase volume fraction (φ), droplet size (Dd) and a set of transport and equilibrium parameters. Out of these parameters, the diffusion coefficient in the continuous phase (D1) and the partition coefficients K1 (dispersed phase/continuous phase) and K2 (continuous phase/SC) are estimated as a function of ointment composition. The diffusion coefficient through SC (D2) is considered to be independent of ointment composition. This model was tested against published human in vivo data for plasma concentration, as a function of time, C4(t). A reasonable agreement (Figure 2) is obtained, after adjusting D2 to 2.3×10-8 cm2/s (a value considerably larger than the initial estimate used). Simulated results shown correspond to our nominal composition, wN, while the exact formulation of the ointment used in the clinical tests, leading to the experimental points, is unknown. In addition, the simulated profile exhibits more accentuated variations than clinical data, although a fair comparison should take in account the high biological variability expressed by error bars. In spite of these limitations, we consider this model reasonably validated and adequate for a preliminary product design stage. Using model predictions, it is then possible to derive a quantitative measure of ointment performance, in terms of drug release. An ideal performance would correspond to having a constant nominal plasma concentration, C4*, and therefore error E(t) = |C4(t) − C4*| seems to provide a good quality measure. However, one also wants patients not to be exposed to high peak plasma concentrations. Therefore, we chose as quality variables the mean value of E(t), designated as y1, and the error at the peak concentration, y2 = C4,max − C4*. We assume C4* = 0.15 ng/cm3 and calculate both y1 and y2 over a 24 h period, with 8-hourly drug administrations. For the nominal ointment composition, with the profile C4(t) of Figure 1, the corresponding values obtained are: y1 = 0.0373 and y2 = 0.0702 ng/cm3. An optimal ointment composition should try to minimize these scores, to be also balanced with other product quality factors, as discussed in the following sections. Plasma concentration (ng/cm3) 0.2 0.15 0.1 0.05
Time (h) 5
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Figure 2. Simulated versus experimental drug plasma concentration following 8-hourly administrations of 0.25 cm3 of 2% ointment, applied over an area of 10 cm2.
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4. Active Ingredient Homogeneity The homogeneity of the drug in the formulation is evaluated immediately after its production as the variability of drug amount in a typical ointment dose, and is largely determined by the mixing process conditions. A plausible model to predict this variability can be based upon a hydrodynamic description of the mixing process, down to the scale of an ointment dose (~1 cm). In this study, we consider that mixing at this scale evolves in parallel with droplet size reduction, which can be reasonably predicted with fewer data about equipment geometry and operating conditions. For the time being, there is no data to correlate droplet size with drug homogeneity, and thus droplet size is considered to be an absolute (inverse) measure of homogeneity. More specifically, our homogeneity index y3 is chosen as the mean Sauter diameter, Dd. The mixing process is considered to take place in a tank equipped with a Rushton turbine impeller, and comprising the following operations: (i) slow addition of the dispersed phase to the lipid phase (previously melted) under agitation, and with both phases at about 50 ºC; (ii) homogenization; (iii) homogenization + cooling until about 25 ºC. During these operations, droplet breakup results from a balance between interfacial tension, that tends to maintain droplets intact, and disruptive forces associated with flow conditions of the mixing process. In our case, we estimate that the Kolmogoroff length scale (size of the smallest eddies) is greater than the final drop size and thus that droplet breakup is dominated by viscous forces inside eddies (Wichterle, 1995). For this regime, the terminal mean drop size (for long mixing times) is proportional to σ/(ρcμcε)1/2, where σ is the interfacial tension, ρc the continuous phase density and μc its viscosity. The power density ε may be estimated based on impeller size and speed N, while transient drop size, Dd, is predicted using an empirical correlation between terminal size and mixing time, tM (Kuriyama et al., 1996). Energy consumption, E, during the mixing process is also estimated and equipment dimensions are calculated for a batch size of M = 2 ton. The complete process model will be discussed in more detail in the conference presentation. Interfacial tension (σ) decreases with surfactant concentration, as more surfactant molecules adsorb at the droplets’ surface. However, above a critical value, known as critical micelle concentration (CMC), surface tension becomes essentially constant, since additional surfactant aggregates in micelles or in multiple layers around droplets. Since only surfactant concentrations above CMC are considered, the limiting minimum surface tension is used to predict droplet size. Based on published experimental results, a conservative overestimate of 0.008 N/m is used.
5. Physical Stability An emulsion is physically stable if its droplet size distribution remains constant, regardless of time or volume element observed (Schubert and Engel, 2004). Instability may occur by droplet sedimentation, aggregation and/or coalescence. The emulsion under study is likely to have a long period of stability (several years), since it incorporates a suitable surfactant, which has a strong tendency to adsorb at the droplets’ surface, and contains a highly viscous continuous phase that inhibits sedimentation. Published results for emulsions similar to the ointment under study indicate that there is a correlation between stability and strength of the surfactant interfacial film, this one increasing with excess surfactant above CMC, as a result of surfactant multilayer formation around droplets. Based on these results, we define our stability index, y4, as the ratio between excess surfactant, above CMC, and the amount of surfactant needed for a monolayer adsorption. A stable emulsion should have a value of y4 significantly
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greater than one, in order to guarantee that enough surfactant is available to strengthen interfacial films. In the study above mentioned, emulsions with y4 = 7 remain stable for 4 weeks in tests performed at 60 ºC. We consider this result as a reference of good emulsion stability. The amount of surfactant required to form a monolayer is calculated from knowing surfactant molecular area (0.346 nm2/molecule) and total interfacial area (6φ/Dd). Surfactant excess is the total amount of surfactant used minus the amount corresponding to CMC, with CMC predicted using a liquid-liquid equilibrium model, where micelles are approximated as a second liquid phase, and employing the UNIFAC method to predict activity coefficients (Flores et al., 2001).
6. Results The above described models predict the set of quality indexes, y1 to y4, as a function of the following design decisions: ointment composition, w, batch size, M, mixing speed, N, and mixing time, tM. Considering for now that w5 is fixed (5%), w1/(w1+w2) = 0.1 and M = 2 ton, there remains a total of 5 degrees of freedom (w1, w3, w6, N and tM), that we will explore. At first, we studied the effect of AI concentration over quality indexes y1 and y2, with w3 and w6 fixed at their nominal values and w4 adjusted to the varying w1 value. Simulation results (Figure 3) indicate that a drug concentration of 2% (current nominal solution) is suboptimal, while a 1.8% ointment seems to be better regarding both quality metrics. Next, we considered a full set of design variables and decision scenarios (Table 1). Scenario A has already been discussed, and exhibits better performance in terms of drug release, when compared with the nominal solution. Since the stability index in A is excessive, scenario B is generated with less surfactant and adjusting y4 to its reference value (~7). Scenario C simulates an increase in the concentration of the more hydrophilic component, C3. A positive impact over quality indexes y1, y2 and also y3 is observed, but much more surfactant is now needed to attain the same stability index, because CMC increases. The more controlled release (smaller y1 and y2) is related to an increase of drug affinity to the ointment continuous phase, and correspondent lower tendency to be transferred to the stratum corneum. This effect here is found to be small, but it can be increased if a more hydrophilic formulation is considered. The decrease in mean droplet size (and associated increase in drug homogeneity) is due to a higher viscosity for the melted lipid phase, leading to an increase in the viscous stresses during emulsification. Error (ng/cm3) 0.1
Error y2 (at peak conc.)
0.08
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Figure 3. Effect of AI concentration over mean and peak errors in plasma concentration profiles.
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Table 1. Product/process design scenarios. w1 (%) w3 (%) w6 (%) N (rpm) tM (min) y1 (ng/cm3) y2 (ng/cm3) y3 (μm) y4 at 60 ºC CMC at 60 ºC (molar fraction) E (kWh)
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A
B
C
D
2 14 2 120 60 0.0373 0.0702 49 232 0.0046 2.26
1.8 14 2 120 60 0.0368 0.0490 49 258 0.0045 2.27
1.8 14 0.62 120 60 0.0368 0.0488 50 6.7 0.0045 2.27
1.8 30 1.30 120 60 0.0363 0.0458 43 6.7 0.0100 2.24
1.8 30 1.34 180 45 0.0363 0.0458 23 7.0 0.0101 5.68
Finally, scenario D corresponds to a different design of the mixing process, with a higher impeller speed but reduced mixing time. A considerable improvement in the homogeneity index y3 is achieved, but obviously at the cost of higher energetic costs. The required surfactant, to keep our stability index at 7, also suffers a slight increase, since smaller droplets offer more surface area to be covered by surfactant molecules.
7. Conclusions We developed an integrated product/process model to support the design of a pharmaceutical formulation. It consistently predicts product quality indexes as a function of composition and mixing process conditions, and therefore it can be used to support optimal product/process design. Although it was derived for a particular ointment product, we believe it can be useful in the development of a larger set of emulsion products, since it is founded on fundamental physico-chemical phenomena, such as mass transfer, interphase equilibria, emulsification and surfactant aggregation. Simulated results obtained for the ointment under study revealed some interesting trends: (i) there is an optimal drug concentration for which the most controlled release is observed during periodical ointment administrations; (ii) a more hydrophilic formulation promotes a more controlled release; (iii) emulsion instability is favored by hydrophilic formulations, due to an increase in surfactant CMC values.
Acknowledgement The authors acknowledge financial support provided by FCT, Portugal, through research project POCI/EQU/59305/2004.
References Bernardo, F. P. and Saraiva, P. M., 2005, Computer-Aided Chemical Engineering, Puigjaner, L. and Espuña, A. (eds.), Elsevier, 20, 1507-1512. Cussler, E. L. and Moggridge, G. D., 2001, Chemical Product Design, Cambridge University Press, Cambridge. Flores, M. V., Voutsas, E. C., Spiliotis, N., Eccleston, G. M., Bell, G., Tassios, D. P. and Halling, P. J., 2001, Journal of Colloid and Interface Science, 240, 277-283. Hill, M, 2004, AIChE J., 50, 1656-1661. Kuriyama, M., Ono, M., Tokanai, H. and Konno, H., 1996, Trans IChemE, Part A, 74, 431-437. Schubert, H. and Engel, R., 2004, Chem. Eng. Res. Des., 82(A9), 1137-1143. Wibowo, C. and Ng, K. M., 2002, AIChE J., 48, 1212-1230. Wichterle, K., 1995, Chem. Eng. Sci., 50, 3581-3586.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Scope for Process Systems Engineering Studies in Proton Exchange Membrane Fuel Cells (PEMFC): A Review of Opportunities R. Madhusudana Raoa , Taehoon Oha , R. Rengaswamy∗
a
a
Process Control, Identification, and Simulation Systems (PROCISS), Department of Chemical Engineering, Clarkson University, Potsdam, New York 13699, USA Proton exchange membrane fuel cells (PEMFCs) are currently in an advanced state of development with promising applications for portable power and power generation. In spite of this, even today, there are several critical issues that researchers are trying to address and resolve. This paper will highlight some of the important technological advancements made in the areas of (i) flow field design, and (ii) water and thermal management. A review of the state-of-art in engineering and design in each of these areas will be summarized. At the same time, we will also highlight the areas where opportunities exist for process systems engineering activities such as, modelling and simulation, optimization, control, diagnostics and fault-tolerant control. The focus of the research work being discussed in this article is the systems engineering of PEMFC with the help of both detailed models and reduced order models (ROMs). The detailed models take into consideration modeling of the reaction and transport processes that occur inside PEMFC. Dynamic models based on flooded-agglomerates for PEMFC as a support tool for systems engineering studies will be discussed. Preliminary results of flow field studies using computational fluid dynamics (CFD) and neural networks (NN) are presented. In addition, optimization results for electrode structure using a steady state spherical flooded-agglomerate model are also presented. Keywords: Fuel cells, PEMFC, dynamic model, process systems engineering, and electrode optimzation 1. Introduction Proton exchange membrane fuel cells (PEMFCs) are currently in an advanced state of development with promising applications for portable power and power generation. In spite of this, even today, there are several critical issues that researchers are trying to address and resolve. A thorough review of several scientific and engineering aspects of PEMFCs is given by Costamagna and Srinivasan [1,2]. This paper will highlight some of the issues in the areas of (i) flow field and electrode design, and (ii) water and thermal management and present CFD and electrode structure optimization studies. ∗
Author to whom all correspondence should be addressed. Email: [email protected]
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Reactant gases are distributed across the fuel cell cross-sectional area using bipolar graphite plates. Gas flow channels are machined in bipolar plates in various configurations. Parallel flow, interdigitated and serpentine channels are some of the designs that been tried. The main aim of the channel design is to achieve uniform gas distribution across the surface of the electrode, at the same time avoid flooding conditions and gas diffusion limitations. Various materials of construction for the bipolar plates are being tried that would allow different geometries of flow channels to be machined at a low cost. Hence, there is a need for flow field simulation and optimization studies based on computational fluid dynamics (CFD) techniques. Such studies will help in not only visualizing flow patterns in various geometries, but also help in achieving the best possible design. CFD studies are computationally intensive and are time consuming if one has to use them for doing flow field design and optimization studies. Hence, we have used neural networks (NN) to develop reduced order models (ROMs) for predicting CFD results in a timeefficient manner. Some results are presented subsequently to highlight the congruence between NN predictions and CFD results. Water and thermal management is another important issue for PEMFCs. There have been several theoretical and experimental studies conducted to understand this problem. In addition, many technological progresses have been made to resolve the above problem. Several operational techniques for inlet gas and membrane humidification, and methods for internal humidification in terms of alternative design of components have been proposed and demonstrated. However, most of the modeling and validation strategies seem to be focused at a single cell level and the dynamics, operational difficulties and also the opportunities that are associated with solving this problem at a stack level seem to have been ignored to a large extent. In this paper, we will review the state-of-art in the solution to this problem as seen in the published literature. Further, it is clear that a good design alone cannot solve this problem. Particularly, in order to address the lifetime thermal and water management for PEMFC, a large number of issues need to be addressed. These can be broadly classified as systems engineering issues. Some of the systems engineering aspects are: (i) detailed first-principles modeling, (ii) diagnostics, (ii) multi-variable control, and (iv) fault-tolerant control. As a part of our investigations in the study of design and optimization of electrode structures in PEM fuel cells, a two-dimensional steady state model for PEMFC cathode with detailed characterization of cathode catalyst layer is being used. Based on our earlier investigations for single spherical agglomerate [3], it was found that oxygen is predominately consumed at the surface of spherical agglomerate. Hence, Pt nanoparticles in the inner core of spherical agglomerate may not be utilized. To minimize the usage of platinum, instead of a uniform distribution of Pt nanoparticles inside the agglomerate, studies were carried with a graded distribution of Pt with more Pt near the surface and less in the core. The effect of such profiles on the i-v curves were studied and optimization studies were carried out for minimizing the amount of platinum. We present some results that suggest optimum loading of catalyst without a significant change in the performance of the i-v curve.
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Figure 1. Pressure drop along the serpentine channels in the anode
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Figure 2. Consumption of H2 along the serpentine channels in the anode
2. CFD studies Computational Fluid Dynamics (CFD) models have the promise of providing detailed information about the operation of fuel cells and have been used in fuel cell modeling. While CFD equipment models provide detailed analysis of the performance, they are very time-consuming to develop and run. The computations become quite complex, especially when such models have to be embedded in flowsheet level optimization. Hence, there has been recent interest in building reduced order models (ROMs), based on the result of detailed CFD simulations that can be routinely used in a number of performance studies. Among the many possibilities for building reduced order models, neural networks (NN) are an attractive choice. Neural networks have been used in a wide range of engineering applications such as, pattern recognition, behavior prediction and function approximations. We have developed reduced-order NN models for quickly predicting the flow of reactants in a PEMFC manifold. A feed-forward, back-propagation neural network was used for the predictions. The data for NN training is generated from the detailed CFD simulations of the manifold using a half-cell model. The inputs to the NN were: channel dimension, inlet velocity, inlet gas temperature and inlet pressure. The output variables are: hydrogen consumption, pressure drop and mean velocity in the channels. Figure 1 shows the comparison of CFD results and NN predictions for pressure drop along the serpentine channel in the anode graphite plate. A comparison of the results for hydrogen consumption is shown in Fig. 2. One can clearly see that there is good congruence between the CFD results and NN predictions. From our investigations, we have found that the NN-based ROM can quickly (≈ 1 sec) predict detailed flow behaviors in the manifold. 3. Water and thermal management Water management is an important issue in proton exchange membrane fuel cells (PEMFCs). There have been several theoretical and experimental studies conducted to understand this problem. In addition, many technological progresses have been made to resolve
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the above problem. Several operational techniques for inlet gas and membrane humidification and methods for internal humidification in terms of alternative design of components have been proposed and demonstrated. However, most of the modelling and validation strategies seem to be focused at a single cell level and the dynamics, operational difficulties and also the opportunities that are associated with solving this problem at a stack level seem to have been ignored to a large extent. Various humidification strategies have been proposed to meet the water retention needs of the membrane [1]. A typical schematic of a stack humidification system consists of humidifying the reactant gases (H2 and O2) externally by liquid water. Temperature of the humidification pots is important and optimization studies carried out have established that humidification temperature has to be 5 − 10o C higher than that of the stack [4]. Several techniques have also been proposed and demonstrated in the past, such as (i) recirculation of gases, (ii) water vapor injection along the gas flow channels, (iii) liquid water injection [5], and (iv) use of wicks for keeping the membrane hydrated [6]. Internal hydration techniques where the water produced at the cathode is used for the hydration of the membrane have also been developed and demonstrated: (i) use of porous carbon blocks for bipolar plates [7], (ii) impregnation of the membrane solution into the electrode [8], forming thin film on the surface of carbon particles, (iii) impregnation of thin Nafion recast membranes with a small amount of nanosize Pt particles [9], and (iv) air plus evaporative cooling [10]. All these methods and design alternatives have been shown to lead to good cell performance. The approach that we would like to address is a predictive control strategy for maintaining a complete water balanced cell for a long periods of time. A detailed dynamic model for a complete PEMFC cell, consisting of anode, cathode and the membrane will be developed. The model will be set up to predict the behavior of the cell under varying load conditions. It is our contention that just the design of a thermal and water device will not solve this problem completely, if one were to desire optimal performance for long periods of time. This is because of the delicate water balance that is needed to achieve the best possible performance. Smart and innovative designs can solve this problem to a large extent, but they have to be developed hand-in-hand with a concomitant operational strategy. 4. Electrode structure optimization It is widely accepted that gas-diffusion electrodes (GDEs) used in fuel cells are threephase electrodes with a complex geometry consisting of conducting solid material for electron transfer, hydrophilic phase for ion transfer and open pores for gas transport. Tantram and Tseung [11] for the first time studied the structure and operation of hydrophobic electrodes under electron micrographs. As GDEs are difficult to characterize, one of the first assumptions that was made to model them was the concept of ”flooded agglomerates” by Giner and Hunter [12]. They have considered cylindrical geometry for the agglomerates. Subsequently, many studies conducted by various researchers with the spherical flooded-agglomerate models were presented for both alkaline fuel cells (AFC) and phosphoric acid fuel cells (PAFC). Similarly, researchers have started studying the effects of various phenomena in the catalyst layer of PEM fuel cells based on flooded-
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Figure 3. Graded distribution of platinum inside spherical agglomerate
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Figure 4. Optimization of platinum loading inside spherical agglomerate
agglomerate structure during the last two years [13–16,16]. Based on our earlier investigations for a single spherical agglomerate [3], it was found that oxygen is completely consumed at the surface of spherical agglomerate for overpotentials less than -0.35 V. Hence, Pt nanoparticles in the inner core of a spherical agglomerate are not utilized. Based on these results we have carried out optimization studies on minimizing the amount of platinum inside the agglomerate. For this, we have considered a graded distribution of platinum inside the agglomerate (Fig. 3) with more platinum near the surface and less towards the center, instead of uniform distribution of platinum throughout the agglomerate. Figure 4 shows a comparison of the i-v curves between the base case and the optimized case. A comparison of the amount of platinum inside the agglomerate for both the cases shows a 50 % reduction in platinum. In spite of a large reduction in the amount of platinum for the optimized case, the performance in terms of i-v curve is not significantly altered. We are currently working on extending the optimization studies for the complete PEM fuel cell. Subsequently, flooded-agglomerates of different geometries and sizes will be considered in cathode catalyst layer and also study their effect on optimizing electrode structures. We will also investigate the benefits these studies can offer in terms of greatly reducing platinum use and hence reducing the cost of fuel cells. Acknowledgements We would like to acknowledge American Chemical Society - Petroleum Research Fund (ACS-PRF) for providing financial support for this work. Grant # PRF 42842-AC9 REFERENCES 1. P. Costamagna and S. Srinivasan. Quantum jumps in the pemfc science and technology from the 1960s to the year 2000. part i. fundamental scientific aspects. J Power
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Sources, 102:242–252, 2001. 2. P. Costamagna and S. Srinivasan. Quantum jumps in the pemfc science and technology from the 1960s to the year 2000. part ii. engineering, technology development and application aspects. J Power Sources, 102:253–269, 2001. 3. R. Madhusudana Rao and R. Rengaswamy. Dynamic characteristics of spherical agglomerate for study of cathode catalyst layers in proton exchange membrane fuel cells (pemfc). J Power Sources, accepted for publication, 2005. 4. E. A. Ticianelli, C. R. Derouin, A. Redondo, and S. Srinivasan. J Electrochem. Soc., 135:2209, 1998. 5. T. V. Nguyen, J. Hedstrom, and N. Vanderborgh. In R. E. White and A. J. Appleby, editors, Proceedings of the Symposium on Fuel Cells, PV 89-14, The Electrochemical Society Proceedings Series, page 39, 1989. 6. M. Watanabe, Y. Satoh, and C. Shimura. Management of the water content in polymer electrolyte membranes with porous fiber wicks. J Electrochem. Soc., 140(11):3190–3193, 1993. 7. S. Miachon and P. Aldebert. Internal hydration h2 /o2 100 cm2 polymer electrolyte membrane fuel cell. J Power Sources, 56:31–36, 1995. 8. H. P. Dhar. US Patent No 5242764, 1993. 9. M. Watanabe, H. Uchida, and M. Emori. Analyses of self-humidification and suppression of gas crossover in pt-dispersed polymer electrolyte membranes for fuel cells. J Electrochem. Soc., 145(4):1137–1141, 1998. 10. R. Mosdale and S. Srinivasan. Analysis of performance and of water and thermal management in proton exchange membrane fuel cells. Electrochim. Acta., 40(4):413– 421, 1995. 11. A. D. Tantram and A. C. C. Tseung. Structure and pewrformance of hydrophobic gas electrodes. Nature, 221:167–168, 1969. 12. J. Giner and C. Hunter. The mechanism of operation of the teflon-bonded gas diffusion electrode: A mathematical model. J Electrochem. Soc., 116(8):1124–1130, 1969. 13. N. P. Siegel, M. W. Ellis, D. J. Nelson, and M. R. von Spakovsky. Single domain pemfc model based on agglomerate catalyst geometry. J Power Sources, 115:81–89, 2003. 14. G. Lin, W. He, and T. V. Nguyen. Modeling liquid water effects in the gas diffusion and catalyst layers of the cathode of a pem fuel cell. J. Electrochem. Soc., 151(12):A1999–A2006, 2004. 15. D. Song, Q. Wang, Z. Liu, T. Navessin, and S. Holdcroft. Numerical study of pem fuel cell cathode with non-uniform catalyst layer. Electrochim. Acta., 50:731–737, 2004. 16. Q. Wang, D. Song, T. Navessin, S. Holdcroft, and Z. Liu. A mathtematical model and optimization of the cathode catalyst layer structure in pem fuel cells. Electrochim. Acta., 50:725–730, 2004.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A framework for innovation in process development for heterogeneously catalysed gasphase reaction systems Daniel Montolio-Rodriguez,a David Linke,b Patrick Linkea a
Centre for Process and Information Systems Engineering, University of Surrey, Guildford, GU2 7XH, U.K. b Institute for Applied Chemistry Berlin-Adlershof (ACA), Richard-Willstaedter-Str. 12, 12489 Berlin, Germany
Abstract There is an increasing need for novel technologies to effectively identify highly efficient chemistries and to enable the quick and reliable development of novel, innovative and sustainable high-performance processes to exploit these. This requires the systematic integration and coordination of R&D activities from chemistry through to process engineering. This paper reports on technology developments in the form of process synthesis methods that will be required to enable the integrated synthesis of heterogeneously catalysed gas-phase reacting processes in parallel to kinetic investigations. The methods are based upon process network representations that can process the information generated in the kinetics investigation activities and embed the vast number of conceptual process design options that are possible for a given system whilst taking into account the practical constraints associated with such systems. The networks are searched using robust optimisation technology in the form of Tabu Search to identify the best processing schemes and operating regions for a given chemistry. This information on the optimal design regions can be constantly fed back to the kinetics development team to guide experimental efforts so that the kinetics match the optimal operating regions of the process. The process synthesis technology developments are illustrated with a complex application in the production of acetic acid. Keywords: process design, decision-support framework, heterogeneous catalysis, acetic acid production.
1. Introduction The current approach to the development and design of heterogeneously catalysed processes is largely sequential. The success at each step is hampered by the lack of systematic support tools to assist the scientists and engineers involved in the design processes to identify innovative solutions reliably and quickly. Methodological shortcomings are exemplified by the lack of coordination of kinetic model development, reactor design and process synthesis. More often than not, kinetic models are developed for operating regions that do not correspond to the optimal conditions identified in the later process design stages. In many cases, the experimental studies have been concluded by the time this information becomes available. The result is then either a compromise design dominated by kinetic model reliability issues or a project delay caused by additional experimental investigations of the kinetics.
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Such problems can be overcome with the use of systematic high-level decision-support methods to coordinate design activities and assist the scientists and engineers involved in the design activities to identify innovative solutions reliably and quickly. Their decisions need to be supported in the context of the performance of the overall process system in which the chemistries are being exploited. The past two decades have seen increased research efforts into how optimal process systems can be systematically identified for a given chemistry. A comprehensive review of these process synthesis methods is given in Linke and Kokossis (2003a). Such methods have recently been identified as key technologies that promise to deliver major improvements in process efficiencies due to their ability to support high-level decisions in process design (Tsoka et al., 2004). However, the methods are very rarely applied in industry, mainly as a result of their heavy computational demands when used on kinetic models of industrial complexity and their inability to (a) handle practical process constraints, (b) evolve high-level conceptual designs into detailed designs that can be implemented in practice, and (c) communicate information to related design activities, mainly kinetic investigations and reaction engineering, in a concise fashion so that design decisions in these areas can be guided in the context of the overall design goal. Our research aims to address these limitations with the goal of realising an integrated approach in which the key process design issues can be addressed in parallel to the investigation of the chemistry from the earliest stage to arrive at the most economically viable and sustainable design via the shortest possible route. In the first instance, we have focused our attention on high-level process synthesis issues of heterogeneously catalysed gasphase reacting systems for their relevance in industrial practice. The following sections outline the process synthesis framework and illustrate the framework with an example in process synthesis for acetic acid production.
2. Process synthesis 2.1. Previous Work and Research Aims A number of optimisation-based general purpose process synthesis tools for integrated reaction-separation systems have been developed to date (Papalexandri and Pistikopoulos 1996; Ismail et al., 2001; Linke and Kokossis 2003a,b). However, these methods fail to address many problems encountered in heterogeneously catalysed gasphase reacting systems for two major reasons. Firstly, the kinetic models are of a complexity beyond the capabilities of these methods so that the superstructure models resulting from the generalised representations cannot be solved reliably with existing optimisation technology. Secondly, practical design issues such as heat management, explosion limits, and complexity controls can not be represented by these methods so that they tend to identify designs that are not practically realisable. Motivated by these shortcomings, the aim of this work is to develop practical representations of process networks for heterogeneously catalysed gas-phase reacting systems that can be reliably optimised. These representations will form the basis of a framework to support highlevel decisions in the process design cycle. 2.2. Process Network Representation and Optimisation We have developed superstructure representations for heterogeneously catalysed gasphase reacting systems. The superstructures account for detailed representations of the reaction system through combinations of generic units that embed options related to mixing (plug flow or backmixed), temperature policies, the mass of catalyst present, as well as constraints related to energy management. The energy management constraints have been derived from physical limits that exist in different heterogeneously catalysed
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reaction systems designs including fixed-bed, tubular, and fluidised bed reactors. Explosion limits are incorporated as design constraints in the network model formulation and can be set with respect to different components present in the system. Heterogeneously catalysed gas-phase reacting systems offer limited scope for reactive separation due to the high operating temperatures. Membranes could in principle be used for separation at high temperature, but developments have not matured enough to allow commercial implementation for most systems. The representation therefore does not consider reactive separation as an option at this point, but this can later be incorporated following the multi-state representation by Linke and Kokossis (2003a). The kinetics of the catalysed reactions tend to be highly complex and the simultaneous solution of detailed reaction-separation models would be an impossible task for most systems. That is why the separation systems are represented in aggregated form to keep the superstructure model at solvable complexity. The separation models assign the components present in the inlet streams to outlet streams according to the order of separation for a given separation operation. Cost expressions are developed using separation systems synthesis models that represent the information on the lowest cost separations for given stream compositions and flow rates. This allows the decoupling of the separation synthesis calculations from the superstructure optimisation. The process superstructures are robustly optimised using Tabu Search, which has found successful applications in the optimisation of reaction (Ashley and Linke, 2004) and reaction-separation (Linke and Kokossis, 2003b) systems. 2.3. Synthesis procedure The process synthesis is performed in three steps to generate maximum design insight and understanding. First, the performances of designs with conventional reactors (usually well-mixed and plug flow reactors) are obtained as base cases. Second, the performance limit that can be attained for the system with an innovative process design is obtained from superstructure optimisation allowing all possible interactions between feed streams and recycle streams with multiple reaction zones each of which can exhibit different mixing, heat management and catalyst mass to be explored. Third, the designs from the second stage are analysed and key design features are identified for which targeted superstructures are optimised to study the impact of individual design features. The synthesis procedure is illustrated in Figure 1.
Base case assessment
Target performance limit
High-level design candidates
Figure 1. Synthesis procedure
Increase search space
Identify improvement in objective with increasing complexity
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3. A Case Study: Acetic Acid production The purpose of the study is to explore design trends and key design features that maximise process performance for a new catalyst. The objective is to explore those designs that maximise the Net Present Value (NPV) of the system. The complex kinetics for the catalyst are given in Baerns et. al (2002) and the reaction paths are shown in Figure 2. Ethane and oxygen are used as the raw materials to produce the desired product acetic acid. The process is naturally structured into a sequential reaction-separation arrangement consisting of a reaction system followed by a flash to separate unreacted ethane, intermediate ethene, and by-product carbon dioxide from acetic acid and water. Carbon dioxide is removed from the ethane/ethene recycle stream using a membrane unit. The acetic acid/water stream is separated in a distillation column. The design decisions relate mainly to the reaction schemes and the resulting superstructure representation is shown in Figure 3. 3.1. Step 1: Identification of base case performances Two processes were optimised to establish the performance to be expected from conventional designs. The optimisation revealed that well-mixed reactors are not an attractive option for this system. In fact, a substantial loss of -7.6M$ can be expected if such a reactor type is chosen. A plug flow reactor with an increasing temperature profile would result in a profit of 8.5M$ per annum. 3.2. Step 2: Performance limit Optimisation of the process superstructure revealed highly profitable schemes that offer an increase in NPV of 150% over the best base case design (22M$/yr). Such a substantial increase in profit warrants further investigation into the design of the system. The multiple designs obtained from the study all showed multiple reaction zones, oxygen side feeding policies and internal recycles. Plug flow was observed for almost all reaction zones, although in few cases well mixed zones co-existed. A few structures also featured ethane feed distribution amongst the reaction zones.
Figure 2. Reaction path for acetic acid production (Baerns et. al, 2002)
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OUTSIDE RECYLCE
Reaction Superstructure
FEED O2
Water Generic unit
Generic unit
Generic unit
INTERNAL RECCYLE
Splitters
Mixers
Acetic Acic
Figure 3. Superstructure representation for acetic acid production 3.3. Step 3: Analysis of key design features All designs identified in step 2 were too complex to be considered for practical implementation. A further step is required to explore the impact of the individual design features that were observed. Reduced superstructures were optimised allowing only variations in operating variables and the few structural options associated with the design features observed in step 2. This study showed that two main features substantially improve the profitability of the process. Firstly, designs with internal recycles between plug flow reactor zones were observed to result in NPVs very close to the performance limit (20 M$/yr). The performance were observed to improve with the internal recycle flows. Secondly, the distribution of oxygen feed between the reaction zones significantly improves the NPV of the system. A system with three reaction zones improves the NPV of the system by about 90% over the best base case (16 M$/yr), whereas a simpler system with only two reaction zones structure still enables an improvement of around 65%. If both these key features are combined, the improvement of the objective function value is even more notable and the performance of the system very closely approaches the performance limit. One such design with two reaction zones, oxygen feed distribution and internal recycle is shown in Figure 4.
Figure 4. Optimised acetic acid process design (performance limit: 22 M$/yr)
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4. Conclusions We have presented a process synthesis framework for heterogeneously catalysed gasphase systems that allows the robust identification of process performance limits and the systematic exploration of design trends. The framework has performed impressively on a case study of a typical complexity encountered in industrial practice. The synthesis framework allows the high-level coordination of process design and kinetic investigation activities. Our current research focuses on identifying (i) efficient ways to integrate the information generated by the synthesis framework to assist kinetic investigations is the focus of our current research, and (ii) multi-scale modelling approaches that enable the efficient evolution of high-level designs into detailed and practical design options.
References V. Ashley, P. Linke. Chemical Engineering Research & Design, 82(2004), 952. S.R. Ismail, P. Proios, E. N. Pistikopoulos, AIChE Journal, 47(2001) 629. D. Linke, D. Wolf, S. Zeiss, U. Dingerdissen, M. Baerns, Journal of Catalysis, 205(2002) 32. D. Linke, M. Baerns, Q. Smejkal, Chemical Engineering and Processing, 44(2005) 421. P. Linke, A.C. Kokossis, AIChE Journal 49(2003a) 1451. P. Linke, A.C. Kokossis. Computers & Chemical Engineering 27(2003b), 733. E. Marcoulaki, P. Linke, A. Kokossis. Chemical Engineering Research & Design 79(2001), 25. V.L. Mehta, A.C. Kokossis, AIChE Journal, 46(2000) 2256. K. P. Papalexandri, E. N. Pistikopoulos, 1996, AIChE Journal, 42(4), 1010. C. Tsoka, W.R. Johns, P. Linke, A. Kokossis. Green Chemistry, 8(2004), 401.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Multi-objective Optimization of Fixed-Bed Ion Exchange Processes for Phytopharmaceutical Production Cláudia M. Silva, Amaro G. Barreto Jr., Evaristo C. Biscaia Jr. Programa de Engenharia Química PEQ/COPPE/UFRJ - Universidade Federal do Rio de Janeiro - Cidade Universitária - CP: 68502 - Rio de Janeiro - 21945-970 RJ - Brasil
Abstract Ion exchange chromatography has been widely employed to produce important ionizable substances. In this contribution, the adsorption stage in an ionic exchange bed is studied in order to isolate phytopharmaceutical compounds. The optimal operation conditions are determined using a multi-objective optimization technique. An improved algorithm comprising new advances in evolutionary strategies is proposed to conduct the optimization of the chromatographic system. A pore diffusion mathematical model is adopted to describe the adsorption stage of the ionic exchange bed. Optimal operating conditions are generated in order to maximize the productivity, recovering yield and process capacity. The scale-up problem of a chromatographic column is addressed by means of a mixed-integer nonlinear programming problem. Simulated results have confirmed the efficiency of the multi-objective technique to find the optimal solution set, offering a viable strategy to solve complex optimization problems. Keywords: multi-objective optimization, ion exchange chromatography, mixed-integer nonlinear programming problems.
1. Introduction Chromatographic methods are frequently used in industrial scale for isolation and purification of phytopharmaceutical products, biomolecules and fine chemicals. In the last decades, pharmaceutical industry has been focused on discovering active substances for treatment of diseases like cancer and AIDS. Progress on this field, however, depends on enhancement of separation technologies, which requires the optimization of extraction, fractionation and purification stages. In this contribution, the adsorption stage in an ionic exchange bed has been studied in order to isolate Lapachol molecules. Lapachol is a naphthoquinone obtained from lapacho tree (Tabebuia avellanedae), which is considered to have antibacterial and antifungal activities, as well as promising anti-tumor properties. Lapachol and its derivative compounds have been intensively investigated for clinical use in cancer and malaria chemotheraphy (Moura et al, 2001). Optimal performance of the adsorption stage entails the maximization of productivity, process capacity and Lapachol recovering yield. In spite of being in conflict, these targets should be satisfied simultaneously and a compromise solution has to be sought. The optimization of such systems has been handled by single objective approaches, combining all targets in a weighted objective function or selecting one target as objective function and the others as constraints. A meaningful optimization approach implies the use of multi-objective concept, which generates a Pareto optimal set, instead of a global optimum.
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An improved multi-objective optimization algorithm based on evolutionary strategies is proposed to solve the formulated problems: optimization of an existing laboratory column and design of a new system. As the latter involves both discrete and continuous variables, a multi-objective mixed-integer nonlinear programming (MOMINLP) formulation was used. Some adaptations were required to make the algorithm capable of treating mixed-integer optimization problems. Simulated results demonstrated the algorithm ability to successfully seek tradeoff surface regions as well as to find the Pareto optimal set, offering a consistent strategy to optimize chromatographic systems.
2. Process Description The isolation of plant active principles in chromatographic columns consists of three operation cycles: raw extract feeding, bed washing with solvent and elution of chemically adsorbed substances. The extraction stage is carried out by percolating a solvent into the adsorbent bed at ambient temperature and atmospheric pressure. In this contribution, the adsorption stage was studied to isolate Lapachol molecules in an ionic exchange bed. Experiments were conducted in a laboratory column, using Lapachol (99%) and formic acid (88%) as extract feeding, the resin Lewatit MP500 (Bayer) as adsorbent and ethanol PA as solvent. Physical properties, operational and equipment parameters are described elsewhere (Barreto Jr., 2005). 2.1. Mathematical Model The pore diffusion model proposed by Barreto Jr. (2005) to describe the adsorption stage in a fixed-bed ion exchange column is adopted. The following hypotheses are considered (Gu, 1997): a) adsorbent particles are spherical with unimodal size distribution and small dispersion; b) fluid phase inside the pores is stagnated; c) liquid film covering the particles is negligible; d) effective axial and intra-particular diffusivities are independent of time and axial and radial coordinates; e) intra-particular transport is due to diffusion on macropores; f) there is an instantaneous local equilibrium between the adsorbed group and the stagnated fluid. The dimensionless mathematical model is expressed as follows: Microscopic mass balance in liquid phase: ∂ci 1 ∂ 2 ci ∂ci (1 − ε) ∂c pi = − −3 η (1) 2 ∂τ ∂z ε Pe ∂z ∂r r =1 Initial and boundary conditions: τ = 0 , ci = c pi = qi = 0
z = 0,
∂ci ∂z
z =0
(
= Pe ci
z =0
− c f (τ)
)
z = 1,
Microscopic mass balance in solid phase: ⎡ 1 ∂ ⎛ 2 ∂c pi ⎞⎤ ∂cTi ⎟⎥ = 0 where cTi = (1 − ε)qi + ε p c pi ⎜r − η⎢ ⎜ 2 ∂τ ∂r ⎟⎠⎦⎥ ⎣⎢ r ∂r ⎝ Boundary conditions: r = 0,
∂c pi ∂r
=0 r =0
r = 1, c pi
r =0
1+
n
∑ b j c pi j =1
z =1
=0
(2)
= ci (τ, z )
ai c pi
Langmuir multi-component adsorption isotherm: qi =
∂ci ∂z
(3)
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The set of partial differential equations (PDE) was spatially discretized into 20 elements along the column axial direction by means of parabolic approximation on finite elements and into 4 points in the particle radial direction, by global orthogonal collocation. The resulting system, consisting of 100 ordinary differential equations (ODE) and 100 non-linear algebraic equations was integrated using the solver DASSL. Regularization exponential functions were adopted to initialise the system (Vieira and Biscaia Jr., 2000).
3. Multiobjective Optimization Evolutionary algorithms are robust stochastic methods founded on the principles of natural genetics, in which the fittest species survive and propagate while the less successful tend to disappear. The evolution process consists of performing a population of individuals with operators to generate the next generation. The basic operators simulate the processes of selection, crossover and mutation, which happen according to pre-established probabilities. Selection is based on the survival potential, expressed by the fitness function. Crossover involves random exchange of characters between pairs of individuals, in order to produce new ones. Mutation is an occasional change in individual’s characters randomly chosen to introduce diversity to the population. Evolutionary methods are able to deal with ill-behaved domains, such as multimodal, discontinuous, time-variant, random and noisy ones. Multi-objective optimization is a special extension of the optimization theory in which multiple opposing targets must be accomplished simultaneously. The search process aims to find solutions that are the best on all objectives. The optimal solution constitutes a family of points, called Pareto front, that equally satisfy the set of objective functions. In the Pareto set, no improvement can be obtained in any objective without deteriorating at least one of the other objectives. Depending on the problem complexity, however, achieving the optimal solution set may be a difficult task. In such cases, efforts are concentrated to reach a solution set as close to the Pareto as possible. Since all objective functions are simultaneously optimized, the solution constitutes a compromise between the conflicting aims. An improved multi-objective optimization approach based on evolutionary strategies has been proposed to deal with the studied problem. It includes a Pareto-set filter, which avoids missing optimal points during the optimization process; an elitism operator, which guides the evolutionary search, propagating the best result of each individual objective function; a niche operator, which prevents genetic drift, maintaining the population uniformly distributed along the optimal set (Chen and Li, 1998). In order to treat multidimensional problems, a new ranking strategy is proposed. The classification procedure, based on the concept of non-dominance, is extended to rank a population of high dimensional solutions. The new strategy does not require any previous knowledge of the relative importance of individual objectives. A fitness function based on each rank population size and rank level is provided to determine the reproduction ratio. Constraints are handled by means of a penalty function method based on fuzzy logic theory (Silva and Biscaia, 2003). 3.1. Formulation of the multi-objective optimization problem Optimization problems regarding the operation and project stages are proposed. The objective functions include the maximization of productivity (mass of adsorbed product per unit mass of adsorbent and operation time), yield (mass of adsorbed product per unit mass of extract fed) and process capacity (flow rate x operation time).
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850 L Rp
(1 − ε p ) ∫
Productivity:
∫r
c s ( z, r , t )drdz 0 0 R 3p Lmr t op
F1 =
L Rp
(1 − ε p ) ∫
Yield:
2
∫r
0 0
F2 =
2
(4)
c s ( z, r , t )drdz
top
(5)
R 3p LQ ∫ c0 (t )dt 0
Process capacity: F3 = Q. top
(6)
3.1.1. Optimization of an Existing System The operation of an existing laboratory column is optimized. The adsorption process in this case is conducted in a column of 15 cm of height and 0.7 cm of diameter, using a resin of particle diameter 500 μm. The objective is to maximize the productivity and Lapachol recovered yield. The decision variables include feed flow rate (Q), outlet limit concentration (Clim) and feed concentration (C0). The ranges of search are: Q = 1 to 20 mL/min, Clim = 1 to 95 % and C0 = 0.001 to 2.5g/L. 3.1.2. Optimization of a System at Design Level The scale-up problem consists of a hybrid discrete-continuous system. Decision variables consisting of operational conditions are continuous, while the project ones are commercially established. The optimization problem is therefore formulated as a multiobjective mixed-integer nonlinear programming (MOMINLP) problem. Three cases are considered: (a) maximization of yield and process capacity, (b) maximization of productivity and process capacity and (c) maximization of the three objectives simultaneously. All cases are submitted to a maximum ΔP for safety operation of 80 psig. Blake-Kozeny correlation for pressure drop across packed bed is adopted: μν L(1 − ε b ) 2 (7) ΔP = 150 d 2p ε 3b The decision variables are interstitial velocity (v), outlet limit concentration (Clim), feed concentration (C0), column height (l), column diameter (Dc) and adsorbent particle diameter (dp). The range of search are v = 1 to 80 cm/min, Clim = 1 to 99 %, C0 = 0.001 to 2.5 g/L, l =10 to 100 cm. The column diameters are commercially defined as 0.7, 5, 25, 50 and 100 cm (Sedgwcik, 2002). The adsorbent particle diameters commercially available are 15, 50, 100 and 500 μm.
4. Results The optimization procedure was exhaustively performed in order to guarantee that the best viable solution set was reached. The optimization parameters were established in a sensitivity study as population size = 20, crossover probability = 90 % and mutation probability = 20 %. 4.1.1. Optimization of an Existing System The Pareto optimal set generated by optimizing the operational conditions of an existing column is presented in Figure 1. Any of these solutions constitutes an optimum and may therefore be selected as the problem solution. The conflicting behavior of the formulated problem is highlighted: the maximum productivity (0.73mg/gmin)
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corresponds to a minimum yield (41%), and the maximum yield (99%) corresponds to a minimum productivity (3x10-4mg/gmin). High values of the flow rate, outlet limit concentration and feed concentration are required to maximize productivity. Low values of such variables result in maximum yield. F2
F1
1
yield (%)
0,9 0,8 0,7 0,6 0,5 0,4 0
0,2
0,4
0,6
0,8
productivity (g/gmin)
3 x 10-4 5 x 10-2 0.18 0.33 0.49 0.58 0.68 0.69 0.70 0.71 0.72 0.73
0.99 0.99 0.99 0.97 0.95 0.91 0.78 0.71 0.64 0.55 0.45 0.41
Q 1.0 1.0 1.0 1.1 1.3 3.4 4.9 6.3 8.4 10.6 16.6 20.0
Clim 0.01 0.01 0.01 0.07 0.12 0.27 0.71 0.79 0.85 0.95 0.95 0.95
C0 1.0 x 10-6 1.6 x 10-4 7.7 x 10-4 1.2 x 10-3 2.2 x 10-3 2.5 x 10-3 2.5 x 10-3 2.5 x 10-3 2.5 x 10-3 2.5 x 10-3 2.5 x 10-3 2.5 x 10-3
Figure 1 - Pareto optimal set for the existing column
4.1.2. Optimization of a System at Design Level The Pareto fronts obtained in cases (a) and (b) of the scale-up problem are illustrated in Figure 2. For the sake of succinctness, only a qualitative analysis of these cases is presented, the numerical results were omitted. It can be noticed that the process capacity in case (a) is twice the one in case (b). When productivity is maximized, the process capacity is very low. When yield is maximized, however, high process capacities can still be obtained for yield values close to 100%. In scale-up problems, the influence of a single decision variable on the objectives may be predictable. The effect of the interaction of several variables, however, is often unknown and may only be verified by optimizing the whole set of objectives. For this reason, case (c) is proposed. Table 1 presents the optimal solutions for this case. By optimizing three objective functions simultaneously, better combinations of results were obtained. A reasonable solution could be yield of 96.8%, productivity of 0.41mg/gmin and process capacity of 1.56x107ml. The results show that, in most cases, high column heights generate better yields. For higher productivities, wide diameters and reduced 2,E+08
process capacity (ml)
process capacity (ml)
4,E+08
3,E+08
2,E+08
1,E+08
0,E+00
1,E+08
0,E+00
0,5
0,6
0,7
0,8
yield (%)
0,9
1
0
0,2
0,4
productivity (g/gmin)
Figure 2 - Pareto optimal set for scale-up problem (cases a and b)
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heights are required. The behaviors of interstitial velocity, outlet limit concentration and feed concentration do not show any specific tendency. The selection of maximum particle diameter for all solutions is probably due to the pressure constraint: larger particles imply less packed bed and, consequently, less pressure drop across the bed. The choice of a solution in the Pareto set requires a subjective decision making criteria, based on additional knowledge of the system. A maximum yield may be prioritized when the recovered product has a high aggregated value. In case of using an expensive resin, higher percentage of ionic exchange capacity may be emphasized. When the substance of interest is in high concentration, lower yield may be acceptable. Table 1 – Pareto optimal set for scale-up problem (case c)
F1
F2
F3
v
Clim
C0
l
Dc
dp
2 x 10-5 2 x 10-4 3 x 10-3 2 x 10-2 3 x 10-2 0.12 0.32 0.41 0.44 0.61 0.67 0.72 0.73
99.5 57.0 73.8 75.5 99.0 93.9 97.1 96.8 95.3 88.7 85.5 47.5 36.4
1.96 x 105 2.46 x 108 1.78 x 108 1.45 x 108 6.34 x 107 6.78 x 107 2.53 x 107 1.56 x 107 1.40 x 107 9.43 x 106 4.74 x 106 4.51 x 106 8.81 x 105
1.0 14.4 15.5 17.4 6.7 32.3 25.2 29.8 43.1 48.2 46.7 80.0 79.7
0.01 0.99 0.90 0.93 0.08 0.49 0.33 0.19 0.28 0.69 0.65 0.99 0.99
0.001 0.001 0.013 0.060 0.201 0.217 1.02 1.19 0.929 1.91 2.50 2.50 2.50
100.0 100.0 99.5 97.2 96.3 90.8 93.2 68.4 50.6 52.5 32.8 16.7 10.0
5 100 100 100 100 100 100 100 100 100 100 100 50
500 500 500 500 500 500 500 500 500 500 500 500 500
5. Conclusions In this contribution, the adsorption stage in an ionic exchange chromatography column has been optimized using an evolutionary multi-objective algorithm and a pore diffusion mathematical model. The formulated problems aim to optimize the operational conditions of an existing column and to scale-up a new system. The targets include enhancing the process productivity and yield, as the product of interest and/or material used may have high aggregated value. In case of scale-up, besides a better quality of the final product, maximum capacity and minimum consumption are always required. Problems involving two and three objective functions were analyzed. A wider insight of the trade-off between the competing goals is gained by optimizing a large number of objective functions. The results demonstrated that improvements can be obtained even when the process targets are in conflict.
References A. G. Barreto Jr., 2005, PhD thesis, COPPE/UFRJ. F. Y. Cheng, D. Li, 1998, AIAA J., 36, 1105. T. Gu, 1997, Ed. Springer-Verlag. K. C. G. Moura et al., 2001, J. Braz. Chem. Soc., 12, 3, 325. V. Sedgwcik, 2002, Amersham Bioscience. C. M. Silva, E. C. Biscaia Jr, 2003, Comp. Chem.Eng, 27, 1329. R. C. Vieira, E.C. Biscaia Jr, 2000, Lat. Amer. Appl. Res., 30, 303.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Computer Aided Methodology for Simultaneous Synthesis, Design & Analysis of Chemical ProductsProcesses Loic d’Anterroches, Rafiqul Gani CAPAC, Department of Chemical Engineering, Technical University of Denmark, DK 2800 Lyngby, Denmark.
Abstract A new combined methodology for computer aided molecular design and process flowsheet design is presented. The methodology is based on the group contribution approach for prediction of molecular properties and design of molecules. Using the same principles, process groups have been developed together with their corresponding flowsheet property models. To represent the process flowsheets in the same way as molecules, a unique but simple notation system has been developed. The methodology has been converted into a prototype software, which has been tested with several case studies covering a wide range of problems. In this paper, only the computer aided flowsheet design related features are presented. Keywords: Molecular design, flowsheet design, product, process, group-contribution
1. Introduction Synthesis, design and analysis of chemical products involves the identification and analysis of molecules and their mixtures with desired (target) properties and analysis of performances of the products, as in drugs, pesticides, solvents, aroma and food products. In each case, molecules that are likely to match the target properties and performances are identified, usually in experiment-based trial and error solution approaches. In product centric process design, also, it is necessary to identify operations and their configuration as a flowsheet with desired (target) performances. Currently, even though design of the product and the process have similarities, they are solved with very different computer-aided methods and tools. The objective of this paper is to present a group contribution (GC) method, which provides the basis for a more efficient and flexible methodology for molecule, mixture as well as process design [1]. That is, in the same way functional groups are defined to represent molecules and to estimate their properties, process groups are also defined to represent process flowsheets and to estimate their flowsheet (operation) properties. That is, once a table of process groups representing a wide range of operations is established, the technique for Computer Aided Molecule-Mixture Design (CAMD) is also applicable for Computer Aided Flowsheet Design (CAFD), so that CAMD and CAFD can incorporate aspects of synthesis, design and model-based analysis for chemical products and processes. Also, since CAMD has the ability to generate and evaluate thousands of molecules within a few seconds of computer time, CAFD would be able to generate numerous process alternatives without any loss of accuracy or application range, also within a few seconds of computer time. Based on the developed GC-based methodology for CAMD and CAFD, a prototype software
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(ProCAFD) has been developed. Several interesting cases studies, highlighting the applicability, flexibility and scope/significance of the CAMD-CAFD methodology, from a wide spectrum of chemical products and processes have been developed. In this paper, one case study is presented. Detailed solutions of all the case studies can be obtained from the authors [2].
2. Description of the CAMD-CAFD Methodology The GC-based methodology for CAFD and CAMD has three main features: representation of molecules and flowsheets with the same chemical notation system; a combined CAMD-CAFD technique (reverse property prediction) to generate and evaluate product-process alternatives; and a reverse (simulation) approach for productprocess design. The CAMD-CAFD methodology is based on the similarities in pattern in the synthesis of molecules and process flowsheets (see Fig. 1).
Fig 1: Similarities in the pattern for product-process synthesis problems A SFILES notation system, based on the SMILES [3] notation system for molecules, has been developed. Here, molecules and flowsheets are represented through a notation system designed originally for modern chemical information processing. This means that the structure of any flowsheet can easily be transferred from one system (or software) to another, just as the molecular structures can be transferred from one system to another and visualized directly from its SFILES strings. The main idea behind the reverse approach is to decompose a synthesis/design problem into two reverse problems. In a reverse simulation problem, instead of determining the unknown process variables through the process model, the property variables are determined. Since the property variables are the unknown variables, the process model in the reverse simulation problem obviously does not need the embedded property models. The calculated property models and/or their functions are then used as design targets for the solution of reverse property problems. Here, given the property variables and/or their functions, the unknown process conditions or the unknown chemical structures and their mixtures are identified. In this step, any number of property models (even experimental data) may be used, as long as the design targets are matched. This significantly increases the search space for the feasible solutions.
3. Overview of the CAMD-CAFD Framework In this paper, only the CAFD-related issues of the combined CAMD-CAFD framework are highlighted. Detailed analysis of the CAMD-related issues can be found
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in Achenie et al. [4]. The CAFD-framework is based on the GC-concept and needs, as in CAMD-framework, flowsheet property models, process group building blocks and process group connectivity rules (including representation of the completed structure). As shown on Fig. 2, the CAFD-framework is composed of eight main steps: the definition of the problem (step-1); the analysis of the problem (step-2); the selection of the matching process group building blocks (step-3); the synthesis and test of the alternatives (step-4); the ranking of the alternatives and selection of the most promising alternatives (step-5); the design of the selected alternatives (step-6); the post analysis of the designed alternatives (step-7) and the final verifications (step-8). The developed prototype software, ProCAFD, also follows these steps. The corresponding work flow and data flow are highlighted on Fig. 2. At each step, the needed information is given by user or generated using a collection of methods and tools, some of which have been specially developed for the combined CAMD-CAFD framework. For example, flowsheet representation and search based on SFILES notation (see section 3.2).
Figure 2: The data-flow and work-flow for the CAFD-Framework
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3.1. Process Groups and Flowsheet Property Model Currently 12 types of process groups (PG) are available – see Table 1. These PGs represent simple distillation column, solvent based azeotropic distillation, flash separation, kinetic-model based reactor, fixed conversion reactor, pressure swing distillation, polar molecular sieve based separation, molecular sieve based separation, liquid membrane based separation, gas membrane based separation, crystallization and adsorption. Each PG is represented by their structure, their input-output concentrations (limiting values) as well as a characteristic parameter (for example, the driving force for separation as well as reaction). Note that the PGs are component independent. Therefore, for any specific application, once the compounds in the system are identified and their corresponding driving forces calculated, the appropriate PGs can be selected. Also, choice of the PGs to represent the flowsheet automatically satisfies the mass balance requirements. That is, for a specified output, the driving force needed to perform the operation can be calculated and used to select the PG. Table 1: List of currently available process-groups (PG) Operation Examples of Process-Groups Distillation column (A/BC), (ABC/DE) Solvent based azeotropic distillation (cycA/B) Flash separation (fABC/BCD) Kinetic-model based reactor (rABC/nE/pABCD) Fixed conversion reactor (rABC/nE/pABCD) Pressure swing distillation (pswA/B) Polar molecular sieve based separation (pmsABC/D) Molecular sieve based separation (msABC/D) Liquid membrane based separation (lmemABC/D) Gas membrane based separation (gmemABC/D) Crystallization (crsABC/D) Adsorption (abEAB/eF/EABF/EF) Currently, parameter tables are available for the prediction of the energy index (Ex) for a process flowsheet as a function of the PG-contributions, the flowsheet structural parameters and the PG characteristic parameter [5]. Ex gives a measure of the energy consumed by the various unit operations present in the flowsheet. The model has been developed (as in the development of GC-based property models for molecules) by fitting collected data for a large number of process flowsheets to a simple mathematical function. The model is predictive because the same groups can be used to predict the properties of other structures not used in the model development (as in molecular property prediction). Note also, that once the PGs and their model parameters are available, numerous alternatives can be generated and their feasibility verified with very few additional calculations. 3.2. SFILES Notation System for Process Flowsheets The SMILES notation describes a molecular structure in the form of a string of characters. Following the same principle, applying the SFILES notation to the flowsheet structure of the well-known HDA-process [6], the following string of characters is obtained: (iA)(rAB/pABCD)<1<2[<(iB)](gmemABC/D)[(oD)](A/BC)1(cycB/C)2(oC)
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The SFILES string is read from left to right. The process groups are delimited by parenthesis, for example, the membrane process group (gmemABC/D). Two consecutive process groups represent a connection from the first process group to the second process group. This is, the case at the start of the SFILES notation with (iA)(rAB/pABCD) representing an inlet process group connected to a reactor process group. Branches are represented using square brackets and recycles using numbers like in the SMILES notation. The difference is that a connection between two process groups is oriented. The orientation, when not following the left to right direction, is clarified with the ”smaller than” character. This is shown with the fragment [<(iB)] where the process-inlet process group (iB) is not connected to an outlet of the reactor process group (rAB/pABCD) but to an inlet of this reactor process group. The recycles are indicated with numbers, one for each recycle. The number indicates the two process groups closing the recycle. For example, the number 1 indicates that one outlet of the distillation process group (A/BC)1 is connected to the inlet of the reactor process group (rAB/pABCD)<1. Detailed description of the rules can be obtained from the authors.
4. Case Study The developed CAMD-CAFD framework together with its corresponding software have been tested against a number of illustrative problems of varying degrees of complexity. For example, optimal sequencing for multicomponent separation by distillation, esterification reaction-separation processes, the HDA-process, use of SFILES to represent network of reaction systems and reverse simulation/design of single-unit separations. In this paper, selected results from the HDA-process are presented as it also serves to validate the results. In principle, new flowsheets can be generated and improved (retrofit) alternatives of existing flowsheets can be generated. The HDA-process involves a catalytic reactor where two competing reactions take place, followed by separation of reactants and products. Additional details of the HDA-process can be found elsewhere [6]. Steps 1-2 involve problem defintion and analysis. The objective here is to generate feasible process flowsheets for the production of benzene by hydro-dealkylation of toluene. Biphenyl is produced as a by-product. Methane enters the system as impurity. This means that unreacted hydrogen and toluene needs to be recycled, methane needs to be purged and benzene needs to be recovered. Based on this information plus other analysis details not discussed here, selection of the PGs and initialization (in terms of their corresponding driving forces) is made (step 3). For separation, in addition to distillation, separation by membrane-based processes (gasseparation as well as liquid-separation), PT-flash and molecular sieve are considered. Feasibility of each of these separation techniques is determined by their corresponding property ratios (for example, for gas separation by membrane, the ratio of the following properties for two key components need to be > 1.07: Van der Waals volume and critical temperature). Based on these information, the PGs are selected (they are listed in Table 2 (A=Hydrogen, B=Methane, C=Benzene, D=Toluene, E=Biphenyl). Step 4 involves generation of feasible flowsheet alternatives and using a similar structure generation algorithm as CAMD, 31 feasible flowsheet alternatives (satisfies all product specifications) are generated, tested (for their individual structural properties as well as the total flowsheet property), and stored as SFILES. The SFILES string given above (in section 3.2) is one of the alternative feasible flowsheets. Examples of other structures rated with respect to their calculated energy index values are listed in Table 3 (step 5). Based on the above, a flowsheet is selected and since for the selected flowsheet, all mass input-output are known together with the corresponding driving forces needed to
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achieve the operation, reverse simulation/design (step 6) is performed to obtain values of all remaining design variables to totally describe the process flowsheet (so that rigorous simulation/optimzation of the process can be performed. Steps 7-8 involve post analysis and final verification. In this process, the results obtained from step 6 have been found to match the results from the rigorous simulation, where, the results from step 6 was used as an initial estimate and the rigorous simulator (PRO-II steady state simulator) converged very rapidly. Table 2: List of selected process-groups for the HDA-process flowsheeet PG-type Process Groups Distillation (ABC/D) ; (ABC/DE); (ABCD/E); (ABD/E); (AC/D); (AC/DE); (ACD/E); (AD/E); (BC/D); (BC/DE); (BCD/E); (BD/E); (C/D); (C/DE); (CD/E); (D/E); (AB/CDE); (AB/CD); (AB/C) Membrane-based (gmemDC/BA); (gmemD/C); (gmemD/CBA); (gmemC/BA); (lmemDC/BA); (lmemD/C); (lmemD/CBA) Molecular-sieve (msC/BA); (msD/CBA); (msDC/BA) Reactor (rAD/pABCDE) Feed (fAB/ABCDE) Table 3: List of feasible process flowsheets (only 4 out of 31 shown here) Process Flowsheet Alternatives Represented by SFILES Notation (iAD)(backbone)1<2<1(ABCD/E)(msD/CBA)2(msC/BA)[(oC)](oAB)](oE) (iAD)(backbone)1<2<1(ABCD/E)(msDC/BA)[(C/D)2(oC)](oAB)](oE) (iAD)(backbone)1<2<1(ABC/DE) (AB/C)[(oAB)](oC)](D/E)2(oE) (iAD)(backbone)1<2<1(AB/CDE) (oAB)](CD/E)[(C/D)2(oC)](oE)
E x 0.0317 0.0987 0.2454 0.4787
5. Conclusions A group contribution based framework for CAFD has been developed and tested with a number of illustrative case studies, one of which has been highlighted in this paper. The CAFD-framework provides insights and capabilities to the design engineer through very quick generation, evaluation and interpretation of all feasible alternatives allowed by different combinations of the available PGs. The calculations are simple and easy to apply. Even though a very large number of problems can already be solved with the current set of PGs, much work is needed to extend the application range by creating a bigger set of PGs and process flowsheet property models. The results, however, indicate that the method has the potential to be trully predictive, provides efficient representations of flowsheet structures and provides an overview of the feasible flowsheet alternatives, together with fast and reliable design calculations.
References [1] L. d’Anterroches, R. Gani, Computer Aided Chemical Engineering, 20 (2005), 643-648. [2] L. d’Anterroches, Process Flowsheet Generation, & Design through a Group Contribution Approach, PhD-Thesis, Technical University of Denmark, Lyngby, Denmark, 2005. [3] D. Weininger, J Chemical Information & Computer Science, 28 (1989), 97-101. [4] L. E. K. Achenie, R. Gani, V. Venkatasubramanian, Computer Aided Molecular Design: Theory & Practice, CACE-12, Elsevier Science b.v., The Netherlands, 2002. [5] ] L. d’Anterroches, R. Gani, Fluid Phase Equilibria, 228-229 (2005), 141-146.
[6] J. M. Douglas, Conceptual Design of Chemical Processes, McGraw-Hill, New York, 1988.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Correlation and Prediction of Drug Molecule Solubility with the NRTL-SAC Model Chau-Chyun Chena and Peter A. Craftsb a
Aspen Technology, Inc., Ten Canal Park, Cambridge, Massachusetts 02141, U.S.A. AstraZeneca Pharmaceuticals Ltd., Process R&D, Macclesfield, Cheshire, SK10 2NA,U.K. b
Abstract The recently proposed Nonrandom Two-Liquid Segment Activity Coefficient model (NRTL-SAC) of Chen and Song (2004) provides a simple and thermodynamically consistent framework to correlate and predict drug solubility in pure solvents and mixed solvents, based on a small initial set of measured solubility data. Used within a process simulator, or through an Excel spreadsheet, the model forms the scientific foundation of an effective solubility modeling tool in support of early stage crystallization process development. The methodology is also applicable to other unit operations where phase equilibrium calculations factor prominently in process design and development. Keywords: Solubility, crystallization, activity coefficient, Nonrandom Two-Liquid theory.
1. Introduction Crystallization is the preferred method of purification in the pharmaceutical industry for both the final drug substance and the isolated intermediates in the synthesis. During the early stages of process development, the quantity of raw material available in support of the laboratory design effort is usually limited, due to the demands of clinical trials, formulation development and the significant cost of manufacture. This problem is compounded by the high rate of drug attrition and the large number of new drug candidates that are concurrently in development and competing for resource. It is common for these factors to constrain the experimental program of solvent selection, which may lead to a sub-optimal design in respect of yield, productivity and manufacturability. Whilst high throughput solubility measurement techniques are improving, they are still time and labor intensive. Where crystallization requires a mixed solvent system it is practically impossible to cover the full range of solvent combinations with sufficient detail to find the optimal solution, even with high throughput techniques. To overcome these obstacles it is highly desirable to have a thermodynamically consistent model to correlate and predict drug solubility in pure and mixed solvent systems based on a small initial data set of measured solubilities. The number of literature references in this area is currently small, particularly with respect to mixed solvent systems where the solubility behavior can be highly non-ideal. Recently Chen and Song (2004) reviewed prior solubility modeling works and proposed the semiempirical NRTL-SAC model as a thermodynamic framework for the correlation and prediction of drug molecule solubility in pure solvent and mixed solvent systems. They presented satisfactory results with NRTL-SAC in correlating drug solubilities in a few representative pure solvents and in subsequent prediction of drug solubilities in other
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pure solvents. Chen and Crafts (2006) further demonstrated robust predictions from NRTL-SAC for drug solubilities in mixed solvent systems. Based on up to four molecule-specific parameters, the model captures the qualitative trends of drug solubilities in common solvents and solvent mixtures. In this short communication we present the semi-empirical NRTL-SAC model and its various industrial applications including support of early stage crystallization process development.
2. Thermodynamic Framework The solubility of a solid organic nonelectrolyte can be described by the expressions:
ln K sp = ln x ISAT γ ISAT = A=
Δ fus S
=
Δ fus S ⎛ Tm ⎞ B ⎜1 − ⎟ = A+ R ⎝ T ⎠ T
(1)
Δ fus H
(2)
R RTm T Δ S Δ H B = − m fus = − fus (3) R R SAT Where K sp is the solubility product constant, xI is the mole fraction of the solute I dissolved in the solvent liquid at saturation,
γ ISAT is the activity coefficient of the solute
I in the solution at saturation, Δ fus S is the entropy of fusion of the solute, Tm is the melting point of the solute, R is the gas constant, T is the temperature, and Δ fus H is the enthalpy of fusion of the solute. Given a polymorph, Δ fus H and Tm are fixed. At a fixed temperature, the solubility is only a function of the activity coefficient of the solute in solution. Clearly, the activity coefficient of the solute in solution plays the key role in determining the solute solubility as the solvent composition changes.
3. NRTL Segment Activity Coefficient Model The semi-empirical NRTL-SAC model computes the activity coefficient for component I from the combinatorial term γ I and the residual term γ I : C
R
ln γ I = ln γ IC + ln γ IR Here
γ
C I
(4)
is calculated with the Flory-Huggins equation for the combinatorial entropy of
mixing and
γ IR
is calculated with the local composition (lc) interaction contribution
γ of the polymer NRTL model (Chen, 1993). The polymer NRTL equation incorporates the segment interaction concept and it computes the activity coefficient for component I in solution by summing up contributions to the activity coefficient from all segments that make up component I. The equation is given as follows: lc I
[
ln γ IR = ln γ Ilc = ∑ rm , I ln Γmlc − ln Γmlc , I m
with
]
(5)
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⎛ ∑j x j G jm 'τ jm ' ⎞⎟ x m ' G mm ' ⎜ ln Γ = +∑ ⎟ ⎜τ mm ' − x k G km ' ⎟ m ' ∑ x k G km ' ⎜ ∑ ∑k xk Gkm k k ⎠ ⎝ ⎛ ∑j x j ,I G jm 'τ jm ' ⎞⎟ ∑j x j ,I G jmτ jm x m ', I G mm ' ⎜ lc , I ln Γm = +∑ ⎟ ⎜τ mm ' − x k , I Gkm ' ⎟ m ' ∑ x k , I G km ' ⎜ ∑k xk ,I Gkm ∑ k k ⎠ ⎝ ∑ xI r j,I xj = I ∑∑ x I ri ,I
∑x G j
I
x j,I =
τ jm
jm
j
lc m
(6)
(7)
(8)
i
r j,I
∑r
(9)
i,I
i
where I is the component index, i, j , k , m, m' are the segment species index, x I is the mole fraction of component I,
x j is the segment-based mole fraction of segment
species j, rm , I is the number of segment species m contained only in component I, Γm
lc
is the activity coefficient of segment species m, and Γmlc , I is the activity coefficient of
segment species m contained only in component I. G and τ in Eqs. 6 and 7 are local binary quantities related to each other by the NRTL non-random factor parameter α : G = exp( −ατ ) (10) Four pre-defined conceptual segments were proposed by Chen and Song (2004) to account for interacting molecular surface of all types: one hydrophobic (x), one polar attractive (y-), one polar repulsive (y+), and one hydrophilic (z). Chen and Song further suggested values for the various binary segment-segment interaction parameters, i.e., τ and α in Eq. 10. The molecular-specific model parameters for all interacting solvents and solutes, i.e., hydrophobicity X, polarity types Y- and Y+, and hydrophilicity Z, correspond to rm , I (m=x, y-, y+, z) in Eq. 5. The NRTL-SAC model has been further extended for the computation of ionic activity coefficients and solubilities of organic salts in common pure solvents and solvent mixtures (Chen and Song, 2005). In addition to the four molecular parameters, an electrolyte parameter is introduced to characterize both local and long-range ion-ion and ion-molecule interactions attributable to the ionized segments of organic electrolytes. In practice, the NRTL-SAC molecular parameters are first identified for common pure solvents from available experimental binary vapor-liquid or liquid-liquid phase equilibrium data, at room temperature or near room temperature. In this process, hexane (with x =1) and water (with z =1) are treated as the reference hydrophobic solvent and hydrophilic solvent, respectively. A databank of molecular segment parameters for the common pure solvents is thus established.
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Experimental solubility data for the drug solute is measured in four to eight solvents with distinctive surface interaction characteristics, at or near room temperature. These solvents should include hydrophobic solvents such as hexane or heptane, hydrophilic solvents such as water and methanol, and polar solvents such as acetone, acetonitrile, DMSO, DMF, etc. From these drug solubility data, we identify drug molecular parameters including the solubility product constant K sp . If drug solubility data are available at multiple temperatures, then the temperature dependency of the solubility product constant K sp can be determined, together with the corresponding values for
Δ fus H and Tm from Eq. 2 and Eq. 3. Alternatively, Δ fus H and Tm can be measured experimentally by differential scanning calorimetry. Given the molecular parameters for solvents and solutes, we perform solid-liquid equilibrium calculations at a given temperature and pressure to identify solute solubilities in any mixture of solutes and solvents. The calculations are performed in a process simulator environment that can be linked to an Excel spreadsheet for increased flexibility and ease of use. The simplicity of NRTL-SAC and its ease of use within a process simulator, linked to an Excel spreadsheet, are contributing to the accelerated acceptance of the model in the pharmaceutical industry. Publications on recent industrial applications of the model have emerged. For example, Tung et al. (2005) reported use of NRTL-SAC and COSMO-SAC models to estimate the solubility of two Statin molecules and two Cox molecules. NRTL-SAC demonstrated superior performance to the ab initio COSMObased model. Crafts (2005) reported the successful implementation and benefits of NRTL-SAC within AstraZeneca.
4. Industrial Applications The use of activity coefficient models in the pharmaceutical industry has been limited to solvent recovery or emission studies due to lack of applicability of existing models to complex pharmaceutical compounds. The UNIFAC model is recognized as perhaps the most successful predictive method available to-date for the chemical industry. However in the pharmaceutical sector UNIFAC suffers from missing functional groups, parameter sets based on data that do not reflect the complexity of pharmaceutical compounds, and the deterioration of the functional group additivity rule for such complex molecules (Gracin et al, 2002). NRTL-SAC offers a promising new activity coefficient model for complex pharmaceutical compounds. By taking solubility measurements in just four to eight representative solvents it is possible to characterize the segment contributions of the drug molecule and then to qualitatively predict the drug solubility in any precharacterized pure solvent or solvent mixture. This method quickly identifies both good solvent and antisolvent candidates for crystallization process design and provides a first estimate of yield and productivity. The average accuracy of NRTL-SAC has been shown to be ~50% which is considered adequate for the task of solvent selection. Applying NRTL-SAC to the data generated by high throughput solubility measurement techniques can help identify experimental errors, for cases where chemical reaction or decomposition are apparent, or where solid state change, due to structural
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polymorphism, or solvate formation may have occurred. Once identified as outliers these data points can be scrutinized in more detail. NRTL-SAC also brings a new perspective to the design of optimal solvent systems. Instead of finding a solvent molecule or solvent mixture with “optimal” functional groups, the optimization would be performed to find the optimal surface interaction characteristics in terms of conceptual segment makeup. The appearance of a solubility peak in mixed solvent systems is an intriguing and common phenomenon that is of high interest to the industry. The solubility peak represents a minimum in the activity coefficient of the solute. The NRTL-SAC method suggests that the solute activity coefficient would approach that of the ideal solution, i.e., unity, should the conceptual segment makeup of the solvent mixture approach that of the solute. It is also possible that the solute activity coefficient would become less than unity if the solvent system exhibits strong attractive interactions with the solute. Predicting exactly which polymorph of a drug molecule will crystallize from a given solvent system is an unsolved problem that is of great interest to the pharmaceutical industry. Predicting solubility is a bounded problem with respect to polymorphism. The most thermodynamically stable polymorphic form of a solute defines the low solubility limit while the upper solubility limit is defined by the amorphous solid state. NRTL-SAC makes it possible to estimate the amorphous phase solubility by calculating the phase equilibrium between a solute rich pseudo-liquid phase and a solvent rich liquid phase. In pharmaceutical process design the number of available polymorphs and their relative stability is usually determined through milligram scale, high throughput screening experiments, in support of the formulation process design. Such information is available to the manufacturing process design team before they make a final selection of the crystallization solvent. Assuming that the most stable polymorph has been found in the screening experiments then calorimetry measured values of Δ fus H and Tm can be used with NRTL-SAC to predict the available solubility window across a range of single and mixed solvent systems. This window is important because it defines the available yield and productivity of the crystallization process. A phase diagram with the solubility of each known polymorph can be generated, by using the respective values of Δ fus H and Tm for each of the known polymorphs. This solubility map helps to identify the best crystallization control strategy to obtain a single desired polymorphic form. In addition to solubility modeling, NRTL-SAC can be used with phase equilibrium calculations of vapor-liquid, liquid-liquid, and vapor-liquid-liquid equilibria. In this way NRTL-SAC parameters developed from solubility data for crystallization design can be used in other industrial applications. For example, dissolved solutes can have a significant and often negative effect on the rate of distillation when trying to remove residual water from an organic solvent, such as toluene, prior to a hydrophobic reaction step. A second example is a liquid-liquid extraction step in a batch reactor, in which an aqueous phase is added to an immiscible organic phase to remove reaction by-products such as inorganic salts and unreacted organic coupling reagents Cleaning between manufacturing campaigns in multi-product plant is a time consuming activity, particularly when changing from high potency to high dose products. The
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cleaning activity can often take several months to complete. The identification of optimal solvent mixtures for cleaning is another area that NRTL-SAC could support. Mixed solvent reflux is often used to remove material from overhead condensers and their associated pipe-work. NRTL-SAC is well suited for predicting solubility in this operation. Product drying is often the bottleneck step in a production process and is notoriously difficult to scale up. NRTL-SAC allows the effect of solute composition on vapor pressure and drying rate to be predicted. Adsorption and capillary effects are still very important towards the end of drying and will require experimental characterization. Chromatography is frequently used in the early stages of process development to recover drug substance that has been contaminated with an unacceptable level of impurities during scale-up. The dissolved solutes must be kept in solution throughout the chromatography operation to prevent blockages. It is common for binary and ternary solvent mixtures to be used in chromatography and the use of NRTL-SAC to predict solubility behavior should make the solvent selection task much quicker. Further applications of NRTL-SAC can be found in the prediction of octanol-water partition coefficients (Log P) to characterize drug bioavailability, solvent selection for formulations, and in drug discovery and pharmacokinetics, where partitioning between the gut, blood and body tissues is of critical importance in predicting dose levels and pharmocological effects.
5. Conclusions Correlation and prediction of drug molecule solubilities play a critical role in the development of pharmaceutical processes, especially in the area of crystallization. The recently developed NRTL-SAC model offers a practical thermodynamic framework for solubility modeling of complex pharmaceutical molecules. Used with process simulators and Excel spreadsheets, the model finds growing acceptance in the industry as an effective engineering tool for crystallization process development. The model may find applications in many other important pharmaceutical operations.
References 1. C.-C. Chen, Y. Song, 2004, “Solubility Modeling with a Non-Random Two-Liquid Segment Activity Coefficient Model,” Ind. Eng. Chem. Res., 43, 8354. 2. C.-C. Chen, P.A. Crafts, 2006, “Correlation and Prediction of Drug Molecule Solubility in Mixed Solvent Systems with the NRTL-SAC Model,” paper submitted for publcation in Ind. Eng. Chem. Res. 3. C.-C. Chen, 1993, “A Segment-Based Local Composition Model for the Gibbs Energy of Polymer Solutions,” Fluid Phase Equilibria, 83, 301. 4. C.-C. Chen, Y. Song, 2005, “Extension of Nonrandom Two-Liquid Segment Activity Coefficient Model for Electrolytes,” Ind. Eng. Chem. Res., 44, 8909. 5. H.-H. Tung, J. Tabora, N. Variankaval, D. Bakken, C.-C. Chen, 2005, “Prediction of Pharmaceuticals Solubility via NRTL-SAC and COSMO,” paper presented at the 16th International Symposium of Industrial Crystallization, Dresden, Germany. 6. P.A. Crafts, 2005, “Solubility Modeling in AstraZeneca with Aspen’s NRTL-SAC Method,” paper presented at Aspen User Group Meeting, Amsterdam, Netherlands. 7. S. Gracin, T. Brinck, A.C. Rasmuson, 2002, “Prediction of Solubility of Solid Organic Compounds in Solvents by UNIFAC,” Ind. Eng. Chem. Res., 41, 5114.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Dynamic Modelling of Complex Batch Distillation Starting from Ambient Conditions Sven Gruetzmanna, Thomas Kapalab, Georg Fiega a
Hamburg University of Technology, Institute of Process and Plant Engineering, Schwarzenbergstr. 95 IV C, 21073 Hamburg, Germany, e-mail: [email protected] b Cognis Deutschland GmbH & Co.KG, 40551 Düsseldorf, Germany
Abstract Batch distillation is an important unit operation for the small-scale production of pharmaceuticals, specialty and fine chemicals and bio chemicals. Due to the increasing demands of the industry, alternative complex batch distillation processes, such as the Middle Vessel Batch Distillation (MVBD), has been proposed. Modelling of the discrete-continuous process is a very challenging task since the process is inherently dynamic and highly non-linear. For the first time, we present a rigorous dynamic model for complex batch distillation columns starting from ambient conditions. It includes the heating of the distillation column and peripheral equipment as well as the formation and propagation of the hydraulic and thermodynamic profiles. These considerations are necessary since the optimal switching of important manipulated variables (e.g. reflux or reboiler duty) is expected to take place before the column reaches its hydraulic steady state. The physical validity of the developed dynamic model is clearly shown by means of comparison between simulation results and industrial experiments. Keywords: batch distillation startup, ambient conditions, dynamic modelling
1. Introduction Batch distillation is a unit operation, which is of increasing interest in industry over the past decade. The reason is a significant change in the market from a quantity-oriented to a quality-oriented demand. Chemical manufacturers also witness a quickly changing demand for high value-added products. This has led to alternative complex batch distillation configurations and operating policies proposed by the scientific community. Because of the frequent startup of a complex batch distillation column, process control during this phase shows a high optimisation potential. However, a detailed analysis of the startup phase has been considered only in few cases (e.g. Wang et al. 2003, Elgue et al. 2004). To the best of our knowledge no publication deals with the analysis and optimisation of a complex batch distillation column taking the startup phase into account. Therefore, the objective of this contribution is to provide a mathematical model that is capable of handling the physical phenomena occurring during the startup of a complex batch distillation column. Since fine and specialty chemicals are often separated in vacuum distillation columns with structured packings as internals, the model must be able to take these constraints into consideration. 1.1. Complex Batch Distillation Columns Batch distillation is favourable in the separation of multicomponent mixtures if the amount of feed is small and high purity products are demanded. Recent investigations focus on complex batch distillation columns, namely, the middle vessel column (Barolo and Botteon 1997, Fieg et al. 2004) and multivessel
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column (Furlonge et al. 1999, Low et al. 2005, Skogestad et al. 1997). The main difference between the conventional and complex batch distillation column is the installation of additional product vessels along the column. Then, the process unit consists of a reboiler, n product vessels, n-1 column sections and a condenser. Fig. 1 illustrates the principle structure of a MVBD.
Fig. 1: Middle Vessel Batch Distillation, principle process behaviour and configuration.
1.2. Alternative Operating Policies An important degree of freedom for the operation of batch distillation is the reflux flow which is a function of time. Słrensen et al. 1997 have presented different concepts to properly adjust the reflux, namely, total reflux, constant reflux, variable reflux and optimal reflux. The cyclic operation with total reflux differs from the others in many ways. During the transient startup phase, the individual product fractions are accumulated in the product vessels, that is, the still pot, the reflux drum and optional side vessels. Then, the column is operated with total reflux until the hydraulic and thermodynamic profiles remain constant. Special process control strategies are used to control the product qualities as desired. If the products meet the quality criteria, the product vessels are emptied. Otherwise, set points have to be changed in order to achieve the desired product compositions (see Fig. 1). The practical application of the cyclic MVBD to industrial relevant mixtures has been proven by Fieg et al. 2004. From an industrial point of view the main advantages are the operation with the highest separation efficiency and simple process control. However, the optimal process control is the result of an extensive process decomposition and optimisation procedure and subject to further investigations. With the development of a rigorous dynamic model that is presented in this contribution, we provide a basis for these challenging studies. 1.3. Startup of Complex Batch Distillation The startup of distillation columns has been discussed in many publications in the open literature. A widely accepted model for the startup of continuous distillation columns has been proposed by Ruiz et al.1988. There, the startup is subdivided into a discontinuous, semi-continuous and continuous period. Indeed, there is also a continuous period in the process of cyclic batch distillation when all state variables remain constant (see Fig. 1). However, experimental studies by Gruetzmann et al. 2005
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recently indicate that the deliberate switching of discrete decision variables (e.g. reflux) can lead to a shorter startup period. Thus, in terms of optimisation a focus on the first and second period, where hydraulic and thermodynamic profiles are formulated, is reasonable. Large changes in the state variables indicate these periods. Fig. 2 illustrates the general process characteristics of a tray temperature during startup.
Fig. 2: Process characteristics of the temperature on tray j (numbering from the top to the bottom).
2. Rigorous Dynamic Modelling Due to the discrete-continuous character of the process, the rigorous dynamic modeling of batch distillation startup is a very challenging task. The high stiffness caused by large changes in the continuous state variables and the discrete state transitions, for example, when vapour enters an empty tray for the first time, result in numerical problems. A well structured model and implicit solvers are necessary to overcome these problems. Since the model that we present describes a column with packing internals, the HETP value has been used to set up a tray-to-tray equilibrium model in the modeling and simulation software Aspen Custom Modeler. Then, for each process unit a proper differential-algebraic equation system of form dx dt
= g ( x, z )
0 = f (x, z )
(1) (2)
has been developed, where x and z denote the time-variant process variables, t the process time and g, f non-linear algebraic functions. The overall complex batch distillation model includes the reboiler, packing sections, a total condenser, the reflux drum, a middle vessel and optional liquid accumulators and distributors. Control models are provided by the software package and can be easily added to the model. The equilibrium stage model is based on the common assumptions with some exceptions. In order to describe the startup behaviour accurately the following features have been included in the model: variable relative volatilities, optional non-ideal vapour-liquid equilibrium, variable molar vapour flow, variable molar liquid holdup on the stages,
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non-adiabatic operation and variable pressure drop. Thus, the well-known MESH equations are extended by the heat accumulation of the process equipment and complemented by algebraic relationships calculating the dynamic tray hydraulics, the maximum molar liquid holdup (as a function of the tray hydraulics), the pressure drop (as a function of the tray hydraulics) and possible heat losses. The necessary physical and thermodynamic properties of the components as well as vapour-liquid equilibrium data are provided by internal procedures from Aspen Properties.
3. Experimental Setup The performance of the developed model is validated by comparison of experiment and simulation. The experimental investigations were carried out in a laboratory column made of glass. The distillation column has a diameter of 70mm and a total height of 4m. It consists of a reboiler, a total condenser and one middle vessel. Both column sections, the one above and the one below the middle vessel contain 1m Sulzer BX packing that is, according to specifications given by the manufacturer and experience gathered by Cognis Deutschland/ Düsseldorf, equivalent to approximately ten theoretical stages. Relevant process variables of the vacuum distillation process have been measured and recorded by a data acquisition system. Control of relevant process variables has been performed by PI-controllers. The product purities have been monitored by taking samples and performing gas chromatography analysis.
4. Simulation Results The objective of this chapter is to verify that the dynamic model represents the entire process, beginning from a column at ambient conditions. First, a consistency check shows that the physical phenomena on a tray are correctly represented. Then, many simulations have been compared to experimental data provided by Cognis Deutschland/ Düsseldorf. One example is presented in this contribution. 4.1. Physical Model Validity Fig. 3 shows a Petri-net representation of the process behaviour on one chosen tray during startup and Fig. 4 the simulation results. At the initial state (stage 0) the column is at ambient conditions. Thus, tray j is empty and no stream goes in or out. During stage 1 a vapour stream Vj+1 enters the empty tray. With the vapour condensing at the column wall and internals, the tray is filled up to its maximum liquid holdup HUmax. If the liquid holdup HUj exceeds the maximum holdup, a liquid stream Lj flows down the column during stage 2. The rising vapour stream still heats the tray up until the liquid is at boiling temperature. In this stage, a vapour-liquid equilibrium is calculated. If the vapour pressure pVL,j is larger than the pressure on the tray pj, a vapour outlet stream Vj is calculated. This indicates the beginning of stage 3. Stage 4 indicates that liquid from the upper tray flows downwards (Lj-1). The simulation results clearly show the physical consistency of the model. pj+1>pj
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Fig. 3: Petri-net representation of the tray hydraulics.
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4.2. Comparison of simulation and experiment One out of many experiments has been chosen to illustrate the model performance. In the experiment a ternary mixture of n-hexanol, n-octanol and n-decanol has been separated in a middle vessel batch distillation column with cyclic operation. The feed (1.65kg) has been charged to the reboiler and the distillation has been performed under vacuum (20mbar). Fig. 5 shows the temperature of different trays as function of time. 140
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Tray 1 and the reboiler represent the top and the bottom of the column, respectively. The temperature measurements that have been used for process control are located on tray 4 and 10. Regarding Fig. 2 one can easily find the different phases during startup again. Note, that the simulation results agree very well with the experiments despite the high level of discontinuities and non-linearity. The deviations can be explained by missing column data (masses of the column sections and equipment) and control parameters. Another important uncertainty is the choice of the inverse HETP value that has been set to 5 for the upper and 6 for the lower column section. This is the reason for the slight deviation of the temperature on tray 4. However, the results prove the validity of the dynamic model.
5. Conclusion In this contribution a rigorous dynamic model, that describes the startup of complex batch distillation columns, has been presented for the first time. At the initial state, the column and the liquid feed are at ambient conditions, the trays are empty. A consistency check has proven the physical validity of the model. Comparison between simulation and experimental results shows that the model can describe the entire process very well. Since an early switching of important variables can improve this innovative process, the model can now be used for dynamic optimisation studies.
6. Acknowledgements We gratefully acknowledge the Forschungsstiftung (MBFSt 2553).
financial
support
from
the
Max-Buchner-
References M. Barolo, F. Botteon, 1997, A Simple Method of Obtaining Pure Products by Batch Distillation, AIChE J., 43, 10, 2601-2604 S. Elgue, L. Prat, M. Cabassud, J.M. Le Lann, J. C´ezerac, 2004, Dynamic Models for Start-up Operations of Batch Distillation Columns with Experimental Validation, Comp. Chem. Eng., 28, 2735-2747 H.I. Furlonge, C.C. Pantelides, E. Sørensen, 1999, Optimal Operation of Multivessel Batch Distillation Columns, AIChE J., 45, 4, 781-801 G. Fieg, Th. Kapala, S. Gruetzmann, 2004, Multivessel Batch Distillation: Experimental Investigations with Respect to Industrial Applications, Proceedings 16th CHISA, electron. resource S. Gruetzmann, M. Temprano, G. Fieg, Th. Kapala, 2005, Führungskonzepte zur Minimierung der Prozessdauer der zyklischen Batch-Rektifikation, Chem. Ing. Techn., 77, 8, 1101-1102 S. Gruetzmann, G. Fieg, Th. Kapala, 2006, Theoretical Analysis and Operating Behaviour of a Middle Vessel Batch Distillation with Cyclic Operation, Chem. Eng. Process., 45, 1, 46-54 K.H. Low, E. Sørensen, 2005, Simultaneous Optimal Configuration, Design and Operation of Batch Distillation, AIChE J., 581, 6, 1700-1713 C.A. Ruiz, I.T. Cameron, R. Gani, 1988, A Generalized Dynamic Model for Distillation Columns - III. Study of Startup Operations, Comp. Chem. Eng., 12, 1-14 S. Skogestad, B. Wittgens, R. Litto, E. Sørensen, 1997, Multivessel Batch Distillation, AIChE J., 43, 4, 971-978 E. Sørensen, 1997, Alternative Ways of Operating a Batch Distillation, Proceedings Distillation and Absorption '97, 643-652 L. Wang, P. Li, G. Wozny, S. Wang, 2003, A Startup Model for Simulation of Batch Distillation Starting from a Cold State, Comp. Chem. Eng., 1485-1497
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Genetic algorithms approach for retrofitting heat exchanger network with standard heat exchangers R. Bochenek, J. M. Jeżowski* Rzeszów University of Technology, Department of Chemical Engineering and Process Control, Al. Powstańców Warszawy 6, 35-959 Rzeszów, Poland
Abstract The work addresses a systematic simultaneous approach to designing heat exchanger network (HEN). The method is tailored for retrofitting HEN consisting of standard heat exchangers, i.e. apparatus with discrete parameters. Note that it can be directly used for non-standard heat exchangers as well. Genetic algorithms were applied as optimization technique. Novel superstructure and structure representations were developed to enhance the optimization. The examples of application proved that the approach allows reaching good local optima solutions. Keywords: heat exchanger network, retrofit, simultaneous approach, genetic algorithms, standard heat exchangers.
1. Introduction Heat integration in a process system requires a centralized HEN. The HEN consists of heat exchangers, mixers and splitters. HEN design problem has been addressed in numerous papers (see the latest review by Furman and Sahinidis, 2002). Many approaches have been developed ranging from insight based with well-known Pinch Technology to simultaneous methods. Most of them are aimed at HEN synthesis. HEN retrofit problem, of even greater industrial significance, has some specific features that are difficult to handle efficiently by synthesis approaches. Here we address HEN retrofit problem. Additionally, the approach is aimed at network consisting of standard heat exchangers.
2. Problem formulation and analysis The general formulation of HEN retrofit problem is as follows. Given a HEN of fixed topology in regards to heat exchangers, mixers and splitters. The parameters of apparatus, initial and final parameters of process streams and utilities are known. The objective is to re-design the HEN at minimum total annual cost. HEN retrofit is performed by a sequence of structural and parameter changes on existing networks as follows: 1. Structural changes: heat exchanger relocations i.e. changing both heat exchanging streams, one of them, shifting an apparatus to another place without changing streams, splitter adding or deleting, inserting a new heat exchanger. 2. Parameter changes: change of split ratio and change of heat exchanger surface area. Note that the number of structural changes - discrete decisions - is high. This results in a highly combinatorial nature of the problem. To account for possible structural changes numerous binary variables have to be applied in simultaneous approaches - e.g. Yee and * Author to whome correspondence should be sent, e-mail: [email protected]
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Grossmann (1991), Ciric and Floudas (1989, 1990), Briones and Kokossis (1999), Sorsak and Kravanja (2004). Moreover, the changes are interrelated, i.e. a specific change often excludes or forces another one. In result, complex logical conditions have to be applied in optimization models. To code them additional binaries are needed. The goal function and also, some functions within constraints are highly nonlinear. The combinatorial complexity of HEN retrofit problem increases significantly if standard heat exchangers, i.e. apparatus of discrete values of heat transfer surface area, are to be accounted for. Sorsak and Kravanja (2004) noted that existing mathematical programming solvers are not able to cope with design problems for HEN involving standard apparatus. Due to highly combinatorial character of HEN retrofit problem such stochastic optimization approaches as genetic algorithms (GA) or simulated annealing (SA) should suit well. The both techniques were applied to HEN synthesis – Dolan et al. (1989, 1990), Lewin et al. (1998), Lewin (1998). We have applied GA since population based approaches such as GA have an opinion of more robust optimizers than single point – based ones such as SA (see e.g. Michalewicz and Fogel, 2002).
3. HEN retrofit method overview 3.1. Basis of the approach The unique feature of evolutionary algorithms is processing a population of solutions represented in form of codes (chromosomes). Though GA can be applied as general purpose optimizer the best results are achieved if GA applies solution space (representation and codes) that are “natural” and specific for a problem. Hence, we applied a chromosome that is a direct representation of network topology. Instead of using classical superstructure model in form of complex equalities and inequalities with large number of binaries, the problem of structure optimization is transformed to processing a single multi-variable representation, which is, in fact the variable in structural GA optimization. The variable encapsulates all topological features of a HEN. It will be referred to as structural matrix SM. It was also necessary to define feasible search space for all possible structural changes in SM matrix. This space ensures that structural changes performed directly on HEN structure, by modifying its codes with specialized genetic operators, generate only structurally feasible solutions. The approach consists in two-level iterative procedure with the use of GA at both levels. Structural optimization is performed at primary level while in the slave level parameters of heat exchangers and splitters are optimized. The second level is executed for each HEN structure generated at the primary level to find values of apparatus parameters. The algorithm starts from the initial parent population, which is generated randomly from the existing HEN. All members of parent populations are defined only at topology level. Then, all the members are processed at the second level. Selected, best members of the parent population are, then, modified by crossover and mutation operators and become new parent population for next cycle of GA. 3.2. Representation and codes for superstructure and structures HEN superstructure concept in this work is primarily based on stream superstructure. Stream superstructure consists of main- and side branches. The latter define potential splitters and mixers. Nodes for matches are, then, imposed on stream superstructure to create a space of heat exchanger superstructure. All branches and nodes in the superstructure are numbered, separately for hot and cold streams. The only rule for numbering of branches is that main branches are given numbers first and, then, side branches. Nodes are given numbers separately for each branch, starting from the inlet of
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a stream to its outlet. Each node is given unique address consisting of two items: branch no. - node no.. A structure is generated from the superstructure by assigning to heat exchangers the addresses of two nodes: a hot stream node and a cold stream node. The code for defining both the superstructure and structure is based on matrix representation. The information on the superstructure is stored in node vectors NODH , NODC and split matrices SPLH , SPLH for hot and cold streams. Vectors NOD and matrices SPL have the following structure: • NOD = [n1, ...,ni,...,nNB]; the vector has NB elements, where NB is equal to total number of branches; element ni equals the number of nodes at i-th branch • SPL = [i=1,..., NS; j=1,2,3]; the matrix has NS rows (where NS is equal to total number of side branches) and three columns (j=1,2,3). The number of main branch from which a side branch is created is in the 1st column (j=1), split node number at main branch in the 2nd column (j=2), mixing node number at main branch is in the 3rd column (j=3). The example of vectors NOD and matrices SPL is shown in Fig. 1. Summing up, HEN retrofit superstructure is built on the basis of the existing HEN topology by adding new potentially possible splits (side branches) and new potentially possible locations of heat exchangers (nodes). 3.3. Generation of structures A selection of structures from parent population is performed proportionally to a value of evaluation criterion with repetitions – see Michalewicz and Fogel (2002). The children population created from the parent one becomes parent population in the next evolution cycle after parameter optimization. All the genetic operations, i.e. mutation and crossover, are performed on codes of structures, i.e. structural matrix SM. Each structure embedded in the retrofit superstructure is created by assigning to each heat exchanger within a structure unique addresses of hot and cold nodes from the superstructure. Structural matrix SM , shown also in Fig.1., is as follows: SM [i=1,...,NA; j=1,...,4]; the matrix has NA rows (where NA equals maximal number of heat exchangers which can exist in a HEN) and four columns (j=1,…,4). The first two columns (j=1,2) involve hot node address while columns 3 and 4 – cold node address. It is important to note that information on splits (i.e. side branches) is extracted from superstructure code (matrices SPLH, SPLC). Hence, matrix SM provides the necessary and sufficient information on HEN structure. The algorithm extracts all other information that is needed for calculations from the superstructure code, i.e. vectors NODH, NODC and matrices SPLH, SPLC. These vectors and matrices are not subjected to any changes. In order to perform structural retrofit changes a set of genetic operators was developed that operate on structural matrix SM. The applied genetic operators allow performing all structural changes that are possible in HEN revamp design. Two crossover operators (single-point and multi-point ones) and seven mutation operators were applied in the GA algorithm. Mutation operators developed allow performing the following changes: relocation of heat exchanger - hot node, relocation of heat exchanger - cold node, relocation of a heat exchanger - both nodes, exchange of hot nodes between two heat exchangers, exchange of cold nodes between two heat exchangers, adding of a new heat exchanger, removing of a heat exchanger. For mutation a type of operator is randomly chosen at first. Assume that the operator acts on heat exchanger hot node only. Then, heat exchanger, i.e. row of SM matrix, is also randomly chosen. Finally, hot node address is mutated by inserting into the chosen row and columns 1 and 2 a new address randomly chosen from addresses of hot empty nodes available in the
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superstructure. In case of single-point crossover operator a crossover point is randomly selected for matrices SM1 and SM2 of randomly chosen parents. Then, two new structures (two new SM matrices) are generated; one consists of upper part of SM1 and lower part of SM2 while the second from lower part of SM1 and upper part of SM2. Because this operation can produce structurally infeasible network a repairing mechanism has been applied. 3.4. Parameter optimization Structures generated at first stage are defined only at level of topology. To select the children population the knowledge of optimization criterion value is necessary. In consequence, basic parameters of apparatus and streams are needed. Notice that costs of structural changes are calculated during first optimization stage. To optimize heat exchanger areas and split ratios we applied the GA method, which is a version of GENCOM software (Bochenek et al., 2001). The program was applied as general-purpose optimizer in equation oriented mode. The optimization model is generated automatically in equation form by the computer program developed for HEN design.
4. Example The developed method was coded into computer software called OPTI-HEN. In this paper we present the application of this program for the simple example taken from Yee and Grossmann (1987) and Ciric & Floudas (1990). Stream parameters in the existing network are given in table 1. The network is shown in Fig.1. together with all vectors and matrices that code the superstructure applied which is shown as background. Notice that the existing HEN has no splitters. The branches in Fig.1 are only in the superstructure. Parameters of existing heat exchangers are gathered in table 2. The annual cost of utilities consumed in the HEN amounts to $/y 158,000. For minimum temperature approach (HRAT) equal to 10 K the minimum utility cost (calculated as the target for the synthesis) is $/y 28,000. According to Ciric & Floudas (1990) the goal function is the annual investment cost of retrofitting the HEN at the condition that utility cost should not be greater than $/y 29,000. Table 1. Stream parameters for the example
Stream H1 H2 HU C1 C2 CU
in
T [K] 443 423 450 293 353 293
T
out
[K] 333 303 450 408 413 313
CP [kW/K] 30 15
Cost [$/(kW y)]
80 20 40 20
The following costs of retrofit changes were taken also from Ciric and Floudas (1990): cost of the addition heat exchanger area [$] 1300 ΔA0,6, cost of area for a new heat exchanger [$] 1300 A0,6, cost of new heat exchanger (match) [$] 3000, cost of heat exchanger relocation [$] 300. We solved the example for standard heat exchangers. We used randomly generated values of heat exchanger surface areas. In order to build the superstructure that does not involve many redundant nodes and side branches we first found process pinches and network pinch using the program CHEMMAP by Jezowski et al. (2002) which is based on network pinch approach of Asante and Zhu (1997). The retrofit superstructure developed shown in Fig. 1 consists of existing apparatus (E-1 to E-5). It has two new potential splitters and mixers: nodes 3, 5
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at H2 and 2, 4 at C1. This gives rise to two branches at these streams that can potentially be included into final solution. Each main and side branch for process stream in the superstructure has at least one additional heat exchanger node to allow for structural changes of heat exchangers. 1
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Fig. 2. Optimal HEN for the example Table 2. Parameters of heat exchangers in HEN from Fig. 2.
Apparatus No. 1 2 3 4 5 6
Existing 2 area, [m ] 46.74 68.72 38.31 40.23 5.40 0.00
Modified 2 area, [m ] 40.00 54.00 32.00 30.11 5.40 160.00
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The best HEN obtained is shown in Fig. 2. The parameters of heat exchangers are gathered in table 2. The HEN features the total investment cost of $/y 32,216 and utility cost of $/y 28,850. The cost parameters of the optimum solution of Ciric and Floudas
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(1990) calculated for non-standard heat exchangers are: $/y 35,784 and $/y 28,744, respectively. Despite the application of standard heat exchangers the developed approach calculated the better solution.
5. Conclusions The simultaneous approach for retrofitting HEN has been developed. The method is based on GA optimization at two-level procedure. In order to enhance solution of HEN retrofit problem with GA we developed novel representations and codes for HEN superstructure and structures. Instead of solving complex MINLP with general purpose optimizer the method operates on the representation tailored for GA. The developed representation is general since it can be applied for HEN synthesis, too. It is of importance that the developed method can efficiently solve problems with standard heat exchangers. The problems solved to date with the approach allow claiming that it is flexible, general and able to calculate good local optimum solutions.
References N.D.K. Asante., X.X. Zhu, 1997, An Automated and interactive approach for heat exchanger network retrofit, Trans IChemE, 75, 349-360 R. Bochenek, J.M. Jeżowski, A. Jeżowska, 2001, Multi-product batch plant optimization using genetic algorithms, 28th International Conference of Slovak Society of Chemical Engineering, Tatranske Matliare, Slovak Republic V. Briones, A.C. Kokossis, 1999, Hypertargets: a Conceptual Programming approach for the optimization of industrial heat exchanger networks - II. Retrofit design. Chem. Eng. Sci., 54, 541-561 A.R. Ciric, C.A. Floudas, 1989, A retrofit approach for heat exchanger networks, Comput. Chem. Eng., 13 (6), 703-715 A.R. Ciric, C.A. Floudas, 1990, A mixed integer nonlinear programming model for retrofitting heat-exchanger networks, Ind. Eng. Chem. Res., 29, 239-251 W.B. Dolan, P.T. Cummings, M.D. Le Van, 1989, Process optimization via simulated annealing: application to network design., AIChE J., 35 (5), 725-736 W.B. Dolan, P.T. Cummings, M.D. Le Van, 1990, Algorithmic efficiency of simulated annealing for heat exchanger network design., Comput. Chem. Eng., 14(10), 1039-1050 K.C. Furman, N.V. Sahinidis, 2002, A critical review and annotated bibliography for heat exchanger network synthesis in the 20th century, Ind. Eng. Chem. Res., 41, 2335-2370. Z. Michalewicz, D.B. Fogel, 2002, How to solve it: Modern heuristics., Springer, Berlin J. Jeżowski, R. Bochenek, K. Wałczyk, A. Jeżowska, 2002, Computer Software for Heat Exchanger Networks Retrofit, 29th International Conference of Slovak Society of Chemical Engineering, , Slovak Republic D.R. Lewin, H. Wang, O. Shalev, 1998, A generalized method for HEN synthesis using stochastic optimization - I. General framework and MER optimal synyhesis, Comput. Chem. Eng., 22 (10), 1503-1515 D.R. Lewin, 1998, A generalized method for HEN synthesis using stochastic optimization - II. The synthesis of cost-optimal network, Comput. Chem. Eng., 22, 1387-1405 A. Sorsak, Z. Kravanja, 2004, MINLP retrofit of heat exchanger networks comprising different exchanger types. Comput. Chem. Eng., 28, 235-251 T.F. Yee, I.E. Grossmann, 1987, Optimization model for structural modifications in the retrofit of heat exchanger networks, Foundations of Computer-Aided Process Operations (Park City, Utah), 653-662 T.F. Yee, I.E. Grossmann, 1991, A screening and optimization approach for the retrofit of heatexchanger networks, Ind. Eng. Chem. Res., 30, 146-162
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Optimization of Nanosized Silver Particles Synthesis via Sequential Pseudo-Uniform Design Method Jyh-Shyong Changa and Yuan-Ping Leeb a
Department of Chemical Engineering, Tatung University, Taiwan Department of Chemical Engineering, Tatung University, Taiwan
b
Abstract A data-driven model based optimization on the synthesis of nanosized silver particles by chemical reduction using formaldehyde in aqueous solution was studied in this work. Effects of the possible processing variables such as the reaction temperature T, the mole ratios of [formaldehyde]/[AgNO3] and [NaOH]/[AgNO3], PVP/AgNO3, and the molecular weight of protective agent PVP (polyvinyl-pyrrolidone) were considered. The colloid dispersion products were mainly characterized for its mean particle size and conversion of silver nitrate. The identified model based on the 44 designed experiments can provide us the optimal conditions for achieving (a) the minimum mean particle size (29nm) with conversion (48%), (b) the desired targets (mean particle size, 39nm and conversion, 97%), and (c) the desired targets (mean particle size, 32nm and conversion, 85%) closely. To accomplish the objectives of this work, the fractional factorial design was first applied to screen the insignificant factor [NaOH]/[AgNO3]. By the contrast experiment done at the near-optimal condition for achieving the minimum particle size of the product, the PVP with MW (10,000) was chosen. A resulting 3 significant factors problem were then solved by the developed sequential pseudo-uniform design (SPUD) method. Keywords: Data-driven, Optimization, Nanosized Silver, Sequential Pseudo-Uniform Design.
1. Introduction In the competitive market, product life cycles are very short. If a new product cannot be made in time to meet the needs on the market, it would be outdated or even no longer wanted; therefore, how to speed up the product or process development is an important issue. This target may be achievable through a reliable process model formulated classically on the basis of mass and energy balance for the processes. On the other hand, the data-driven model approach is found to be limited in the research field relating to the nanoparticle production. Artificial neural networks (ANNs) which offer a data-driven modeling approach are well suited for the above-mentioned purpose. Presently the most widely used ANN type is a multi-layer artificial feed forward neural network (FNN), which is trained by the back propagation (BP) algorithm. The application of FNNs to data is claimed to be one of the powerful data-driven models, since the FNNs have an ability to learn and extract the inputs/outputs relationships from the presentation of a set of training samples. Their flexibility has been a valuable quality compared with parametric techniques that require the assumption of a specific hard model form. Therefore, if the data-driven
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model approach can be applied in the design and operation of unknown or complex systems, the expenditures of time would be much reduced in a practical application. In this study, batch nanosized silver particles synthesis was adopted as the testing process for verification of the proposed optimization method which was developed on the basis of the FNNs coupled with sequential pseudo-uniform design (SPUD) (Chang and Lin, 2004). The proposed optimization method could provide us with the optimal operating conditions for achieving diverse objective effectively. To build these FNNs, learning and validation procedures are, in general, required. The interpolation and/or extrapolation capabilities of an FNN are directly related to the training data. To achieve robust performance, data must be distributed across all of the regions of the input space that are of interest. Therefore, the developed SPUD method meets the requirement and will be applied to locate limited but sufficient experiments for gathering the experimental data. The developed SPUD method is an extended version of the uniform design (UD) method (Fang, 1980; Fang and Ma, 200) with a special feature that could locate experiments uniformly and sequentially in the investigated domain. With respect to details of the SPUD method, one could refer to our previous work (Chang and Lin, 2004). 2. Optimization of Products and Process via the SPUD Method The propos feature that could locate experiments uniformly and sequentially in the investigated domain. With respect to details of the SPUD method, one could refer to our previous work (Chang and Lin, 2004).ed algorithm for determining the optimum operating conditions for an unknown process via the SPUD method are: (1) Choose a suitable level q0 for each control factor using the available UD table. (2) Build an FNN model based on the experimental data from step 1. (3) Check the adequacy of the identified FNN model by the interpolated experiments chosen according to the SPUD method (recycle from step 4) or the UD method (recycle from step 6). (4) Find the optimal conditions by the random search method if the identified model is adequate; go to step 5. Otherwise, perform the chosen experiments provided by the SPUD method and identify a new FNN model from the available experiment data; go to step 3. (5) Check the adequacy of the optimal conditions experimentally; if the optimal conditions determined are reliable, stop the procedure. (6) If the condition given in step 5 cannot be met, carry out more experiments around the assumed optimal conditions from the UD table again and identify a new FNN model; go to step 3. 3. Experimental System As described in the work of Chou and Ren (2000), production of nanosized silver particles could be carried out as follows. Silver nitrate (Showa) solution
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of 0.01M was used as the precursor. A predetermined quantity of the protecting agent PVP (molecular weight (MW) = 10,000, Sigma) or PVP (molecular weight (MW) = 40,000, Sigma) was then added to AgNO3, using formaldehyde (37% solution, Acros) as reducing agent. However, if only formaldehyde was used, the reduction rate would be too slow at room temperature due to low pH. Suitable quantities of alkaline solution consisting of sodium hydroxide (Showa) were added. The formation of silver colloids can be achieved in a suitable time course. Products of silver colloids were characterized mainly on its particle size distribution using the dynamic light scattering submicron particle size distribution analyzer (LB-500, Horiba). TEM (H-7100, HITACHI) was also applied to obtain information on the morphology of samples colloids. Variations in size distribution were followed along with time. In order to analyze the conversion of silver nitrate, the flame atomic absorption spectrometer (AA) (SpectrAA 220, Varian) was used to measure the concentration of silver ion in the reacting medium. XRD (XRD-6000, Shimadzu) was also applied to obtain information on the structure of silver particles. UV-Vis (V-500, Jasco) measure absorbance during the reaction, small volumes of samples were taken at different times and immediately diluted with deionized water. Effects of the possible processing variables such as the reaction temperature T (20-60℃), the molar ratios of [formaldehyde]/[AgNO3] (1-10 mol/mol),[NaOH]/[AgNO3] (0.5-1.5 mol/mol), PVP/AgNO3 (5-10g/g), and the molecular weight of protective agent PVP (polyvinyl-pyrrolidone) (10000 or 40000 g/mol) on the particle size and the conversion of silver nitrate were considered in this study. The following step by step experiments were carried out : (P1) Survey of the possible control factors affecting the process. (P2) Screening experiments using 25-1 fractional factorial design (FFD). (P3) Identification of a 2nd order regression model based on the initial U10 10 3 experiments. (P4) Testing of the identified model from (P3) using the partial experimental data provided by the SPUD 9 19 3 experiments. (P5) Identification of a 2nd order regression model based on the based on the initial U10 10 3 experimental data and the testing data given in (P4). (P6) Testing of the identified model from (P5) using the remaining experimental data provided by the SPUD 9 19 3 experiments. (P7) Identification of a 2nd order regression model and FNN models with different nodes based on the based on the experimental data gathered experiments, and partial from the U10 10 3 , SPUD 9 19 3
( )
( )
( )
( )
( )
( )
( )
SPUD18 37 3 experiments. (P8)
Testing of the identified models from (P7) using the remaining experimental data provided by the SPUD18 37 3 experiments. The
( )
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(P9)
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identified FNN model with 4 hidden nodes was assumed to be reliable. The optimal condition for achieving minimum mean particle size was estimated. However, there still existed some mismatch between the measurements and the predicted model outputs at this optimal condition. Identification of the FNN model with 4 hidden nodes based on the experimental data provided by U10 10 3 experiments, partial
( )
( ) experiments and the additional boundary experiments
SPUD18 37
3
provided by the 23 factorial design (FD). (P10) Testing of the identified models from (P9) using the remaining experimental data provided by the SPUD18 37 3 experiments. The testing results were acceptable, however, the measurements collected at the assumed optimal condition determined in (P8) still existed some mismatch compared to the model outputs. (P11) Identification of the FNN model with 4 hidden nodes based on the experimental data gathered in (P9) and the additional U 4 4 3 experiments near the assumed optimal condition determined in (P8). In this time, the testing points provided by the partial SPUD18 37 3 experiments and the measurements collected at the assumed optimal condition determined in (P8) were much closed to the model outputs identified in (P11). The identified FNN model at this time was checked to be acceptable. Optimal operating conditions for different objective functions were achieved based on the identified FNN model given in (P11).
( )
( )
( )
4. Statistics Analysis of the Identified Model from the Designed Experiments The experimental data collected via the modified SPUD method were used to identify a 2nd order regression model and FNN models with different hidden nodes. An identified model is chosen to be most acceptable if the performance indices including (a) the smallest estimated standard deviation σˆ (b) the smallest of MSLOF (c) the largest R2 (Table 1) and (d) the identified model is suitable fitted (－) or over fitted (＋) examined by model outputs calculations given a random testing set. Based on the given criterion, the FNN model with 4 hidden nodes was chosen to be the most acceptable model for mean particle size ( σˆ =2.88, MSLOF=20.45, R2=0.95, and the model was not over fitted); Another FNN model with 4 hidden nodes was chosen to be the most acceptable model for conversion ( σˆ =1.816, MSLOF=7.65, R2=0.99, and the model was not over fitted). Based on the identified model, the following three objectives were achieved: (a) the objective chosen to minimum mean particle size (experimental ζ =28.96nm, 29.42nm; model ζ =28.63nm) at the conditions T =20( ), mole ratios of [NaOH]/[AgNO3]=0.5 (mole/mole), and
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PVP/AgNO3=8.842 (g/g) determined by the identified model. Meanwhile, the conversion was 47.94 % (b) the objective chosen to be the composed desirability objective function (experimental ζ =36.72nm, 37.68nm, X=97.3%, 97.9% ; model ζ =38.8, X=97.41) at the conditions T =29.6 ( ), mole ratios of [NaOH]/[AgNO3]= 1.5 (mole/mole), PVP/AgNO3=7.97 (g/g) determined by the identified model (c) the objective chosen to be the minimum mean particle size at a assigned conversion (experimental ζ =32.76nm, 31.78nm, X=86.9%, 87% ; model ζ =32.66, X=85%) at the conditions T =20 ( ), mole ratios of [NaOH]/[AgNO3]= 1.43 (mole/mole), PVP/AgNO3=5.76 (g/g) determined by the identified model (Table 2 for example). Form the experiments described (step P3 to P11, shown in section 3), one can find that only least experiments (44 experiments for identifying a reliable model and 5 experiment for checking the identified model) were required to obtain a reliable model, however, the identified model seem to give inaccurate predictions around the boundaries of the control factors, because less experiments were arranged by the SPUD method. Therefore, in addition to the experiments located by the SPUD method, the 23 FD experiments and the U4(43) experiments were augmented to give a reliable model to achieve three optimization problems described. 5. Conclusions The application of UD method to nonlinear processes by an FNN model or a regression model can be used to build a model for an unknown process efficiently because it allows many levels for each factor. If the cost of each experiment is high, low partitioned levels are usually proposed first to carry out the experiments. However, if a reliable model cannot be obtained from the designed experiments, the developed SPUD method in our laboratory can be employed to locate additional experiments in the experimental region. Once the identified model is verified as reliable based on the statistical analysis, the optimal operating conditions can be determined to guide the process to the desired objective and were demonstrated experimentally in this work. References 1. Chang, J. S.; Lin, J. P. Product and Process Development via Sequential Pseudo-Uniform Design. Ind. Eng. Chem. Res. 2004, 43, 4278. 2. Chou, K. S.; Chiang, Y. R. Synthesis of Nanosized Silver Particles by Chemical Reduction Method. Materials Chemistry and Physics. 2000, 64, 241. 3. Fang, K. T. Uniform Design: Application of Number-Theoretic Methods in Experimental Design. Acta Math. Appl. Sin. 1980, 3, 363. 4. Fang, K. T. and Ma, Z. X. Orthogonal and Uniform Experimental Design; Hong Kong Baptist University: HK, 2000.
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Table 1. Analysis of Variance for Significance of Model Source of
Sum of Squares
Degrees of
Mean
Freedom
Square
k (= p-1)
SSmodel k
− yˆ i )
N-p
SS E N−p
− yˆ i ) 2
m-p
SS LOF (m − p)
∑∑ ( y
− yi ) 2
N-m
SS PE ( N − m)
∑∑ (y
2
Variation Model
2
∑∑ (yˆij − y.. ) m
n
i =1 j =1
Error
2
∑∑ (y m
n
ij
i =1 j =1 m
Lack of Fit
∑n (y i =1
m
Pure Error
i
ni
i =1 j =1
m
Total
i
ij
n
i =1 j =1
ij
− y.. )
F0
MSmodel MS E
MS LOF MS PE
N-1
Table 2. Comparison of ζ and X for the Measurements and the Model Outputs for Achieving the Objective (Minimum Mean Particle Size) Run Experiment (1) Experiment (2) Model output
T (℃)
[NaOH]/[AgNO3] PVP/AgNO3 (mole/mole) (g/g)
ζ
(nm)
X (%)
20
0.5
8.842
28.96
47.6
20
0.5
8.842
29.42
48.5
20
0.5
8.842
28.63
47.94
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Multiclass Molecular Knowledge Framework for Product and Process Design a,b
a
a
Mourad Korichi, Vincent Gerbaud, Pascal Floquet, A.-Hassan Meniai, d a Saci Nacef, and Xavier Joulia
c
a Laboratoire de Génie Chimique, 118 route de Narbonne, F-31077 Toulouse France b LVPRS, Université de Ouargla, BP 511 Ouargla, Algeria c LIPE, Université de Constantine, Constantine, Algeria d LGC, Université de Sétif, Sétif, Algeria Abstract Computer Aided Product Design (CAPD) is widely used in process system engineering as a powerful tool for searching novel chemicals. The crucial steps in CAPD are the generation of candidate molecules and the estimation of properties, especially when complex molecular structures like flavors are sought. In this paper, we present a multiclass molecular knowledge framework which is based on chemical graph theory and chemical knowledge. Three kinds of functional groups are defined: elementary, basic and composed groups. These serve to generate four classes of knowledge that can be useful for property estimation and molecular design. An Input/output structure basing on XML language is defined to favor the interoperability between softwares. Keywords: Molecular graph; Group contribution; Property prediction, CAMD
1. Introduction Design activities in chemical engineering range from design of chemical product to the design of process. In chemical product design specially computer aided molecular design (CAMD), we try to find a chemical product that exhibits certain desirable or specified properties. These must be calculated from models describing the chemical structure-property relationship, like QSAR or group contribution methods. Such group contributions are also often used in the steps devoted to the computational searching of chemical product (Harper and Gani, 2000; Achenie and Sinha, 2003). Furthermore, refining new molecule candidates often forces to consider the molecular level where detailed advanced properties originate. In process design, a crucial step is physical and thermodynamic property values of a single component or multi-component process streams. Experimental measurements are always the preferred source of these values, but because of the expense and difficulty of measurements and the nearly infinite combination of compounds and conditions encountered, estimation techniques are preferred. The role of molecular class knowledge is then essential as it provides the right information for the right estimation techniques or requests at the right level (gross formula, expanded formula, atomic coordinates …). In this paper attention is focused on the importance of molecular knowledge for product and process design activities based on chemical graph theory and chemical knowledge. Three kinds of functional groups are defined: elementary, basic and composed groups. We defined the relation between them to generate different classes of knowledge that can be used in property estimation techniques and molecular design.
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2. Molecular representation The description of molecules should include all information that a researcher will be working with. Molecule can be represented by many useful chemical representations. The description must first facilitate the manipulation of structure, second not be limited by molecular size or topological complexity, and finally, not violate elementary rules like the chemical valence rules. In general, two molecular representation families are widely used in chemistry: • Line based extensions include atomic species and adjacency list. The line notation of a compound is an ordered list of symbols representing atoms, bonds and charge. There are several line notation methods available, such as the SMILES (Weininger, 1988) and the WLN (Smith, E.G., 1968). • Graph or object oriented representation where the bonds and atoms are described by detailed features. This is more suitable for the representation and manipulation of molecular structures. There are a various types of this representation include atom tables, bond tables and many more. In our work, we are motivated to use the simple molecular graph representation with some modification in the structure of the graph for two reasons. First, the use of molecular graph assumes the real connection between atoms in the molecule. It is not intended to be compact, and can be easily expanded to provide information that may be frequently needed in analyzing the molecular structure (connection of atoms, rings, stereo-chemical and many more). Second, It enables to use classical properties estimations techniques based on group contribution methods but also on molecular descriptors (QSPR) as these seemingly coarse techniques are improving and may partially substitute group contribution estimation techniques in future software packages for Computer Aided Molecular Design (Bünz et al., 1998). The Molecular graph MG = (V, E) is defined as mathematical representation of a molecule where atoms and bonds correspond to vertices (V) and edges (E) respectively. The MG used in this paper is undirected and unweighted and has no multiple edges between vertices (nodes) and no self-loops at any vertices, so this is called simple graph (Pogliani, 2000). MG can be represented by a variety of matrices such as vertex adjacency matrix, edge adjacency matrix, incidence matrix, cycle matrix or distance matrix. In this paper, the molecular structure is stored temporarily in the computer as an encoded chemical graph connectivity matrix which is composed of the encoded atoms in a suitable type of connection table. The element of an encoded adjacency matrix expresses the connection relation. Zero indicates no connected elements. The proposed molecular graph (A) is an N x N matrix, where N is the number of atoms in the molecule (Eqn. nr.1.). The off diagonal elements represent the connections between pairs of atoms. All bonds in the molecule are coded by 1. The diagonal elements represent the elementary groups present in the molecule described in section §3. A = (aij )
⎧ ⎧1 ⎪i ≠ j aij ⎨ aij = ⎨ ⎩0 ⎪i = j a ii ⎩
if atom i is connected to atom j otherwise informations about atom i
(Eqn. nr. 1)
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3. Automatic decomposition algorithm of molecule to groups (ADAM) Automatic decomposition of molecule into groups is an important step for the prediction of physicochemical properties from diverse estimation techniques based on molecular descriptors, like Group Contribution (Joback and Reid, 1987; Constantinou and Gani, 1994) and QSPR. In addition, this is very important for process simulation when molecules are often not yet integrated in the simulator database, and can help to integrate a wide range of estimation techniques mentioned above. There are few publications concerning the decomposition of molecules into functional groups as input for property estimation packages based on the line notation widely used in chemistry (Qu et al., 1992; Jeremy Rowley et al., 2001). Figure 1 shows the three kinds of groups defined in this paper, namely elementary, basic and composed groups.
0 0 0 0 ⎤ ⎡61000 1 ⎢ 1 62000 1 0 0 1 ⎥⎥ ⎢ ⎢ 0 1 82100 1 0 0 ⎥ ⎢ ⎥ 0 1 61000 1 0 ⎥ ⎢ 0 ⎢ 0 0 0 1 61000 0 ⎥ ⎢ ⎥ 1 0 0 0 8200⎦⎥0 ⎣⎢ 0
Figure 1. Molecular graph and kinds of groups for ethyl acetate.
Our framework is decomposed into four classes: 1. The elementary class represents molecules by a simple graph and identifies the connection between atoms. The atoms and their neighbour’s information are described by elementary groups coded as vectors on the diagonal. Each vector contains the following information. P1
P2
P3
P4
P5
……………….
PN-1
PN
• Atom identifier (P1): Except hydrogen, each atom is set independently by its atomic number. • Bond identifier (P2): Hydrogen bond is not considered but only bonds between carbon-carbon, carbon-non carbon and non carbon-non carbon atoms. Single, double, triple and fourth bond are identified by number code 1, 2, 3 and 4 respectively. • Characteristic identifier (P3): It enables to respect the octal rule and is used in equation (2) to calculate the number of hydrogen attached to the central atom in the elementary groups. • Ring/Non ring identifier (P4): This number takes the value 0 if the atom is non ring, 1 in non aromatic rings and 2 or more otherwise (heterocyclic …).
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• Others identifiers (P5…PN): The user can add any others important information to identify some specificity to the elementary groups. For example the position of OH in primary or secondary in the structure. 2. The second class involves the transformation of the first class molecular graph into a vector of basic groups. Basic groups are subdivided into two categories: simple basic and composed basic groups. • A simple basic group is a single elementary group with all hydrogen atoms attached to it, for example CH3, CH and OH. This is represented by a couple (X, Y), where X characterizes the elementary group and Y the number of hydrogen atoms connected to the central atom calculated by equation (2), where the NHi is the number of hydrogen atom attached to atom i, ViSTD is the standard valence of principal atom i, aij the connection between atom i and atom j. The constants Ci(P2) and Ci(P3) are the bond and characteristic identifiers respectively of atom i. Ethyl acetate (figure 1 & table 1) contains three simple basic groups: two CH3 and one CH2 described by (61000,3) and (61000,2) and the data structure is two [1,(61000,3)] and one [1,(61000,2)] respectively.
NH i = Vi STD −
Natoms
∑a
j =1, j ≠ i
ij
(Eqn. nr. 2)
− C i( P 2) + C i( P 3) + 1
• A composed basic group is built of several elementary groups with all hydrogen atoms. Composed basic groups are in limited number and deal with chemical functionality of the molecules: HCOO-, -COO-, -COOH, -C=O, HC=O, -C≡N and NO2. The structure information for this category is written according to equation (3), where the Index_EG refers to the number of elementary groups and (X, Y)i refers to the ith elementary groups. The ethyl acetate molecule has one composed basic group –COO- built from three elementary groups C=, O= and –O-.This group become [3,(62000,0), (82100,0), (82000,0)] (see figure 1 & table 1).
[Index _ EG , ( X , Y ) , ( X , Y ) 1
2
,..., ( X , Y ) Index _ EG
]
(Eqn. nr. 3)
3. In the third class, composed groups are built from data structures information in primary and secondary classes using the group profile principle and heuristic rules. Such composed groups are used in multi-order groups contribution methods and UNIFAC models. They are therefore not unique but should rather be generated knowing the group contribution method to be used. As an illustration, we take the UNIFAC group CH3COO-. This composed group can be formed by the simple basic group CH3 and the composed basic group -COO-. In general, the connection between basic groups generate the composed groups must be assumed. The data structure of this class is written according to equation (4), where BG1, BG2,…, BGN are the data structure presented in equation (3) (see table 1).
[Index _ BG , BG , BG ,..., BG 1
2
Index _ BG
]
(Eqn. nr. 4)
4. In the fourth class, and for specific applications in product and process design, like flavour industry where isomer have to be differentiated by using the molecular simulation, the molecular structure information generated in classes 1, 2 and 3 can be refined to three-dimensional representations using atomic coordinates and other partial charges features, readily usable in molecular simulation packages.
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Table 1. Data structure of the tree kinds of groups CH3
CH3
[1, (61000,3)]
C= CH3COO-
COO-
O=
[3, (62040,0),(82000,0),(82100,0)]
-O{2, [1, (61000,3)], [3, (62040,0),(82000,0),(82100,0)]}
4. Input / Output formalism The eXtensible Markup Language (XML) is a universal method to represent structured data according to normalized syntaxes and can permit the exchange and storage of data. As input/output to our framework, two standard XML formalisms are used in this work, the Chemical Markup Language (CML) (Murray-Rust and Rzepa, 2001) for molecules and ThermoML (Frenkel et al., 2003) for experimental properties. In the first step, we exploit some essential definition in CML format, elements name like molecule, atom and bond and elements attributs like atomId, order and atomRef. Some modifications are proposed in the CML format such as the substitution of atomArray by others elements name like elementary group, basic group and composed group. <XmlEntry> <molecule id="Ethyl_Acetate"> <elementaryGroupArray> <elementaryGroup id="a1" atomType="6" bondType="1" hydrogenCount="0" ring="0" others="0"/> ……………………………………………… <elementaryGroupArray> ……………………………………………… ……………………………………………… <property> <propertyType id="Pure"/> <propertyName id="BoilingTemppreture"/> <propertyClass PropertyType="groupContribution" MethodName="GC.Jobakc.Ried.1987"/> Figure 2. XML for ethyl acetate
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This XML format is parsed to generate all information in the four classes. In the second step, the Compound and PureOrMixtureData blocks from ThermoML are used. The Coumpound block is defined as CML format presented in the first step and PureOrMixtureData block is modified to support the group contribution methods for property estimation, not exist in the originally ThermoML structure (Frenkel et al., 2003).
5. Illustration example Figure 3 illustrates briefly two major steps of the decomposition algorithm for 2-methyl butanoic acid. In the first step (01), the graph decomposition and the graph analysis procedures are performed. The molecule – based on the nature of chemical bonds – is parsed in simply groups. In the second step (02), group profile principle and heuristic rules are applied to locate different type of groups, especially the composed basic and composed basic groups.
Figure 3. Decomposition procedure for 2-methyl butanoic acid.
6. Summary A new methodology for the multiclass molecular knowledge product and process design is presented. The approach is decomposed into four classes to cope with different level of molecular knowledge. Any molecule is decomposed in elementary, basic and composed groups suitable for any property estimation method and Computer Aided Molecular Design. This decomposition approach is important when atoms and bonds are manipulated in the molecular generating level in the CAMD. An XML input/output format is briefly presented to facilitate storage and exchange of data.
References Achenie, L. E. K., and Sinha, M., Advances in Environmental Research, 8 (2003) 213-227. Bünz, A. P., Braun, B. and Janowsky, R., Ind. Eng. Chem. Res. , 37(1998), 3043-3051. Constantinou, L. and Gani, R., AIChE J., 40 (1994) 1697-1710. Frenkel, M., Chirico, R. D., Diky, V. V., Dong, Q., Frenkel, S., Franchois, P. R., Embry, D. L., Teague, T. L., Marsh, K. N. and Wilhoit, R. C. , J. Chem. Eng. Data, 48 ( 2003), 2-13. Harper, P. M. and Gani, R., Computers and Chemical Engineering, 24 (2000) 677-683. Joback, K. G.; Reid, R. C., Chemical Engineering Communication, 57 (1987) 233-243. Murray-Rust P. and Rzepa, H. S., J. Chem. Inf. Comput. Sci., 41(2001), 1113-1123. Pogliani, L., Chem. Rev., 100 (2000), 3827-3858. Qu,D., Su, J., Muraki,M. and Hayakawat,T.J. Chem. Inf. Comput. Sci, 32 (1992), 448-452. Jeremy Rowley, R, Oscarson, J. L., Rowley, R. L. and Wilding, W. V., J. Chem. Eng. Data, 46 (2001), 1110-1113. Smith, E.G.,Wiswesser Line-Formula Chemical Notation Method, McGraw-Hill, NY,1968. Weininger, D., J. Chem. Inf. Comput. Sci., 28 (1988), 31-36.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Quantitative Structure – Odor Relationship: Using of Multidimensional Data Analysis and Neural Network Approaches. a,b
a
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Mourad Korichi, Vincent Gerbaud, Pascal Floquet, A.-Hassan Meniai, Saci d a Nacef, and Xavier Joulia a Laboratoire de Génie Chimique, 118 route de Narbonne, F-31077 Toulouse France b LVPRS, Université de Ouargla, BP 511 Ouargla, Algeria c LIPE, Université de Constantine, Constantine, Algeria d LGC, Université de Sétif, Sétif, Algeria Abstract Structure – odor relationships (SOR) are key issues for the synthesis of new odorant molecules. But, this relation is hard to model, due to limited understanding of olfaction phenomena and the subjectivity of odor quantity and quality as stated in Rossitier’s review (1996). Many molecular descriptors are used to correlate molecule’s odor, but no universal rules emerge in this field. In this paper, we focus on the use of molecular descriptors as an alternative approach in the prediction of odors, by the mean of regression techniques. Principal Component Analysis (PCA) and Stepwise Collinearity Diagnosis (SCD) techniques are used to reduce the dimensionality of data, by the identification of significant molecular descriptors. Then, the chosen molecular descriptors are used with a neural networks algorithm to correlate the structure to molecular odor quality. The results are validated on balsamic flavor. Keywords: Molecular descriptors, Neural network, SOR.
1. Introduction Odorant compounds are found in a wide variety of products ranging from foods, perfumes, health care products and medicines. Either combined or alone, flavor and fragrance compounds are used to induce consumers to associate favorable impressions with a given product. In some cases, products have one predominant component which provides the characteristic odor. However, in most cases, products containing odors include a complex mixture of fragrant compounds. Some of them are classified within REACH, a European Community document, regulating the use of chemicals in terms of environment and toxicity. Structure – Odor relationships (SOR) are very important for the synthesis of new odorant molecules. This relation is difficult to model due to the subjectivity of the odor quantity and quality. Olfaction phenomenon is not yet completely understood and odor measurements are often inaccurate (Amboni et al., 2000). Research has been oriented to the use of structural, topological, geometrical, electronic, and physicochemical parameters as descriptors, to generate odor predictive equations. Therefore, a number of computational techniques have been used successfully. Artificial Neural Networks (ANN’s) are one of these promising techniques
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readily suited for the assessment of poorly understood properties like odor. In this paper, we aim to use molecular descriptors as an alternative approach in the prediction of molecule’s odor by the mean of regression techniques. Principal Component Analysis (PCA) and Pairwise Collinearity Diagnosis techniques are used to reduce the dimensionality of data, by the identification of significant molecular descriptors. Then, the chosen molecular descriptors are used with a neural networks algorithm to correlate the structure to molecular odor quality. Figure 1 summarizes the methodology. Molecular Descriptors Selection 28 Molecular Descriptors: Constitutional, topological and connectivity. Mw, Aromaticity, nBz, nR5, nR6, SiK, XiA,
582 Molecular Descriptors: Constitutional, topological, walk and path counts, connectivity , information, 2D autocorrelations, edge adjacency, burden eigenvalues and topological charge.
Complete Correlation Analysis 16 linearly independent descriptors
140 linearly independent descriptors Principal Component Analysis
4 Principal components
10 Principal components Classification & regression
Discriminant Analysis
Neural Network
Figure 1. Schematic representation of methodology on structure – odor relationships.
2. Molecular Descriptor Selection Molecular descriptors accounts for a particular aspect of the molecule structure. As examples, simply count of atoms, functional groups and characteristic fragments are some of the constitutional descriptors family of the studied structure. Topological descriptors are related to the two-dimensional representation of the molecular structure. Molecular descriptors are the most significant common features of molecular structure that can be used to develop Structure - property relationships. In our case, the property is the odor of a molecule. Our input data set contains 121 molecules of balsamic odor splited in 5 sub-notes of typical odors (see Table 1): anise, balsam, honey, vanilla and sweet (Aldrich Flavors and Fragrances catalog, 2005). The dragon software (TALETE, 2005) is used to calculate up to 582 molecular descriptors of the input data set. According to figure 1, two cases are explored, first, 34 simple descriptors (molecular
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weight, aromatic ratio, number of benzene-like rings, connectivity indices, Kier symmetry index and n-path Kier alpha-modified shape index) used in the different relations described in Rossitier’s review on structure – odor works since 1930 (1996) are calculated. In the second case, 582 molecular descriptors are considered: constitutional descriptors, topological indices, walk and path counts, connectivity and information indices, 2D autocorrelations, edge adjacency indices, burden eigenvalues and topological charge indices. All descriptors are calculated from the 2D molecular representation. Table 1. Input data set of molecular structure Odor type Anise Balsam Honey Vanilla Sweet
Number of compounds 10 18 21 15 58
Arbitrary continuous Odor codification 0 to 0.15 0.25 to 0.35 0.45 to 0.55 0.60 to 0.75 0.85 to 0.95
Arbitrary discontinuous Odor codification 0.15 0.35 0.55 0.75 0.95
2.1. Complete Correlation Analysis The complete correlation analysis is used to select a subset of linearly independent descriptors. Descriptor dependency is evaluated using the Dragon software by setting a predefined value Rmax (In this work, Rmax = 0.97) below which descriptors are considered linearly independent. 2.2. Principal component analysis Principal component analysis is one of the oldest, but still most widely used techniques of multivariate analysis. The basic idea of the method is to try to describe the variation of the variables in a set of multivariate data, as parsimoniously as possible using a set of uncorrelated variables, each of which is a particular linear combination of those in the original data. This enables us to reduce the molecular descriptors dimensionality, by the identification of the principal components that can be used in the structure - odor relationship. All eigenvalues greater than 1 are retained to describe the principal axes. In the first case, four principal components are kept to describe the 16 molecular descriptors, widely used in the correlation of the structure – odor. In the second case, ten principal components are retained to represent the 140 molecular descriptors.
3. Artificial Neural Networks (ANN) Approach The ANN (Dreyfus et al., 2004) trained by back-propagation (BP) network algorithm has a two layers architecture: the first layer is called the hidden layer. The number of hidden neuron is a variable X between 7 and 9. The output layer consists of one neuron, namely the odor quality. The network configuration is m-X-1, where m represents the number of principal components encapsulating the maximum of information of the linearly independent molecular descriptors (§ 2.2). The ANN has a feed forward layered structure with connections allowed only between adjacent layers. The balsamic odor sub-notes as output, are represented by arbitrary continuous codification described in table 1. Input and output data are normalized, and hyperbolic tangent sigmoid transfer
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function is used in the hidden layer. The output layer is a linear function. In this work, the training and the validation sets are generated randomly, corresponding respectively to 70% and 30% of the input data set of 121 molecules. After several training sessions, an optimal number of hidden neurons X equal 8 and 7 is retained for case one and two respectively. The network is trained for 500 epochs with a gradient algorithm. The performance goal is 0.001.
4. Discriminant Analysis Discriminant analysis is an analytical technique, whereby a multivariate data set containing m variables is separated into a number (k) of pre-defined groups, using discriminant functions (Z) which are linear combinations of the variables. Two cases are studied to discriminate the molecules into different odors based on the results of PCA described on the paragraph (§2.2). 4.1. Discriminant Analysis based on the first PCA study (four principal components) In the first case there are four principal components. Results are presented in the table 3 with an overall 69.4% of the molecules in the data set, are well classified. The molecules of vanilla odor have the highest correctly classification per cent. Table 3. Discriminant analysis based on the first PCA study. Groups Anise Balsam Honey Vanilla Sweet
Anise 8 1 0 0 3
Predicted groups Balsam Honey Vanilla 0 0 2 14 1 1 3 15 0 2 0 12 8 8 4
Sweet 0 1 3 0 35
molecules 10 18 21 14 58
Correctly classified 0.800 0.778 0.714 0.857 0.603
4.2. Discriminant Analysis based on the second PCA study (ten principal components) In the second case, there are ten principal components. 83.4% of 121 molecules in the data set are well classified, with the honey molecules having the highest classification, and the sweet molecules having the lowest, like in the first case. Table 4. Discriminant analysis based on the second PCA study Groups Anise Balsam Honey Vanilla Sweet
Anise 8 0 0 0 3
Predicted groups Balsam Honey Vanilla 0 0 2 17 0 0 0 20 0 1 0 13 2 5 4
Sweet 0 1 1 0 44
molecules 10 18 21 14 58
Correctly classified 0.800 0.944 0.952 0.929 0.759
5. Results and discussions 5.1. Artificial Neural Networks Approach It is well-known that ANN performance depends on many variables, as the number of hidden neurons, the degree of homology between the training and the validation sets and
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the input variables (Principal components in this case). In figure 2, results are represented as the variation of the odor quality versus the molecule identification code, (a) for the first case and (b) for the second case. In the first case (a), the ANN does not converge as shown by the similarity of the response for all molecules despite their initial differences. In the second case (b), the training set is well represented, but almost all the validation set is not. This clearly shows the non predictive capacity of the ANN approach. Kovatcheva et al. (2004) on a kNN approach for modeling structure ambergris odor relationship suggest to use division procedures based on sphereexclusion algorithms and demonstrate a predictive capacity. But the ambergris odor is due to well known chemical structures, unlike the balsamic odor where molecule structure is more heterogeneous with several odor sub-notes. Also, Chastrette et al., (1995 & 1996), Cherqaoui et al. (1998) and Zakarya et al. (1999) do not consider subnotes, only, requesting a discrete response. Odor
Calculated odor
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Figure 2. Reference odor (Aldrich, 2005) and calculated odor versus molecular identification.
5.2. Discriminant Analysis Approach 69.4% and 83.4% of molecules are well discriminated in the two cases respectively, especially anise and vanilla molecules groups. This is better than the ANN approach.
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Case 2 has more odor discriminant, because it incorporates more numerous and diverse molecular descriptors. Among the molecules that are not discriminated, the two molecules from anise, classified in the vanilla group bear similar molecular structure with vanilla type molecules, which have three oxygen atoms, high Kier symmetry index and n-path Kier alpha-modified shape index. For case 2, in balsam and honey sub-note odors, the molecule wrongly classified is considered differently depending on the referential nomenclature (we used Aldrich’s) that both belong to balsamic and/or rose main odors. In the group of vanilla, one molecule is discriminated as balsam sub-note. From the referential nomenclature, we can say that the molecule has different odor types. In the sweet sub-note, fourteen molecules are distributed into other sub-notes. The low discrimination of the sweet odor may be attributed to the subjectivity of this sub-note, unlike vanilla or anise. Indeed, sweet is not considered as a typical odor type in the reputed referential chart “the field of odors” of Jaubert et al. (1995).
6. Conclusion and perspectives In this work we present different ways to estimate and discriminate odors of molecules, based on molecular descriptors using multidimensional data analysis, and neural network applied to balsamic odors. The multidimensional data analysis is a powerful tool to reduce data sets and encapsulate the maximum of molecule’s structure information. Discriminant analysis results using only 2D molecular representation are encouraging. Further work using 3D representation molecular descriptors may improve the results. The neural network satisfactorily correlates the molecules with their assigned odor, based on sufficiently numerous and diverse molecular descriptors. But it is unable to predict balsamic odor and its sub-notes. Compared with literature, successful results in ANN approach are due to the well known families of odor. The heterogeneous nature of the molecules assigned to balsamic odor and the absence of evident structure – odor relationship, forces us to request a continuous discrimination between sub-notes. References Aldrich Inc., Flavors and Fragrances catalog, , http://www.sigmaaldrich.com/, 2005. Amboni, R. D. C., Junkes, B., Yunes R. A. and Heinzen, V. E. F., J. Agric. Food Chem., 48 (2000) 3517-3521. Chastrette, M., Aïdi, C. E. and Peyraud, J. F., Eur. J. Med. Chem., 30 (1995), 679-686. Chastrette, M., D. Cretin, D. and Aïdi, C. E., J. Chem. Inf. Comput. Sci., 36 (1996), 108-113. Cherqaoui, D., Essefar M., Villemin, D., Cense, J.-M., Chastrette, M., and Zakarya, D., New J. Chem, 1998, 839- 843. Dreyfus, G., Martinez, J.-M., Samuelides, M., Gordon, A. B., Badran, F., Thiria, S, Hérault, L, Réseaux de neurones :: méthodologie et applications, Paris , Eyrolles, 2004. Jaubert, J. N., Tapiero, C. And Dore, J. C. Perfumer Flavorist, 20 (1995), 1-16. Kovatcheva, A., Golbraikh, A., Oloff, S. Xiao, Y_D., Zheng, W., Wolschann, P. Buchbauer, G., and Tropsha, A., J. Chem. Inf. Comput. Sci., 44 (2004), 582-595. Rossitier, K. J., Chem. Rev., 96 (1996), 3201 - 3240. Talete srl, Dragon Profesional Software, Via V.Pisani, 13-20124 Milano(ITALY), 2005. Zakarya, D., Chastrette, M., Tollabi, M. and Fkih-Tetouani, S., Chemometrics and Intelligent Laboratory Systems, 48 (1999), 35–46.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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SIMULATION AND OPTIMIZATION IN 1,3-BUTADIENE PROCESS FROM C4-CUT USING GENETIC ALGORITHM Farhang Jalali*, Raheleh Saffari, Chem. Eng. Dept., Faculty of Eng., Univ. of Tehran, Iran [email protected]
ABSTRACT Separation of 1,3-butadiene from the C4-cut is not possible by conventional distillation due to formation of several azeotropic mixtures and very close boiling points of the components. The BASF 1,3-butadiene extraction process takes advantage of highly improved relative volatilities of most of the components in the presence of a selective solvent. The solvent is n-methylpyrrolidone (NMP) which contains approximately 8.3% water. In the present work, first,1,3-butadiene extraction process is simulated in steady-state conditions. The results of the steady-state simulation are compared with plant data and show a good agreement between these values. It was found that the NRTL equation of state is able to predict the experimental data satisfactorily throughout the process. The binary interaction coefficients of the components were tuned in this study such that the equation of state best fits the real equilibrium data. An optimization framework is proposed in this work for a synthesis of extractive distillation sequence, based on a modified genetic algorithm coupled with a sequential process simulator. In the methodology developed here simulation models are automatically generated through a process and are evaluated for various candidate configuration of the system. These candidates are suggested by a genetic algorithm that automatically guides the system towards better solutions. Keywords: Simulation, Optimization, 1,3-Butadiene
1
INTRODUCTION
1,3-Butadiene is a colorless, non-corrosive gas with mild aromatic or gasoline-like odor with the boiling point of –4.4° C at atmospheric pressure and liquid density 611 Kg/m3 at 20°C. It is used primarily as a co-monomer for producing styrene-butadiene rubber. Separation of 1,3butadiene from C4-cut by conventional distillation is not possible due to the formation of several azeotropes and very close boiling points of most of the components. The alternative process for separation of 1,3-butadiene from the C4-cut is extractive distillation. Extractive distillation is defined as distillation in the presence of a miscible, high boiling, relatively nonvolatile compound (i.e., the solvent) which forms no azeotrope with the other components in the
mixture. This separation technique is widely used in chemical and petrochemical industries for separating azeotropic, close-boiling and low relative volatility components from the mixture. In extractive distillation, the solvent is chosen to interact differently with the components of the original mixture, thereby, altering their relative volatilities. Since these interactions occur predominantly in the liquid phase, the solvent is continuously added close to the top of the extractive distillation column such that an appreciable amount of solvent is present in the liquid phase on all the trays below. N-Methylpyrrolidone (NMP), a solvent with high solubility and selectivity for unsaturated compounds which has been used successfully in numerous industrial plants, proved its merits for 1,3-butadiene extraction. The BASF process for the recovery of high purity 1,3-butadiene from C4-cut employs NMP as the selective solvent. Table 1 gives the selectivity of NMP in different mixtures. The high selectivity of NMP for 1,3-butadiene versus the less soluble butanes and because and more readily soluble acetylene compounds versus butadiene in the solvent makes NMP an ideal aid in the recovery of butadiene with optimum purity. The selectivity of NMP is also sufficient to separate 1,2-butadiene from 1,3-butadiene. Selectivity of NMP for propyne is relatively low. However, the difference between the boiling points is large enough to separate propyne from 1,3-butadiene down to the permissible level. In addition to the selectivity, other properties of NMP such as low vapor pressure, stability and proper solubility for acetylenes has made this solvent the best for extraction of 1,3-butadiene. Table 1: Selectivity of NMP (40° C, 760 mm Hg) Mixture
Selectivity
1,3-Butadiene / i- Butane
8.52
1,3-Butadiene / n- Butane
4.37
1,3-Butadiene / 1- Butene
2.66
1,3-Butadiene / 2-cis- Butene
1.65
1,3-Butadiene / Propyne
1.09
1,2-Butadiene / 1,3-Butadiene
1.88
1-Butyne / 1,3-Butadiene
2.46
Vinyl acetylene / 1,3-Butadiene
5.44
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2
PROCESS DESCRIPTION
The block diagram of BASF butadiene extraction process is shown in Figure 1 and comprises of two major process steps: • The extractive distillation section • The distillation section Butanes and Butense, which have the smallest solubility in NMP, are separated as the overhead product (raffinate) of the extractive distillation. A vapor stream consisting mainly of 1,3-butadiene and C4-acetylenes is sent to the second extractive distillation step. In this tower the more soluble acetylenic components are removed by means of fresh solvent, NMP. Crude butadiene is withdrawn as the overhead product of the second extractive distillation step. The C4-acetylenes are fully absorbed by NMP and are withdrawn from the degassing section. In the degassing section dissolved hydrocarbons are completely removed from the solvent. Crude butadiene obtained from the
Raffinate
Propyne
Extractive Distillation
Distillation
1,3-butadiene
C4-Cut
C4-acetylenes
C4/C5-HC
Figure 1- Block Diagram The extractive distillation section comprises the following subsections: 1. The extractive distillation section • Extractive Distillation I • Extractive Distillation • Degassing 2. The distillation section The distillation section comprises the following subsections: • Propyne Distillation Column • Final Distillation Column extractive distillation is further purified in two subsequent distillation towers. In the first distillation tower propyne together with some 1,3-butadiene for dilution is withdrawn
as the overhead product. In the second distillation tower a mixture containing 1,2-butadiene and C5-hydrocarbons are separated as the bottom product. The 1,3-butadiene product is withdrawn from the overhead of the final distillation tower.
3
STEADY-STATE SIMULATION
The first step in process simulation is to determine the suitable thermodynamic system, i.e., a proper equation of state with attention to the components and process conditions. For polar or non-ideal chemical systems usually binary thermodynamic systems are employed. In this way, an equation of state (such as ideal gas, Peng-Robinson, SoaveRedlich-Kwang, etc.) is used to predict the vapor fugacity coefficient while another equation of state (usually based on the activity model) is employed for the liquid phase. In the present study, NRTL is used as the main equation of state, for liquid phase with its binary interaction coefficients being modified in order to best represent the vapourliquid equilibrium behaviour of the system. In two distillation towers because of absence of NMP, Peng Robinson is used as the thermodynamic model. The vapor phase is assumed to be ideal in this work. The flow diagram of process is shown in Figure (2). According to the PFD. The liquid C4-fraction from battery limit is fed to the feed surge drum where fluctuations of the flow to the process can be compensated. The reboilers which are heated up with hot NMP that is routed from the bottom of the degassing tower make the feed to be vapored.. The required flow rate of C4-hydrocarbons to the main washer is measured in the vapor line connecting to drum with mainwasher and adjusted by flow control. The vaporized C4-cut enters the main washer at the bottom together with the top gas from the rectifier and is washed counter-currently with the NMP-solvent. This packed column is simulated with a tray tower that the number of trays is calculated in attention with the type of the packing and HETP. In this way 1,3-butadiene is almost completely absorbed. The overhead product contains the bulk of propane, propene, propadiene, butanes and butanes present in the C4-feed stock of the plant. The solvent N-methylpyrrolidone (NMP) with approx. 8.3% (w/w) water, is fed on the top bed of the main washer. The solvent flow is controlled by FIC. The solvent withdrawn from the bottom of main washer is pumped on the top of the rectifier. The level controller of main washer acts on a flow controller to ensure that the flow of solvent can be kept sufficiently constant. In the upper part of rectifier the less soluble i.e. more volatile butene are stripped from the solvent by a counter-current vapor stream of the more readily soluble butadiene rising from the bottom. The gasous mixture of 1,3-butadiene and butene is leaving the top of the rectifier and is fed back to the bottom of the main washer. The concentration of 1,3-butadiene in
Simulation and Optimization in 1,3-butadiene Process from C4-cut the vapor rises to its maximum between the rectifiers upper and lower part. From this location a butadiene-rich side-stream is withdrawn and fed into the bottom of the after-washer . The bottom of rectifier is divided into two compartments . the solvent loaded with hydrocarbons is drawn off from one compartment and routed to heat exchanger where it is heated up on the tube side of the exchanger train. Then it is flashed into the other compartment of the rectifier. By means of the pre-degassing a considerable amount of the dissolved hydrocarbons is vaporized. Butadiene still containing C4-acetylene is withdrawn as a gasous side-stream and fed into the bottom of after-washer. In this column butenyne (vinylacetylene) and 1-butyne (ethylacetylene) are removed from 1,3butadiene. The C4-acetylene are more soluble in NMP e.g. less volatile then 1,3-butadiene and are removed in counter-current with fresh solvent that is fed on the top of the column. The C4-acetylene absorbed in the solvent is
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drawn from the bottom of after-washer and transferred with after washer pump to the rectifier. Water which had been present in the overhead vapor is separated from the organic phase in the bootleg of an accumulator. the pre degassing solvent from the rectifier is heated up again in solvent heater and flashed into the degassing tower. The hydrocarbons dissolved in the loaded solvent from rectifier have to be separated completely before the solvent is recycled to the extractive distillation. The degassing of the solvent is carried out in the degassing tower by stripping with solvent and water evaporated in reboiler. The crude butadiene is fed to the propyne distillation column and the bottom product of the propyne column is fed into the final distillation column. Here components with lower volatility than 1,3-butadiene are separated as the bottom product while 1,3-butadiene is withdrawn as the overhead products.
Figure 2: Process Flow Diagram
The results of the steady state simulation for some main parameters of the process are shown in Tables 2 to 6. The corresponding experimental values are also given in the table. As could be seen from this figures, there is a good agreement between the simulation and actual plant data. Table 2: Comparison between design and simulation values in main washer column Parameter
Simulation value
Design value
Mass fraction of 1,3butadiene in overhead stream
0.18
0.19
Parameter
Simulation value
Design value
Mass fraction of i-butene in overhead stream
53.14
49.96
69.75
73.18
8.27
7.54
42.4
43.3
Mass fraction of 1,3butadiene in bottom stream Mass fraction of i-butene in bottom stream Temperature of bottom stream (°C)
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Table 6: Comparison between design and simulation values in final distillation tower
Table 3: Comparison between design and simulation values in Rectifier tower Parameter Mass fraction of 1,3butadiene in overhead stream Mass fraction of 1-butene in overhead stream Mass fraction of i-butene in overhead stream Mass fraction of butane in overhead stream
Simulation value
Design value
69.64
68.92
3.4
3.6
16.7
17.28
1.8
1.93
Mass fraction of 1,3butadiene in overhead stream
4
Mass fraction of 1,3butadiene in overhead stream Mass fraction of 1butyne in overhead stream Mass fraction of butenyne in overhead stream Mass fraction of 1,2butadiene in overhead stream
Simulation value
Design value
98.7
99.08
0.04
0.00
0.03
0.00
0.41
0.43
Table 5: Comparison between design and simulation values in degassing tower Simulation value
Design value
of 1,3overhead
81.15
80.79
of 1,2overhead
6.48
6.39
99.14
99.4
Parameter Mass fraction butadiene in stream Mass fraction butadiene in stream Temperature of stream (°C)
overhead
Simulation value
Design value
Mass fraction of 1,3butadiene in outletstream
99.46
99.7
Mass fraction of i-butene in outlet stream
0.0
0.045
Mass fraction of transe-2butene in outlet stream
0.022
0.21
Mass fraction of cis-2butene in outlet stream
0.30
0.22
78.9
Table 4: Comparison between design and simulation values in After washer tower Parameter
Parameter
OPTIMIZATION
Evolutionary algorithms (EAs) are based on the paradigm of evolution. Of course, the mechanisms of the nature for improving the extremely complex biological organisms can not easily be transferred to an algorithm for the optimization of technical systems. But several approaches with different degrees of complexity are developed as discussed for instance by Baeck and Schwefel (1993). In the terms of EAs a representation of the technical system to be optimized is called an individual. A population is a set of individuals competing with each other with respect to their target function values (fitness) Starting from an initial random population of individuals, new populations are iteratively created and their target function values are calculated by respective invocations of an evaluation function. Only the best individuals are selected as the basis (parents) for the next population. The process of generating new individuals from two randomly chosen parents is modeled by the genetic operators recombination, shown in figure 1 and mutation, a further stochastic variation for each single parameter setting of the newly created individual. The definition of these variations is strongly dependent on the representation of the individual, corresponding to different types of EAs. Genetic algorithms are stochastic methods based on the idea of evolution and survival of the fittest. In a GA, a set of values of the optimization variables forms an individual, usually codified in a chromosome through a series of bits (0–1). The algorithm starts generating a random population (a group of individuals), and then repetitively evolving it with three basic genetic operators: selection, crossover, and mutation. For a detailed explanation on genetic algorithms and operators. In the present work, Genetic Algorithm which is one of the most important of EAs is implemented to optimize some operation conditions and parameters in Butadiene extraction plant. It is calculated that Uniform crossover was preferred over single point, with 0.75 as the best
Simulation and Optimization in 1,3-butadiene Process from C4-cut value for the crossover probability .Generations of 25 individuals gave the best results in comparison with larger or smaller populations and the best values for the probability of the mutation is 0.005 which is a governing factor in the performance of the GA.
Optimal design Model construction
user
Codification and paremeters
5 IMPLEMENTION OF GENETIC ALGORITHM WITH A COMMERCIAL SIMULATOR Distillation is still one of the most important operations in process engineering, The modeling and formulation of such models, however, is difficult and time consuming. In addition to the time and expertise needed to formulate these models, the synthesis and design of distillation sequences pose other difficulties for their solution. The use of rigorous design and thermodynamic models leads to very large non-convex models and very difficult to converge. Moreover, taking into account structural and design decisions, such as the existence of stages, columns, condensers and reboilers, have lead to the inclusion of integer variables further increasing the difficulty of solving the model. To compensate for these difficulties, it is often necessary to supply initial values for the optimization variables very close to the actual solution, something that is not always an easy task, and even recent works have used simplifications for the design model, thermodynamics, hydraulics, or the cost functions to obtain feasible solutions or to examine complex superstructures in synthesis problems . An alternative to reduce some of these problems is the use of commercial simulators. These tools include a variety of highly efficient rigorous thermodynamic and design models that allow the process engineer to evaluate different flowsheets and modeling options in an easy way. However, the computational structure of modular simulators has not allowed the complete incorporation of the latest improvements in mathematical optimization algorithms, usually based on the evaluation of the gradients of the model, information that is not directly available in the simulator. A commercial simulation software is used as the simulator to evaluate the individuals generated by the GA. The values of the optimization variables are passed to the simulator through a program implemented in Visual BASIC to control the simulator. Through the client object the simulated file is open, closed, and manipulated to modify the structure of the flowchart process. The input and output data in the simulation are organized where the values of the variables within the simulation are read and modified. In the architecture used in this work, the user interacts with the GA defining the parameters of the algorithm, explicit constraints and convergence options, with the simulator, to select the mathematical models. This process is depicted in Fig. 3.
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Codified data, simulation input
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Simulation output, fitness
Fig3. Implemented architecture between simulator and the GA
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CONCLUSION
In this section, the optimization on operation conditions in an extractive distillation systems is presented to illustrate the advantages of the proposed framework. We have considered the After washer tower which is one of the main parts of this plant that the feed of distillation section is withdrawn from this tower. The objective function in this case is to maximize the mass percent of 1,3-Butadiene that is the overhead stream.
CASE 1 The aim is to optimize the mass percent of 1,3-Butadiene in the over head stream of after washer tower. The number of stages, the feeding stage and the reflux ratio are constant. The pressure at the top plate(the stage of solvent entrance) is 490 kPa and at the bottom( the stage of feeding) is 510 kPa. The solvent flow enters at 42°C and the mass flow of solvent is 33000Kg/hr. The column was simulated in the simulation software program. The thermodynamic properties were calculated with the equation of state of NRTL which were modified. Several numerical experiments were studied to analyze the behavior of the GA and the proposed strategies under different conditions, the variables in this case are the temperature of the solvent and the pressure of feed that should be optimize which were codified in the genetic algorithm resulting in a chromosome. The population size of the GA was set to 25 individuals and 80, 100 and 200 generations were defined. The table (7) shows the results of this optimization. The results shows that the mass percent of Butadiene in overhead stream will be increased by decreasing the temperature of entrance solvent and increasing the pressure of the feed. Its clear that because the solution of hydrocarbons in NMP is exothermic, the reduction of temperature and pressure increasing will maximize the mass percent of Butadiene in overhead stream.
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Generation
1 2 3
80 100 200
Solvent temperature (°C) 38.5 38 38
Feed pressure (Kpa) 513 513 513
1,3-Butadiene (%) 0.9759 0.9760 0.9760
Time (s) 25 45 130
Table 7: The result of operation conditions in after washer column(case 1)
CASE 2 In this case, again, the aim is to optimize the mass percent of 1,3-Butadiene in the overhead stream of after washer tower. But this time the variables are temperature and mass flow of solvent. Like case 1, The column was simulated in the simulation software program. The thermodynamic properties were calculated with the equation of state of NRTL which were modified. The population size
Item
Generation
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80 100 200
Solvent temperature (°C) 38.5 38 38
of the GA was set to 25 individuals and 80, 100 and 200 generations were defined. The table (8) declares the results of this optimization. The results shows that the mass percent of Butadiene in overhead stream will be increased by increasing the solvent temperature and mass flow.
Solvent mass flow (Kg/hr) 38935 38950 38950
1,3-Butadiene (%) 0.974 0.975 0.975
Time (s) 40 70 150
Table 8: The result of operation conditions in after washer column(case 2)
ACKNOWLEDGMENTS The research described in this paper is supported by Amir Kabir petrochemical company. The authors would like to be gratefully for technical and financial supports.
REFRENCES Amir Kabir petrochemical company Documents. William L.Luyben. 1990. Process modeling, simulation and control for chemical engineers. McGraw-Hill International Editions. William D.Mccain,Jr. 1990. The properties of petroleum fluids. PennWell publishing company. J,Vladimir de Oliveria,A.M.Cohen Uller, 1995, Solubility of pure 1,3 butadiene an methyl propene and their mixture in pure n-methyl-2-pyrrolidone and its aqueos solutions; Fluid Phase Equilibria 118(1996) 113-14. Mario Llano-Restrepo, Jaime Aguilar-Arias, 2002 Modeling and simulation of saline extractive distillation columns for the production of absolute ethanol, Computer and Chemical Engineering 27(2003) 527-549.
Goldberg, David E. “Genetic Algorithms in Search ,Optimization and Machine Learning”. AddisonWesley Pub. Co. 1989. David A Coley, “An Introduction to Genetic Algorithms for Scientists and Engineers”, World Scientific, 1998. Goldberg, David E. “The Design of Innovation : Lessons from and for competent Genetic Algorithms”, Boston, MA: Kluwer Academic Publishers, 2002. Perry R.H & Chilton C.H. “Chemical Engineering Handbook”, 5 th ed, MC Graw Hill, 1973. SRI(Stanford Research Institue), document NO.247, Butadiene with NMP, Page 60-64 Kefeng Wang, Yu Qian!, Yi Yuan, Pingjing Yao, “Synthesis and optimization of heat integrated distillation systems using an improved genetic algorithm”, Computers and Chemical Engineering 23 (1998) 125-136. Jose Leboreiro, Joaquin Acevedo, “Processes synthesis and design of distillation sequences using modular simulators”, a genetic algorithm framework”, Computers and Chemical Engineering , 28 (2004) 1223– 1236.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Property Clustering and Group Contribution for Process and Molecular Design Fadwa Eljacka, Mario Edena, Vasiliki Kazantzib, Mahmoud El-Halwagib a
Dept. of Chemical Eng., Auburn University, Auburn, AL 36849, USA Dept. of Chemical Eng., Texas A&M University, College Station, TX 77843, USA
b
Abstract In this work, property clustering techniques and group contribution methods are combined to enable simultaneous consideration of process performance requirements and molecular property constraints. Using this methodology, the process design problem is solved to identify the property targets corresponding to the desired process performance. A significant advantage of the developed methodology is that for problems that can be satisfactorily described by three properties, the process and molecular design problems can be simultaneously solved visually, irrespective of how many molecular fragments are included in the search space. On the ternary cluster diagram, the target properties are represented as individual points if given as discrete values or as a region if given as intervals. The structure and identity of candidate components is then identified by combining or “mixing” molecular fragments until the resulting properties match the targets. Keywords: Property integration, group contribution, process and molecular design
1. Introduction Product synthesis and design problems involve identification and selection of compounds or mixtures capable of performing certain tasks or possess certain physical properties. As the feasibility of using a given compound is dictated by its properties, it would seem appropriate to employ a property driven solution methodology for the molecular synthesis. Numerous contributions have been made in the areas of molecular synthesis and Computer Aided Molecular Design (CAMD) [1,2], however in order to utilize these techniques the desired component properties must be specified ahead of design. Doing so may lead to suboptimal designs, as the property targets for a new compound inherently will be dictated by the process, where it is to be employed. Thus there is a critical need for a systematic methodology capable of addressing both problems simultaneously, i.e. identify the target properties of a new compound from the process design problem and then synthesize molecular structures that match the targets.
2. Property Integration Framework The property integration framework enables representation of processes and products from a properties perspective [3-5]. The framework enables identification of the desired component properties by targeting optimum process performance without committing to any components during the solution step [6]. The identified property targets can then be used for solving a molecular design problem, which returns the corresponding components. To provide a unifying methodology for handling both process and molecular design problems, the property integration framework is extended to include Group Contribution Methods (GCM), which allow for prediction of physical properties
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from structural information [7-9]. By combining the two approaches, a framework is obtained that enables representation of processes from a properties perspective, while utilizing proven methods for synthesizing molecular structures with certain properties.
3. Property Clusters for Molecular Design Property clusters are defined as conserved surrogate properties that are functions of the unconserved properties. The clustering technique utilizes property operators, which are functions of the original raw physical properties [3]. Although the operators themselves may be highly non-linear, they are tailored to possess linear mixing rules, e.g. density does not exhibit a linear mixing rule, however the reciprocal value of density follows a linear mixing rule [3-5]. It is clear that the operator expressions will invariably be different for molecular fragments and process streams, however as they represent that same property, it is possible to visualize them in similar fashion. Extending the original property integration framework to include GCM for molecular design requires the introduction of molecular property operators. Fortunately, the equations employed in GCM are similar to the original property operator formulation, i.e. the properties are described by linear additive rules for the individual molecular fragments [7-9]: Ng
ψ Mj ( Pj ) = ∑ ng ⋅ Pjg
(1)
g =1
In Eq. (1), ψMj (Pj) is the molecular property operator of the jth property. The RHS of the equation is always in the form of summation of the number of occurrences of each group (ng) multiplied by the contribution to property j from group g (Pjg). Some properties are not predicted directly, but are estimated as functions of other properties that can be predicted using GCM, e.g. vapor pressure is estimated from the boiling point, which is a property described by GCM [10]: 1.7
⎛T ⎞ log VP = 5.58 − 2.7 ⎜ bp ⎟ ⎝ T ⎠
⎛T ⎝ tb 0
ψ M (Tbp ) = exp ⎜
⎞ Ng ⎟ = ∑ ng ⋅ tbg ⎠ g =1
(2)
It should be noted that although the molecular property operator expression can be very complex, the mixing rules for the molecular fragments are simple linear additive rules. This attractive feature enables simple formulation of molecules on the ternary cluster diagram. Next, the molecular property operators can be converted to clusters according to the procedures developed for the original property clusters [3-5]. To ensure that the operator values are in the same order of magnitude, the molecular property operators are normalized by dividing by a reference operator. The Augmented Property index AUPM for each molecule is defined as the sum of all the NP dimensionless property operators (ΩM), and finally the property cluster Cj is obtained as the ratio of ΩM and AUPM:
Ω Mji =
ψ M ( Pji ) j
ψ ref j ( Pji )
NP
AU P M = ∑ Ω Mj j =1
Cj =
Ω Mj AU P M
(3)
In the original cluster formulation for process design, mixing of two sources is a straight line, i.e. the mixing operation can be optimized using lever-arm analysis. Analogously,
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combining or “mixing” two molecular fragments in the molecular cluster domain follows a straight line (an illustrative example is given in Figure 1 below). Furthermore, when the property targets are given as intervals or ranges of property values, the feasibility region can be described by the same six unique points as for the original clusters [4,5]. Now that the process and molecular design problems are both described in terms of clusters, a unifying framework exists for simultaneous solution of property driven design problems. In addition, the clustering technique reduces the dimensionality of both problems, thus it is possible to visually identify the solutions, which is a significant advantage of this approach. Design and optimization rules have been developed for property based process design problems [4,5], and in the following similar rules are presented for property based molecular design problems. Rule 1: Two groups, G1 and G2, are added linearly on the ternary diagram, where the visualization arm β1, describes the location of G1-G2 molecule.
β1 =
n1 ⋅ AUP1 n1 ⋅ AUP1 + n2 ⋅ AUP2
(4)
Rule 2: More groups can be added as long as the Free Bond Number (FBN) is not zero.
⎡ Ng ⎤ ⎡ Ng ⎤ FBN = ⎢ ∑ ng ⋅ FBN g ⎥ − ⎢ ∑ ng − 1⎥ − 2 ⋅ NORings ⎣ g =1 ⎦ ⎣ g =1 ⎦
(5)
FBN is the free molecular bond number of the formulation, ng is the number of occurrences of group g, FBNg is the unique free bond number associated with group g, and NORings is the number of rings in the formulation. Rule 3: Location of the final formulation is independent of the order of group addition. Rule 4: For completeness, the final formulation must not have any free bonds, i.e. FBN has to be equal to zero. Given a completed molecular formulation, three conditions must be satisfied for the designed molecule to be a valid solution to the process and molecular design problem. Rules 5 and 6 are the necessary conditions, while rule 7 is the sufficient condition: Rule 5: The cluster value of the formulation must be contained within the feasibility region of the sink on the ternary molecular cluster diagram. Rule 6: The AUP value of the designed molecule must be within the range of the target. If the AUP value falls outside the range of the sink, the designed molecule is not a feasible solution. Rule 7: For the designed molecule to match the target properties, the AUP value of the molecule has to match the AUP value of the sink at the same cluster location. It should be emphasized, that any molecular design that is obtained from GCM is only as good as the accuracy of the prediction method. This is a general shortcoming, which
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is not unique to the implementation in the property integration framework. Therefore, once a candidate formulation of fragments is translated to actual chemicals, it has to be checked that the properties of these chemicals match all the property targets, including properties not directly estimated from GCM. C2 0,9
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Figure 1. Example of visual molecular synthesis.
4. Case Study – Molecular Design of Metal Degreasing Solvent This case study has been used in earlier works to highlight different process design aspects of the property integration framework [3-5]. The case study deals with exploring the possibility of condensing off-gas VOCs to reduce the use of a fresh solvent. In this work, the focus is on the molecular design of a fresh solvent matching the property targets identified in previous works. Three properties are used to evaluate the suitability of a given organic process fluid for use in the degreaser: • • •
Sulfur content (S) - for corrosion consideration, expressed as weight percent. Molar Volume (Vm) - for hydrodynamic and pumping aspects. Vapor Pressure (VP) – for volatility, makeup and regeneration.
The synthesized solvents will be pure components; hence there is no need to include the sulfur content in the molecular design problem. Therefore an additional property constraint, i.e. heat of vaporization, is placed on the candidate solutions. The target property ranges obtained from solving the process design problem [3-5] are outlined in Table 1. The target vapor pressure range is converted to the corresponding boiling point range according to Eq. (2). Property Hv (kJ/kg) VP (mmHg) Vm (cm3/mol) Tb (K)
Lower Bound 50 1825.4 90.1 418.01
Upper Bound 100 3878.7 720.8 457.16
Table 1: Target property ranges for degreaser solvent.
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The molecular property operators are identified from the group contribution equations given below [8,11]:
Vm = d + ∑ ng ⋅ vg
ΔH v = hv 0 + ∑ ng ⋅ hv g
Tb = tb 0 ⋅ ln ∑ ng ⋅ tbg
g
(6)
g
For visualization purposes and to illustrate the methodology, only the eight molecular groups given in Figure 2 are used to synthesize candidate molecules. Converting the property contributions of the individual fragments to cluster values results in eight discrete points, while the property targets in Table 1 are visualized as a feasibility region [3-5]. The resulting ternary cluster diagram is shown in Figure 2, where the dotted line outlines the feasibility region. C2
Molecular Groups
0.9
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0.6
G1: CH3 G2: CH2 0.7 G3: CH2O G4: CH2N 0.6 G5: CH3N G6: CH3CO 0.5 G7: COOH G8: CCl 0.4
G5 G3
0.7
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G6
G7 0.2
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0.9
C1
Figure 2: Ternary diagram used to represent molecular synthesis problem
Seven candidates molecules, denoted M1-M7 in Figure 3, are identified as possible solutions to the molecular design problem. They all satisfy the necessary condition of being located within the feasibility region (Rule 5). When comparing the AUP values of the molecules to the AUP range of the feasibility region (Rule 6), M5 and M6, fail to satisfy this constraint. Next step is to compare the AUP values of each molecule to the AUP value of the feasibility region at the same cluster point (Rule 7). This is a necessary condition for ensuring that the property values will match the original property targets when mapped back to the property domain. Candidate M3 fails to satisfy this criterion. Although candidate M7 satisfies the three properties estimated using GCM, i.e. Hv, Vm and Tb, when calculating the corresponding vapor pressure it fails to satisfy the property target. Consequently, the only feasible candidates are M1, M2, and M4, which correspond to 2-octanone, 2,5-hexadione, and butanoic acid, respectively [12]. When the candidate solvents are mapped back into the process design
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domain (not shown in this paper), the optimal solution 2,5-hexadione is identified visually using lever arm analysis to maximize the usage of the recycled condensate. C2 0.9
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CH3-(CH2)5-CH3CO 0.7 CH3CO-(CH2)2-CH3CO (CH3)3-(CH2)5-CH2N 0.6 CH3-(CH2)2-COOH (CH3)2-CH30.5CO-CCL -(CH2O)5- ring CH3-(CH2)2-CH 0.43N-COOH
M6 M3 M1
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Figure 3: Candidate formulated molecules
5. Conclusions In this work, a systematic property based framework for simultaneous solution of process and molecular design problems has been presented. The recently introduced property integration framework has been extended to include group contribution methods for molecular design. Using property clusters, the process design problem is solved to identify the property targets corresponding to desired process performance. The molecular design problem is solved to systematically generate structures that match the targets. Ternary cluster diagrams enable visualization of the problem and solution.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
P. M. Harper and R. Gani, Comp. & Chem. Eng., 24, 2000 E. C. Marcoulaki and A. C. Kokossis, Comp. & Chem. Eng., 22, 1998 M. D. Shelley and M. M. El-Halwagi, Comp. & Chem. Eng., 24, 2000 M. R. Eden, S. B. Jørgensen, R. Gani and M. M. El-Halwagi, Chem. Eng. & Proc., 43, 2004 M. M. El-Halwagi, I. M. Glasgow, M. R. Eden, X. Qin, AIChE Journal, 50, 2004 M. R. Eden, S. B. Jørgensen, R. Gani and M. M. El-Halwagi, Computer Aided Chemical Engineering, 15A, 2003 K. G. Joback and R. C. Reid, Chemical Engineering Communication, 57, 1983 L. Constantinou and R. Gani, AIChE Journal, 40, 1994 J. Marrero and R. Gani, Fluid Phase Equilibria, 182-183, 2001 M. Sinha and L. E. K. Achenie, Advances in Environmental Research, 5, 2001 L. Constantinou, R. Gani and J. P. O’Connell, Fluid Phase Equilibria, 103, 1995 Computer Aided Process Engineering Center (CAPEC), ICAS User Guide, 2005
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A combinatorial formulation for optimal sizing, scheduling and shift policy in designing the milling section of a ceramic tile industrial plant B.P.M. Duarte,a,b L.O. Santos,b J.S. Mariano,a a
Dep. of Chemical Engineering, ISEC, Polytechnic School of Engineering, R. Pedro Nunes, 3030-199 Coimbra,Portugal b GEPSI - PSE Group, CIEPQPF, Dep. of Chemical Engineering, University of Coimbra, Pólo II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal
Abstract A modelling approach to optimally design batch labor intensive units is proposed. It stands on the trade-off between the depreciation cost of the equipment to install and the cost of resources, such as energy and manpower. The plant schedule is modelled employing a discrete Resources Task Network framework. It allows the inclusion of resources availability constraints in the design problem formulation, such that the problem of finding the optimal shift policy is inclosed. The resulting modelling strategy is applied to the design of the grinding section of a ceramic tile industrial plant. Keywords: Capacity planning, Resources allocation, MILP, Ceramic tile industry.
1. Introduction The optimal design of batch units, such as the sizing of equipment to install and of the connections to build between pieces of equipment, is nowadays state of the art [Yeh,1987; Ravemark, 1998]. Typical sizing problems stand on the minimization of the installation cost of the equipment subject to operational/control/safety constraints. On the other hand, the optimal plant scheduling stands on the minimization of the operation cost or similar criteria, such as the profit maximization or the tardiness minimization, subject to the resources availability and the plant production sequence. In the later years the integration of design and scheduling of multipurpose/multiproduct batch units into a single framework deserved large attention [Lin, 2001; Castro, 2005]. In fact, the optimal design, particularly the sizing, of labor intensive or semi-labor intensive (LI/SLI) batch units stands on the trade-off between the equipment installation cost and the plant operation cost. Such a trade-off requires that both these two cost components are taken in account into a single time basis. This leads to the replacement of the installation cost by the depreciation cost of equipment providing that its life time is fixed a priori. The resources involved in the production, such as manpower, may strongly interact with plant design. Indeed, the design obtained from considering only the equipment cost may lead to non-optimal/infeasible solutions with respect to both components of the cost function and resources availability constraints. Since one of the features of LI/SLI units is the dependency on labor, the optimal design must address all the decisions concerning the size of equipment, commonly available in standard sizes, the plant schedule, the manpower schedule, and the shift policy to be adopted. Typically, ceramic tile industrial plants feature continuous operating units such as furnaces and involve some degree of labor-intensive batch units as well, such as the
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grinding units. In general, the grinding section of the plant is devoted to the production of a single product (ground clay) which is later fed to formation units. The tasks involved in the grinding section are performed according to a given operation sequence and require manpower and energy resources. Since the design problem involves the determination of the scheduling policy given the available manpower, this leads to a problem of the class of the LI/SLI design problems
2. Problem Modeling The design problem introduced in the previous section comprises decisions regarding: the number of units to install; the size of each unit, assuming that only standard sizes are available; the optimal schedule of the plant, which leads to a resources schedule; the shift policy to implement for manpower. The short-term scheduling problem is modelled employing a discrete Resource Task Network framework [Pantelides, 1994]. The process is described by a set of topological entities comprising two types of nodes: tasks and resources. The horizon, H, employed for short-term scheduling is discretized into time slots bounded by event points where process changes occur. Table 1. Parameters and variables of the optimization problem
D i I k K m M
Total demand Unit type Set of unit types Task Set of tasks Unit number Set of unit numbers
p P r R s S
Shift policy t Time event Set of shift policies T Set of time events Resource Slot interval Δt Set of resources Ω k Duration of task k θ State Time index Set of states Z* Optimum Cost of resource r at time t Q r ,t
sf
Task that releases final product
τk
Ci Vi S p,m,i , s ,t
Number of time intervals of task k Depreciation cost of unit type i Volume of the unit of type i Binary variable that assigns state s at unit m of type i at t, shift policy p
N p ,m,i,k ,t
Binary variable that assigns task k at unit m of type i at t, shift policy p
wk , s ,t
Binary variable that assigns the consumption of state s in task k at time t
R p,r ,t
y p,m,i
Binary variable that assigns the availability of resource r at t, shift policy p Binary variable that assigns the consumption/release of r at t, shift policy p Binary variable that assigns the unit m to type i, shift policy p
xp
Binary variable that assigns the shift policy p
u Τ p, r
Set of time events at which resource r is not available for shift policy p
u p , r ,t
The resulting model, based on resources and process states balances at the event points, has the form of a MILP problem. In the later years the literature focused specially on continuous RTN formulations and these definitely overcame the number of applications using a discrete RTN representation [Zhang, 1996]. Nevertheless, in this work the
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discrete representation is employed to account for the specificity of the resources involved in the grinding process, and for its time dependent availability. Indeed, manpower and energy usage must satisfy labor contracts enforced by some industrial sectors, and must take into account the energy commercialization policies as well. To handle these features the initial point of the horizon H, t 0 , is fixed a priori and the resources availability constraints are imposed accordingly. The resulting combinatorial formulation stands on the assumption that the plant topology is only partially known a priori, because the sequence of operations of each piece of equipment devoted to each task is known, not the number of pieces of equipment needed. The depreciation cost is determined based on the equipment cost and its corresponding life time [Turton, 1998], and the sizes of the equipment to buy are standard. The parameters and variables involved in the formulation are listed in Table 1. The model is given by: Z * = min
∑ ∑ ∑ y p,m,i C i + ∑ ∑ ∑ u p,r ,θ Qr ,θ p∈P r∈R θ ∈T
p∈P m∈M i∈I
τk
S p,m,i, s ,t = S p,m,i, s ,t −1 + ∑ ∑ w k , s,θ N p,m,i,k ,t −θ k∈K θ =0
τk
(Eqn. 1 )
(Eqn. 2 )
R p ,r ,t = R p,r ,t −1 + ∑ ∑ ∑ ∑ u p ,r ,θ N p ,m,i,k ,t −θ
(Eqn. 3 )
N p ,m,i,k ,t ≤ y p ,m,i , ∀p ∈ P, m ∈ M, i ∈ I, k ∈ K, t ∈ T
(Eqn. 4 )
∑ ∑ y p,m,i ≤ 1, ∀ m ∈ M
(Eqn. 5 )
∑ y p , m ,i ≤ x p ∀ m ∈ M , p ∈ P
(Eqn. 6 )
∑ xp =1
(Eqn. 7 )
m∈M i∈I k∈K θ =0
p∈P i∈I
i∈I
i∈I
∑ ∑ ∑ N p,m,i, s f ,t Vi ≥ D, ∀ p ∈ P
(Eqn. 8 )
m∈M i∈I t∈T
∑ y p,m,iVi ≤ ∑ y p,m−1,iVi , ∀ m > 1, p ∈ P
i∈I
(Eqn. 9 )
i∈I
S p ,m,i, s ,0 = 0, ∀ p ∈ P, m ∈ M , i ∈ I , s ∈ S
(Eqn. 10 )
R p ,r ,t = 1, ∀ p ∈ P, r ∈ R, t ∈ T pu,r
(Eqn. 11 )
⎛ Ωk ⎝ Δt
τ k = int ⎜⎜
⎞ ⎟⎟, ∀ k ∈ K ⎠
(Eqn. 12 )
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with y, N , S , R, x ∈ {0,1} . Eqn. 1 is the total cost accounting for both the costs of the equipment depreciation and of the resources. This formulation assumes that there are no limitations on the availability of raw materials and other plant utilities. Eqn. 2 and 3 represent the states and the resources balances at each event point, respectively. Eqn. 4 represents the allocation of unit m of ith discrete size whenever at least one task takes place on it. Eqn.5 aims at reducing the number of nodes to search in the integer problem solution by stating that each unit can only have one discrete capacity i ∈ I for each shift policy p ∈ P to implement. Eqn. 6 states that shift policy p is assigned whenever there is at least an unit operated by following it. Eqn. 7 states that only one shift policy p ∈ P is to be implemented. Eqn. 8 represents the production demand satisfaction condition. This demand can be either internal, if the product is to be processed by another plant section, or external, in the case the product is to be sold. Eqn. 9 enforces that the process units needed are numbered in a decreasing order of their capacities. The purpose of this condition is also to reduce the integrality gap of the resulting MILP problem presented to the solver. Eqn. 10 and 11 represent the availability of states and resources at the beginning of the time horizon, respectively. Finally, Eqn. 12 determines the number of event points to complete each task.
3. Application The modelling approach presented in Section 2 is applied to the problem of the optimal design of the grinding section of a ceramic tile industrial plant. This section runs in batch mode and involves a sequence of three tasks, k=1,2,3: • Feed the mill with raw material. This task changes the state of the mill from empty to filled, and needs one operator to control the feed operation during 1 hour; • Grind the material filled. This task changes the state of raw material to ground clay, requires the allocation of a mill unit during 13 hours and consumes energy; • Release the ground clay. Here the mill state is changed from filled to empty, and the mill is made ready for a new load. This requires also an operator to control the discharge and clean the mill during 1 hour. The grinding section is operated based on an intermediate finite storage capacity transfer policy, since the mills can be programmed to start later the grinding operation, and the ground clay can wait inside the mill after the grinding, until the manpower is available to proceed with the discharge. The standard sizes of mills to install available in the market and its cost data is presented in Table 2 The depreciation cost is determined based on equipment cost assuming a life time of 9.5 years \cite{Turton1998}. The energy resource has a lower cost at weekend and during the night period (Table 3). The manpower cost is estimated from the average salary practiced by the Portuguese ceramic industrial sector. The alternative shift policies, Pi, i = 1, " ,3 to implement are set according to the Portuguese Labor Contract work policy for the ceramic industrial sector: P1: 1 worker, 5 days a week; working period: 8:30-12:30 & 14:00-18:00; P2: 2 workers, 5 days a week; working period: 7:00-15:00 & 14:00-22:00; P3: 5 workers, 7 days a week covering 24 hours. The MILP problem formed by Eqn. 1-12 is solved with GAMS/CPLEX 9.0.2, using a tolerance of 10 −6 . The short-term horizon H comprises 180 hours (7.5 days). The initial time of the horizon, t 0 , is set to 0:00 of Monday, and the time slot is 1 hour.
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Table 2. Discrete net capacities and costs of mill units available in the market. Designation
Net capacity (l)
Depreciation (€/week)
Energy consumption (kWh)
M1 M2 M3 M4
50000 35000 25000 15000
1084.44 867.55 759.11 672.35
105.0 92.5 80.0 67.5
Table 3. Energy cost data (€/kWh). Day
Period 1
Period 2
Period 3
Mon.-Fri. Sat. Sun.
0.2162
0.0945
0.0481
Period 4
Period 5
0.0945
0.0481
Period 6
0.0481
Period 1: 9:30-12:00 & 18:30-21:00; Period 2: 7:00-9:30, 12:00-18:30 & 21:00-24:00; Period 3: 0:00-7:00; Period 4: 9:30-13:30 & 18:30-22:00; Period 5: 0:00-9:30, 13:30-18:30 & 22:00-24:00; Period 6: 0:00-24:00
Table 4 presents the results for six demand scenarios, ranging from a very low demand (1000 m2/day) to a large demand (10000 m2/day). The optimal shift policy for demand scenarios that require more than 1 mill unit is P3. Indeed, the cost to increase manpower availability is lower than the depreciation cost of an additional grinding unit. Table 4. Optimal solutions Scenario
D+
M1
M2
M3
M4
Shift policy
Z* (€)
CPU time# (s)
S1 S2 S3 S4 S5 S6
1000 1500 2000 3000 6500 10000
1 1 2 3
1 1 -
-
-
P1 P1 P3 P3 P3 P3
1624.866 1922.271 2707.026 3291.026 6014.384 8769.689
497.11 84.59 138.28 293.17 794.21 748.69
+
Market demand (m2/day).
#
On a Windows Pentium 2.80 GHz computer.
The CPU time (Table 4) shows a non-regular behavior as the market demand increases. A demand scenario may require a combination of units/sizes that enables to fulfill the target, even operating the units to install in sub-intensive modes. Indeed, some of those units are operated based on a sequence where they produce less batches than the maximum they can possibly produce during the time horizon considered. This feature leads to degenerated solutions. That is, some batches and tasks can be assigned to different time events without any major difference in the objective function. When this situation occurs the final assignment is fully dictated by the energy cost policy. Figures 1 and 2 present the plant schedule for demand scenarios S1 and S5, respectively. Scenario S1 is based on one-man shift policy, and one may see that the mill grinds a batch during the weekend in order to take advantage of the lower cost of energy. That is also the reason for running the grinding task during the night, a strategy which is adopted independently of the demand target. For scenario S1 the filling and discharge tasks are concentrated during the period the manpower resource is available (during the day). On the opposite, the grinding operation starts later in order to use cheaper energy. To fulfill the demand of scenario S5, one may observe that mill 2 is not
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operated at maximum rate, since there is room to produce an additional batch, similarly to unit 1.
Figure 1. Schedule of the grinding section for demand scenario S1.
Figure 2. Schedule of the grinding section for demand scenario S5.
4. Conclusions A modeling approach to determine the optimal design of batch labor intensive units based on the trade-off between the depreciation and operation costs is presented. The model is developed using a discrete Resources Task Network framework. It accounts explicitly for manpower availability constraints arising from the Labor Contract legislation, and from the commercial policies practiced by the energy supplier. The optimal design comprises decisions with respect to the number/size of units, the plant schedule and its shift policy. This modelling approach is applied to the design of the grinding section of a ceramic tile industrial plant, which is a typical example of semilabor intensive plants. The results provide a good and rational basis to optimally install and run the plant.
References N.C. Yeh, G.V. Reklaitis, (1987), “Synthesis and sizing of batch/semicontinuous processes”, Computers Chem. Engng., No. 11, 639. D.E. Ravemark, D.W.T. Rippin, (1998), “Optimal design of a multi-product batch plant”, Computers Chem. Engng., No. 22, 177. X. Lin, C.A. Floudas, (2001), “Design, synthesis and scheduling of multipurpose batch plants via an effective continuous-time formulation”, Computers Chem. Engng., No. 25, 665. P.M. Castro, A.P. Barbosa-Póvoa, A.Q. Novais, (2005), “Simultaneous design and scheduling of multipurpose plants using Resource Task Network based continuous-time formulations”, Ind. Eng. Chem., No. 44, 343. C.C. Pantelides, (1994), “Unified frameworks for optimal process planning and scheduling”, Foundations of Computer Aided Process Operations, D.W.T. Rippin, J.C. Hale, J.F. Davis (eds.), 253. X. Zhang, R. Sargent, (1996), “The optimal operation of mixed production facilities General formulation and some approaches for the solution”, Computers Chem. Engng., No. 20, 897. R. Turton, R.C. Bailie, W.C. Whiting, J.A. Shaeiwitz, (1998), Analysis, synthesis and design of chemical processes, Prentice-Hall, Inc., New Jersey.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
An automated method for synthesizing a multistream heat exchanger network based on stream pseudo-temperature Dongwen YUAN,a Yao WANG*,a Wu XIAO,a Pingjing YAO,a Xing LUO,b Wilfried ROETZELb a
Institute of Process Systems Engineering, Dalian University of Technology,Dalian 116012, P.R.China b Institute of Thermodynamics,University of the Federal Armed Forces Hamburg,D22039 Hamburg,Germany
Abstract This paper proposes an approach for synthesizing multi-stream heat exchanger network based on the effective temperature levels of streams named “stream pseudo-temperature”. Pseudo-temperature is obtained by the stream heat transfer temperature difference contribution value which is optimized with Genetic/Simulated annealing algorithm. Pinch point and utilities are presented, and a temperature-enthalpy diagram is constructed based on the stream pseudo-temperature. An automated method for multi-stream heat exchanger network synthesis using the temperature-enthalpy diagram is given. The performance of the proposed approach is demonstrated using an example and better solution is obtained compared with literatures. Keywords: multi-stream heat exchanger network; pinch; temperature-enthalpy graph; temperature difference contribution value
1. Introduction Heat exchanger network synthesis(HENS) is a complex combinatorial problem. The conventional Pinch Design Method(PDM) employs a single allowable minimum heat transfer temperature difference (ΔTmin) . As ΔTmin increases, the relative position of the hot and cold composite curves on the Temperature Enthalpy (T-H) diagram are apart from each other, hence all temperature differences throughout the heat exchanger system increase, resulting in a reduced heat transfer area and capital cost, while at the same time, the utility requirements and their cost increase. As ΔTmin decreases, vice versa. At a value of ΔTmin, ΔTopt ,there will be a minimum in the total cost. The optimal ΔTmin, ΔTopt,can be used to the HENS.The solution achieved through the conventional PDM is based on some assumptions: all the exchangers in the network have the same heat transfer coefficient;the same materials of construction and the uniform type; The global optimal cost network can not be reached due to these assumptions. From the industrial point of view, this paper describes an approach for multi-stream heat exchanger network (MSHEN) synthesis that uses the heat transfer temperature difference contribution values of the streams. The calculation of the heat transfer
*E-mail: [email protected] Supported by the Deutsche Forschungsgemeinschaft (DFG NO. RO 294/9)
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temperature difference contribution values considers the differences between streams in heat transfer coefficient and materials of construction etc..
2. Stream Pseudo-temperature To have an insight into the stream heat transfer ability level in the heat transfer process, the stream apparent temperature (indicated by thermometer) should be shifted as stream pseudo-temperature. The stream pseudo-temperature is defined as: (1) T p = T ∓ ΔTc where, “－” is for hot stream; “＋” is for cold stream. Tp － stream pseudo-temperature, K; T － stream apparent temperature, K; ΔTc － stream temperature difference contribution value, K. The temperature difference between any two streams in heat exchange is expressed as the sum of two parts, either from each stream. ΔTm = ΔThot ,c + ΔTcold ,c (2) The suitable ΔTm is used for determining the matches between hot and cold streams in a network. 2.1. Effect of the film heat transfer coefficient on Tp Stream film heat transfer coefficient (h) (including film, wall and fouling contribution) is the crucial physical nature in the heat transfer process. When h-values vary significantly, the ”vertical match” against the composite curves can not give the minimum area network,therefore the “non-vertical” or “ criss-crossing” match, which is more complicated, may be reasonable. Because the stream with lower h-value needs a larger heat transfer temperature difference to reduce the area requirement, a stream with a higher film heat transfer coefficient can exchange heat with smaller approach temperature. Ahmad[1] presented equation (3) to determine the stream heat transfer temperature difference contribution value: ΔTi , c = C / hi (3) Where, hi －the heat transfer film coefficient of stream i, kW•m-2•K-1. C is a constant that is adjusted until the heat recovery level obtained by using the temperature difference contributions is the same as the one using a single global minimum approach temperature. 2.2. Effect of the cost per unit heat transfer area on Tp A corrosive stream that requires special materials of construction will have a greater contribution to capital cost than a non-corrosive stream. Consequently, in order to reduce the total network capital cost, the special material area from the contribution of corrosive stream must be reduced by sharing out a larger heat transfer temperature difference, and at the same time, the area from the contribution of non-corrosive stream will be forced to increase due to its heat transfer temperature difference becoming smaller. In this case, the minimum area network is not equivalent to the minimum capital cost network. In order to share out heat transfer temperature differences in a network for various streams that require mixed materials of construction for minimizing the total network capital cost, Nishimura[2] presented equation (4) which showed the relationship between the heat transfer temperature difference with the cost per unit heat transfer area. It gives the solution to minimize the heat transfer area per unit heat exchanging duty.
An Automated Method for Synthesizing a Multi-Stream Heat Exchanger Network
(T1 − t )
Where,
a1 a a a = (T2 − t ) 2 = … = (Ti − t ) i = (Tn − t ) n U1 U2 Ui Un
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(4)
Ti － the temperature of hot stream i, K; t － temperature of cold stream, K; ai － capital cost per unit area of heat exchanger, $•m-2; Ui － the heat transfer film coefficient, KW•m-2•K-1.
2.3. Effect of the stream thermodynamic mean temperature on Tp In the energy balance, work and heat are counted the same. In the exergy balance, work and heat are not counted the same. All work input increases the exergy of streams flowing through the process. As to the heat, only a portion of the heat transferred into a system is available to increase the exergy of flowing streams. The heat is degraded by a coefficient 1–(To/T), where To and T refer to environment and heat source temperature (K) respectively. Consider a steady state heat transfer between the high-and lowtemperature reservoirs in Fig.1
T1 Q
T2
Fig.1 Heat exchange between two reservoirs The loss of exergy(lost work, Lw) is determined as following:
Lw = (1 −
T0 T T T T −T ΔT )Q + (1 − 0 )(−Q) = Q( 0 − 0 ) = QT0 ( 1 2 ) = QT0 ( ) (5) T1 T2 T2 T1 T2 ⋅ T1 T2 ⋅ T1
and then the approach temperature
Lw ⋅ T1 ⋅ T2 Lw Lw = T1 (T1 − ΔT ) = (T12 − T1 ⋅ ΔT ) (6-a) Q ⋅ T0 Q ⋅ T0 Q ⋅ T0 Lw Lw (T22 + T2 ⋅ ΔT ) (6-b) ΔT = T2 (T2 + ΔT ) = Q ⋅ T0 Q ⋅ T0
ΔT = or,
In general, T1 ⋅ ΔT << T1 , T2 ⋅ ΔT << T2 (rough treatment here). 2
2
Lw 2 Hence, ΔT = Lw T 2 or ΔT = T2 (7) 1 Q ⋅ T0 Q ⋅ T0 It can be seen that, for a given rate of heat transfer Q and a given rate of exergy loss(Lw),the approach temperature(ΔT ) increases almost proportionally with the increase in the square of the temperature level(T1 or T2 ). This explains the necessity to use a very small approach temperature, on the order of 1℃ in the cold boxes of cryogenic process. If the ΔT is not reduced, there would be a large increase in the energy required to operate the process. According to the discussion above, the heat transfer temperature difference contribution value of a stream can approximately be expressed as: T c = C ⋅ h −1 / 2 ⋅ a 1 / 2 ⋅ T 2 (8) ΔTc — heat transfer temperature difference contribution value, K;
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C — the coefficient obtained by regression technique; h —the heat transfer film coefficient,KW•m-2•K-1; a — capital cost per unit area of heat exchanger , $•m-2; T — the thermodynamic mean temperature of each stream, K. After selecting a set of suitable heat exchanger data which reflect the effects of h, a, and T on ΔTc, adequately, the value of coefficient C in equation (8) is obtained by using regression technique. The value of ΔTc obtained by equation (8) is rude because the error of reference streams data are not avoidable, so ΔTc should span an indefinite interval, [ΔTc–ΔTk, ΔTc+ΔTk], for example, substituting 0.2ΔTc for ΔTk. Given any group of ΔTc –values in this interval, Corresponding network can always be attained by relying on the strategy represented as follows. The annual cost is regarded as an objective function and ΔTc–values are taken as decision variables simultaneously. The optimization of ΔTc– values is a complex combinatorial problem which can be solved by Genetic/simulated annealing algorithm (GA/SA). Optimum ΔTc–values attained by GA/SA mean the corresponding network is the cost-optimum solution.
3. Automatic Generation Strategy of MSHEN Stream pseudo-temperature is calculated by using equation (8) and (1). Construct hot and cold composite curves on the basis of the pseudo-temperature problem table algorithm. The pinch point is located and utilities are determined. Divide the hot and cold composite curves into several blocks by cutting vertically at every point where the composite curves bend (kink). Namely, wherever a change in slope occurs in either composite curve. The segments that require hot or cold utilities in both composite curves are left out in Fig. 2. Given an adjustable constant K which is a parameter dependent on the engineer’s experiences, the interval whose heat load is less than K is regarded as a small block (block 2, 6, 7 in Fig.2 ), which needs to be merged into a big one in order to prevent some small-duty exchangers. With that conjoin all the proximate small blocks (block 6 and 7 become a new one in Fig.2) until the left of small blocks are all independent. If the new block from 6 and 7 also belong to the small-duty, put it into the only neighbour block because it is terminal (put 6+7 into 5 in Fig.2). If the small-duty block has two neighbours (block 2 in Fig.2), put it into the one which has the larger mean heat transfer temperature difference to prevent the inverse heat transfer when the exit temperature of cold stream is larger than the inlet temperature of hot stream (put block 2 into block1 in Fig.2). The pseudo-temperature of each stream is reverted to type. Two stream heat exchangers network are obtained from the first enthalpy interval to the last. The matches between hot and cold streams follow the heuristic guidelines [3]. In each interval the amount of heat load of hot streams and the heat load of cold streams are balanceable, and the heat transfer temperature difference is not less than the allowable values. Several branches of a stream in the same enthalpy interval merged into a multi-stream heat exchanger. Each interval corresponds to several multi-stream or two stream heat exchangers. A group of multi-stream heat exchangers in series is obtained. The procedure is presented as follows: Step 1: Given stream data, calculate ΔTc for each stream according to the equation (8) Step 2: Set the indefinite interval [ΔTc-ΔTk, ΔTc+ΔTk] for each ΔTc, select a ΔTc from individual randomly Step 3: Determine the pseudo-temperature for each stream using equation (1)
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Step 4: Locate the pinch and construct the T—H diagram Step 5: Divide the hot and cold composite curves into enthalpy intervals Step 6: Determine a two-stream heat exchanger network on the T—H diagram Step 7: Construct multi-stream heat exchangers from two stream heat exchangers Step 8: Calculate the annual cost of the gotten network in step (6) Step 9: Go to step 3.Regenerate a set of ΔTc in their individual intervals with GA/SA to optimize ΔTc-values until the best solution 260
hot composite curve
240
cold composite curve
200
o
T( C)
220
180 160 140 120 100
1 1000
2 1500
4
3 2000
6 7
5 2500
H(KW)
3000
3500
Fig.2 Combination of small interval
4. Example This example is taken from Ciric[4]. It involves a system of four hot streams and three cold streams, with heating and cooling utilities provided by steam and cooling water. Problem data is shown in Table1. The synthesis results are shown in Fig.3 and Table2. Results attained by superstructure in literatures are also shown in table 2 for comparison. The figure underlined in Fig.3 is not the load of a stream but a heat exchanger. Table 1 Stream data Stream H1 H2 H3 H4 C1 C2 C3 S CW
Tin( ) 160 249 227 271 96 115 140 300 70
Tout( ) 110 138 106 146 160 217 250 300 90
Fcp(kW· -1) 7.032 8.44 11.816 7.0 9.144 7.296 18 ---
Cost($•kW-1•a-1) -------80 20
The spaghetti design in this paper cuts the streams into several segment, the number of enthalpy intervals is larger than the number of stages of superstructure. Hence a larger number of exchangers are required in this paper. In Ciric or Wei’s superstructure, ΔTmin is a crucial factor. When ΔTmin becomes smaller, the recovery energy increases sharply correspondingly, as a result, heat transfer area increases inevitably, however, there is inferior limit determined by feasibility.The stream temperature difference contribution value proposed in this paper guarantees that the temperature differences are large enough between two streams exchanging heat within any exchanger, and as a
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result, the total area of the whole network is smaller compared with the Ciric or Wei’s model. Furthermore, the Pinch Design Method has much more recovery energy, so the total annual cost is less. Table 2 Comparison of the solution with literature results Item [4]
Ciric Wei[5]
This paper
Total area (m2) 281 301 241
Equipment cost* (＄•a-1) 91104 75678 86183
Number of units （number） 12 7 12
Utility cost (＄•a-1) 23356 23846 11998
Total annual cost (＄•a-1) 114460 99524 98180
Equipment cost:1300[area(m2)]0.6 $•a-1; Annual cost= Equipment cost+ Utility cost 434.598 194.17 H1 160℃
7 1030.28
2
H2 249℃
6
242.584
192.00 ℃
173.16℃
3
152.75℃
5 H3 227℃
429.84
2
190.45℃
138℃ 534.20
171.62℃
3
110℃
124.50
7
150.62
279.03
149.68℃
151.21℃
6
1
106℃ 53.19
2
H4 271℃
192.84℃
3
174.01℃
153.60℃
6
150.62
146℃
4 5
C1 160℃
1
C2 217℃
158.01 ℃
4
143.53℃
141.61 ℃
6
96℃
7
115℃
347.13 C3 250℃
226.43℃
2
156.47℃
3
140℃
Fig.3 Synthesis result
5. Conclusion In this paper, an automatic generation strategy for synthesizing multi-stream heat exchanger network based on stream pseudo-temperature is proposed. The automatic procedure is used to obtain a family of multi-stream heat exchangers in serial. This method is more feasible and effective than mathematical model.
References [1] Ahmad S., 1985, Heat exchanger networks: cost trade-offs in energy and capital, Ph. D Thesis, University of Manchester Inst. of Sci. and Technol. [2] Nishimula H., 1980, A theory for the optimum synthesis of heat exchange systems,J. Optimization Theory Applic., 30,423-450 [3] Pingjing Yao, 1995, Total Process Energy Integration.Dalian, Dalian University of Technology Press, 46-53 [4] Ciric A. R. and Floudas C. A., 1991, Heat exchanger networks synthesis without decomposition, Computers and Chemical Engineering, 15, 10, 385-396 [5] Guanfeng Wei,Pingjing Yao,Xing Luo, Wilfried Roetzel,2004,Study on multistream heat exchanger network synthesis with parallel Genetic/simulated Annealing algorithm,Chinese J.Chem.Eng.,12,1,66-77
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Flexible Heat Exchanger Network Design for Chemical Processes with Operation Mode Changes Masaru Noda and Hirokazu Nishitani Graduate School of Information Science, Nara Institute of Science and Technology 8916-5 Takayama-cho, Ikoma, NARA 630-0192 JAPAN
Abstract A new synthesis approach is proposed for a flexible heat exchanger network (HEN) design for chemical processes with operation mode changes. The proposed approach derives an optimal HEN, including heat transfer areas and operation patterns for process streams, which minimizes the total utility cost required for transferring the state of the system’s main devices during operation mode changes. The effectiveness of the proposed approach is demonstrated through a case study. Keywords: heat exchanger network (HEN), dynamic transshipment model, non-steady state operation, optimal design
1. INTRODUCTION At an early stage of chemical process design, much less concern is given to nonsteady state operations, such as start-up, shutdown, and load change, than to steady state operations. However, the recent increase in the number of chemical processes operated on a demand basis, such as a fuel cell system, has led to a new design problem; both steady state and non-steady state operations need to be considered simultaneously in the optimal design problems. Since the 1980s, many authors have investigated HEN with regard to flexibility and controllability [1-5]. These studies, however, only focus on the flexibility and controllability of HEN around certain steady state conditions. The synthesis problem discussed in this paper focuses not only on the design of an optimal HEN but also operation patterns during an operation mode change. The objective function of this problem is to minimize the total utility cost required for dynamically transferring the state of process streams and main devices in a process from an initial condition to the specified final condition. Here, a main device is defined as the dominant facility that must be installed for the process function to perform properly. Heat exchangers, heaters, and coolers are not included in the main devices. This paper is organized as follows: First, we propose a new HEN model that can handle the dynamic operation of a process by extending the conventional HEN model used in pinch technology. Second, we apply the proposed method to a case study and draw some conclusions in the last section.
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2. OPTIMIZATION APPROACH FOR FLEXIBLE HEN DESIGN The transshipment model is an algorithmic optimization model that can automatically synthesize an optimal HEN [6]. This model includes all the possibilities of heat transfer between streams. One of the key features of the transshipment model is temperature intervals that are introduced so that each temperature interval has a different set of process streams crossing it. However, this model does not take into account any of the transitional behaviors of a process, which are very important when addressing the optimal design problem in non-steady state operations. We developed a new multi-period mode called a dynamic transshipment model by extending the transshipment model in the following way. Figure 1 shows a schematic diagram of a heat balance model at interval k in sub-period t. First, the whole nonsteady state operation period is divided into a set of sub-periods, and in each sub-period, the process is assumed to be in a pseudo-steady state. By introducing this assumption, the non-steady state operation is represented by a discrete model. Second, for each main device q, the ideal temperature profile T*q(t) is defined as a sub-period function that calculates the amount of heat accumulated in the main device during each sub-period. T*q(t) is usually obtained at the preliminary process design stage, where a HEN structure is not considered. Third, interval temperatures Tk(t) are defined for each sub-period based on the inlet and outlet temperatures of the process streams and the temperatures of available utilities. Here, the interval temperature is a boundary temperature between two connected temperature intervals. Below is some additional information necessary for formulating a dynamical design problem: 1. A set of hot process streams and a set of cold process streams that can be used as the heating and cooling mediums 2. Inlet and outlet temperatures of all process streams 3. Initial and final temperatures of the main devices in the process 4. Heat transfer areas and heat capacities of the main process devices 5. Available utilities and their temperatures In this example, as the main device’s temperature T*q(t) is between Tk(t) and Tk+1(t), the amount of heat transfer into main device q is located at interval k so as to minimize the temperature difference between the process streams and the main device. Ri ,k - 1 ( t ) Rm , k - 1 ( t ) Tk ( t ) + 'Tmin Hot utility
Tk ( t )
S
Qmjk ( t )
Qm ( t )
Qijk ( t )
H Hot stream Qik (t )
Cold stream Q Cjk ( t ) Qmqk ( t ) Qiqk ( t )
Qink ( t ) Tk +1 ( t ) + 'Tmin
Tk 1 (t ) Ri , k ( t )
Main device Qqk (t ) Cold utility W Qn ( t )
Rm , k ( t )
Fig. 1 Heat transfer model at interval k in sub-period t
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The dynamical HEN design problem using dynamic transshipment model is formulated as the LP problem given in Eq.(1-a)-Eq.(1-f). Here, Eq.(1-a) is the objective function, and Eq.(1-b)-Eq.(1-f) denotes the heat balance models. As this model is an LP model, the optimal solution can be obtained relatively easily. Tf
§
¦ ¨© ¦ c
Min
t 0
s.t.
Rik (t )
¦Q
ijk
¦Q
(t )
ink
jCk
Rmk (t )
¦Q
mjk
(t )
jCk
mjk
mSk
ink
iH k
Ri1 (t )
RiK (t )
(1-a)
(t )
¦Q
(1-b)
iqk
(t )
Ri ,k 1 (t ) QikH (t ), i H k
qEk
¦Q
mqk
(t )
Rm ,k 1 (t ) QmS (t ), m S k
(1-c)
qEk
¦Q
¦Q
mS
nWk
· QmS (t ) ¦ cnQnW (t ) ¸ nW ¹
m
(t ) ¦ Qijk (t ) Q Cjk (t ),
(1-d)
j Ck
iH k
(t ) QnW (t ), n Wk , Qqk (t )
¦Q
mqk
mS k
(t )
¦Q
iqk
(t )
(1-e)
iH k
0 , Rik (t ), Rmk (t ), Qijk (t ), Qmjk (t ) t 0 , Qink (t ), QnS (t ), QnW (t ) t 0 (1-f)
Given parameters: QHik : Available heat amount of hot stream i at interval k QCjk : Available heat amount of cold stream j at interval k Qqk : Amount of heat accumulated in main device q at interval k : Utility cost of hot utility m cm : Utility cost of cold utility n cn : Interval temperature between interval k-1 and interval k Tk 'Tmin : Minimum temperature difference between hot stream and cold stream : Total operation time Tf Unknown variables: QSm : Heat amount of hot utility m QWn : Heat amount of cold utility n Qmjk : Heat amount transferred from hot utility m to cold stream j at interval k Qijk : Heat amount transferred from hot stream i to cold stream j at interval k Qink : Heat amount transferred from hot stream i to cold utility n at interval k Qiqk : Heat amount transferred from hot stream i to main device q at interval k Qmqk : Heat amount transferred from hot utility m to main device q at interval k Rik : Amount of heat residual of hot stream i exiting interval k Rmk : Amount of heat residual of hot utility m exiting interval k Index sets: Sk = {m | hot utility m is present at or above interval k} Wk = {n | cold utility n extracts heat from interval k} Hk = {i | hot process stream i is present at or above interval k} Ck = {j | cold process stream j demands heat from interval k} Ek = {q | main device q existing at interval k}
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The heat transfer area Aijk(t) of a heat exchanger in sub-period t is calculated by Eq.(2).
Aijk (t ) Qijk (t ) / U'Tijklm (t )
(2)
Here, ǻTlm represents the logarithmic mean temperature difference. In this model, heat transfer areas of every match are allowed to change independently in each sub-period. Therefore, in the case of sub-periods with a smaller heat transfer area Aijk(t) than the maximum heat transfer area Amaxijk, Amaxijk is used instead of Aijk(t). In this study, we take a two-step optimization approach to solve a HEN synthesis problem under non-steady state operations. In the 1st step, an optimal HEN is derived using the dynamic transshipment model. In the 2nd step, the operation patterns of the process streams and utilities are optimized so as to minimize the error between the ideal and calculated temperature patterns of the main device using gPROMSTM, where the structure of HEN and Amaxijk are fixed. The dynamic optimization problem in the 2nd step is formulated as shown in Eq.(3). Tf
Min u (t )
¦ T
* q
(t ) Tq (t )
2
t 0
s.t. f ( x, x , u )
0 xL d x d xU , uL d u d uU
(3)
Here, Tq(t) is the simulated temperature pattern of the main device q, f = 0 is a physical process model, x and u are state and control variables respectively. Subscripts L and U denotes the minimum and maximum values.
3. CASE STUDY We apply the proposed approach to an optimal HEN design problem of a chemical process with operation mode changes in this process, here is one hot stream H to be cooled, one cold stream C to be heated, and one main device q. Table 1 shows the inlet and outlet temperatures of streams H and C, and T*q(t) in three operation modes. The flow rates of streams H and C, which are optimized in the 2nd step, are assumed to be 1.0 kmol/h. The specific heats of streams H and C is 1.0 kJ/kmol, and the heat capacity of the main device q is 2.0 kJ/ ºC/h in this case study. Table 1 Dynamical HEN design conditions Stream H C Device q
Mode 1 (t = 0h) Tin [ºC] Tout [ºC] 300 120 160 300 T*q(0) [ºC] 280
Mode 2 (t = 4h) Tin [ºC] Tout [ºC] 380 120 160 380 T*q(4) [ºC] 360
Mode 3 (t = 7h) Tin [ºC] Tout [ºC] 320 120 160 320 T*q(7) [ºC] 300
In this case study, the state of the process gradually changes from mode 1 to mode 2 in 4 hours, and then from mode 2 to mode 3 in 3 hours by changing operation variables in the process. During these mode changes, the inlet temperatures of streams H and C are assumed to be kept at the exact values outlined in Table 1. The 1st step is to derive an optimal HEN with the heat transfer areas, and the heating duties of burners and coolers so as to minimize the total utility cost during the entire operation. The objective function, which is given by Eq.(4), is the weighted total utility cost during the whole operation period (Tf = 7h).
Flexible Heat Exchanger Network Design for Chemical Processes Tf
§
¦ ¨© ¦ 8Q
Min
t 0
S m
mS
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· (t ) ¦ 2QnW (t ) ¸ nW ¹
(4)
According to the proposed two-step approach, we first define T*q(t) as a function of time to calculate the amount of the heat accumulated into the main device q in each subperiod. In this case study, the linear temperature profile given by Eq.(5) is assumed as T*q(t) .
280 20t (t Tq* (t ) ® ¯440 20t (t
0 4h ) 4 7 h)
(5)
Figure 2 shows the four temperature intervals defined for this case study. The interval temperatures of the process streams T1 – T7 are defined for each mode as shown in Table 2. A heat sink of the main device q is located in the temperature interval 2 because T*q(t) exists there. The minimum temperature difference is set to 20ºC. Table 2 Interval temperature settings t = 0-4h t = 4-7h
T1 160 160
T2 220+20t 380-20t
T3 280+20t 440-20t
T4 300+20t 460-20t
T5 240+20t 400-20t
T6 180 180
T7 120 120
Figure 3 shows the optimal design result obtained in the 1st step, where there are two burners B1 and B2 and two heat exchangers Hx1 and Hx2. The maximum heat exchanger areas of Hx1 and Hx2 are 6.0 m2 and 3.0 m2, respectively. In the optimal design, the cold stream C is heated by Hx1 and B1 and fed to the main device to heat it up. After being heated up by B2, stream C is discharged. The hot stream H is fed into the main device and is cooled by Hx1 and Hx2. In the dynamic transshipment model, we don’t consider the heat transfer rate model in relation to the size of the heat exchagers. Therefore, when the operation patterns obtained in the 1st step are applied to the detailed physical process model, including the heat transfer rate model, the simulated temperature pattern Tq(t) doesn’t follow T*q(t) as shown in Fig. 4. T4 = Tout
T4 (t )
㬍
T4 ( t ) T5 (t )
Hot stream H 㬍 㬍
Interval 1 Main device q
B2
180
㬍
T7
120
㬍
T3
Hot stream H
Interval 2
T2 (t )
㬍
㬍 T1 160 Cold stream C (T1 = Tin) Interval 4
(T7 = Tout)
Fig. 2 Defined Temperature Intervals
T2 Main Device
T3 (t )
㬍
Interval 3
T6
Q = 20 kJ/h
T4
Tq(t)
T7 (120 ºC)
Q = 0䌾40 kJ/h
B1 T5 A = 6.0 m 2
Hx1
T6 Hx2
T1 (160 ºC)
A = 3.0 m2
Air Cold stream C
Fig. 3 Optimal HEN in 1st Step
In the 2nd step, the flow rates of process streams and heat duties of the burners are optimized as functions of time using heat transfer rate equations, where the HEN structure and heat transfer areas obtained in the 1st step are fixed. The left graph in Fig.5 shows the optimal operation patterns obtained in the 2nd step. When the flow rates of process streams are optimized in the 2nd step, Tq(t) follows T*q(t) exactly as shown in the right graph of Fig. 5.
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40
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30 1.0
20 Heat duty of B1 Flow rate of H&C
0.5 0
1
2
3
10 4
5
6
7
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340 320 300 Ideal temperature pattern Simulated temperature pattern
280 260
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Fig. 4 Operation patterns (left) and Tq(t) (right) in the 1st Step 360
30 20
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Flow rate [kmol/h]
2.0
340 320 300
Ideal temperature pattern Simulated temperature pattern
280 260
0
1
2
3 4 Time [h]
5
6
7
Fig. 5 Operation patterns (left) and Tq(t) (right) in the 2nd Step
4. CONCLUSION We proposed a process synthesis method, which considers a non-steady state operation for chemical processes. For this purpose, we developed a dynamic transshipment model that describes heat accumulation in the main devices during the transitional operation, and which is not included in the conventional transshipment model. The proposed method only requires information related to the temperature profiles of process streams and heat capacities and heat transfer areas of main devices to derive an optimal HEN when considering non-steady state operations. Those design specifications can usually be obtained at an early stage of the process design. The obtained design result can be used as the initial design for a more precise optimization step. The proposed two-step optimization method, which considers non-steady state operations, can also be applied to other chemical processes where the transitional operation is essential.
[REFERENCES] [1] Aaltolra, J.; “Simultaneous Synthesis of Flexible Heat Exchanger Network,” Applied thermal engineering, 22, 907-918 (2002) [2] Biegler, L. T., I. E. Grossmann, and A. W. Westerberg; “Systematic Methods of Chemical Process Design,” Prentice Hall PTR (1997) [3] Furman, K. C. and N. V. Sahinidis; “A Critical Review and Annotated Bibliography for Heat Exchanger Network Synthesis in the 20th Century,” Ind. Eng. Chem. Res., 41, 2335-2370 (2002) [4] Glemmenstad, B., S. Skogestad, and T. Gundersen; “Optimal Operation of Heat Exchanger Networks,” Comp. and Chem. Eng., 23, 509-522 (1999) [5] Kotjabasakis, E., and B. Linnhoff; “Sensitivity Tables for the Design of Flexible Processes (1) – How much contingency in heat exchanger networks is cost-effective,” Chem. Eng. Res., 64, 199-211 (1986) [6] Papoulias, S. A., and I. E. Grossmann; “A Structural Optimization Approach to Process Synthesis-II. Heat Recovery Networks,” Comp. and Chem. Eng., 7, 707 (1983)
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Molecular design based on enhanced topological descriptors Anton A. Kiss,a Mircea V. Diudeab a University of Amsterdam, Nieuwe Achtergracht 166, 1018WV Amsterdam, Netherlands b“ Babes-Bolyai” University, Arany Janos 11, 400028 Cluj-Napoca, Romania
1. Abstract Topological characterization of chemical structures allows the design of novel molecular structures with desired properties. Simpler yet powerful molecular descriptors such as Szeged topological indices were used to demonstrate their prediction capability in the molecular design of organic compounds. Two sets of explosives and barbiturates were used for QSPR and QSAR correlations, respectively. In addition to simplicity and rapidity of computation, we show that the proposed descriptors can be successfully used in molecular design. 2. Introduction The molecular descriptors can be correlated with activities by using Quantitative / Qualitative Structure – Property / Activity Relationships (QSPR/ QSAR). These physico-chemical descriptors that include parameters accounting for topology, electro-negativity and steric effects, are determined empirically or by computational methods. Properties and activities used in QSPR/QSAR include physico-chemical measurements and biological tests.1 The problem is that most of the molecular descriptors are complex and require long computing times, making them unsuitable for product design of large molecules. To solve this problem we use simpler yet powerful molecular descriptors in terms of topological indices – single numbers representing a chemical structure. In spite of the considerable loss of information by the projection of a structure into a single number, such descriptors have broad applications in predicting molecular properties. When a topological descriptor correlates with a molecular property, it can be denominated as a molecular index or topological index. In this work we use Szeged topological indices2 to demonstrate their prediction capability in the molecular design of organic compounds. These indices account for the molecular topology, as well as multiple bonds, heteroatoms, electronegativity and other properties. This study proposes a molecular design methodology based on QSPR/QSAR methods that can be applied to accurately predict certain physico-chemical properties or activities, without using more expensive experimental methods.
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3. Methodology QSPR / QSAR correlations can be developed based on existing structure – properties databases. Then, novel molecular structures are generated and their molecular descriptors are calculated. These descriptors are validated against the QSPR/QSAR correlations. If the results are promising the new structure can be synthesised and tested experimentally (Figure 1). This way, the number of trial experiments can be reduced by several orders of magnitude.
Molecular structure generator
StructureStructure-properties database
QSPR/QSAR correlations
Molecular descriptor calculator No Validation Yes
Synthesis and experimental test
Figure 1. Molecular design methodology based on QSPR/QSAR.
Definitions. Szeged index is defined by SZe = ∑e Ni,(i,j) Nj,(i,j), where Ni,(i,j) and Nj,(i,j) denote the number of vertices lying on the two sides of the edge/path (having the endpoints i and j).2 When the Szeged matrix is defined on paths, the index calculated on it is the hyper-Szeged index, SZp. Szeged unsymmetric matrix, SZu, is a square array of dimensions NxN that allows the construction of the symmetric matrices SZe and SZp by [SZe/p ]ij = [SZu ]ij [SZu]ji. In order to account for heteroatoms and multiple bonds in molecular graphs, we introduced the Szeged property matrices, [USZP]ij = Pi,p. The above indices (calculated by equation 1) are illustrated for the graph G1 (2,3-Dimethylpentane) in Figure 2. Ie/p = ∑e/p [SZuP]ij [SZuP]ji ; I = SZ; SZA; SZX 6
2
SZu; Pv = 1; m = 1
G1 3
4 5
1 7
(1)
0 6 4 4 2 1 4
1 0 4 2 2 1 1
1 3 0 2 1 1 1
3 3 5 0 1 3 1
3 5 5 6 0 3 5
1 6 4 4 2 0 4
3 3 6 2 2 3 0
SZuA; Pv = Av; m = 1/12 SZeA = 66.74; SZpA = 217.73 SZuX; Pv = EVGv; m = 1 SZeX = 5.54; SZpX = 19.38
SZe = 46; SZp = 151 Figure 2. Szeged matrices, vertex and fragmental indices for graph G1 (2,3-Dimethylpentane).
Correlating studies. The mathematical models of a certain property are performed by MLR (Multiple Linear Regression). Then, the model is validated by the leave-one-out and leave-one-third-out cross-validation procedures.3
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For n observations and m independent variables, MLR is represented by: m
Yi = b0 + ∑ bij X ij
(2)
j
or, in matrix form as Y=bX, where Y is the vector of responses, X is the matrix of independent variables and b is the vector of regression coefficients. The regression coefficients can be determined by the least-squares solution of b=(XTX)-1XTY. With b calculated, equation 2 can be used for estimating the chosen property for other chemical structures, usually from generated libraries. 4. Results and discussion The proposed molecular design methodology was applied on two sets of explosives and barbiturates, using QSPR and QSAR correlations. Whereas the properties of these compounds are available in literature, the topological indices were calculated based on their molecular formula. Single and multiple linear regressions (SLR and MLR) were developed using the Szeged type of molecular indices. These correlations were confirmed latter by cross-validation procedures. In the leave-one-third-out test, one-third of the dataset is not used for regression hence the property prediction is based on a smaller dataset. This test is obviously more powerful than the typical leave-one-out cross validation. QSPR. A set of explosives (Table 1) was tested for the correlation of two important properties, with the topological descriptors shown in Table 2. The dispersion coefficients in water and air are closely related to the explosion performances of these compounds. Statistics of variable regressions for both as well as the cross validation tests are given in Table 3. Table 1. Name and properties of the explosive compounds. Graph 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
DC water *10-6 (cm2/s) Trinitrotoluene 6.71 2,4-Dinitrotoluene 7.31 2,6- Dinitrotoluene 7.31 1,3- Dinitrobenzene 7.94 1,3,5- Trinitrobenzene 7.20 Hexahydro-1,3,5-trinitro-1,3,5-triazine 7.15 Octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine 6.02 N-2,4,6-Tetranitro-N-methylaniline 5.99 Picric acid 7.03 Pentaerythrytol tetranitrate 5.61 Nitroglycerin 6.95 Nitroguanidine 10.40 Ethylene glycol dinitrate 8.72 Diethylene glycol dinitrate 7.05 Propylene glycol dinitrate 7.93 Name
DC air (cm2/s) 0.0639 0.0670 0.0670 0.0729 0.0679 0.0739 0.0629 0.0590 0.0660 0.0570 0.0700 0.1019 0.0839 0.0689 0.0769
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Table 2. Topological indices for the explosives dataset (Table 1). Graph 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
SZp 4348 2050 1993 1542 3450 3450 11794 10342 4348 11514 3677 159 827 2518 1153
SZe 594 360 348 296 516 516 1156 1014 594 968 424 48 151 344 184
SZeA 601.049 378.451 366.007 303.292 509.917 555.083 1228.833 1022.236 613.896 1083.000 475.729 61.389 172.771 408.569 214.278
SZpA 4169.958 2049.049 1978.160 1500.896 3239.271 3528.729 12083.000 9969.632 4248.014 12479.986 3961.931 194.514 904.382 2913.472 1274.937
SZeX 74.949 57.524 49.430 49.076 66.837 68.038 100.785 99.907 78.289 94.923 68.805 36.330 43.192 45.670 50.820
SZpX 510.786 333.915 290.101 268.231 431.457 439.675 824.478 820.577 528.620 832.585 455.598 133.953 206.946 270.017 262.697
Table 3 indicates a good estimative and predictive ability of the regression equations. The fragmental descriptors are more suitable for QSPR studies, proving that Szeged indices are indeed very good tools in correlating studies. Table 3. Statistics of mono variable regression (Y=a+bX) and cross validation test for explosives. No 1 2 3 4 5 6 7 8
Y CDair
CDwater
X ln SZe 1/ln SZeA 1/ln SZpA 1/W ln SZe ln SZeA ln SZpA 1/SZpX
a
b
0.00073
0.4157
15.60561 - 1.3934 16.31334 - 1.4915 15.82862 -1.0766
r 0.943 0.960 0.954 0.942 0.973 0.979 0.984 0.961
11
CDwater = 15.828 – 1.076 ln SZpA r = 0.984
2
9
-6
CDwater / [10 cm /s]
10
8 7 6 5 5
6
7
8
9
ln SZpA
Figure 3. Correlation of CDwater property with ln SZpA.
10
s 0.0037 0.0031 0.0034 0.0038 0.2801 0.2443 0.2123 0.3321
F 105.09 155.21 133.61 102.87 231.72 310.57 413.59 160.95
rcv
scv
0.951 0.0035 0.960 0.968 0.975
0.338 0.302 0.266
Figure 3 shows the excellent monovariate correlation of CDwater with ln SZpA. In addition, the cross-validation tests (leave-one-out and the more powerful leave-onethird-out) confirm the high capability of these indices to predict the property of components not used for regression, but also other novel components.
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QSAR. A set of 25 barbiturates (Table 4) was correlated with an important property that is in connection with the drug membrane transport phenomena.4 Table 4. Topological Indices and logP for the barbiturates dataset (G1 and G2 in Figure 4). No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25.
R1 Methyl Ethyl Propyl Butyl Methyl Ethyl Propyl Butyl Isobutyl Amyl Isoamyl Methyl Ethyl Propyl Isopropyl Methyl Ethyl Propyl Isopropyl Methyl Ethyl Methyl Methyl Ethyl Ethyl
R2 1-methyl,1-propenyl 1-methyl,1-propenyl 1-methyl,1-propenyl 1-methyl,1-propenyl 1-methylvinyl 1-methylvinyl 1-methylvinyl 1-methylvinyl 1-methylvinyl 1-methylvinyl 1-methylvinyl Ethyl Ethyl Ethyl Ethyl Methyl Methyl Methyl Methyl Propyl Propyl Isopropyl Butyl Butyl Ethyl
R3 Methyl Methyl Methyl Methyl Ethyl Ethyl Ethyl Ethyl Methyl Methyl Methyl Methyl Methyl Propyl
SZp 2521 3088 3943 5094 1821 2273 3022 4010 3828 5357 4971 4414 5311 6436 6429 4564 5451 6640 6555 5597 6656 5627 7148 8435 8638
SZe 402 468 549 646 334 395 470 560 536 666 618 548 624 717 702 557 634 728 713 640 722 626 749 838 834
SZpA 3466.39 4253.24 5437.50 7027.09 2504.36 3134.29 4172.92 5538.61 5288.00 7392.34 6865.62 5175.90 6287.16 7717.54 7678.24 5343.83 6436.92 7936.94 7804.57 6737.13 8071.52 6733.33 8769.94 10411.75 10525.17
SZeA 560.77 653.22 766.09 900.74 465.41 550.86 655.36 780.29 748.01 926.99 862.43 663.04 762.59 883.91 865.05 676.65 777.56 900.25 881.38 784.36 892.27 766.86 927.47 1045.10 1045.49
logP 0.65 1.15 1.65 2.15 0.15 0.65 1.15 1.65 1.45 2.15 1.95 1.15 1.65 2.15 1.95 1.15 1.65 2.15 1.95 1.65 2.15 1.45 2.15 2.65 2.65
Table 5. Statistics of mono- and bi-variable regression for data in Table 4: log P = a + ∑biXi No. 1 2 3 4 5 6
Xi SZp SZe SZpA SZeA SZpX SZeX
b 0.0003 0.0045 0.0002 0.0041 0.0136 0.1397
a 0.0604 -1.1072 -0.1949 0.9934 -0.8170 -1.6967
r 0.9085 0.9562 0.9541 0.9935 0.8310 0.7075
s 0.2627 0.1840 0.1883 0.0718 0.3498 0.4445
cv (%) 15.924 11.154 11.415 4.354 21.202 26.941
F 108.82 245.66 233.53 1740.58 51.34 23.03
7
SZp SZe SZp SZeA SZe SZeX SZpA SZeA
-0.0005 0.0121 0.0000 0.0044 0.0066 -0.1021 -0.00001 0.00486
-2.9098
0.9872
0.1023
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423.46
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0.9938
0.0717
4.351
871.96
0.0401
0.9895
0.0931
5.643
513.80
-1.8626
0.9943
0.0685
4.157
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R1
log P
1.5
G1
1
R1
O HN
O HN
600
700
800
SZeA
900
1000
1100
R3
NH O
R2
O
0.5
500
H
C
O
2
0 400
R2
G2
NH O
Figure 4. Correlation of log P with SZeA (left). Structure of barbiturates as generic graphs (rigth).
Statistics of the correlations included in Table 4 are given in Table 5. Clearly, the Szeged property index SZeA is the best choice in modeling logP. No significant improvement was recorded in bivariate regression. The crossvalidation tests (r=0.9913, s=0.0824, cv=4.999% for monovariate and r=0.9921, s=0.0721, cv=4.796% for bivariate regression) show no significant drop in the r-value, thus confirming the good predicting ability of the correlating equations. The QSPR and QSAR results shown here surpass those reported in literature.4 5. Conclusions Statistics of single variable regressions and cross validation test indicate excellent estimative and predictive ability of the regression equations. The Szeged indices show a good ability in modeling some important physicochemical properties, and the recorded results surpass those reported in literature. In addition to their simplicity and rapidity of computation, the proposed topological descriptors can be successfully used in molecular design. A reliable molecular design methodology was proposed, based on QSPR/QSAR methods that make use of enhanced topological indices. The QSPR/QSAR methods described in this study are able to accurately predict the properties of novel components, thus reducing dramatically the number of required experiments. References 1. 2. 3. 4.
Estrada E., SAR and QSAR in Environmental Research, 11, 55-73, 2000. Diudea M.V., Gutman I., Croatica Chemica Acta, 71, 21-51, 1998. Diudea M.V., Kiss A.A., Estrada E., Guevara N., Croatica Chem. Acta, 73, 367-381, 2000. Guo M., Xu L., Hu C.Y., Yu S.M., Commun. Math. Comput. Chem., 35, 185-197, 1997.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Effects of Catalyst Activity Profiles on the Scale-up of Polymerization Reactors Sandor Nemetha, Janos Abonyia, Balazs Feila, Peter Arvaa, Janos Tolvethb, Akos Janecskab, Gabor Nagyb a
University of Veszprem, Department of Process Engineering, P.O. Box 158, Veszprem, H-8201, Hungary b TVK Ltd, Product and Application Development, P.O. Box 20, Tiszaujvaros, H-3581, Hungary
Abstract The aim of this paper is to demonstrate a method that is able to transform information observed by laboratory experiments with catalysts possessing changing activity to continuous industrial reactor. The method is based on the fact that the change of catalyst activity and quality properties of the polymer can be determined as functions of the catalyst age based on laboratory polymerization experiments carried out for different time periods. This information integrated with the residence time distribution of the industrial reactor can be used to estimate the average properties of the polymer powder produced in the continuous industrial reactor. The method is demonstrated in case of ethylene polymerization. Keywords: olefin polymerization, activity profiles, residence time, scale-up, polymer properties
1. Introduction The wide application of polyolefin products requires different and various technical properties from the polymers. This is why polymerization processes are carried out with different types of catalysts in different types of reactors. In case of high-density polyethylene the polymerization is usually catalyzed by Ziegler-Natta (ZN) or Phillipstype (PT) catalysts in stirred slurry-phase, loop or gas-phase fluidized bed reactors. In case of these catalysts the mechanisms of the elementary reactions and the rate constants of the partial reactions significantly differ from each other, which allows the production of polyolefin products with different properties. The activity profile of a catalyst is basically determined by these elementary steps of polymerization reactions such as activation of catalyst, chain initiation, chain propagation, chain transfer, and chain deactivation (Table 1). Hence, with the use of different types of catalysts different activity profiles can be observed in time. The catalysts used for high-density polyethylene (HDPE) production can be divided into two groups from activity profile point of view (Keii, 1972): a ‘buildup’ type and a ‘decay’ type. The so-called Phillips-type Cr-oxide catalysts belong to the buildup group whose characteristics are activating after a so-called induction period and after a short steady-state period a slow deactivation follows. Heterogeneous metallocene and ZNtype catalysts form the decay group, e.g. TiCl4 on silica carrier with triethylaluminium cocatalyst, by which quick activating followed by deactivating process can be observed. These activity profiles can also be obtained by laboratory experiments performed under isotherm conditions and applying constant component concentrations (Calabro, 1988,
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S. Nemeth et al.
McDaniel 1991). Besides the type of metal atom, catalyst activity profile strongly depends on the type of the catalyst carrier, the preparation way, and the quality and quantity of the present co-catalyst, which are deeply analyzed by numerous works (Keii and Soga, 1986). Therefore, the developed activity profile can be influenced by several ways, so the most advantageous profile, which means specific mass of the polymer produced during the time period 0
2. Experimental Set-Up 2.1. Polymerization reactors Data acquisition and control of the laboratory polymerization reactor was performed using FIX MMI software and Eurotherm T103 PLC controllers. The polymerization reactor module consists of a 20 litres reactor, heating/cooling system, feedstock preparation modules and controllers. The reactor was provided with an impeller type stirrer (magnetic coupled) to assure good mixing, stirrer speed is manually adjustable between 0-1150 rpm. The reactor could be operated between 20-50 barg and temperatures between 30-120 °C. For temperature control and safety a dual temperature element was placed inside the reactor. The industrial system is the Phillips Petroleum Co. Particle Form suspension ethylene polymerization process. The slurry polymerization takes place at a temperature between 85-110 °C. The catalyst and the inert solvent are introduced into the loop reactor where ethylene, hydrogen and 1-hexene are circulating. The reactor consists of a folded loop containing four long runs of pipe 1 m in diameter, connected by short horizontal lengths of 5 m. The slurry of HDPE and catalyst particles circulates through the loop at a velocity between 5-12 m/s. The polymer is concentrated in settling legs to about 6070% by weight slurry and continuously removed. The solvent is recovered by hot flashing and rectification. The polymer powder is dried and pelletized.
Effects of Catalyst Activity Profiles on the Scale-up of Polymerization Reactors
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2.2. Sample preparation and characterization Materials (ethylene, 1-hexene, hydrogen, iso-butene) were treated by molecular sieves (13X) and alumina (A204) because the catalysts systems are very sensitive to poisons (CO, CO2, H2O, and for other electron donor compounds). Before the polymerization the laboratory reactor was inertisated. After several cleaning procedures the catalyst, iso-butane and hydrogen were filled into the reactor. The reactor was heated up to reach the polymerization temperature, while the required 1-hexane and ethylene was filled in. The reaction was performed at constant temperature and pressure. The reactor pressure was controlled by ethylene feeding. The properties of polymer samples were determined after the polymerization process. The melt index and density of the polymer were measured according to ISO/DIS 1133 and EN ISO 1183-3 standards. MWD of the polymer samples were determined by GPC, while -CH3 group number was measured by FTIR method.
3. First principle model of ethylene polymerization Table 1 shows the detailed reaction mechanism of ethylene polymerization process. The potentially active element (Cr) of Phillips type catalysts is activated by monomer or comonomer molecules. Since the potentially active element (Ck) is located on the surface of solid carrier, active centers with different activity (k=[1, 2, …, i, …]) will be arisen that are able to be transformed into each other. The active centers (P0k) after the reaction with momoner (M) or comonomer (CM) are able to start polymerization and chains ended with monomer (P1,0) or comonomer (Q0,1) with polymerization grade equal to one are arisen that are able to grow. Polymer chains are growing if monomer or comonomer molecules are built in (Pi,j, Qi,j), or they may be deactivated (Di,j) simultaneously by the reaction with monomer, comonomer or chain transfer agent (H2) molecules. During chain transfer reaction, the active center leaves the polymer chain and is able to build another chain, while during deactivation the active center loses its activity. It can be seen also from Table 1 that the reaction system is complex and the parameters of reaction rate equations can be determined only by numerous experiments. However, discovery of the detailed polymerization kinetics fells out of the scope by preliminary qualification of catalysts. During the preliminary examinations, the catalyst productivity and the spectrum of potentially producible products have to be determined. These tests are carried out in laboratory reactors. By these laboratory experiments enough information has to be obtained to be able to estimate the product quantity and quality in industrial reactors. The method presented in Escape 15 conference (Nemeth, 2005) provides an opportunity to qualify the catalysts preliminarily because it is able to determine the change of the catalyst activity on the basis of monomer consumption measured during the laboratory experiments. In the laboratory reactor the age of catalyst and polymer particles are the same while in continuous reactors the particles can be characterized by their age distribution that is in connection with the flow conditions of the polymer phase. Therefore the age distribution of catalyst and polymer particles has to be determined in case of continuous reactors (Nemeth, 2005). In case of copolymerization, the reaction rate of polymerization can be expressed in the following form:
R p (τ ) = g (τ ) ⋅ k p ⋅ C (τ ) ⋅ M ⋅ M w
(1)
S. Nemeth et al.
940
where g (τ ) is the density function of the activity profile, τ age of the catalyst,
M = M + CM , M w avarage molecular weight of monomer, comonomer system, k p pseudo-kinetic rate constant of propagation reaction (Xie, 1993). Table 1. Chemical Reactions for Polymerization of Ethylene Site activation
k
k act , k , cocat
by co-mon.
Chain initiation
k 0
C k + CM ⎯⎯ ⎯⎯→ P
ract, k ,CM = kact, k , M ⋅ Ck ⋅ CM
C k ⎯⎯ ⎯→ C l
rtr , k = ktr , k ⋅ Ck
k tr , k
-
Site transform.
ract, k , M = kact,k , M ⋅ Ck ⋅ M
act , k , M C k + M ⎯⎯ ⎯ ⎯→ P0k
by mon.
k I ,k ,M
k 0
rI ,k ,M = k I ,k ,M ⋅ P0k ⋅ M
k 1, 0
by mon.
P + M ⎯⎯⎯→ P
by co-mon.
I , k , CM P0k + CM ⎯⎯ ⎯→ Q0k,1
rI ,k ,CM = k I ,k ,CM ⋅ P0k ⋅ CM
k
k p ,k ,M ,M
Propagation
Pi ,kj + M ⎯⎯ ⎯ ⎯→ Pi +k1, j
rp ,k ,M ,M = k p ,k ,M ,M ⋅ Pi ,kj ⋅ M
Qik, j + M ⎯⎯ ⎯⎯→ Pi +k1, j
rp ,k ,CM ,M = k p ,k ,CM ,M ⋅ Qik, j ⋅ M
k p , k ,CM , M
k p ,k , M ,CM
k i, j k i, j
k i , j +1 k i , j +1
P + CM ⎯⎯ ⎯⎯→ Q k p , k ,CM ,CM
Q + CM ⎯⎯ ⎯⎯→ Q Chain transfer
k
tr ,k , M , H 2 Pi ,kj + H 2 ⎯⎯ ⎯ ⎯→ Dik + P0k
by H2
k
tr , k ,CM , H 2 Qik, j + H 2 ⎯⎯ ⎯⎯→ Dik + P0k
rtr , k ,CM , M = ktr , k ,CM , M ⋅ Qik, j ⋅ M
k
rtr , k , M ,CM = ktr , k , M ,CM ⋅ Pi ,kj ⋅ CM
k
rtr , k ,CM ,CM = k tr , k ,CM ,CM ⋅ Qik, j ⋅ CM
tr , k , M ,CM Pi ,kj + CM ⎯⎯ ⎯⎯→ Dik + P0k
k
rtr , k , M = ktr , k , M ⋅ Pi ,kj
k
rtr , k ,CM = ktr , k ,CM ⋅ Qik, j
k
rd , k , M = k d , k , M ⋅ Pi ,kj
k
rd , k ,CM = k d , k ,CM ⋅ Qik, j
tr , k , M Pi ,kj ⎯⎯ ⎯→ Dik + P0k tr , k ,CM Qik, j ⎯⎯ ⎯→ Dik + P0k
Site deactivation
spontaneous
rtr , k ,CM , H 2 = k tr , k ,CM , H 2 ⋅ Qik, j ⋅ H 2
k
tr ,k ,CM ,CM Qik, j + CM ⎯⎯ ⎯⎯→ Dik + P0k
spontaneous
rtr , k , M , H 2 = k tr , k , M , H 2 ⋅ Pi ,kj ⋅ H 2 rtr , k , M , M = k tr , k , M , M ⋅ Pi ,kj ⋅ M
tr ,k ,CM , M Qik, j + M ⎯⎯ ⎯⎯→ Dik + P0k
by co-mon.
rp ,k ,CM ,CM = k p ,k ,CM ,CM ⋅ Qik, j ⋅ CM
k
tr , k , M , M Pi ,kj + M ⎯⎯ ⎯ ⎯→ Dik + P0k
by mon.
rp ,k ,M ,CM = k p ,k ,M ,CM ⋅ Pi ,kj ⋅ CM
d ,k , M Pi ,kj ⎯⎯ ⎯→ Dik d ,k ,CM Qik, j ⎯⎯ ⎯→ Dik
The evolved molecular structure is determined by the relative rate between the chain transfer and propagation reactions. It can be formed according to the assumed reaction mechanism shown in Table 1 with different catalyst age:
β (τ ) =
Rchain transfer R propagation
=
ktr , H 2 ⋅ H 2 + ktr , M ⋅ M + ktr + kd g (τ ) ⋅ k p ⋅ M
,
(2)
and the molecular mass distribution of the produced polymer: ∞
w(r ,τ ) = r ⋅ β ⋅ exp(− r ⋅ β ); 2
w(r ) =
∫ R (τ )⋅ w(r ,τ )⋅ dτ p
0
∞
∫ R (τ )⋅ dτ p
0
from which Mn, Mw, PD can be calculated (Soares et al, 1997).
(3)
Effects of Catalyst Activity Profiles on the Scale-up of Polymerization Reactors
941
The change of the size of the polymer particle produced by catalyst particle with size can be formed in the following:
dr (t ,τ , rcat ) g (τ ) ⋅ k p ⋅ M ⋅ M W ⋅ C (t ,τ , rcat ) = ; t =τ = 0 2 dt 4 ⋅ r (t ,τ , rcat ) ⋅ π ⋅ ρ pol
r = rcat ,
(4)
and the average size of polymer particle in a given time instant is: ∞
PS (t ) =
⎡rcat ,max
∫ D(t ,τ ) ⋅ ⎢⎢ ∫ w(r ) ⋅ r (t ,τ , r ) ⋅ dr 0
⎣ rcat ,min
cat
cat
cat
∞
⎤ ⎥ ⋅ dτ ⎥⎦
(5)
∫ D(t ,τ ) ⋅ dτ 0
where D(t,τ) is the mass of polymer particles in the reactor with the age τ (Nemeth, 2005), and w(rcat) is the size distribution of catalyst particles.
4. Application of the method In Figure 1, polymerization experiments with the same temperature and concentration conditions can be seen. In this figure the reaction rates are presented that can be calculated by the ethylene input to hold the pressure constant. 7
Reaction rate (kg/ g Cat. h)
6
5
4
3
2
1
0 0
10
20
30
40
50
60
70
80
Time (min)
Figure 1. Polymerization experiments by the same conditions but for different time.
It can be determined that the catalyst activity is changing significantly during the experiment. The pseudo-kinetic constants can be determined by these diagrams and by the analysis of polymer powder after the experiments (measurements of Mn, Mw by GPC, comonomer content of polymer based on FTIR analysis). The results of simulations using the identified kinetic constants can be found in Table 2. In this table, the measured and simulated value of average molecular mass (Mn, Mw) and comonomer content of the polymer can be seen.
S. Nemeth et al.
942 Table 2. Experimental and calculated results Time (min)
Mn
Mn (calc.)
Mw
Mw (calc.)
20 27 39 50 55 75
13150 18500 16200 17200 17400 21100
14000 15100 15600 16900 16400 20200
86600 80250 84600 102000 106000 112000
75000 72000 75200 89000 92300 98600
Co-mon. (w%) 2.01 1.85 0.87 0.8 0.78 0.41
Co-mon. (w%) (calc.) 1.91 1.72 0.91 0.85 0.82 0.5
From the results it can be determined that the presented model is able to describe the polymerization processes accurately enough to qualify the catalyst in advance. The operation of the industrial reactor was also simulated using the identified kinetic constants. The catalyst is used for polymerization also in industrial conditions, therefore it is possible to analyse the scale-up effects. The computational results can be compared with the measured ones by the industrial reactor if the reactor model is solved using the measured 4 w% ethylene and 0.08 w% 1-hexene concentrations and 70 minute residence time. During the analysed time period the polymer was produced with catalyst intput equal to 1.6-1.8 kg/h. Due to the information based on the laboratory reactor 2.5 kg/h catalyst is needed to produce the same amount of polymer, which is about 38% difference. The measured average molecular mass of the polymer is Mn=23100, Mw=107500, while these values are 21800 and 95300 based on the calculation, respectively. The hexene content of the polymer is 0.35% from the measurement, and 0.45% from the calculation. From these results it can be determined that the developed model is able to describe the processes within the laboratory and also the industrial reactor accurately enough to qualify the catalysts in advance. However, to solve the scale-up problem there is a need to identify the detailed model. The proposed method is able to provide information to design the experiments needed for that.
5. Conclusion The presented method can be used to describe the polymerization also with catalysts whose activity is continuously changing. Based on that, the method can be used to qualify the catalysts preliminary, to estimate the product spectrum producible in the industrial reactor and to design the experiments to identify detailed kinetic constans.
References Calabro, D.C. and F.Y. Lo, 1988, Transition Metal Catalyzed Polymerization, Qirk, R.P ed. Cambridge Univ. Press, Cambridge. Keii, T., 1972, Kinetics of Ziegler-Natta Polymerization, Kodansha Scientific Books, Tokyo. Keii, T., and K. Soga 1986, Catalytic Polymerization of Olefins, Kodansha LTD, Tokyo. McDaniel, M.P. and Martin, S.J., 1991, Poisoning studies on Cr/Silica 2. Carbon Monoxide, J. Phys. Chem., 95, 3289. Nemeth, S., B. Feil, P. Arva and J. Abonyi, 2005, Effect of catalyst activity profiles on the operation conditions of an industrial polymerization reactor, Puigjaner L., Espuna A. (eds) Computer-Aided Chemical Engineering, 20A, 667 Soares, J.B.P, J.D. Kim and G.L. Rempel, 1997, Analysis and Control of the Molecular Weight and Chemical Composition Distribution of Polyolefins Made with Metallocene and ZieglerNatta Catalysts, Ind. Eng. Chem. Res., 36, 1144 Xie, T., A.E. Hamielec, 1993, Modelling free-radical copolymerization kinetics - evaluation of pseudo-kinetic rate constant method, Makromol. Chem Theory Simul., 2, 421
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Optimization-based Root Cause Analysis Eyal Dassau and Daniel Lewin PSE Research Group, Chemical Engineering, Technion, Haifa, Israel Abstract A systematic approach to yield enhancement was recently proposed by Dassau et al [1], which combines six-sigma with design and control to improve estimated yields in the process design stage. After identifying the critical-to-quality variables, the key step involves the analysis of process measurements to identify the root cause for low quality or yield. The availability of a model of the process permits a significant improvement to this step, through the incorporation of formal optimization as a means of automating root cause analysis. The problem is formulated as a mixed-integer nonlinear program, whose system variables include the possible perturbations that affect low quality and low yield, and whose decision variables are all of the possible process improvements. The potential of proposed optimization-based root cause analysis to enhancing yield is demonstrated on the process of penicillin production, involving both fermentation and downstream purification. Keywords: Root Cause Analysis; Bioprocessing; Six-sigma; Yield Enhancement
1. Introduction Batch process design is an iterative activity, involving significant capital and human resources. Identifying bottlenecks and other constraints to enhanced process performance is an important aspect of incremental improvement, which can be assisted by employing systematic root cause analysis. In multi-step processes, this calls first for the detection of the most problematic step and then the identification of the combination of the associated degrees-of-freedom that contribute to the problem. This is often carried out in practice by applying Design of Experiments (DOE) in the process itself. As will be shown in this paper, a more efficient approach is one relying on process simulation and optimization. As pointed out by Bogle et al [2], a way to overcome development and manufacturing problems is to adopt a plantwide stance in the design and optimization of the process, which calls for a systematic method, such as Six-sigma (6σ), that can serve both to identify sources of poor performance and as the driving force for continuous improvement. Poor performance could be the outcome of either a poorly designed process, or its control system, or a combination of the two. By improving the most significant drawbacks, one will generally improve the process controllability and resiliency leading to increased sigma levels and to a superior process [3]. In this paper, we extend the approach of Dassau et al [1], showing how an optimization-based approach can be used for systematic root cause analysis. After some background on root cause analysis and 6σ, we present our proposed approach and then demonstrate its capabilities in the improvement of a process for penicillin manufacture. Root cause analysis is routinely used to identify a failure, investigate its causes, and suggest corrective actions. This is traditionally carried out by a descriptive statement of each failure followed by the suggestion of its probable causes and their investigation,
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often carried out in practice by brainstorming. Subsequently, the most probable causes are identified and recommendations are made [4, 5]. Six-sigma (6σ) is an iterative five-step procedure to progressively improve product quality. The five steps are: Define, Measure, Analyze, Improve, and Control, referred to by the acronym, DMAIC. Initially, the DMAIC procedure is applied to quantify the base-case conditions. Then, cycles of the procedure are implemented to iteratively improve the process [6]. In itself, 6σ is a systematic way to enhance process performance, with the key step in the DMAIC procedure being the identification of the root cause of inadequate performance. However, this will only detect the weakest link in the production sequence, whereas a more useful outcome would be to identify the degrees of freedom in a critical unit’s design and/or control system that need to be manipulated to improve the process. This can be done by optimization-based root cause analysis, as described next
2. Optimization-based root cause analysis A new perspective to root cause analysis is introduced as a solution of an optimization problem, formulated as a mixed-integer nonlinear program (MINLP), whose system variables include the possible perturbations that affect low quality (Critical to quality, CTQ) and low yield (Critical to Productivity, CTP), some of whom are decision variables. Following Lewin et al [7], the MINLP is formally defined as: min J { x,u,d , θ} u∈U,θ
Subject to:
x = f ( x,u ,d , θ )
y = g ( x,u ,d , θ ) , y ∈ Y
z = h { x,u,d , θ} , z ∈ Z (1) where J is a objective function accounting for benefits and costs associated with process improvements, x is a vector of process states, y is a vector of measured process outputs where the output variables are required to lie inside a hypercube, Y, rather than meet specific setpoints, z is a vector of process quality variables (CTQ and CTP) or attributes of the manufactured product that needs to meet specifications within the hypercube Z, u is vector of manipulated variables, d, is a vector of uncontrolled disturbances, and θ, a vector of continuous and discrete design variables. As demonstrated in the example that follows, after appropriately defining the objective function of the above MINLP, its solution identifies the combination of manipulated variables that are the root cause of inadequate process performance. Figure 1 presents a summary of operations that are executed to lower the DPMO level of the entire plant, through the adjustment the CTQ and CTP variables to acceptable values. First, the CTQ and CTP variables are selected, and then a preliminary simulation is invoked based on the current design and initial conditions. The process performance is diagnosed using the simulation results, meaning that the values of the CTQ and CTP variables are collected and the defects per million opportunities (DPMO) computed for each unit, that is, the number of observations of each CTQ or CTP variables that are outside their specification windows. The DPMO value can either be estimated by counting the actual off-specification measurements in the production trajectory over time, or can be computed on the basis of the observed mean, μ, and standard deviation, σ, of the CTQ variable. It should be noted that the main trigger for RCA is a management decision and could be the DPMO level, as in the following example, or the production time or any other target. Next, the most problematic unit
Optimization-based Root Cause Analysis
945
operation is selected as a target for detailed root cause analysis. In this framework, root cause analysis involves the formulation of a MINLP and its robust solution obtained using a Genetic Algorithm (GA, [8]), that manipulates design and control variables (both discrete and continuous) of the unit to identify the root cause or causes and to suggest improved values for those variables. This instigates the next DMAIC improvement cycle, where the improvements are implemented and performance is again diagnosed. The procedure is repeated until acceptable DPMO values or CTQ/CTP variables are obtained in each process unit, or until economic considerations dictate its termination. Note that since the GA maintains a population of potential solutions, it not only provides a solution to the MINLP of Eq. (1) but also has the ability to trace the source of the poor performance by analyzing the variance of the solution population.
Fig. 1. Optimal root cause analysis working sequence.
The proposed methodology is demonstrated on a simplified process for the production of penicillin, considering only the fermentation and the first down-stream processing step as shown in Figure 2. Each unit operation, together with its control system, is modeled, calibrated and implemented, using Matlab® and Simulink®. For details, see Dassau et al [1]. Penicillium chrysogenum Substrates
Penicillin (product) Reaction / Fermentation
Primary Recovery
Intermediate Recovery
Fig. 2 Schematic of simplified penicillin process.
Final Purification
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E. Dassau and D. Lewin
3. Demonstrative example The DMAIC procedure is applied on the penicillin simulation to define the base-case conditions as summarized in Table 1. Subsequently, cycles of the procedure are implemented to iteratively improve the process, noting that improvements at each cycle are implemented in the triggered unit exhibiting the highest DPMO value or if its CTQ or CTP variables are off-specification. Table 1 - Summary of control limits, DPMO and TY for the base case LCL UCL DPMO Fermentor pH 4.9 5.1 45,445 Temperature 22 28 465 Reactive Extractor - TY = 73% pH 4.8 5.2 462,456 6.75×10-5 Cx (mole/liter) Reactive Re-Extractor - TY = 86% pH 7 9 31,264 Cx (mole/liter) 4.2×10-5 Total Production Time (hr) Total TY %
Production Time (hr) 422 5 5 432 63
Cycle I. As clearly indicated in Table 1, the first unit operation that needs to be selected for improvement is the reactive extractor, since in the base-case design, it exhibits the highest value of DPMO. Analysis shows that the degree of extraction reaches only 73% after 5 hours and that the pH value is clearly not on its set point of 5. Moreover, the value of Cx, which represents degradation products of penicillin, is rising constantly during the batch. The total throughput yield for the sequence is 63% and the production time is 432 hours. Ideally, we would like to simultaneously reduce the values of DPMO and Cx, to increase the throughput yield and to minimize the production time. As a step to overcome such poor performance the root cause analysis mechanism is invoked, which involves the solution of the following specific MINLP: 1 ⎞ ⎛ min ⎜ α1 i DPMO + α 2 iCx + α 3 i ⎟ w.r .t TY ⎝ ⎠ C 0 , t , CL a
( )
s.t. x = f ( t , x, u ) , x ( 0 ) = x0 DPMOU ≤ DPMO ≤ DPMOU CxL ≤ Cx ≤ CxU , TY L ≤ TY CaL ( 0 ) ≤ Ca ( 0 ) ≤ CaU ( 0 ) , t L ≤ t ≤ t U CL ∈ [ 0,1]
(2) where, α1, α2 and α3 are objective function weights noting that in this case we do not include costs associated with improvements, the superscript L and U are lower and upper limit respectively, and the manipulated variables are the initial concentration of LA-2 in the organic phase, Ca(0), the extractor processing time, t, and the status of the possible pH controller, CL (open- or closed-loop). Careful selection of manipulated variables based on process understanding and base-case results can help in choosing DOFs for optimization. In this case, the DOFs were selected in two steps. First, only the design parameter CL was selected as a DOF, leading to promising results (lower DPMO in Cx). Then, the two additional variables were added as DOFs. A more general way to arrive to the optimal solution is to allow the GA to simultaneously manipulate all of the
Optimization-based Root Cause Analysis
947
DOFs of the system and investigate which of them are important (i.e., belong to the subset of the root cause) by analyzing their variance and the magnitude of the computed change from their initial values. The solution of the MINLP in Eq. (2) indicates that the root cause is a combination of problems: (a) the absence of a pH control system in the reactive extractor, leading to increase in the amount of impurities, which has a negative affect on the downstream units; (b) an insufficient amount of LA-2 in the feed; (c) insufficient processing time. By introducing a control system to maintain the pH at its set point of 5, increasing Ca(0) from 0.01 to 0.044 mole/liter and increasing the processing time from 5 to 5.54 hr, this achieves not only excellent pH control, but also reduces the amount of impurities by 84%, and increases the TY of the unit from 73 to 97%, leading to an increase in the overall TY from 63 to 80%. It should be noted that each of the components of the objective function are normalized to avoid bias. Furthermore, since the GA produces a population of possible solutions, the analysis of the variance in the values of the DOFs manipulated allows the key variables to be identified. In this case, it is apparent that the absence of a control system and the amount of LA-2 are more important root causes to inferior performance than the short production time. Cycle II. Having improved the extractor operation, the DMAIC procedure is repeated to further improve the process. We note that the improvement implemented in Cycle I improves the quality of the feed to the reactive re-extractor, this reduces the DPMO for that unit from 31,264 to 4,092 before having made any additional improvements (see Table 2). Moreover, noting that the fermentation part now exhibits the highest DPMO level and dominates the overall production time, its reduction would provide a means to increasing the overall productivity of the process. Once again the root cause analysis mechanism was invoked involving the MINLP: min (α1 i DPMOT + α 2 i DPMO pH + α 3 it ) w.r .t
GT , f g , Pw
s.t. x = f ( t , x, u ) , x ( 0 ) = x0 DPMOiL ≤ DPMOi ≤ DPMOiU , i = T and pH t ≤ t U , C ps ≤ C p , f gL ≤ f g ≤ f gU , GTL ≤ GT ≤ GTU , PwL ≤ Pw ≤ PwU
(3) where GT is the threshold glucose concentration in the fermentor at which additional substrate is added, fg is the oxygen flow rate, Pw is the agitator power setting (affect oxygen mass transfer), and Cp is the local penicillin concentration, and C ps is the production specification. The solution of Eq. (3) leads to the following recommendations: (a) changing GT from 0.3 to 49 g/l; (b) changing fg from 8.6 to 8.3 l/h; and (c) changing Pw from 29.9 to 39 W. Implementation of these recommendations reduces the fermentation time for a peak penicillin concentration of 1.5 g/l from 422 to 264 hours This reduced production time is achieved at a price of temperature distributions with a higher variance than in the base case, with a DPMO level of 1,754, which however, needs to be weighed against the resulting reduction in batch time of about 40%. Note also that these have no effect on the total TY. Cycle III. The last cycle of the DMAIC procedure is invoked on the re-extractor to reduce penicillin degradation which is rather high. This situation is improved by introducing a pH controller in this unit also, with the most important outcome being a decrease of 45% in the concentration of impurities in this unit. The down-side is a slight decrease in the degree of extraction from 83% to 81%, reducing the total TY to 79%. This improvement should be evaluated with respect to what is more critical to the
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management: 45% decrease in impurity level is achieved at a cost of a 1% decrease in yield. Moreover, it is important to note that what are considered as acceptable ranges for DPMO values and CTQ/CTP variables are specific to each processing unit and variable. 5. Conclusions We have shown that formulating root cause analysis as an optimization problem and implementing it as part of the DMAIC procedure can locate the root cause and generate better and more comprehensive solutions than could be achieved by conventional brainstorming. However, automated root cause analysis cannot replace process understanding, and should be seen as a means of assisting in the generation of more creative solutions to production problems. For example, in the process analysis demonstrated here, the proposed approach achieves a 37% reduction in batch time, accompanied by a 25% increase in throughput yield and a 45% reduction in impurities. Evidently, this systematic approach can make a substantial impact in the pharmaceutical industry, through improved overall process yield, quality and return on investment. Table 2 - Record of improvements using the proposed procedure. Base-case Cycle 1 Cycle 2 Fermentor DPMO - pH DPMO - Temperature Reactive Extractor DPMO - pH Cx (mole/liter) Reactive Re-Extractor DPMO - pH Cx (mole/liter) TY % Production Time (hr)
Cycle 3
45,445 465
45,445 465
22,942 1,754
22,942 1,754
462,456 6.75×10-5
<1 1.1×10-5
<1 1.1×10-5
<1 1.1×10-5
31,264 4.2×10-5 63 432
4,092 2.0×10-5 80 432.4
4,092 2.0×10-5 80 273.5
1,883 1.1×10-5 79 273.5
References [1] Dassau, E., I. Zadok, and D.R. Lewin, "Combining Six-Sigma with Integrated Design and Control for Yield Enhancement in Bioprocessing," submitted to I&EC Research (2005). [2] Bogle, I. D. L., A. R. Cockshott, M. Bulmer, N. Thornhill, M. Gregory, and M. Dehghani, "A Process Systems Engineering View of Biochemical Process Operations," Computers & Chemical Engineering, 20(6-7) 943-949 (1996). [3] Seider, W. D., J. D. Seader, and D. R. Lewin, Product and Process Design Principles: Synthesis, Analysis, and Evaluation. 2nd ed. John Wiley and Sons, New York (2004). [4] Diesselhorst, T. and E. Klaui, "Root Cause Analysis of Operational Induced Vibrations in a Feedwater System," Nuclear Engineering and Design, 206(2-3) 129-137 (2001). [5] Elleithy, R. H., "Root Cause Analysis; Fundamentals and Applications," Annual Technical Conference - Society of Plastics Engineers, 60(3) 3082-3087 (2002). [6] Rath and Strong, Six Sigma Pocket Guide. Rath & Strong Management Consultants (2000). [7] Lewin, D. R., W. D. Seider, and J. D. Seader, "Towards Integrated Design and Control for Defect-free Products," Chapter D3 in Integration of Process Design and Control, eds. P.E. Seferlis and M.C.E. Georgiadis, Elsevier Science: San Diego. 533-554 (2004). [8] Lewin, D. R., "Multivariable Feedforward Control Design Using Disturbance Cost Maps and a Genetic Algorithm," Computers & Chemical Engineering, 20(12) 1477-89 (1996).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A synthesis procedure for the design of semicontinuous reactive distillation for specialty chemicals. Thomas A. Adams II, Warren D. Seidera a
Department of Chemical and Biomolecular Engineering, University of Pennsylvania Philadelphia, PA 19104
Abstract Semicontinuous reactive distillation (SRD) is a novel processing technique that tightly integrates reactions and separations together using forced cyclic principles. Previous simulations of SRD have shown that it is not only feasible, but also economically superior to equivalent batch and continuous processes for a wide range of production rates. Although the SRD process is complex, the suggested synthesis algorithm provides a method for generating an SRD schema from continuous or batch designs. Keywords: Semicontinuous, reactive distillation, synthesis algorithm.
1. Introduction At the earliest stages of design, process engineers typically choose between using batch or continuous processing to achieve their design objective. The largest factor in this decision is often the desired output capacity; continuous processing for large outputs, and batch processing for small ones. However, growth in the fine and specialty chemical industries is causing an increasing demand for capacities at intermediate production rates where the decision between batch and continuous operation is not as clear. To address this issue, semicontinuous processes have been developed that can be the optimal design strategy for those intermediate capacities. Several semicontinuous processes have been explored for a variety of design scenarios. These include semicontinuous distillation (Monroy-Loperena, 2004, Phimister, 2000a, Phimister, 2000b, Phimister, 2001), semicontinuous extractive distillation (Phimister, 2000c), semicontinuous pressure-swing azeotropic distillation (Phimister, 2000d), and most recently, semicontinuous reactive distillation (Adams, 2005b, Adams, 2005c). The SRD system has been shown to be the economically optimal design strategy for intermediate production rates when comparing it to equivalent batch and continuous processes. However, the system is complex, and so a synthesis strategy is presented here to aid in the design of such systems.
2. Example The general synthesis strategy is best illustrated by example. Consider the reversible, exothermic reaction of acetaldehyde (Tnbp = 21°C) and propylene glycol (Tnbp = 181°C) to form water and the specialty chemical 2,4-dimethyl-1,3-dioxolane (Tnbp = 93°C) , abbreviated “DMD”: This system is complicated by a low-boiling azeotrope between water and DMD (86°C at 1 bar), and the formation of an additional, three-species azeotrope between water,
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DMD, and acetaldehyde at pressures above 2.1 bar. Furthermore, the kinetics of the reverse reaction are fast, such that the equilibrium conversion is only 70 to 80 percent within workable temperature ranges. It is desired to create a mixture of water and DMD in equimolar amounts with no more than 2.0 mole percent impurities.
3. First Step: Consider Process Designs with Common Solutions 3.1. Continuous Processing Designing a continuous process to achieve our design objective is a simple procedure using well-known heuristics and rules of thumb. The highly reversible nature of the reaction system means that both reaction and separation steps will be required to achieve our products in the desired purity. Reactive distillation (RD) in the traditional sense is an unattractive method for several reasons. First, because the two reaction products have intermediate boiling points, and the reaction is highly reversible, it is difficult to recover the products in a sidedraw. Second, because one of the reagents leaves in the distillate (acetaldehyde), and one must leave in the bottoms product (propylene glycol), the reaction is shifted to the left. To counter this effect, a large excess of one reagent can be used, with high recirculation costs. Third, in reactive distillation, the temperatures in the column are thermodynamically related to the pressure, which cannot exceed 2.1 bar for the case study. Hence, temperature cannot be adjusted to control the rate and extent of the reaction. Furthermore, a coupled external reactor is often more economically attractive for exothermic, reversible A + B C + D reactions such as this (Kaymak, 2004). Therefore, the reaction should be performed in an external reactor. Because the reactor effluent will contain all four species, separation will be required to remove the reagents from the effluent mixture, as shown in Figure 1. Since it is usually energetically optimal to remove the lightest species first in a distillation train, the feed is sent to a distillation column where acetaldehyde is removed in the distillate and recycled to the reactor. The remaining three species exit in the bottoms product and are sent to a second distillation column. In this column, propylene glycol is removed in the bottoms product and recycled to the reactor, and the water and DMD exit in the distillate. Considering total mole balances, if we would like equimolar water and DMD to exit the system, we will need to feed the reactor with equimolar amounts of acetaldehyde and propylene glycol.
Figure 1: Continuous process. A = Acetaldehyde, P = Prop. Glycol, D = DMD, and W = Water.
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3.2. Batch Processing A batch process to perform the desired reaction and separation can be designed through similar thinking, as shown in Figure 2. Following the conclusions for the continuous process, the reaction will be performed outside of the separation unit. For a batch design, however, only one distillation column is needed. The four-species reactor effluent near chemical equilibrium is charged to the column sump. Then, the lowest boiling point species, in this case acetaldehyde, is removed first and recycled to the reactor unit. Once it has been collected, the next lightest species, DMD and water, are seen in the distillate. These are collected in a receiving tank until the equimolar purity has been reached and all of the product has been collected. Note that during the cycle, the azeotrope between the two products will cause both species to appear in the distillate together in varying amounts during this step. When this has been completed, the propylene glycol that remains in the column and sump can be reused for another reaction. The column heating ceases, allowing the propylene glycol to fall into the sump, which is then pumped back into the reactor. For efficient processing, the reactor and distillation column operate in parallel.
Figure 2: Batch process. A = Acetaldehyde, P = Propylene Glycol, D = DMD, and W = Water.
4. Second Step: Design the Semicontinuous Reactive Distillation System 4.1. Overall Design Semicontinuous separation processes utilize forced-cyclic methods to perform multiple types of separations inside the same distillation column during a cycle. For example, in the case of SD, three species can be separated with only one column. Similarly, with SRD, two different separation steps can also be separated with only one column. This reduces capital costs (compared to continuous designs) and has a positive effect on the economics of the system. To do this, consider each step in the separation train of the continuous process as one phase of operation. In the previous example, the first step was to remove acetaldehyde, and the second was to remove propylene glycol. The semicontinuous method uses a column to first remove acetaldehyde from the fourspecies reactor effluent, and then uses the column to remove propylene glycol from the resulting three-species mixture. Then, the cycle is repeated.
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An auxiliary storage vessel will be needed to hold the three-species mixture as it exits from the bottoms during the first phase until it is ready to be used for the second phase. The reactor vessel is not suitable for this purpose because the product-rich mixture will start to react in the reverse direction. Therefore, a second vessel is needed. The reactor vessel, however, is a good destination for the acetaldehyde-rich distillate in the first phase, and the propylene glycol-rich bottoms product in the second phase, since these can be utilized to create more product. The distillate in the second phase will contain water and DMD in near-equimolar amounts. It might be possible to collect this simply as the desired product; however, the controllability of collecting both DMD and water in the distillate at a consistent equimolar mixture is a significant issue (See Section 4.3). Instead, the DMD and water in the second phase are recycled to the feeding vessel, causing a gradual purification of this vessel as it moves from a threespecies mixture to having only equimolar amounts of DMD and water. Thus, the overall design scheme is such that in each phase, the vessel feeding the column also receives the distillate returned to it, and the bottoms product is sent to the other vessel, as shown in Figure 3.
Figure 3: SRD process. A = Acetaldehyde, P = Propylene Glycol, D = DMD, and W = Water. The operating modes in which a stream is activated are annotated.
4.2. Transitioning Between Phases To change from one phase to the next, some transitional steps are needed to ensure proper functionality. For example, at the end of the first phase, the distillate is acetaldehyde rich, and the bottoms contains the other three species. If the second phase immediately preceded the first, the distillate would be pumped into the auxiliary tank, contaminating it with undesired acetaldehyde and reducing the product purity. Furthermore, the bottoms stream, containing the reactor product, would be pumped into the reactor, causing undesired backward reaction. To correct this problem, two transitional steps are needed between each phase. First, the column operates in a typical batch distillation; the feed to the column is stopped, no bottoms product is collected, and the distillate continues to be recycled to its originating vessel. This allows the light
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species to be “purged” from the column, and can be stopped once the light species is no longer detected in the distillate. Second, the column is fed with liquid from the other tank, with the distillate now transferred to that tank as well. The column continues to operate in batch mode until column profiles shift such that the compositions in the distillate and sump are such that the new phase can begin without impurities being collected in the distillate and bottoms product. The resulting cyclic policy is summarized in Table 1. An animated video of this process is also available (Adams, 2005a). Table 1. SRD cyclic policy. A = Acetaldehyde, P = Prop. Glycol, D = DMD, W = Water. Mode 1 Purify 2 Purge 3 Shift 4 Purify 5 Purge 6 Shift
Feed to Column
Species in Feed A,D,W, P
Distillate Destination
Distillate Species
Bottoms Destination
Bottoms Species
Reactor
A
Tank 2
D,W,P
--
--
Reactor
A
--
--
Tank 2
D,W,P
Tank 2
D,W
--
--
Tank 2
D,W,P
Tank 2
D,W
Reactor
P
--
--
Tank 2
D,W
--
--
Reactor
A,D,W, P
Reactor
A
--
--
Reactor
Stopping Criteria Reactor has no P,D,W left No more A seen in distillate No more D,W seen in sump Tank 2 has no P left No more D,W seen in distillate P,D,W seen in sump
4.3. Controllability and Operability The SRD system requires good dynamic control to successfully handle several modes of operation. This will require a supervisory control system that is capable of knowing the current mode of operation and when to switch between them. One common solution is to design a separate set of controllers for each operating mode and activate the appropriate system when it is needed (Kordon, 1999). In many cases, one controller (e.g., a reflux ratio controller) may be used for many or all of the different modes, but the tuning parameters and set-point may be different from mode to mode. Because of the semicontinuous nature of the system, steady-state operation will never be reached. For the purification steps (Modes 1 and 4), the composition of the feed to the column changes drastically over the course of the mode, and maintenance of the distillate or bottoms purity cannot easily be achieved with a typical feedback PID controller. Instead, a model-based feed-forward controller that manipulates the boilup rate in response to changes in the feed composition is effective in maintaining the product purity (other solutions are possible). An additional feedback loop handles process-model mismatch errors. For the transition modes (Modes 2, 3, 5, and 6), the column operates in batch operation. In these modes, traditional control methods for batch distillation can be utilized. Because of the dynamic nature of the purification modes, it may be infeasible to maintain product purities in both the distillate and bottoms streams. However, the control system can be designed such that maintaining the purity of the bottoms stream is most important, and maintaining the purity of the distillate is a secondary priority. Because of the feed-recycle loops, when the distillate purity is not achieved, the impurities are merely returned to the tank from which they originated. This will lengthen the time to complete the purification of that tank, but it will not affect the overall final product purity.
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The internal flow rates in the column must be fairly constant throughout the cycle so that weeping and flooding do not occur. Thus, an additional control system that manipulates the feed rate or distillate rate may be necessary. The flooding can be prevented by using an algorithm such as the Fair method (Wankat, 1988) to calculate a minimum column diameter at regular intervals over the cycle. When this calculated diameter has small deviations and remains close to (but not above) the specified diameter for the column, both flooding and weeping will be prevented. The control system, its tunings, set-points, and the triggers that determine when a mode switch should occur are subject to optimization. For example, it may be more economical to have lower distillate purities in the recycle loops and longer cycle times than to have higher purities, shorter cycle times, but a higher rate of energy expenditure.
5. Conclusions The design of a semicontinuous reactive distillation system at intermediate production rates can be derived from equivalent continuous and batch systems that achieve the same objective. In general, each stage in a separation train of the continuous system becomes a phase of operation in the semicontinuous one. Transitional modes that clear the column of the lightest species and shift the column profiles to their desired state are needed between major phases of operation. A different control system can be activated during each of the modes to achieve the objectives of that mode. The feed-recycle loops allow for feasible controllability of the system by focusing purity demands on the bottom of the column without overall loss of product purity.
References
T.A. Adams, 2005a, Animation of semicontinuous reactive distillation case study, case 280, www.seas.upenn.edu/~adamsta/case280.avi T.A. Adams and W.D. Seider, 2005b, A novel concept: Semicontinuous reactive distillation, in 2005 AIChE Spring Nat'l. Meet. American Institute of Chemical Engineers, New York, NY 10016-5991, United States: Atlanta, GA, United States. p. 1735. T.A. Adams and W.D. Seider, 2005c, Semicontinuous reactive distillation for specialty chemicals production: Economic comparison with batch and continuous processing, in 2005 AIChE Annual Meet. American Institute of Chemical Engineers, New York, NY 10016-5991, United States: Cincinnati, OH, United States. D.B. Kaymak and W.L. Luyben, 2004, Design of distillation columns with external side reactors, Ind. Eng. Chem. Res., 43, 25, 8049-8056. A. Kordon, P.S. Dhurjati, Y.O. Fuentes and B.A. Ogunnaike, 1999, An intelligent parallel control system structure for plants with multiple operating regimes, J. Proc. Cont., 9, 453-460. R. Monroy-Loperena and J. Alvarez-Ramirez, 2004, Some aspects of the operation of semicontinuous, middle-vessel distillation columns, Chem. Eng. Commun., 191, 11, 1437-1455. J.R. Phimister and W.D. Seider, 2000a, Distillate-bottoms control of middle-vessel distillation columns, Ind. Eng. Chem. Res., 39, 6, 1840-1849. J.R. Phimister and W.D. Seider, 2000b, Semicontinuous, middle-vessel distillation of ternary mixtures, AIChE J., 46, 8, 1508-1520. J.R. Phimister and W.D. Seider, 2000c, Semicontinuous, middle-vessel, extractive distillation, Comput. Chem. Eng., 24, 2-7, 879-885. J.R. Phimister and W.D. Seider, 2000d, Semicontinuous, pressure-swing distillation, Ind. Eng. Chem. Res., 39, 1, 122-130. J.R. Phimister and W.D. Seider, 2001, Bridge the gap with semicontinuous distillation, Chem. Eng. Prog., 97, 8, 72-78. P.C. Wankat, 1988, Equilibrium staged separations: Separations in chemical engineering, 707.
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Multi-objective optimisation of batch distillation processes Tajalasfia M. Barakat, Eric S. Fraga and Eva Sorensen1 Centre for Process Systems Engineering, Department of Chemical Engineering, UCL (University College London), Torrington Place, London WC1E 7JE, U. K.
Abstract This paper considers for the first time the simultaneous multi-objective optimisation of design and operation of batch distillation processes. The overall problem is formulated as a mixed integer dynamic multi-objective optimisation (MIDMO) problem. The optimisation strategy comprises of economics indices that reflect capital investment, operating costs and production revenues. A case study for the separation of a homogeneous tangent-pinch (acetone-water) binary mixture is presented for a dualcriteria optimisation case of minimising capital investment whilst also minimizing production costs. It is found that the proposed genetic algorithm based multi-objective framework can be implemented to multi-dimensional engineering problems to obtain the Pareto-optimal set. The presented algorithm was found to successfully handle the multi-objective nature of the simultaneous design and operation of batch distillation columns. Key words: Multi-objective optimisation, Genetic algorithm, Batch distillation
1.
Introduction
Batch processing has received renewed attention over the past decade, particularly in the low-volume, high-value-added fine chemical and pharmaceutical industries. The importance of batch processes is increasing due to increasing cost pressures associated with over-capacity of high volume continuous plants and the ongoing preference of custom-made rather than commodity chemicals. Within these industries, batch distillation remains the most commonly used technique for separating liquid mixtures despite being an energy and capital intensive process. The optimal design and operation of batch distillation columns has therefore received considerable interest in recent years, not only in terms of novel column configurations and operating procedures, but also of how the optimal designs and operations are formulated as optimisation problems and how these problems are solved using efficient solution techniques (Low & Sorensen, 2003). Batch distillation processes, as most batch processes, are highly complex in nature and involve a number of operational and economical objectives that need to be considered. The performance of these processes depends on a number of different criteria that are 1
Author to whom correspondence should be addressed. E-mail: [email protected]
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often conflicting (e.g. revenue & cost). An effective optimisation of such systems therefore requires the consideration of multi-criteria approaches to accommodate for the multi-objective nature of the problem and to effectively evaluate and optimise the performance while meeting the objectives. Although the optimisation of design and operation of batch distillation processes has been attempted before, this is the first time the simultaneous multi-objective optimisation of design and operation of such processes is considered. The existence of multiple performance measures in a problem results in a set of optimal solutions known as Pareto-optimal solutions, instead of a single optimal solution (Deb et al., 2002). Solutions in the Pareto-optimal set are all equivalent, therefore in the absence of user defined decision making information, no solution can be said to be better than the other. Solutions in the Pareto-optimal set are to assist in the decision making process by highlighting the tradeoffs involved in the problem studied. Depending on which objective is regarded as more important than others (e.g. capital investment savings favoured over energy savings), the decision will clearly have an impact of some extent on other decision variables. Conventionally, multiple objectives are combined into a single objective function composed of the weighted sum of individual objectives, thereby allowing for solutions using existing single-objective optimisation techniques. Other methods suggest using one objective function at a time and combining the solution results of the different optimisation runs in order to obtain the Pareto-optimal set. Many authors have over the last decade suggested a number of multi-objective algorithms, details of these methods and the Pareto dominance concepts can be found in Deb et al. (2002), Dedieu et al. (2003) and Silva & Biscaia (2003). The objective of this work is to propose a genetic algorithm based multi-criteria optimisation procedure that allows the determination of the Pareto-optimal set for multiobjective optimisation of design and operation of batch distillation processes. The effectiveness of the novel method is illustrated by application to the design of a batch distillation process for the separation of a tangent-pinch mixture (acetone-water).
2.
The multi-objective batch distillation problem
Problem definition In batch separation processes, there is a trade-off between capital investment in terms of equipment and performance on one hand and operational decisions and performance on the other. For instance, it is possible to design a distillation column with a low number of trays operating at high reflux ratio, or alternatively, design the column with more trays and operate at lower reflux ratio while still achieving the same separation requirements. High product recovery generally leads to longer batch times but generally incurs higher operating costs. Thus the multi-criteria balance between capital investment and operating costs must be considered. Objective functions The problem of optimal design and operation of batch separation processes, as it is considered in this paper, is to determine the Pareto-optimal solution set that will satisfy all specified separation requirements and constraints. The multiple objectives to consider are 1) the investment cost (Low and Sørensen, 2003):
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f 1 = CC c = K 1 N t0.802V 0.53 + K 2V 0.65
957
(1)
and 2) the annualised operating costs (Low and Sørensen, 2003):
f 2 = AOCc = Cutility × (Qreboiler + Qcondenser )
(2)
Formally, given a mixture Mfeed with number of components NC to be separated, minimum product purities xmin,i, minimum product recoveries Mi,f , price structure of feed and products, Cfeed & Ci , and total production time available per annum TA ; determine the set of design variables ud and operation variables uo to minimise CCC while minimising AOCc:
Minud ,uo CCc
&
Minud ,uo AOCc
subject to: (3)
f ( x , x , t , u d , u o ) = 0 ∀ i = 1,..., n C
c i ( t f ) ≥ c imin u dmin ≤ u d ≤ u dmax
&
u omin ≤ u o ≤ u omax
(4) (5)
Equation (3) represents the mathematical process model of the batch separation process where x is a vector of process state variables. Equation (4) represents the final product recovery and purity constraints imposed that must be satisfied at the end of the batch. Equation (5) represents the physical and optimisation bounds of the design and operating control variables, respectively. For the batch distillation process, the set of operating variables include vapour boilup rate and column reflux ratio profile, i.e. uo = {V, RC}. The vapour boilup rate can subsequently be used to determine the diameter of the column (e.g. using Guthrie’s correlation) as well as the reboiler and condenser heat loads. Design variables include the optimal number of trays Nt, i.e. ud = {Nt}. Process Models The batch distillation model is based on the approach of Low and Sørensen (2003). This model disposes of some of the common modelling assumptions, such as negligible tray holdup and constant molal overflow that may otherwise have a significant impact on the optimal solution. The main features of the model are: Dynamic mass and energy balances and rigorous thermodynamics through the use of liquid and vapour fugacities. The assumptions retained in this work include no entrainment effects, no downcomer dynamics, adiabatic column operation, phase equilibrium and perfect mixing. Solution Methodology Simultaneous multi-objective consideration of optimal design and operation of batch separation processes translates into an optimisation problem with both discrete (e.g.
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number of trays) and continuous variables (e.g. reflux ratio). Furthermore, the nonlinear dynamic models used here, as well as the nonlinear objective functions defined, transform the problem into a complex mixed integer dynamic multi-objective optimisation (MIDMO) problem. The proposed multi-objective GA optimisation framework works through the conventional genetic algorithm operators of which further details can be found in Goldberg (1989). In this work, a given solution set consists of all decision variables that are represented in the genome as direct real values instead of converted binary bits and mapping which has been found to be less efficient (Coley, 1999). The initial population is created randomly. The objectives and constraints of each individual in this population are evaluated using the gPROMS simulation software (Process Systems Enterprise Ltd., 2005). A penalty function procedure is applied (Eq. 6) when necessary. Solutions are assigned a fitness score based on whether an individual is infeasible, dominated or non-dominated with respect to the rest of the population. The Pareto domination concept was used in order to define dominated and non-dominated solutions. The fitness score, as defined in Equations 6 and 7, rewards non-dominated solutions at the expense of the others with the aim of encouraging the GA to drive the population towards the generation of successively better Pareto fronts.
⎧⎡ c min − c i ( t f ⎪⎪ ⎢1 − i ki = ⎨⎢ c imin ⎪⎣ ⎪⎩1
⎧Π in=c1 k i ⎪ f ( g ) = ⎨2 ⎪3 ⎩
)⎤ ⎥ ⎥ ⎦
pt
min if ci (t f ) < ci
∀i = 1,..., nc
(6)
otherwise
if solution is infeasible if solution is feasible but dominated
(7)
if solution is feasible and non-dominated
The GA procedure uses a tournament selection procedure (with a tournament size of 2), 75% crossover rate, 10% mutation rate and a stopping criterion based on the number of generations. A replacement population strategy is implemented. The procedure has been implemented using the GALib genetic algorithm library (Wall, 1999).
3.
Results and Discussion
The procedure has been applied to the two criteria optimisation problem of minimizing capital investment (f1) and minimizing operating costs (f2) for the separation of a tangent-pinch mixture of acetone and water. The batch distillation specifications are shown in Table 1. Figure 1 illustrates the evolution of the Pareto-optimal set generated. It also shows the generated Pareto-optimal sets of representative generations. The population of the 44th genetic generation is the first generation to have produced a feasible Pareto set as all
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previous populations consist solely of infeasible solutions. This illustrates the complexity of the simultaneous multi-objective optimisation of batch distillation processes. The figure shows the progression of the Pareto front to the final population obtained after 200 generations, with the fastest convergence achieved between generations 100 and 120. The Pareto optimal set of generation 200 was found to contain four points compared to two in previous sample generations. This indicates the ability of the algorithm to obtain more inclusive optimal sets as the generation progresses. The confidence in the Pareto optimal set obtained, however, will depend on the number of genetic generations. Generation overlap, where a Pareto-front in a higher generation is worse than that of a lower one, was found to exist. This is due to the lack of an elitism operator that can be incorporated into the suggested procedure to favour the selection of Pareto points to the next generations and avoid any possible loss of these points. This is, however, accounted for in this work by the crossover rate of 75% allowing for 25% of the current population to be carried unchanged to the next evolution, although a future inclusion of an appropriate elitism operator can ensure that all good solutions, i.e. points in the Pareto-optimal set, will be retained in future generations. Figure 1 not only addresses the fundamental tradeoff between batch distillation capital investment and operating costs, it also confirms the ability of the proposed algorithm to evolve towards the Pareto optimal solution set if applied to any other type of multidimensional optimisation problems.
2.20
6
CapCost (10 ´£)
2.00
Generation 44 Generation 100 Generation 120 Generation 200
1.80
1.60
1.40 110.00
130.00
150.00
170.00
190.00
3
OperCost (10 ´£/y) Figure 1. Graph of the Pareto fronts for the populations at four different generations.
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Property
Value
Property
Value
Feed conc., xi,feed (mol frac.) Acetone, Water Batch size, Mfeed (mol) Tray/Cond. holdup (mol)
0.70, 0.30 20,000 0.1
Production time TA (hr) Batch setup time, ts (min) Products purity, xi,f (mol frac.) Product recoveries, Mi,f
7920 30 ≥ 0.97 ≥ 0.70
Table 1, case study specifications
4.
Conclusions
This paper focuses on the development of a multi-objective genetic algorithm for constrained multi-criteria engineering problems. The algorithm has been applied to the optimisation of batch distillation columns for a bi-criteria case. It has been demonstrated that the suggested algorithm can be applied successfully to generate the optimal Pareto set. The convergence of such sets will, however, depend on a sufficient penalty being applied to drive the solution towards feasibility and eventually Pareto non-dominance as well as the termination criterion chosen. The performance will also depend on the complexity of the engineering model implemented and whether an elitism operator is applied.
References 1. 2. 3. 4. 5. 6. 7. 8.
Coley, D., An introduction to genetic algorithm for scientists and engineers, World Scientific Publishing, Singapore, 1st ed., 1999. Deb K., A. Pratap, S. Agarwal and T. Meyarivan, A Fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computations, 6(2), 182-197, 2002. Dedieu, S., L. Pibouleau, C. Azzaro-Pantel and S. Domenech, Design and retrofit of multiobjective batch plants via a multicriteria genetic algorithm, Computers & Chemical Engineering, 27(12), 17231740, 2003. Goldberg, David E., Genetic algorithms in search, optimization and machine learning, Boston ; London : Addison-Wesley, 1989. Low, K.H. and E. Sørensen, Simultaneous optimal design and operation of multivessel batch distillation, AIChE J., 49(10), 2564-2576, 2003. Process Systems Enterprise Ltd., gPROMS User Guide, 2005. Silva, C. and E. Biscaia, Jr, Genetic algorithm development for multi-objective optimization of batch free-radical polymerization reactors, Computers & Chemical Engineering, 27(8-9), 1329-1344, 2003. Wall, M., GAlib: C++ library of Genetic Algorithm components, version 2.4.5, 1999, available from http://lancet.mit.edu./ga.
Nomenclature AOCc ci Ci fi f(g) ki Mfeed Mi,f nc Nt Qi
Annualised operating costs (£/yr) Problems constraints Cost of i Function i Fitness of function g Penalty term Batch feed size (mol) Final product i recovery Number of components Number of trays Heat duty of i
Rc tf ts TA u ud uo V x xi xmin,i
Reflux ratio Total batch processing time (min) Batch setup time (min) Total production time available per annum Vector of control variables Vector of design variables Vector of operation variables Vapour boilup rate Vector of state variables Composition of component i in mixture Minimum composition of component i in mixture
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Sustainable Production of Industrial Chemical Products from Bioresources Jeffrey Seaya,b, Mario Edenb, Robert D’Alessandroa, Christoph Weckbeckerc a
Degussa Corporation, 4301 Degussa Road, Mobile, AL 36590, USA Department of Chemical Engineering, Auburn University, AL 36849, USA c Degussa AG, Rodenbacker Chaussee 4, Hanau-Wolfgang, D-63457, Germany b
Abstract In this work, we present the preliminary results from a study of the development of processes to produce industrially important chemicals from sustainable, bio-based glycerol. The objective of this work is to identify potential industrial uses for the glycerol produced as a side product of the manufacture of biodiesel from fatty acids. Systematic process systems engineering tools such as process simulation and pinch analysis are employed to target economically viable production pathways. In this way, potential process and product options can be evaluated so that further research and development can be focused on candidates with the greatest economic potential. Commercially available software packages are used to develop mass and energy balances for the proposed conceptual processes, determine the minimum utility requirements using thermal pinch analysis and analyze the economic potential of switching production from crude oil derived to sustainable, bio-based feed stocks. Keywords: Sustainable production, renewable bioresources, process integration
1. Introduction At the current rate of consumption, conservative estimates predict the depletion of fossil fuel resources in 80-120 years [1]. As prices for crude oil increase, a similar increase is seen in the price of crude oil derived chemical feed stocks. Thus there is a growing need for the development of novel production processes, which are based on cost effective, renewable raw materials. In addition to the economic benefit, the use of renewable biobased feed stocks decreases greenhouse gas emissions as carbon sources are switched from crude oil to agricultural products [2]. Additionally, the development of industrial processes utilizing bio-based feed stocks will enhance the markets for these products, thus encouraging the development of more consistent supplies. The objective of this work is to utilize systematic, process systems engineering methods to screen candidate conceptual processes for the production of important chemicals and chemical intermediates from bio-based glycerol. This screening procedure allows the process with the greatest economic potential to be selected before beginning detailed process design studies. Historically, glycerol has been produced as a byproduct of the manufacture of soap by hydrolysis of animal fats. However more recently, glycerol has been generated as a byproduct of biodiesel manufacture. Due to its high viscosity, glycerol must be removed from the biodiesel product, thus reducing the carbon utilization. Therefore, the identification of novel industrial uses for glycerol can improve the overall carbon utilization and possibly the profitability of the biodiesel process [2].
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2. Feasible Reaction Pathways The purpose of this investigation is to evaluate the potential for switching the production pathways of three industrially important chemicals from utilizing crude oil derived raw material feed stocks to bio-based glycerol. Currently, each of the products chosen for consideration is primarily produced by the catalytic partial oxidation of propylene, although other pathways utilizing crude oil derived raw materials are also used [3]. The subject industrial chemicals are as follows: • • •
Acrylic acid, which is used in the manufacture of acrylate ester polymers. These polymers have application in the coatings industry, as non-woven fabric binders, as textile and leather finishers as well as in the oil additives industry [3]. Acrolein, which is a widely used intermediate in the production of building materials, herbicides, algaecides, amino acids, and water treatment chemicals [3]. 1,3 Propanediol (PDO), which is used in the manufacture of 1,3-Propanediol Terephthalate. This fiber molecule is used for textiles in the garment industry [4].
Each of these products can also be manufactured via the catalytic dehydration of glycerol [5,6]. The identity and yield of the product is determined by the catalyst used. The case study presented in this paper provides a generic framework for analyzing the economic potential for switching from propylene to glycerol as a feed stock.
3. Sustainable Production Case Study Current production schemes for the chemicals targeted in this study are based on catalytic partial oxidation of propylene. The identity and yield of the product of interest is primarily dependent on the catalyst used; however, the side products of these reactions will generally include carbon oxides, light aldehydes and organic acids [3]. The propylene oxidation reaction is exothermic and is carried out at low pressure. As an alternative reaction pathway, the catalytic dehydration of glycerol is proposed. The dehydration reaction is endothermic and potential side products include light aldehydes, mid-boiling ketones and polyglycerols [7]. Recent studies have indicated that dehydration of glycerol can be carried out in either the liquid or vapor phase [7], however the complete kinetics of these reactions have not yet been identified.
4. Conceptual Process Development To illustrate the methodology for determining the economic potential of switching from production pathways based on crude oil derived feed stocks to bio-based glycerol, a case study will be developed to compare processes based on propylene oxidation with processes based on glycerol dehydration. As the reaction pathway of the glycerol dehydration reaction is unknown, a number of pseudo components were included to represent the actual byproducts. To ensure a realistic process, the pseudo components were chosen so low and high boiling side products are present in the reactor effluent. 4.1. Process Simulation Models For this case study, both liquid and vapor phase production pathways are considered. Multiple reaction temperatures and conversions were considered to cover a wide range of possible operations. Various recycle options and side product purification options are also considered. In total, five liquid phase (Case L1–L5) and five vapor phase (Case V1–V5) glycerol based production schemes are investigated and compared with the
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traditional manufacturing process (Case S1). For the glycerol processes, the options are based on the temperature, pressure and compositions reported in previous studies [7], along with various recycle options for unreacted glycerol and excess water. Process simulation models are developed for all cases and subsequently optimized by process integration methods to provide a systematic framework for evaluating the technical, economic and environmental feasibility of switching to a bio-based feedstock. Figure 1 illustrates the base conceptual process for the vapor phase reaction. Since essentially complete conversion (+99.9%) of glycerol can be achieved in the vapor phase reaction process, only two columns are required to purify the product [7].
Light Ends
Product
Glycerol
Water
Vapor Phase Reactor
Light Ends Column
Product Column
Waste Water
Figure 1. Schematic representation of a conceptual vapor phase glycerol process
Waste Water
Light Ends
Product
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Water
Liquid Phase Reactor
Crude Product Column
Glycerol Column
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Figure 2. Schematic representation of a conceptual liquid phase glycerol process
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Analogously, Figure 2 illustrates the base conceptual process for the liquid phase reaction. Because less than 25% glycerol conversion can be achieved in the case considered for study, three columns are required to purify the product [7]. A primary difference between the liquid phase and vapor phase processes is that in the liquid phase, the product is collected as a bottoms product, while it is an overhead product in the vapor phase process. The effect on product quality must be carefully considered in future stages of conceptual process development. 4.2. Process Integration Analysis In order to ensure optimal resource utilization, it is imperative to apply holistic methods like pinch analysis to optimize any proposed conceptual processes. Increased energy and mass integration enable higher utilization of raw materials and minimizes the use of external utilities [8,9]. The minimum utility requirement for each of the candidate conceptual processes used for this case study was evaluated using Aspen HX-Net software [10]. The minimum hot and cold utility requirements were calculated based on the utilities anticipated to be available in an existing multiple user industrial facility. The minimum utility operating costs are based on the coldest cold utility and the hottest hot utility available. This cost, along with the estimated capital cost of the designed heat exchanger network for each conceptual design is used in the full economic analysis. The normalized variable costs per unit of product, based on the process simulation and pinch studies are reported in Figure 3, where the base case process, S1, is used to normalize all other calculated values. These results indicate that the variable costs of the vapor phase glycerol processes are roughly three times the variable cost of the standard case. Furthermore, the variable costs of the liquid phase glycerol processes are seven to sixteen times higher than the standard case. Although there is a significant reduction in the variable cost of the liquid phase processes at minimum utility, the variable costs are still five to thirteen times higher than the standard case. Normalized Variable Cost Index 18
Cost Index
15 12 9 6 3 0 S1
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Figure 3. Normalized variable cost index
For the conditions chosen for this case study, further consideration of the liquid phase process is shown to be unnecessary. The reason for the large cost difference between the
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liquid and vapor phase reaction schemes can be illustrated using the cost breakdown of the key contributions to the total variable cost, which is presented in Figure 4. Each simulation case is based on an equal quantity of raw material feed. This means that the absolute cost of the raw material is the same for each of the glycerol based conceptual processes. It is clear from the results shown in Figure 4 that the contribution of utility costs in the liquid phase processes is responsible for the resulting high variable cost. This increased contribution of utility cost to the total variable cost for the liquid phase processes can be attributed to several factors, i.e. the very high pressures required and the relatively low conversion of glycerol (less than 25%), which results in high recycle rates and thus large utility cost. Therefore, for the conditions of the case study, ongoing efforts are focused on optimization of the vapor phase processes, rather than the liquid phase process. However, it should be noted that development of new catalysts or reaction techniques that improve the overall conversion of the liquid phase glycerol process could result in this pathway being preferred. Costs as a Percentage of Total Variable Cost
100% 80% 60% 40% 20% 0% S1
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Waste Disposal Cost
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Figure 4. Variable Cost Analysis Breakdown
4.3. Economic Analysis For the next step of the analysis, Aspen Icarus Process Evaluator (IPE) is employed to estimate the total project cost and the fixed capital costs for the vapor phase and standard processes [10]. For the purpose of screening potential conceptual processes, the economic analysis generally is based on IPE default parameters. Exceptions include the equipment metallurgy, the basis of project contingency, the plant lifespan for depreciation and the projected plant location. The results of the total manufacturing cost analysis indicate that the variable costs account for more than 85% of the total manufacturing costs in the vapor phase and standard process cases. Therefore, for the conditions chosen for this case study, it is currently not attractive to switch the production from being petroleum based to being bio-based. The glycerol based production process with the lowest total product cost is still prohibitively expensive when compared with the traditional, crude oil based process.
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5. Conclusions and Future Work Through the application of process systems engineering methods and tools, multiple candidate conceptual processes for the production of industrially important chemicals from dehydration of renewable, bio-based glycerol have been identified. Simulation models of the candidate conceptual processes were used to generate performance data which was used to perform a comparative economic analysis. Given the parameters considered in the case study presented in this work, current prices do not merit switching the production from crude oil derived feed stocks to a renewable bio-based raw material. However, a continued increase in crude oil prices coupled with a decrease in glycerol cost will have a major impact on the profitability of the new processes. Additionally, this analysis illustrates that developing a reaction mechanism with a product yield equivalent or better than the current technology is critical to economic viability. Otherwise, the cost of separation is insurmountably high. In conclusion, the developed models provide a means of estimating the raw material cost targets that would merit the switch in feedstock. Additionally, if crude glycerol could be utilized as a feed stock as opposed to refined glycerol, the economic viability of sustainable production is greatly improved. The next phase of this research involves experimental identification of the actual byproducts and the detailed reaction kinetics for the vapor phase dehydration of glycerol. Throughout the development of more detailed designs, the environmental impact of the glycerol based production schemes will be evaluated and compared with the traditional manufacturing process. The WAste Reduction (WAR) algorithm developed by US-EPA [11,12] will be used to ensure that processes with the highest potential for sustainability are chosen from the economically viable process options. Finally, the preliminary simulation models and economic analysis framework will be continuously updated as new knowledge becomes available to guide future efforts towards the process with highest technical, economical and environmental feasibility.
6. Acknowledgements The authors greatly appreciate the financial support and facilities provided by Degussa Corporation. In addition, the authors want to acknowledge Degussa intern students Maria Schley and Astrid Roesner for their hard work on this project.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
US Department of Energy, World Energy Report, 2003 US Department of Agriculture, EU: Biodiesel Industry Expanding Use of Oilseeds, 2003 J. McKetta, Encyclopedia of Chemical Processing and Design, 1976 DuPont Company, Sorona Polymer Website, 2005 C. S. Miner and N. N. Dalton, Glycerol, 1953 S. Ramayya, A. Brittain, C. DeAlmeide, W. Mok and M. Antal, Jr., Fuel, Vol. 66, 1987 A. Neher, T.Haas, A. Dietrich, H. Klenk and W. Girke, U.S. Patent 5387720 M. M. El-Halwagi and H. D. Spriggs, Chem. Eng. Prog.,Vol. 94, 1998 B. Linnhoff, Trans. IChemE. Chem. Eng. Res. Des., 71, Part A5, 1993 Aspen Technology, Aspen Engineering Suite User Manual, 2005 D. M. Young and H. Cabezas, Computers and Chemical Engineering, 23, 1999 D. M. Young, R. Scharp, and H. Cabezas, Waste Management, 20, 2000
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Cooling Crystallization: a Process-Product Perspective Caliane Bastos Borba Costa, Rubens Maciel Filho LOPCA, Chemical Engineering School (FEQ), UNICAMP, CP 6066, CEP 13081-970, Campinas – SP, Brazil
Abstract This paper presents a CAPE tool that focus on a wide perspective to the process and product design in a batch cooling crystallization operation. The tool comprises modules for process evaluation, design of optimal operation policy, in a process-product perspective, and the adjustment of modeling (parameter estimation). Some issues concerned to each module are discussed, like the available methods for modeling solution and the optimization, as well as the challenges in the particulate product design field. Keywords: crystallization, product design, process simulation, optimization, modeling
1. Introduction The fast demanding market requires the development and increasing research in the area of product design, which devotes efforts in the design of specific products, for very specific purposes, such as drugs with controlled release of active ingredients or functional feed, in which additives may stimulate imune system or help in the cholesterol level control. In the particulate area, massively present in the high-value added chemicals and pharmaceutical industry, the product design nowadays demands not just purity, but also specific particle size distribution (PSD), particle shape and morphology, which represents an increase in the chemistry complexity (Ng, 2001). In the process design, on the other hand, model-based approaches must necessarily be able to drive to process designs that maximize productivity and reduce reprocessing/clean-up costs and time. Efforts should be driven to integrate both process and product requirements. In this scenario, the present paper presents a computer-aided tool to integrate process and product design for batch cooling crystallization, dealing with the issue of modeling and optimization as well as of the understanding of the productprocess complexities.
2. Crystallization: Process and Product Design Concerns The process design concerns the determination of the operating conditions (cooling rate, seeding policy and mixing) to produce the crystals, given the identity of the crystallizing product and the required yield. But the crystallization system does not exist in isolation and it does have an influence on the downstream processing system in which crystals are separated from solution and dried. Bearing the exposed in mind, not just yield is required, but also crystals features that are satisfactory for the downstream processing. In the present work, the process design is solved with CAPE methods, which integrate energy, mass and population balances, composing first principles understanding of materials and process to assure correct process representation
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(modeling). Mathematical programming problems are formulated and solved in order to determine operating conditions that drive to achievement of required process yield. On the other hand, the product design requires a look backwards in the process, in order to produce crystals with very specific features for the active ingredient to work properly (e.g. controlled release). In a lot of applications, PSDs often need to be maintened in specific ranges to ensure good quality indices. The proper PSD is, therefore, essential in product desings, being the critical link between the product quality indices and the operating process variables. Indeed, the achieved crystal-size distribution (CSD) in crystallization processes controls the end product quality and influences significantly the necessary liquid-solid separation. The population balance used in the modeling of the process can be used to accurately describe, analyze and control the CSD (Shi et al., 2006). However, when a product is being designed from the beginning, a product is sought to meet a market need and then ideas are generated. The product must have a good performance (fulfill market need) and be convenient (ease of handling, minimal environmental impact) (Gani, 2004). If the product is a crystal, however, neither population balance nor mass and energy balances are able of predicting, for example, particle morphology when different solvents are used or the important solubility dependence on temperature. This sort of problem can be solved with molecular modeling, a research area of increasing interest, able of predicting product properties. It is still a challenge to integrate product and process design in a complex and complete CAPE tool (multi-level modeling tool), which could not only design a process to meet required yield and CSD of known properties, but also design the crystals properties. According to Gani (2004), the CAPE community has been an user/implementer of property models in various computer-aided application tools. For the current and future products, however, it is necessary to develop a new class of computer-aided methods and tools that is systematic but flexible, that is simple but accurate and most important, that can “create” the necessary models for a given problem. In the present work a powerful and useful tool is presented, which itegrates the process and product designs of a crystal with known properties (solubility data, crystal morphology, and so on). Molecular issues are not covered by the tool; the properties are supplied as input data. The tool is able to cover various product-process crystallization design problem formulations, and to identify methods and policies to drive to optimal product design, constrained to economical or process/product issues.
3. Crystallization Modeling and Inter-relations in a Batch Crystallizer The modeling of the process involves mass, energy and population balances. For a batch crystallization, these balances are represented by Eq. (1)-(3) respectively: ∞
dC = −3ρc kv ∫ G ( L, t )n( L, t ) L2 dL dt 0
ρ C pVsusp
(1)
∞
dT = −ΔH c 3 ρc kvV ∫ nL2 GdL − UAc (T − T j ) dt 0
∂ ( Gn ) ∂n =− − D ( L) + B ( L) ∂t ∂L
(2)
(3)
Cooling Crystallization: A Process-Product Perspective where D(L) and B(L) denote death and birth phenomena rate, which include agglomeration and breakage of existing crystals and nucleation of new ones. G(L,t) represents the growth rate. Ac is the heat transfer area, C the solute concentration, Cp represents the specific heat capacity, kv the volume shape factor; L is the particle characteristic dimension, n is the density distribution of particles, T and Tj stand for the crystallizer and coolant temperatures, respectively. U denotes the global heat transfer coefficient, V and Vsusp are the particle and suspension volumes, ΔHc is the crystallization heat and ρ and ρc represent respectively suspension and particle densities. Among the kinetic mechanisms taking place in the crystallization process, nucleation and growth are the dominant ones, but in many systems, agglomeration and breakage are present to a certain extent, that must be considered into the modeling. The developed CAPE tool deals with nucleation, growth and agglomeration. Fig. 1 depicts the general framework for the batch crystallization evolution.
Fig. 1: General framework for the inter-relations among the conservative equations in a batch crystallizer.
Mass and energy balances are easily handled by numerical methods of ODEs solution. The population balance equation (PBE), Eq. (3), however, is a partial hyperbolic differential equation, with no analytical solution and requires development and adaptation of numerical techniques. The developed CAPE tool makes use of the Method of Classes (Marchal et al., 1988; David et al., 2003) and the vastly known Orthogonal Collocation Method to solve the PBE. Both methods transform the partial differential equation in a set of ODE, solvable by public recognized codes, like DASSL.
4. CAPE Tool for Batch Crystallization The process-product design integration is achieved in the developed CAPE tool by incorporating the stages of both process and product problems into one integrated structure. It turns possible also cooling crystallization processes evaluation, i.e., perform an analysis of the present operating conditions and trial of new alternatives in order to improve sustainability indices based on product design.
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The optimal process design, defined in terms of product feature requirements, may be obtained through optimization in terms of minimizing (or maximizing) both single or a multiparametric performance function. The step of optimizing a product design to meet a set of requirements of diferent production stages is an increasing area of interest and research, which is supported by CAPE tools development. Good understanding of the target product properties is essential to achieve optimal design. Optimizing crystallization process, with focus on the integration of process-poduct design, is dealt in the developed CAPE tool both with Successive Quadratic Programming (SQP) and Genetic Algorithms (GAs). The latter is very attractive in simultaneously evaluating extremely different process conditions, in order to detect the region of global optimum operation. Fig. 2. brings, in a schematic way, the working structure of the CAPE tool, with three main modules: Process Evaluation, ProcessProduct Perspective, Modeling Adjustment. Due to the limitation of pages, it is not possible to get into details of each module unit. The code for each module is independent but make use of common sub-routines, like the one that contains the process modeling or the optimization codes
Fig. 2: Crystallizer CAPE tool and its modules structure.
Both modules of Process-Product Perspective and of Modeling Adjustment make use of optimization methods, that is, the problems covered by these modules are translated into optimization ones. For example, looking through a process-product perspective, the optimization is formulated with an objective function (actually, it is possible to define just one objective function, characterizing a single objective problem, or two or more objectives, composing a multi-objective optimization problem) that translates both product and process goals (for example, minimization of the coefficient of variation of the CSD and maximization of the mean crystal size). This objective function is constrained to a set of equality and inequality constraints related to process design
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specification (such as physical operation ones and minimum acceptable yield) and a set of equality constraints representing the process model equations (mass, energy and population balances). Process-product design problems may become too complex to solve if the model is highly non-linear and discontinuous, although in principle, a global solution could be obtained. Actually, this is the case of models dealing with PBE. Generally, optimization problems can be solved through deterministic and heursitic methods. The developed tool makes use of both of them, with Successive Quadratic Programming, SQP, (deterministic method) and Genetic Algorithms, GA, (an evolutionary algorithm, part of the heuristic procedures). Fig. 3, in which maxgen stands for the maximum number of generations, brings the main steps in each one of these methods. SQP is an iterative procedure, which makes use of the objective function and its derivatives in order to generate a new (better) solution. On the other hand, GA initializes a population of possible solutions and evolves the individuals with the genetic operators (selection, crossover, mutation) during some generations in order to get “fitter” solutions.
Fig. 3: Optimization Module, used both in Process-Product Perspective and in Modeling Adjustment Modules.
The developed tool provides evaluation of existing processes, understanding of the product-process complexities, evaluation of obtained products features (CSD) in extremely different kinetic systems with different operating policies, as well as optimizing both product and process in order to obtain the optimal policy to be implemented in the batch process. Fig. 4 brings examples of modules responses. On the left, two optimal cooling profiles are presented (for minimizing the coefficient of variation of the final CSD), as well as the non-optimized cooling policy. It is an illustrative example of two optimal responses from the Process-Product Perspective Module, each one using one type of method (deterministic and heuristic ones). The SQP response is dependent on the initial estimate and the optimal policy for the two methods were different. The CSDs depicted on the right of Fig. 4 illustrates how the system can
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produce completely different responses, reflected in terms of final obtained CSDs, depending on the employed cooling policy. As a last powerful option, the user can obtain the kinetic parameters of a solute-solvent crystallizing system, given the employed policies and the the system response (experimental / industrial data).
Fig. 4: CAPE tool response for two different problems: optimization of product-process (left) and processes simulations (right) in order to evaluate obtainable CSDs with different cooling and seeding policies.
5. Conclusions The development of tools for process-product design integration is essential to carry out extensive evaluations so that decisions can be taken in an early stage of design leading to robust and relatively easy to operate process. Bearing this in mind, in this work it was developed a computer-aided tool able to integrate process and product design for batch cooling crystallization, dealing with the issue of modeling and optimization taking into account the product-process interactions. This was achieved through the incorporation of the stages of both process and product problems into one integrated structure. To do so, the process design is solved with CAPE methods, which integrate energy, mass and population balances and mathematical programming problems are formulated and solved in order to determine operating conditions that drive to achievement of required process yield. The product quality specifications are considered trough the particle size distribution maintained in specific ranges to ensure good quality indices.
References David, R., Paulaime, A-M., Espitalier, F., Rouleau, L., 2003, Modeling of Multiple-mechanism Agglomeration in a Crystallization Process, Powder Technology, 130, 338-344 Gani, R., 2004, Chemical product design: challenges and opportunities, Computers and Chemical Engineering, 28, 2441-2457. Marchal, P., David, R., Klein, J. P., Villermaux, J., 1988, Crystallization and Precipitation Engineering – I. An Efficient Method for Solving Population Balance in Crystallization with Agglomeration, Chemical Engineering Science, 43, 59-67 Ng, K. M., 2001, A Multiscale-Multifaceted Approach to Process Synthesis and Development, Proceedings of the 11th European Symposium on Computer Aided Process Engineering, Elsevier, Kolding, Denmark, pp.41-54 Shi, D., El-Farra, N.H., Li, M., Mhaskar, P. D. Christofides, 2006, Predictive control of particle size distribution in particulate processes, Chemical Engineering Science, 61, 268-281.
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A Systematic Approach for Automated Reaction Network Generation Shuo-Huan Hsua, Balachandra Krishnamurthya, Prathima Raob, Chunhua Zhaoa, Suresh Jagannathanb, James Caruthersa and Venkat Venkatasubramaniana,* a
School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA Department of Computer Science, Purdue University, West Lafayette, IN 47907, USA
b
Abstract In this work, we propose a systematic approach to gather reaction mechanism knowledge and automatically generate reaction networks based on this knowledge. Ontologies are created to model all related information and knowledge. Ontologies for molecule patterns, as well as elementary reaction operations have been created. The reasoning capability provided for an ontology is used to classify molecules or fragments to predefined molecule patterns. A reaction mechanism is modeled as a set of individuals of the defined ontology. The semantic consistency between the elementary steps in reaction mechanism is also validated using the reasoning capability. An execution engine has been developed to automatically generate reaction network, given the reaction mechanisms and molecules. The reaction mechanism knowledge stored in the system can also be easily reused to create new reaction mechanisms. Keywords: Reaction Network, Reaction Mechanism Knowledge, Ontology.
1. Introduction New chemical/biological compound development is a challenge for scientists and engineers. Traditionally, researchers have discovered new materials by trial and error experiments. The first step is to propose new reaction pathways based on the knowledge from textbooks, existing databases, and intuition. The second step is to look for suitable catalysts to realize the pathway and perform experiments to verify their hypothesis. This entire process is time-consuming and tedious. Hence considerable resources of the enterprise have to be expended for product development. To accelerate this discovery process, computational approaches are required. An important requirement of this approach is the systematic management of chemistry knowledge. The reaction knowledge can be divided into three levels in order of increasing generality: overall reactions (observed reactions from the experiments), elementary reactions (building blocks of overall reactions), and reaction mechanisms (abstract reaction specification describing the elementary steps). Several commercial chemical reaction databases are available to design new pathways by searching through the entire database, but impossible to design new chemical compounds since the desired compound is not in the database. The elementary reaction network is useful for studying the thermodynamic and kinetic behaviors of the overall reactions because most of computational chemistry data supports the elementary reactions only. It helps to design new catalysts and find the optimal operating conditions, e.g., temperature and pressure, but not new compounds. In order to effectively use computational approach for new product design, the reaction knowledge in the abstract level of reaction mechanisms should be the correct design *
corresponding author. E-mail address: [email protected].
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base. The reaction mechanisms are not restricted to specific reactants or products, and therefore it is general enough to predict the existence of new compounds. NetGen (Broadbelt et al. 1996) has been successfully used to generate reaction mechanisms for several distinct reaction chemistries using BE and R matrix representations (Ugi et al. 1993). This approach generates a compact mechanistic model. The result might, however, miss some reaction mechanisms as the network generation process is dependent on process conditions, but it is still useful in capturing the overall behavior of the reaction network. NetGen focused on kinetic analysis, not product design, because the elementary reactions cannot be reused for new compounds or new operating conditions. In this work, we propose an intelligent Correctness and Consistency Checking framework using ontology and semantic web approach, shown in Figure 1, to Reaction systematically gather, manage and reuse the Network knowledge about reaction pathways, which Generator may come from the theories, experiments or superstructure search on reaction database Molecule Proposed Reaction Classification Mechanism Classification (Wang et al. 2001; Zhu et al. 2005). The gathered knowledge has to be shared and Reaction searched in order for it to be useful. To Molecule/Pattern Reaction Mechanism Ontology Ontology facilitate sharing knowledge has to be Chemistry Ontology Knowledge represented in a self describing and Chemistry Ontology Base structured format. Ontology provides this capability. The chemistry ontology defined Graphical User Interface in this system consists of three different Figure 1 Overview of the proposed system ontologies to describe molecules/patterns, reaction mechanisms and reactions. The molecule/pattern ontology defines elementary concepts such as atoms, bonds, atom patterns etc, which are essential for describing any reaction, reaction mechanism or other chemical phenomena. The reaction mechanism ontology defines concepts such as transformation operators that enable the representation of reaction mechanisms at an abstract level so that the expert can rapidly test the hypothesis by simply combining several instances stored in the knowledge base. The reaction ontology mainly builds on the molecule ontology to describe the reactions, generated for a given mechanism, that can be used for further analyses such as thermodynamic analysis, kinetic analysis, etc. Generating a reaction network based on the reaction mechanisms for given set of reactants is an important application of the knowledge base. For small systems it is straightforward to generate the elementary reactions from reaction mechanisms. But for large systems, such as catalytic systems with many different reaction types, it is very difficult for experts to generate the reactions. The reaction network generator is a tool that automates reaction network generation using the reaction mechanisms in the knowledge base. The reaction network generated by this tool can be used in quantitative model building to obtain the kinetic parameters based on experimental data (Katare et al. 2004). Given the large size of the knowledge base and various sources, it is important to ensure the integrity of the knowledge.The correctness and consistency checking component of this system provides this feature, It uses the classification capabilities available for the ontologies in order to achieve this. In particular, it uses the molecule classification and reaction classification which are based on molecule and reaction ontologies respectively.
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2. Knowledge Representation - Ontology Based Approach In this work, we use ontology (Gruber 1993) to represent the knowledge formally and explicitly. Ontology defines the common vocabulary that needs to be shared in a domain as concepts and also relations between these concepts. Web Ontology Language (OWL) is used to encode the ontology. The information which is described in a semantically rich way in an ontology can be easily accessed by other tools, such as an editor (to assist the development of ontology), visualization applications (to provide an intuitive way to access the information by users as well as make knowledge collection easier). It also stores the raw information like the common information repository, which enables information sharing between tools. The Web Ontology Language is description logic (McGuinness and van Harmelen 2004) based and hence supports automated reasoning. A reasoner can be used to classify the concepts hierarchy, find inconsistencies in the hierarchy, as well as classify the individuals to the right concepts. The reasoning capability can assist in developing an ontology, and also in gathering the knowledge from domain experts without exposing them to the complex concept hierarchy. Ontology is the foundation for Semantic Web (Berners-Lee et al. 2001), in which the information is more machine processable, and more intelligent applications could be developed to use and share the information. Semantic web advocates the methodology of separation of information and the applications that use the information. In this work we have adopted this technique in the development of the reaction network generator. 2.1. Representation of Reaction Mechanisms The chemists usually sketch the mechanisms on the RC + R RC + R C paper, as illustrated in Figure 2. We need to start H C H H H H H defining the vocabulary by defining the abstract S elementary reaction. Briefly speaking, the reaction S mechanism is a transformation between input and output Figure 2 The reaction mechanism for molecule patterns, i.e., a group of molecules with the the de-hydrogenation on the same characteristics. To fully describe a reaction carbonium ion mechanism, the input/output patterns and the transformation steps need to be described. Prickett and Mavrovouniotis (1997), proposed a procedural language approach, called Reaction Description Language (RDL) to describe the reaction mechanism. RDL describes a mechanism in detail by a sequence of statements. It is, however, not intuitive for the users. Also due to the lack of semantics in the language, it is very difficult to manage the knowledge stored in RDL. Furthermore, since the statements that describe a molecule pattern are not unique, the efficiency of pattern identification depends on the experience of the users. To overcome these limitations of RDL, we adopt the Reaction ontology approach, to describe different aspects of the Mechanism Molecule Atom reaction knowledge (http://www.w3.org/TR/owl-ref). & Molecule Pattern Electron Transform Onto-logy for molecule and patterns, as well as reaction Container Operators mechanisms have been created and encoded in OWL. Conceptually, these ontologies are developed in a Reaction hierarchical manner, shown in Figure 3. The molecule ontology is in the bottom level, and becomes the basic component of the reaction mechanism and reaction Figure 3 Relationship among the ontologies. The properties for describing the reactions are chemistry ontology
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completely different from those for describing the molecules. Thus, different types of ontologies are needed. 2.2. Molecule and Pattern Ontology Molecule ontology defines the most basic concepts, such as atom and electron container, i.e., bonds, lone pair electrons, and single electrons. The class-subclass relation in ontology is used to describe the hierarchy of the concepts. For example, Atom is defined as abstract concept, and carbon, hydrogen, oxygen, etc. are defined as subclasses of Atom. The pattern ontology is defined based on molecule ontology to describe general patterns in molecules. 1. Ontology for describing atoms: The concept Atom is defined by a set of physical and chemical properties, such as atomic number, charge, mass number, and symbol, etc., to fully describe the basic structure of an atom. Different types of atoms have different constraints on some of the values. For example, for a carbon atom, the atomic number is 6. We can define the concept Neutral Carbon by adding one more constraint: the charge property is No Charge. 2. Ontology for describing bonds: The electron container is the superclass of bond, lone pair electrons, and single electron since all of them describe the electrons connecting the atoms. The electron container consists of two properties, one atom array with maximum size 2, and one integer which stores the number of electrons in the container. Similar to defining subclasses of Atom, we can construct the concepts of Bond, Lone Pair electrons and Single Electron. 3. Ontology for describing molecules and molecule patterns: The molecules are composed of a list of atoms and a list of bonds. Some of the similar concepts such as ions and radicals have the same structure. The patterns can be defined as an abstract molecule with different types of constraints. The carbonium ion shown in Figure 2 (the reactant) can be described as a hydrocarbon with a positive carbon with 5 bonds associated with it, so called carbonium atom. The experts can create the concepts of hydrocarbon and carbonium atom in the molecule pattern ontology and use them to describe the reaction mechanism. 2.3. Reaction and Reaction Mechanism Ontology The Reaction concept consists of reactants, products, reversibility, thermodynamic and kinetic parameters. The reaction ontology can also be used to interface with existing reaction databases, and to use the knowledge stored there. The concept of reaction mechanism is more complicated. It includes the patterns of reactants and products, and the transformation operators, such as charge addition and subtraction, bond addition or deletion, and so on. Essentially, the ontologies defined in the previous sections are used as the components of the reaction and reaction mechanism ontologies. 2.4. Knowledge Management Given the large scale of the knowledge base, the disparity of the knowledge sources, and the community approach to create the knowledge, without an automatic and intelligent mechanism to maintain the integrity of the knowledge base, it is easy for inconsistent, incorrect or duplicated knowledge to be populated in the system. Therefore, we need tools to identify and correct the illformed knowledge. One advantage of using a knowledge base is that we can use a reasoner to compute the correct hierarchy of the concepts, find duplicates, and therefore make the knowledge consistent. For example, if the pattern of secondary carbon is defined as a subclass of class “pattern”, the reasoner is able to check the definition of the secondary carbon and discover that it is a subclass of class “carbon”. Also, the reasoner can correctly classify the instances to the most appropriate
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concept that they belong to. For example, three individuals, methane, ethane and salt are created using the concept “molecule”. The reasoner can tell us that the methane and ethane are hydrocarbons, and the salt is an ionic compound. Basic physical, chemical and thermodynamic laws can be applied to ensure the correctness of the input knowledge, such as charge balance, material balance and thermodynamic stability. Such constraints can be captured using rule languages such as Semantic Web Rule Language (SWRL), based on the concepts defined in molecule, pattern, and reaction mechanism ontology.
3. Generating Reaction Network The knowledge gathered in the knowledge base is used to generate reaction networks. The reaction network generation tool takes the mechanism assembled from the reaction knowledge and generates a detailed reaction network. This automation tool is necessary given complexity and large size of the reaction network. For example, in zeolite catalytic systems, there are at least hundreds of reactions occurring simultaneously; in the metabolic system of E. coli, about thousand reactions run to keep it surviving. Traditionally the experts write down the reactions by hand. This process however, is error prone. The reaction network generator automates this process and provides an efficient and reliable approach. The reaction network generated unimoleStart Initial Put input molecules cular and bimolecular elementary reactions molecule into Queue UnProcessedMol following the exhaustive search process Pop out a molecule from UnProcessedMol presented in Figure 4. All given and produced Add the popped molecule molecules are fed into the hypothesized into SetProcessedMol reaction mechanism to generation unimolecular Proposed Run the reaction mechanism Reaction using the popped molecule reactions as well as all the pairs of those Mechanism as the reactant molecules are investigated to generate Push products Products are in bimolecular reactions. The reaction network No into Queue UnProcessedMol UnProcessedMol or ProcessedMol? generator has been developed and tested using the propane aromatization on HZSM-5 zeolite Yes (Bhan et al. 2005) as a case study. As reported No UnProcessedMol in the paper, 12 various reaction types in carboempty? nium and carbenium chemistry are used to Yes End describe the mechanism, such as adsorption, desorption, dehydrogenation, β-scission, oli- Figure 4 Overall control scheme of the network generator gomerization, cyclization, aromatization, and so forth. 72 different gas and surface species and 312 reactions are included in the microkinetic model without isomers. Based on the result, this mechanism generates 4,453 different molecules including structural isomers which results in more molecule combinations that need to be explored, and thus about a million reactions are generated. Additional knowledge is required for post-processing and simplifying the network. It can be lumping the isomers due to the experimental limitation, thermodynamic constraints, etc. The network generation can lead to combinatorial explosion in the number of molecules and hence the reactions. It seems that there is no way to get rid of the exhaustive search since it is a combinatorial problem. Applying the exhaustive search approach in the reaction network generator, the worst case complexity is O(n2), where n is the number of molecules in the system. However, n is usually big for a complex reaction network, and the running time is therefore prohibitively long for a large system.
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4. Summary and Future Work The reaction mechanism knowledge needs to be systematically captured, managed and reused to support chemical product discovery and development. In this work we propose an ontology based approach for this purpose. Ontology has been developed for molecule, pattern as well as reaction mechanisms. We also investigate the use of reasoning on ontology to ensure the correctness and consistency of the knowledge base. A tool for generating reaction networks based on the knowledge has been developed. This demonstrates the applicability of using the information models captured in ontology as a foundation for tool development. The ultimate goal of this work is to develop a system to provide an intuitive way to capture the reaction mechanism knowledge, to manage this knowledge, and to provide support tools for new product discovery and development. Currently we are extending the ontology to include different types of chemistry knowledge, such as polymer chemistry, biochemistry, etc. The open and extensibility of the proposed ontology-driven approach makes it possible. Searching and querying the knowledge base is another important feature that we are investigating. The substructure (Stobaugh 1985) and superstructure search methodologies could be incorporated in the system so that the experts can obtain the molecules or reactions with common characteristics. The combinatorial problems encountered in network generator need to be addressed in order for the system to handle large reaction systems. Furthermore, the user-friendly graphical interface and visualization tools are currently under development to present the information and knowledge stored in the system to users and assist knowledge collection and analysis.
References Berners-Lee, T., Hendler, J., and Lassila, O. (2001). "The Semantic Web." Scientific American. Bhan, A., Hsu, S.-H., Blau, G., Caruthers, J. M., Venkatasubramanian, V., and Delgass, W. N. (2005). "Microkinetic Modeling of Propane Aromatization over HZSM-5." Journal of Catalysis, 235(1), 35-51. Broadbelt, L. J., Stark, S. M., and Klein, M. T. (1996). "Computer generated reaction modelling: decomposition and encoding algorithms for determine species uniqueness." computers and chemical engineering, 20(2), 113-129. Gruber, T. R. (1993). "A Translation Approach to Portable Ontology Specification." Knowledge Acquisition, 5, 199-220. Katare, S., Caruthers, J. M., Delgass, W. N., and Venkatasubramanian, V. (2004). "An Intelligent System for Reaction Kinetic Modeling and Catalyst Design." Industrial and Engineering Chemistry Research, 43(14), 3484-3512. McGuinness, D. L., and van Harmelen, F. (2004). "OWL Web Ontology Language Overview." W3C Recommendation. Prickett, S. E., and Mavrovouniotis, M. L. (1997). "Construction of complex reaction systems --- I. Reaction description language." Comp. Chem. Eng., 21(11), 1219-1235. Stobaugh, R. E. (1985). "Chemical Substrcture Searching." J. Chem. Inf. Comput. Sci., 25, 271-275. Ugi, I., Bauer, J., Bley, K., Alf, D., Dietz, A., Fortain, E., Gruber, B., Herges, R., Knauer, M., Reitsam, K., and Stein, N. (1993). "Computer-Assisted Solution of Chemical Problems --- The Historical Development and Present State of the Art of a New Discipline of Chemistry." Angew. Chem. Int. Ed. Engl., 32, 201-227. Wang, K., Wang, L., Yuan, Q., Luo, S., Yao, J., Yuan, S., Zheng, C., and Brandt, J. (2001). "Construction of a generic reaction knowledge base by reaction data mining." Journal of Molecular Graphics and Modelling, 19, 427-433. Zhu, Q., Yao, J., Yuan, S., Li, F., Chen, H., Cai, W., and Liao, Q. (2005). "Superstructure Searching Algorithm for Generic Reaction Retrieval." J. Chem. Inf. Model., 45, 1214-1222.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A hybrid methodology for detailed heat exchanger design in the optimal synthesis of heat exchanger networks J.M. García, J.M. Ponce and M. Serna Facultad de Ingeniería Química - Universidad Michoacana de San Nicolás de Hidalgo. Morelia, Michoacán, México
Abstract This paper presents a hybrid method for the synthesis and optimization of heat exchanger networks, which includes detailed design of heat exchangers. This task is achieved by combining the pinch design method with mathematical programming techniques, together with an optimal design algorithm of shell and tube heat exchangers based on the rigorous Bell-Delaware method. As result, the stream pressure drops are treated as optimization variables. Thus, the capital cost of the pumping devices and the electricity cost to run these equipments are considered in this problem together with the costs for heat exchanger area and utility consumption. The problem is decomposed as a binary tree, where each node is categorized as either capital-dominant or energydominant problem. Subsequent decomposition of each node is determined by this dominance. The final design is obtained recursively applying a design algorithm from child nodes to their parent node. The match-selection procedure is a hybrid method that exhibits some of the features of both evolutionary and mathematical programming methods. The method starts allocating matches using an IP assignment model. This step is then followed by an evolutionary procedure in which the remaining selections of the design are treated as new problems. The process is repeated until no savings can be discovered. The method avoids the solution of complex MINLP models, and consequently it is possible to solve large problems. Furthermore, it readily copes with typical constraints, such as forbidden matches and imposed matches. Therefore, safety and layout considerations are easily incorporated into the design. Keywords: Heat exchanger network synthesis, shell and tube heat exchangers, BellDelaware method, detailed design.
1. Introduction The synthesis of heat exchanger networks (HENs) has been the subject of significant amount of research over the last three decades, promoted heavily during the 1970s because of the rising costs of energy observed. The two most common methods for grassroots HENs designs are broadly classified in two categories, namely, the pinch design method and mathematical programming methods (Furman and Sahinidis, 2002). In the pinch design method the system is normally partitioned using a pinch decomposition strategy, and, matches are selected by application of heuristic rules (Linnhoff and Hindmarsh, 1983). This approach is time consuming and the quality of the results is determined by the designer’s experience. In the second kind of methods, a mixed-integer nonlinear programming (MINLP) model is applied for the development of a HEN based on a network superstructure solving for the minimum total annual cost.
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Thus, mathematical programming methods have a more rigorous approach to find a solution, but lack of user interaction. Until recently, however, with exception of an attempt to synthesize heat and power integrated HENs using detailed MINLP models that was done by Sorsak and Kravanja (1999), both methods have neglected the effects of detailed heat exchanger design on HENs synthesis; for example, film heat transfer coefficients for the streams are assumed constant. To outline appropriately interactions between the consumption of power, the capital cost and the energy recovery level of the network, recently Mizutani et al. (2003) proposed a MINLP model for the HENs synthesis including detailed design considering shell and tubes units. This formulation represents great advance in the appropriate treatment of the problem. However, since the heat exchangers are represented rigorously using the Bell-Delaware method (Taborek, 1983), it is expected that the function of the total annual cost to minimize is large and irregular. In addition, the complexity of the resultant MINLP model is of such magnitude that Mizutani et al. (2003) had to impose certain geometric limitations to the units in order to facilitate the obtaining of feasible solutions. Consequently, these two aspects of the approach of these authors reduce their possibilities to find a good solution. This paper presents a hybrid method for HENs synthesis and optimization including detailed design of heat exchangers. This task is achieved by combining a tree decomposition method with a match-selection model, together with an optimal design algorithm of shell and tube heat exchangers based on the rigorous Bell-Delaware method. Thus, the capital cost of the pumping devices and the electricity cost to run these equipments are considered in this problem together with the costs for heat exchanger area and utility consumption. This approach simplifies notably the problem of detailed HENs design, reduces the computation effort significantly and, therefore, it allows solving large-scale problems.
2. Recursive method for the synthesis of heat exchanger networks The network structure is generated using a recursive method based on a tree decomposition strategy and a match-selection model proposed by Ren et al. (2001). This approach decomposes the synthesis problem as a binary tree and calculates the cost of each node, which is then categorized as either a capital-dominant or an energydominant problem. Subsequent decomposition of the system depends on this dominance. For problems dominated by capital cost, any reduction in capital investment will decrease the overall cost. To reduce the investment capital, it is possible to use fewer units and/or improve heat exchanger driving forces. When reducing capital costs, energy consumption usually is increased and, therefore, energy costs; however, the total annual cost will fall because the capital cost is the dominant factor. Hence, such systems are best treated as a single entity, thereby avoiding further decomposition. An inverse case is presented in problems where the main component of the total annual cost is the utility cost; here energy saving becomes important, and decomposition might reduce the utility consumption. The decomposition is carried out in the pinch point; then the pinch point design rules are used to generate maximum energy recovery networks. The fractional contribution of utility cost to the total annual cost is defined as: d=
utility cost total annual cost
(1)
The value of d for a network can be used to determine whether energy or capital dominates the total annual cost. According to the experience, Ren et al. (2001) suggest
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that a fractional value of 0.5 represents the transition point. Therefore, if d > 0.5, the utility cost is the main contribution for the total cost; otherwise the main contribution for the total annual cost is the capital cost. Usually, the dominant-cost component for the problem is unknown prior to design; consequently, the root node (the initial design) is considered as capital-dominant problem. For such problems, decomposition to secondary nodes is unnecessary. A single root node exists in the binary tree; however, it is possible that the design of the root node is dominated by its utility cost, then decomposition is required, which allows an expansion of the binary tree in two branches, or secondary nodes, and the process is repeated. 2.1. Match selection model A simplified superstructure was proposed to determine a basic topology for the initial design in each node assuming that the matches between a hot stream i and a cold stream j is independent of any other one and, consequently, that follows the pattern shown in Fig. 1. This simplifying assumption identifies all possible matches. It should be noticed that in this approach the interactions among the diverse matches are rejected. Thus, it is easy to obtain the optimum detailed design of each match independently using the algorithm of Serna and Jiménez (2005) and, as a result, the total annual costs associated to all possible matches. Therefore, the match selection model developed by Ren et al. (2001) is enhanced through the incorporation of the optimum detailed design of individual heat exchangers. Aci
HOT STREAM i
Aij
ΔPi
ΔPc
COLD UTILITY STREAM
ΔPj
COLD STREAM j
ΔPci P-1
COOLER
A hj
ΔPhj HEAT EXCHANGER
HOT UTILITY STREAM
ΔPh HEATER
Fig. 1. Matching model for single hot and single cold process streams
For each match, the optimal cost (Costij) can be defined by:
Costij = CostHX ij + CostH hu + CostCcu
(2)
where the term CostHXij is the optimal cost for the detailed heat exchanger, which include the capital cost for the exchanger and the capital and operational costs for the two pumps. The others two terms, CostHhu and CostCcu, are the heater and cooler optimal costs, respectively, which are defined in similar form as the heat exchanger, only it is necessary to add the utility costs. It is important to highlight that the pumping costs for the hot utility service is considered only for the case that this is in liquid phase. A fundamental part of the recursive method is the inclusion of a match matrix, where the optimal total annual costs associated to all possible matches are stored. The optimal cost of the match between a hot stream i and a cold stream j is stored as the element aij
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of the matrix. In the simplified superstructure, the match matrix is always square; when the number of hot or cold streams is different, utility streams are included to balance the matrix. Therefore, the model for the optimal matches selection in the initial design settles down as an integer programming problem (IP) as: min
l
l
∑∑ a x i =1 j =1
l
∑x
s.t.
ij
i =1
l
∑x j =1
ij
ij ij
(3)
= 1 ∀j = 1, l
= 1 ∀i = 1, l
l = max(m, n, )
The solution of this problem can be easily obtained even for large size problems. Using this methodology, the difficulties associated to the solution of complex NLP, MINLP or MILP problems are avoided. 2.2. Remainder section analysis In the initial design each stream is restricted to participate in a single match; however, following the evaluation of the remainder sections several cold streams (or hot) can be coupled with more than one hot stream (or cold). When this situation is presented, it is possible to analyze the convenience of stream splitting to modify the driving forces and, consequently, to reduce the total annual cost. To simplify the match selection mathematical model, the partition temperatures in the simplified superstructure are assumed constant. Hence, an optimization of partition temperatures is performed following the combination of secondary nodes. This optimization repeats every time that a node breaks down.
3. Shell and tubes heat exchanger rigorous design In order to determine the elements of the match matrix, it is necessary to carry out the optimal design of the heat exchangers which result of proposing the transfer of heat between each hot with each cold stream according to the model shown in Fig. 1. To achieve this task, this work proposes networks constituted only by shell and tubes heat exchangers. The algorithm proposed by Serna and Jiménez (2005) provides the detailed optimal design of these units. The development of this algorithm is based on two compact formulations combined with the basic design equation to represent the thermohydraulic behavior of the fluids. The function to minimize consists of the capital cost of the exchanger and the two pumps, plus the power cost necessary to operate these pumps, and can be defined as: c3 c6 c3 c6 ⎡ ⎡ ⎛ Q ⎞ ⎛1 ⎛ KTQQ ⎞ ⎛1 Dt ⎞ ⎤ Dt ⎞ n c6 ⎤ T ⎥ ⎢ + + + + + + TAC= NKF ⎢c1 +c2 ⎜ R K c c R ⎟ ⎟ hT ⎥ ⎟ ⎜ ⎟ ⎜ dw F 4 dw 5⎜ Dh Dh ⎢⎣ ⎢⎣ ⎥⎦ ⎝ NFTΔTML ⎠ ⎝ hS ⎝ FTΔTML ⎠ ⎝ hS ti T ⎠ ⎥ ti T ⎠ ⎦ c9 c9 ⎡ ⎛ K QQ⎞ ⎛ 1 D ⎞ n ⎛ H C K QQ⎞⎛ 1 D ⎞ m D ⎞ m c9 ⎤ ⎛ H C K QQ⎞⎛ 1 +KF ⎢c7 +c8 ⎜ S S ⎟ ⎜ +Rdw + t ⎟ hS ⎥ +⎜ Y T T T ⎟⎜ +Rdw + t ⎟hT +⎜ Y S S S ⎟⎜ +Rdw + t ⎟hS Dh Dh Dh ⎢⎣ ⎥⎦ ⎝ ηT FTΔTML ⎠⎝ hS ti T ⎠ ti T ⎠ ti T ⎠ ⎝ FTΔTML ⎠ ⎝ hS ⎝ ηSFTΔTML ⎠⎝ hS
( )
( )
(4)
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Here the values for the parameters KT and KS depend on the exchanger geometry, the physical properties and the volumetric flow for the tube and shell fluids, respectively (Serna and Jiménez, 2005).
4. Results The example used for the illustration of the hybrid method is a problem reported by Mizutani et al. (2003). The stream data for this problem is presented in Table 1. A minimum temperature difference of 10°K is also specified. Mizutani et al. (2003) give the cost data and the physical properties for the streams. The hybrid method gives the final network shown in Fig. 2. The performance of the network is given in Table 2 together to that of the network obtained by Mizutani et al. (2003) for the purpose of comparison. From Table 2, we can see clearly that the proposed methodology reduces significantly (72%) the total annual cost of the final design. It is appreciated, therefore, that the MINLP model of Mizutani et al. (2003) fails in the search of the best solution, due to the enormous complexity that the detailed design of the shell and tubes heat exchangers incorporate to the synthesis of heat exchanger networks problem. Thus, the result from the example demonstrates that the hybrid methodology is suitable for optimal synthesis of HENs that include the detailed design of heat exchanger. Table 1. Streams data for Example Stream
m (kg/s)
Tin (°K)
Tout (°K)
H1 H2 H3 C1 C2 C3 CW S
16.3 65.2 32.6 20.4 24.4 65.2
426 363 454 293 293 283 300 700
333 333 433 398 373 288 320 700
E4
H1
426 °K
H2
363 °K
H3
454 °K
E2 P-8
333 °K
E3 358 °K
E5 338.561 °K
E1
398 °K
333 °K
433 °K
E6 385.083 °K
646.629 kW 373 °K
310.775 °K 3720.019 kW
344.943 °K 1680.008 kW
3110.2 kW
288 °K
293°K
C1
293 °K
C2
889.82 kW
283 °K
C3 800.0004 kW
Figure. 2. Final network for Example
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This work
Exchangers investment cost
$12,388/yr
$10,920.88/yr
Utility cost
$173,456/yr
$38,797.74/yr
Pumping costs
$17,076/yr
$6,236.96/yr
Total annual cost
$202,920/yr
$55,955.58/yr
5. Conclusions A combined heuristic and mathematical programming approach is presented for the detailed design of shell and tube heat exchangers in the optimal synthesis of HENs. Thus, the proposed hybrid methodology considers the pressure drop together with area and utility costs in the network design. The synthesis problem is recursively solved using a tree decomposition strategy and a match-selection model (Ren et al., 2001). Using the assumption of independent matches, the optimum detailed design of individual heat exchangers was obtained by the algorithm proposed by Serna and Jiménez (2005). The heat exchanger design model is based on the Bell-Delaware method for the shell-side fluid flow (Taborek, 1983). A very important feature of the hybrid method is that it reduces significantly the computation effort required to solve the problem of detailed design of heat exchanger networks. This is due to the good selection of matches among streams in the diverse stages of the recursive method by using an IP model instead of a MINLP model, which is more complex and more difficult to solve. For the example solved in this work, the hybrid method improves significantly the results obtained previously by Mizutani et al. (2003).
Acknowledgements The authors would like to acknowledge the support received from CIC-UMSNH, México (grant 20.1) for the development of this work.
References Furman, K.C., Sahinidis, N.V., 2002. A critical review and annotated bibliography for heat exchanger network synthesis in the 20th Century, Industrial & Engineering Chemistry Research 41, 2335-2370. Linnhoff, B., Hindmarsh, E.C., 1983. The pinch design method for heat exchanger networks. Chemical Engineering Science 38, 745-763. Mizutani, F.T., Pessoa, F.L.P., Queiroz, E.M., Hauan, S., Grossmann, I.E., 2003. Mathematical programming model for heat-exchanger network synthesis including detailed heat exchanger designs, 2. Network synthesis. Industrial & Engineering Chemistry Research 42, 4019-4027. Ren, Y., O’Neill, B. K., Roach, J.R., 2001. A recursive synthesis method for heat exchanger networks, I. Algorithm. Industrial & Engineering Chemistry Research 40, 1168-1175. Serna, M., Jiménez, A., 2005. A compact formulation of the Bell-Delaware method for heat exchanger design and optimization. Chemical Engineering Research & Design 83(5), 539-550. Sorsak, A., Kravanja, Z., 1999. Simultaneous MINLP synthesis of heat and power integrated heat exchanger networks. Computers & Chemical Engineers 23(Supplement), S143-S147. Taborek, J., 1983. Shell-and-tube exchangers: Single-phase flow. Heat exchangers design handbook, (E. U. Schlunder, ed.), Vol. 3, Section 3.3. Hemisphere Publishing Corp., Washington, DC.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Optimal design of shell-and-tube heat exchangers using genetic algorithms J.M. Ponce1, M. Serna1, V. Rico2 and A. Jiménez2 1 Facultad de Ingeniería Química - Universidad Michoacana de San Nicolás de Hidalgo. Morelia, Michoacán, México 2 Departamento de Ingeniería Química - Instituto Tecnológico de Celaya,Celaya, Guanajuato, México. Abstract This paper presents an approach based on genetic algorithms for the optimal design of shell-and-tube heat exchangers. The proposed approach uses a compact formulation of the Bell-Delaware method to describe the shell-side flow pattern. The optimization procedure involves the determination of suitable values of major geometric parameters such as the number of tubes passes, standard internal and external tube diameters, tube layout and pitch, type of head, fluid allocation, number of sealing strips, inlet and outlet baffle spacing, and shell-side and tube-side pressure drops. The proposed methodology takes into account several geometric and operational constraints typically recommended by design codes, and may provide global optimum solutions as opposed to local optimum solutions that are typically obtained with many other optimization methods. An example previously solved with a disjunctive programming method is used to show the application of the proposed approach. The results show how the previous design was significantly improved through the use of the optimization approach based on genetic algorithms. Keywords: heat exchanger, shell and tube, Bell-Delaware, genetic algorithms.
1. Introduction Shell and tube heat exchangers are widely used in many industrial power generation units, chemical, petrochemical, and petroleum industries. These types of heat exchangers are robust units that work for wide ranges of pressures, flows and temperatures (Taborek, 1983). A typical optimization problem for heat exchangers consists of finding a unit that meets a given heat duty at the minimum total annual cost, subject to a given set of constraints. The total annual cost should include the annualized capital cost of the exchanger plus two pumping devices (one for the tube-side fluid and another one for the sell-side fluid), and the operating (power) costs of such pumps. The traditional design algorithm for shell and tube heat exchanger consists of rating a large number of different exchanger geometries to identify those that meet the given heat duty and certain geometric and thermo-hydraulic constraints. This approach is not only time-consuming, but it is also restricted in terms of ensuring an optimal design (Muralikrishna and Shenoy, 2000). Recently, novel algorithms for the optimal design of shell and tube heat exchangers have been proposed (Serna and Jimenez, 2005). However, most of such algorithms use standard optimization techniques based on gradient methods and, as a consequence, they may be trapped at local optimum solutions because of the nonconvexities of the design model. Moreover, in these algorithms the following geometric and operational
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variables of the exchanger are fixed as design data by the designer: the number of tube passes, tube internal and external diameters, layout pattern and pitch, as well as type of head construction and fluid flow allocation (i.e., the allocation of the fluid streams to the shell or tube side). As a result, the optimization problem is oversimplified since several potential parameters that may be optimized are regarded as constant. To overcome the above limitations, this work presents an approach based on genetic algorithms for the optimal design of shell and tube heat exchangers. The proposed approach uses a compact formulation of the Bell-Delaware method to describe the shellside flow with no simplification; this approach, therefore, has the same degree of accuracy as the full Bell-Delaware method, and can incorporate the entire range of geometric parameters of practical interest. The optimization procedure involves the selection of suitable values for major geometric parameters such as the number of tube passes, standard internal and external tube diameters, tube layout and pitch, type of head, fluid allocation, number of sealing strips, inlet and outlet baffle spacing, and shellside and tube-side pressure drops. The proposed methodology ensures that a global optimum and/or a set of excellent near-optimal solutions are obtained.
2. Fundamental relationships The heat exchanger area must satisfy the basic design equation:
A=
⎞ Q ⎛1 Dt ⎛ Dt ⎞ Dt D + t Rdt ⎟⎟ ln ⎜ ⎜⎜ + Rds + ⎟+ FT ΔTML ⎝ hs 2k w ⎝ Dti ⎠ Dti hT Dti ⎠
(1)
Two additional equations are needed that relate the exchanger area to the film coefficients and the allowable pressure drops. The compact formulations developed by Serna and Jimenez (2004) are used for that purpose. For the tube side one obtains:
ΔPT = KT A(hT ) n
(2)
while for the shell-side fluid, the following compact formulation based on the BellDelaware method is used,
ΔPS = K S A(hS ) m
(3)
where KS, KT, m and n depend on the geometric parameters of the exchanger and the fluid physical properties. It must be emphasized that these compact formulations are the consequence of an analytical treatment of the original equations, not empirical correlations. The problems with this formulation is that the parameters KS, KT, m and n depend on the exchanger geometry, which is not initially known. To develop an efficient algorithm, we use such parameters as search variables, and decoupled the equations that contain the unknown variables, as shown by Serna and Jiménez (2004).
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2.1. Constraints for the model To get a practical design, the shell and tube heat exchanger must satisfy the given heat duty and the following operational and geometric constraints:
ΔPt ≤ ΔPt ,max ΔPs ≤ ΔPs ,max vt ,min ≤ vt ≤ vt ,max vs ,min ≤ vs ≤ vs ,max
(4)
Ds ≤ Ds ,max L ≤ Lmax Rbs ,min ≤ Rbs ≤ Rbs ,max Rsmsw,min ≤ Sm Sw ≤ Rsmsw,max
where ΔP is the pressure drop, v the velocity, Ds the shell diameter, L the total length, Rbs the ratio baffle spacing to shell diameter, Rsmsw the ratio cross flow area to window flow area, Sm the cross flow area and Sw the window area. The first four equations are thermo-hydraulic constraints and the last four equations represent geometric constraints. Typical design limits to be used in this set of constraints are given by Muralikrishna and Shenoy (2000). 2.2. Objective function The objective function consists of the minimization of the total annual cost. A typical total cost includes five components: the capital cost of the exchanger, the capital cost for two pumps, and the operating cost of the two pumps,
{
TAC = A f C a + Cb A c + C e + C f (M t ΔPt ρ t ) + C e + C f (M s ΔPs ρ s ) + C pow H η{M t ΔPt ρ t + M s ΔPs ρ s }
g
g
}
(5)
Also, for a proper degree of accuracy, the estimation of the heat exchanger capital cost must include the costs for component parts and manufacturing procedures. For such estimation, we use in this work the relations reported by Purohit (1983). 2.3. Optimization variables The problem includes the following optimization variables: tube-side and shell-side pressures drops, baffle cut, number of tube passes, standard inside and outside tube diameters, tube pitch, tube pattern arrangement, fluid allocation, and sealing strings.
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3. Optimization model The model consists of minimizing Eq. (5), subject to the design equation, the compact formulations for the pressure drops, the implicit relations for the exchanger geometry and the correction factors for the Bell-Delaware method, as well as the explicit constraints given by Eq. (4). It is well known that the non-convexities of these types of design models may affect the application of typical solution algorithms, with potential convergence problems and the possibility of getting trapped into a local optimum solution. To overcome this problem, we use in this work a genetic algorithm for the solution of the optimization problem. Genetic algorithms are search methods based on the combination of natural selections and genetics. They are based on the principle of survival of the fittest, and provide a search method that is extremely efficient and virtually independent of the non-convexities of mathematical models (Goldberg, 1989). Fig. 1 shows the main steps of the proposed approach; notice how Eq. (4) is implemented trough the use of a penalty term. Initial population Set optimization variables
Shell and tube heat exchanger designs TAC and implicit variables for each design
New generation
Tests constraint Evaluate penalty term Selection Fitness function TAC+penalty term
End
Yes
Optimum?
New generation Mutation Crossover No
Fig. 1. Solution strategy for the optimization problem
4. Results and Discussion The example presented here was reported by Mizutani et al. (2003). The design data are shown in Fig. 2. For the solution of this example, a constraint in the tube length of 4.88 m was imposed. A summary of the results obtained with the proposed model is given in Table 1, where the solution reported by Mizutani et al. (2003) is also shown for comparison. Mizutani et al. (2003) used a disjunctive programming optimization method to solve this problem. It can be observed that the proposed algorithm provided a better solution than the one obtained by Mizutani et al. (2003). This is a clear case in which the solution provided by a standard optimization approach was trapped into a local optimum value. In contrast, the proper use of genetic algorithms can provide a global optimum (or an
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excellent near-optimal) solution. For the solution of this problem we used a population size of 20 individuals, and obtained the final solution in 30 min of real time. Cold Stream: Tcin=298.15K Tc out=313.15K mc=68.88 kg/s kc =0.59 W/(m K) ρc =995 kg/m3 Cpc=4200 J/(kg K) µc=0.0008 kg/(m s)
Hot Stream: Thin=368.15K Thout=313.15K mh=27.78 kg/s kh=0.190 W/(m K) ρh=995 kg/m3 Cph=2840 J/(kg K) µ h=0.00034 kg/(m s)
Area cost=123A0.59 Pumping cost=1.31(ΔPt mt/ρt+ΔPs ms /ρs)
Fig 2. Data for the example Table 1. Example results Mizutani et al.
This work
202
242.881
2
U (W/m K)
860
714.511
Number of tubes
832
653
Tube layout
square
triangular
Number of tube passes
2
6
Din (mm)
12.6
22.918
Dout (mm)
15.9
25.4
Number of baffles
8
8
Head type
fixed
pull floating
Hot fluid allocation
shell
tube
FT *ΔTLM Shell diameter (m)
24.9
25.004
0.687
1.10534
Tube length (m)
4.88
4.88
Area (m2)
Baffle spacing (m)
0.542
0.516275
ΔPt (Pa)
22676
10981.3
ΔPs (Pa)
7494
4714.28
Pumping cost ($/year)
2424
960.361
Area cost ($/year)
2826
3142.59
Total annual cost ($/year)
5250
4102.95
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5. Conclusions In this work, an optimization model for the design of a shell and tube heat exchanger has been proposed. The model includes the Bell-Delaware correlations for the shellside fluid, which provides a suitable representation of the fluid flow pattern within the shell. The optimization strategy is based on the use of a genetic algorithm, in which the geometric and operational constraints have been implemented through the use of a penalty function. Genetic algorithms are in general more efficient in terms of providing excellent optimum solutions than other standard optimization methods, which frequently get trapped into local optimum solutions when applied to nonconvex models. The results of the case study show how a reported optimum solution using a disjunctive programming technique was noticeably improved with the use of the proposed method. The main limitation of the proposed algorithm is its high demand of CPU time, but this problem is overcome satisfactorily with the advance in the speed of the current computers.
References Goldberg, D. E., 1989. Genetic algorithms in search, optimization and machine learning. Ed. Addison Wesley Longman. Mizutani, F.T., Pessoa, F.L.P., Queiroz, E.M., Hauan, S., Grossmann, I.E., 2003. Mathematical programming Model for heat exchanger network synthesis including detailed heat exchanger designs. Industrial & Engineering Chemestry Research (42), 4009-4018. Muralikrishna, K., Shenoy, U. V., 2000. Heat exchanger design targets for minimum area and cost. Trans IChemE, Part A, Chemical Engineering Research & Design, (78), 161-167. Purohit, G.P., 1983. Estimating costs of shell and tube heat exchangers. Chemical Engineering 90 (17), 57-67. Serna, M., Jiménez, A., 2004. An efficient method for the design of shell and tube heat exchangers. Heat Transfer Engineering. 25 (2), 1-12. Serna, M., Jiménez, A., 2005. A compact formulation of the Bell-Delaware method for heat exchanger design and optimization. Chemical Engineering Research & Design. (83), Issue A5, 539-550. Taborek, J., 1983. Shell-and-tube exchangers: Single-phase flow, in: E. U. Schlunder (Ed.), Heat exchangers design handbook, Vol. 3, Section 3.3, Hemisphere Publishing Corp.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Integration of Process Design and Operation for Chemical Product Development with Implementation of a Kilo-plant Yu Qian*, Zhihui Wu, Yanbin Jiang School of Chemical Engineering, South China University of Technology Guangzhou, 510640, P. R. China *Corresponding author: [email protected]
Abstract Presented in this paper is an integrated approach of computer aided product development, process design and operation analysis based on a kilo-plant. It supports to find solutions for the complicated technical problems in chemical product development, as well as in the design and operation of the manufacture processes. As a research platform, the kilo-plant was specifically designed for batch manufacture of fine and specialties chemicals in kilogram-scale. The characteristics of product synthesis, process operation, and product quality control are investigated in coupled with computer aided monitoring, online modeling, simulation and operation optimization. As a consequence, it provides a systematic methodology through the integration of chemical product discovery, process design and operation, which aims at responding to the rapid changes in the marketplaces to the new products demands. Keywords: Integration, Kilo-plant, Product development, Process Operation, Process analysis 1. Introduction
The chemical process industry faces very challenging economic and social issues. Severe market competition needs fast responses from product innovation, process development, to production, especially for high value-added fine and specialty chemicals. The challenges of process systems engineering is concerned with the improvement of decision making processes for the creation and operation of the chemical supply chain. It deals with the discovery, design, manufacture and distribution of chemical products in the context of many conflicting and multi-attribute goals. The target are to develop human and
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environment friendly, resource conservative chemical products and production processes, and to research the related complex problems and technology for product innovation, reaction synthesis, optimization design, and operation control (Grossmann, 2000; 2004). Nowadays, chemical engineering is developing from process oriented to product oriented. Chemical product engineering is to study the relation between product structure and quality as the centre contents. On the macroscale level, the integration of the product design and process design, agile manufacturing, system integration, and kilo-lab, etc. has represented the front and wide prospect of this field too (Vaidyraman, 1999; Qian, 2004). Chemical product design and process development are usually consists of several stages: products performance analysis, molecular structure and formulation, laboratory test, pilot plant, trial production, and commercial production. Most fine chemicals and specialty chemicals with high value-added are produced in small volume, even in kilogram scale, but the related manufacture process may still be complicated. A pilot plant is a typical verification tool in the development of a new product/process, which is, however, time-consuming and capital-expensive. To address this requirement, the concept of mini-plant is proposed. A mini-plant consists of a series of functional operation units, which are assembled according to production flow process. It can carry out product synthesis, separation and purification in a certain amount of volume. The process from testing stage to production operation stage is a complicated research task, which involves the aspects of quality, dynamics, thermodynamics etc. A mini-plant assembled from those main classic operation units can help to solve the production industrialization relevant factors (Wörz，1995). In this paper, an integrated approach of computer aided product development, process design and operation analysis based on a kilo-plant, is presented. 2. A kilo-plant for integration development of product and process A computer integrated process experiment platform, kilo-plant as the core facility, has been set up in author’s laboratory. The structure is shown in Figure 1. The kilo-plant consists of the main features of a complete process system, the kilo-plant includes reactors, separation units, heating and cooling utilities, instrumentation, control system and recycles. Several process parameters, e.g., temperature, pressure, flow rate, rotation speed, pH, etc., are precisely controlled. It allows for changes in the process flowsheet (feed point location, recycle, etc.) for investigation of process alternatives. The kilo-plant provides researchers the knowledge of the process operation based on the information of the real experiment. The kilo-plant integrates the main components in chemical process operation to help us to adopt a ‘systems approach’ to engineering problem solving. The structure of the experimental platform is shown in Figure 2.
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Vacuum Pump
Pipeline
Condenser 1
Signal Wire
Condenser 2 Product Vessels
Partial Condenser 1
Light Composition
Partial Condenser 2 Nitrogen/ Oxygen
0
Heavy Composition
Heater
Raw Material
Reactor (10 l) Supply Vessels
10l
Distillation Column
2l
Reactor (2 l) Centrifuge
HeatingCooling System
Operator Station
Figure 1. Process diagram of the kilo-plant Fault Detection and Diagnosis G2
Scheduling and Optimization GAMS
Safety … Evaluation
Quality Control
Process Simulation Aspen plus, Pro/II
Process Monitoring Data Acquisition and Rectification DataCON
Database
Process Control CS-1000 Process Simulator Aspen Dynamic, Lab CVI
Process Equipments Kilo-plant
Figure 2. Structure of the experimental platform of process operation system
The experimental platform consists of three main layers: (1) A kilo-plant and a number of process simulators, including ASPEN Plus, Dynamics, Simulink, SuperPro, and the real-time platform G2, to facilitate integrating a variety of tasks of chemical process operations. They are used as experimental objects in the platform. (2) A control system, with a data acquisition and rectification interface, and a database. The control system used is CENTUM-CS1000. Opening Database Connection (ODBC) and Common Object Request Broker
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Agent (CORBA) are followed to realize communication among these software systems in the platform. The Standard for exchange of product data (STEP) is used for information integration. (3) Advanced applications, including process monitoring, dynamic simulation of the process, fault diagnosis, production scheduling, advanced control, online optimization, and product quality control. Integration of process conditions, operation and control strategies are studied. In the basis level, Aspen Dynamic process simulator and SIMULINK toolbox in MATLAB are applied to help extract the principle and specialty of process operation that run in the kilo-plant. In the mediate level, the PLC controller and CENTUMCS100, a small distributed control system (DCS) are used for data acquisition and information transmission. With the application of DataCON, noise to the process parameters is eliminated. We developed a real-time expert system for the process and equipment operation on G2 system. Aspen Plus, Pro/II are used to simulate the process for parameter optimization by comparing with the operation state of kilo-plant. Thus, the kilo-plant provides an effective platform for chemical process development and product design. It is applied to determinate the product raw material, manufacture method, the material flowsheet in the operation units and the units assembling sequence. 3. Integration of product and process design With the scope of chemical product engineering, among others, chemical product development, design, manufacture and delivering, are of the key technical problems. With the platform of the kilo-plant, the practical and key concepts of product engineering technology concerning the product design and process development can be developed and examined in a systematic way, as illustrated in Figure 3. Creation of the chemical supply chain, global life cycle design (LFD) = life cycle assessment (LCA) + life cycle costing (LCC)
Market needs for product performance, formulation
Computer aided molecular design (CAMD); Product quantitative structure-activity relation (QSAD)
Product
Process and operation integration, optimization
Product manufacture, process synthesis, and quality control on the Kilo-plant
Figure 3. A roadmap for the product centered triple “molecule-product-process” engineering
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On the microcosmic level, chemical product engineering investigates the basic models of product quality influenced by active materials with complicated molecular structures, interaction between molecules, and their quantitative relations, such as reaction activity, solubility etc.. On the mesoscale level, the integration of product design and process design make it easier to adjust the product structure and its properties which will accelerate the development process. In the process design and product manufacture, the influence of the operation parameters and processing ways on product micro-structure and properties can be considered. For example, in reaction system, concentration ratio of the raw material and product is influenced by temperature, pressure, and flow rate etc. Different catalysts also affect the reaction selectivity during the process experiment. As to the separation process, the behaviors of the phase equilibria should be cleared during the process design, such as the equilibrium relations of gas-liquid phases, liquid-liquid phases. At the macroscale level, product design is also an intriguing area of research (Cussler and Wei, 2003). The emphasis in research is on the time of new products to market needs, and the systematic exploration of alternatives for developing new products, which is typically a multidisciplinary task that needs both scientists, engineers of different disciplines and business people. A problem here that has not received much attention is the integration of product and process design. 4. A case study of product and process development A case for the production of Nipagin ethyl ester was carried out in the kilo-plant. It is a new anti-mildew and antiseptic agent which has the best inhibiting effect on aflatoxin in food, drink, cosmetics and medicine. The synthesis of Nipagin ethyl ester is carried out with the kilo-plant to illustrate the application of computer-aided system integration in product quality control in batch process. Nipagin ethyl ester is produced through the reaction of ethanol with Nipagin acid. The related equation is as follows: HO
COOH
CH3CH2OH
DBSA
HO
COOCH2CH3
H2O
The operation of the kilo-plant is monitored with 15 sensors of a variety of process variables, such as temperature, pressure, flow rate, pH value, and the rotation speed of stirrer etc. In the Nipagin ethyl ester synthesis case, the reaction conversion, reaction dynamics, kinetics analysis, and the process conditions adjustment, control strategy for product design and process design integration are invididually and/or integrated investigated. The product and process model were built for the optimization of the process. pH value and dynamic temperature trajectory are selected as key control parameter of the process operation. In terms of the product quality and purity, the process
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development and the associated operating conditions modifications in the kiloplant can't be easy to realize in ordinary chemical laboratories. For the system integration technology research, the feeding control, synthesis, heat exchange, and crystallization separation are facilitated to design the process cooperation running. It makes improvement of the production. 5. Conclusions Chemical product design and process development are among the most important research areas of chemical product engineering. The implementation of the kilo-plant experiment platform will not only make it more effective to realize flexible operation, but also reflect the dynamic change of the process to adopt an advanced control strategy and system integration method. The feasible scheme screened will accelerate the development of the manufacture process for the product. Because it is more convenient to adjust the developing route of the new product, on the other hand, the problems likely occurred in the actual industrial production can be tested, discovered and solved a prior. These make chemical engineering research respond faster to the technical difficulty of the chemical product manufacture and fulfill the market demands for a variety of chemical products. 6. Acknowledgements Financial supports from the National Natural Science Foundation of China (No.20476033, 20376025, 20536020) and Guangdong Province Science Fund (04020121) are gratefully acknowledged. 7. References Cussler, E.L., J. Wei, 2003. Chemical product engineering, AIChE Journal, 49(5), 1072-1075. Grossmann, I.E., A.E., 2002. Westerberg. Research challenges in process systems engineering. AIChE Journal, 46(9), 1700-1703. Grossmann, I.E., 2004. Challenges in the new millennium: product discovery and design, enterprise and supply chain optimization, global life cycle assessment. Computers & Chem. Eng., 29, 29-39. Qian Y, J.Z. Pan, L. J. Zhang, 2004. Structure-property relationships and models of controlled drug delivery of biodegradable PLA microspheres, Chinese J. Chem Eng. 12(6), 869-876. Vaidyaraman, S., C.D. Maranas, 1999. Optimal refrigeration cycle synthesis and refrigerant selection, AIChE Journal 45, 997-1017. Wörz, O., 1995. Process development via a miniplant. Comp. & Chem. Eng., 34, 261-268.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Importance of the selection of feed tray location on the optimum design of a heterogeneous azeotropic distillation column with p-xylene feed impurity I-Lung Chiena*, Hao-Yeh Leeb, Tang-Kai Gaub, and Hsiao-Ping Huangb a
Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan. b Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan.
Abstract In the production of aromatic acid, such as terephthalic acid, tiny amounts of one reactant (in this study, p-xylene) may also enter into the acetic acid dehydration column through the feed stream. In this work, the process design flowsheets with and without this tiny impurity are both considered. For the case with this tiny impurity in the feed stream, a side stream is necessary to purge out this impurity, otherwise, accumulation of this impurity will occur inside the column. If only optimum side-stream location and its flow rate are considered in the optimization search, it is found that the TAC (total annual cost) and the operating cost of this acetic acid dehydration column become much higher by just adding extremely small amount of p-xylene in the feed stream. With careful selection of the feed tray location, significant TAC as well as energy savings for the operation of this column can be realized. Keywords: Acetic Acid, Heterogeneous Azeotropic Distillation, Column Design, Side Stream, Feed Tray Location
1. Introduction Acetic acid (HAc) dehydration is an important operation in the production of aromatic acid, such as terephthalic acid, or in the manufacture of cellulose acetate. To make the separation easier, an entrainer is often introduced into the system. In a review paper, Othmer [1] described an azeotropic distillation system containing a dehydrating column, a decanter, and a water column for the separation of HAc and water. The entrainer used before 1932 was ethylene dichloride, and later n-propyl acetate and nbutyl acetate were used to reduce the organic reflux and heat duty used in the dehydrating column. In a paper by Pham and Doherty [2], examples of using ethyl acetate (cf. Tanake and Yamada [3]), n-propyl acetate (cf. Othmer [4]), or n-butyl acetate (cf. above [3] and [4]) as the entrainer were listed in a table of examples of heterogeneous azeotropic separations. Siirola [5] uses HAc dehydration as an example to demonstrate a systematic process synthesis technique to the conceptual design of a process flowsheet. Ethyl acetate was used as the entrainer in that paper to design a complete HAc dehydration process with multiple-effect azeotropic distillation and heat integration. Wasylkiewicz et al. [6] proposed using a geometric method for the optimum design of a HAc dehydrating column with n-butyl acetate as the entrainer. Recently, Chien, et al. [7] studied the design and control of HAc dehydration system via heterogeneous azeotropic distillation. A suitable entrainer of isobutyl acetate (IBA) was selected from three candidate acetates by TAC analysis. ____________________ * Corresponding author. I-Lung Chien, Tel: 886-2-2737-6652; Fax: 886-2-2737-6644., E-mail: [email protected]
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In the previous studies, only three components (HAc, H2O, and the entrainer) in the heterogeneous distillation column were focused. However, in the production of aromatic acid, such as terephthalic acid, tiny amounts of one reactant may also enter into the acetic acid dehydration column through the feed stream. Chien, et al. [8] studied the design and operation of an industrial column for HAc dehydration with five feed streams. The entrainer used for this industrial column to aid the separation is also IBA. In that paper, optimum design of the side-stream location and its flow rate were performed and an automatic purging strategy was proposed to prevent accumulation of an impurity (not specified for proprietary reason) inside the column. However, the effect of changing the feed tray location was not investigated. In this work, HAc dehydration column via heterogeneous azeotropic column system with and without the impurity in the feed stream will be thoroughly studied. The entrainer used in the study is also isobutyl acetate and the impurity in the feed stream is assumed to be p-xylene which is a reactant commonly used in the terephthalic acid plant. Three optimum design processes will be studied. The first one assumes that the tiny amount of p-xylene is not considered in the feed stream. The optimized design and operating variables include: the column total stages, feed tray location, and the aqueous reflux ratio. The column total stages and the feed tray location obtained from this optimum search will be used in the second study with tiny amounts of p-xylene entering into the column through the feed stream. The optimal side-stream location and its flow rate will be investigated. In the third design also with the impurity in the feed stream, besides the side-stream location and its flow rate, the feed tray location will also be adjusted in this case. The results of the above three cases will be compared.
2. Steady-state design with no feed impurity In this study, the feed stream is assumed to contain equal molar of acetic acid (HAc) and water (H2O). The entrainer (isobutyl acetate, IBA) is introduced into the column through organic reflux stream. Because the addition of the entrainer IBA will form a heterogeneous azeotrope with minimum temperature of the system, thus this azeotrope will ideally go out of the system from column top. After cooling of this stream to 40 ºC at decanter, this stream will naturally split into two liquid phases. The organic phase containing mostly IBA can be refluxed back to the column. Part of the aqueous phase is also refluxed back to the column for high purity reason with the rest going out of the system as waste water stream. The process flowsheet can be seen in Figure 1. The product specifications are 99.9 mol% of HAc to be recovered from the column bottoms and high purity water stream containing only 0.1 mol% HAc to be withdrawn from the decanter aqueous outlet stream. In the process simulation, the bottom product specification is set by varying the reboiler duty and the aqueous outlet product specification is set by varying the entrainer makeup flow rate. There are three design and operating variables in this flowsheet which can be optimized. They are: the column total stages, feed tray location, and the aqueous reflux fraction. The optimization procedure is to find the minimized total annual cost (TAC) at particular column total stages by varying the feed tray location and aqueous reflux fraction. For each simulation run, process simulation tool of Aspen Plus [9] was used to obtain the simulation result. Figure 2 shows the optimized result with the column total stages fixed at 39 (not counting reboiler). With the similar optimization search at other total stages, the overall minimized TAC can be found at column total stages at 39 with feed tray location at 9th stage and aqueous reflux fraction set at 0.11. The optimized base case condition is summarized in Figure 3.
Importance of the Selection of Feed Tray Location
999
IBA Makeup Organic Reflux Feed
Decanter
(HAC+H2O mixture)
Aqueous Product (H2O)
Steam
Reboiler Bottom Product (HAC)
Fig. 1. Process flowsheet without feed impurity
TAC($1000/year)
Stages = 39
381 380 Feed Stage =8 Feed Stage =9 379 Feed Stage =10 378 377 376 375 374 373 372 371 370 369 0.100 0.105 0.110 0.115 0.120 0.125 0.130 0.135 0.140
Aqueous Reflux Fraction
Fig. 2. Optimized results with column total stages at 39.
1 Feed 100 kmol/hr HAc : 50 mol% Water : 50 mol%
9
Makeup : IBA 0.0582 kmol/hr
Organic Reflux 36.96 kmol/hr HAc : 0.19 mol% Water : 7.98 mol% IBA : 91.83 mol%
Decanter
Aqueous Product 50.058 kmol/hr HAc : 0.1 mol% Water : 99.79 mol% IBA : 0.109 mol%
Aqueous Reflux 6.187 kmol/hr
Reboiler
Steam Duty =1113.681 kW
40 Product 50 kmol/hr HAc : 99.9 mol% Water : 9.25e-02 mol% IBA : 7.47e-03 mol%
Fig. 3. Optimized base case condition for no impurity case.
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3. Steady-state design with feed impurity 3.1. Side-Stream Location and Its Flow Rate as the Optimized Variables. In the production of aromatic acid, such as terephthalic acid, tiny amounts of one reactant (in this study, p-xylene) may also enter into this acetic acid dehydration column through the feed stream. Thus, a side stream is necessary to purge out this impurity, otherwise, accumulation of this impurity will occur inside the column. In this second case with feed impurity, feed composition of 50 mol% water, 49.9 mol% acetic acid, and 0.1 mol% p-xylene is assumed. Intuitive design thinking is to just use the optimum total stages and feed tray location obtained from the above no impurity case and then investigating the optimal side-draw flow rate and side stream location via TAC analysis. The merit of this design thinking is that the same column design can be suited for both the no impurity case and with tiny impurity case. Stages = 39 Feed Location = 9 Sidedraw Location = 21
TAC($1000/year)
580 575 570 Sidedraw Flow Rate = 0.6 kmol/hr Sidedraw Flow Rate = 0.7 kmol/hr Sidedraw Flow Rate = 0.8 kmol/hr
565 560
0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48
Aqueous Reflux Fraction
Fig. 4. Optimized results with sidedraw location at 21st stage.
Feed 100 kmol/hr HAc : 49.9 mol% Water : 50mol% PX : 0.1mol%
1
Organic Reflux 53.839 kmol/hr HAc : 0.17 mol% Water : 6.36 mol% IBA : 86.21 mol% PX : 7.25 mol%
Decanter
Aqueous Reflux 36.027 kmol/hr
9
21
Makeup IBA: 0.0675 kmol/hr
Sidedraw 0.7 kmol/hr HAc : 40.65 mol% Water : 42.9 mol% IBA : 2.2 mol% PX : 14.26 mol%
Aqueous Product 49.752 kmol/hr HAc : 0.1 mol% Water : 99.8 mol% IBA : 0.1 mol% PX : 4.18e-4 mol%
Steam Duty = 1710.8 kW 40 Reboiler
Product 49.615 kmol/hr HAc : 99.9 mol% Water : 0.098 mol% IBA : 4.21e-03 mol% PX : 1.97e-07 mol%
Fig. 5. Optimized base case condition with feed impurity (not changing feed tray location).
The optimization procedure is to fix the total stages at 39 (not counting reboiler), feed tray location at 9th stage, and to vary the sidedraw location and sidedraw flow rate. Again, the bottom product specification is set by varying the reboiler duty and the aqueous outlet product specification is set by varying the entrainer makeup flow rate. For each simulation run, the aqueous reflux fraction gives the lowest TAC is selected. Figure 4 is an example of such simulation run with sidedraw location at 21st stage.
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From the figure, aqueous reflux fraction of 0.42 and sidedraw flow rate of 0.7 kmol/h will be selected. Doing similar runs at other sidedraw location, the overall optimized result is found. The optimized base case condition is summarized in Figure 5. Comparing the base case conditions of Figs. 3 and 5, it is found that the reboiler duty is changed from 1114 KW to 1711 KW, a 54% increase in this energy cost. The TAC is changed from $370.2×103 to $559.5×103 (a 51% increase), and the operating cost is changed from $106.7×103 to $163.8×103 (a 54% increase). This shows a dramatic difference in the base case conditions by just adding 0.1 mol% of p-xylene feed impurity into this column. 3.2. With Addition of the Feed Tray Location as the Optimized Variables. If additionally feed tray location can be considered as another optimized variable in the overall optimization procedure, the base case condition is very different from that of Fig. 5. By considering feed tray location as another optimized variable does not defeat the original purpose of operating this column under both “no feed impurity” and “with feed impurity” cases. The optimization procedure will be more complicated by adding one more variable in the search algorithm. Basically the overall optimization procedure as in Section 3.1 can be followed to find the design condition with the lowest TAC at particular feed tray location. Then, this procedure is repeated to find the design condition with the lowest TAC at another feed tray location. By collecting all the results at many possible feed tray locations, the design condition with the minimized TAC can be obtained. The final optimized base case condition is summarized in Figure 6 with feed tray location changes from 9th stage as in Fig. 5 to 24th stage. The sidedraw location is also changed from 21st stage as in Fig. 5 to 18th stage. Notice also that the sidedraw flow rate is further reduced from 0.7 kmol/h to 0.2 kmol/h because of much richer p-xylene can be established inside the column at this new base case condition. The liquid composition profile for the base case of Fig. 6 is show in Figure 7.
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Steam Duty = 1393.476 kW 40 Reboiler
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Fig. 6. Optimized base case condition with feed impurity (changing feed tray location)
By comparing Fig. 6 with earlier Fig. 5, it is found that the reboiler heat duty can be reduced from 1711 KW to 1393 KW (a 18.6% reduction). The TAC of the system is reduced from $559.5×103 to $438.3×103 (21.7% reduction), and the operating cost is also reduced from $163.8×103 to $133.4×103 (18.6% reduction). Hence, by just adding feed tray location as another variable in the optimization search, significant
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savings in TAC and operating energy can be realized. This base case design will be used in a following-up control study to investigate the proper overall control strategy to hold the bottom and top product purities despite feed water composition changes or the changes in p-xylene impurity in feed stream. HAc W ater IBA PX
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4. Conclusions In this paper, the process flowsheets with and without tiny impurity of p-xylene are both considered in the optimal design of acetic acid dehydration column via heterogeneous azeotropic distillation. The simulation results show that dramatic difference in the base case conditions are found by just adding 0.1 mol% of p-xylene feed impurity into this column. Over 50% more TAC and operating energy are needed to operate this column with sidedraw under “with feed impurity” case. However, by considering feed tray location as an additional optimized variable in the optimization search, significant savings in TAC with a reduction of 21.7% can be realized. Note particularly with this change of the feed tray location from 9th stage to 24th stage, significant saving of the operating energy with a reduction of 18.6% can also be achieved.
Acknowledgements This work is supported by the National Science Council of the R. O. C. under grant No: NSC 90-2214-E-011-013
References [1] D. F. Othmer, Chem. Eng. Prog., 59, (1963) 67-78. [2] H. N. Pham, and M. F. Doherty, Chem. Eng. Sci., 45, (1990) 1845-1854.. [3] S. Tanake, and J. Yamada, J. Chem. Eng. Jpn., 5, (1972) 20-26. [4] D. F. Othmer, Chem. Metall. Eng., 40, (1941) 91-95. [5] J. J. Siirola,In An Industrial Perspective on Process Synthesis; L. T. Bieglar, and M. F. Doherty, (eds.); AIChE Symposium Series 304; CACHE: Austin, TX, Vol. 91, (1995) pp 222233. [6] S. K. Wasylkiewicz, L. C. Kobylka, and F. J. L. Castillo, Chem. Eng. J., 79, (2000) 219-227. [7] I. L. Chien, K. L. Zeng, H. Y. Chao, J. H. Liu, Chem. Eng. Sci., 59, (2004) 4547-4567. [8] I. L. Chien, H. P. Huang, T. K. Gau, and C. H. Wang, Ind. Eng. Chem. Res., 44, (2005) 35103521. [9] Aspen Technology, Inc. Aspen Plus User’s Manual 11.1, Aspen Technology, Inc, Cambridge, MA, 2002.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Supporting Waste Minimization Studies by Integrating Expert System with Process Simulators Iskandar Halima and Rajagopalan Srinivasana,b* a
Institute of Chemical and Engineering Sciences (ICES), 1 Pesek Road, Jurong Island, Singapore 627833 b Department of Chemical and Biomolecular Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Abstract Waste minimization of large-scale chemical processes poses a multi-dimensional challenge. In this paper, we present an integrated solution framework consisting of an expert system and process simulator. The proposed framework has been developed by capitalizing on the CAPE-OPEN capability of HYSYS simulator through XML data exchange. First, a qualitative analysis is carried out by the expert system to diagnose the sources of wastes and to derive heuristics solutions. This is followed by quantitative assessment at process variable level using HYSYS modelling engine. Stochastic optimization of process variables for minimizing waste and maximising product flow is also performed. We illustrate the framework using an ammonia synthesis case study. Keywords: Intelligent system; Process simulation; Cleaner production.
1. Introduction Design of a chemical plant always involves a combination of synthesis, analysis and evaluation of different design alternatives. Such activities have been traditionally driven by economic factors first, followed by engineering, safety and environmental considerations. However, the situation has changed much during the past decades. Increased public awareness, stricter regulations and cost competition have left the chemical industries with tremendous pressure to eliminate or at least minimize its waste discharge − in most cases, this demands for fundamental changes to the initial design and operation of the plant. The role of process simulators is thus becoming more important for achieving the required environmental objectives. Through simulation, different process alternatives can be evaluated within a short time without the need for extensive experimentation or pilot plant testing. There is abundant literature on the use of commercial process simulators as environmental design support tools. Cabezas et al (1999) applied Waste Reduction (WAR) methodology using CHEMCAD simulator to compare the environmental performances of various process modifications. Dantus and High (1999) applied compromise programming method and stochastic annealing algorithm into ASPEN PLUS simulator to simultaneously maximize the profit and minimize the environmental impact. Chen and Shonnard (2004) used HYSYS simulator as a screening tool to evaluate the economic and environmental viability of a process. While these simulators are effective for solving the waste problems, their use is still not a straightforward task *
Corresponding author. Email: [email protected]
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as it demands for considerable know-how, skill and expertise in the part of the user to identify the units and the variables that control the overall waste feature in the plant. For such tasks, very little help is actually provided by simulators. This shortcoming had indeed been highlighted during a joint workshop organized by the USEPA, the Department of Energy (DOE) and the Center for Waste Reduction Technologies (CWRT) with a recommendation to develop a coupled expert system and process simulator (Eisenhauer and McQueen, 1993). But to date, no such development has been reported. Previously, we have reported our attempts to integrate an expert system with a simulator (Halim and Srinivasan, 2002a). The expert system, called ENVOPExpert, is developed using G2 expert system shell and automatically identifies the sources of waste, proposes process design changes and calculates both the environmental impact and the plant profitability. However, to achieve the above tasks, the user has to manually interpret and transfer the results of the expert system to the simulator and vice versa. In this paper, we present a further evolution that capitalizes on the CAPE-OPEN functionality of the HYSYS simulator for XML data transfer. In this case, ENVOPExpert provides decision support by automatically accessing the available physical property, process chemistry, and flowsheet information of the process from HYSYS.
2. Integrated Waste Minimization Framework Figure 1 describes the proposed integrated framework. It is assumed that a steady state simulation model of the process is available a priori. First, the process description in HYSYS is converted into XML data format in preparation for exporting to ENVOPExpert. Other waste minimization related information that is not available in HYSYS such as material impact factors and plant economics data such as capital and operating costs are supplied by the user through Microsoft Excel. Next, this information is exported to ENVOPExpert for analysis through an ActiveX and Excel-VBA interface. Waste minimization analysis is next performed in G2. The outcomes from ENVOPExpert would be qualitative waste minimization solutions and some suggestions for design and operation changes. ENVOPExpert then uses HYSYS to simulate these changes and evaluate the quantitative improvement in the waste impact as well as the costs involved. In the following section, we illustrate the various elements of this methodology using an industrial case study.
3. Case Study: Ammonia Synthesis Figure 2 describes ammonia production from synthesis gas. The feed stream (Stream 5) containing nitrogen, hydrogen, and impurities (argon and methane) is passed through a series of reactors. The products from the reactors are condensed using a coolant and then separated in a flash separator, where ammonia liquid is collected as product (Stream 21) and the vapour mixture is recycled back to the reactor. To prevent the build-up of impurities within the process, a purge stream (Stream 28), which becomes the only waste stream in this process, is used. Table 1 shows the WAR based environmental impacts of each process material. This case study has been used by Cabezas et al (1999) to illustrate the WAR algorithm for deriving waste minimization solutions. In this paper, we illustrate the proposed integrated methodology, especially the capability to simultaneously address waste source reduction and material recovery.
Supporting Waste Minimization Studies
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Figure 1. Integrated Expert System and Process Simulator
Figure 2. Ammonia Synthesis Process
Table 1. Environmental Impact Factors
Component Methane Ammonia Nitrogen, Hydrogen and Argon
Environmental impact 0.0074 0.93 0
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4. P-graph Analysis for Qualitative Solutions Diagnosis of the material sources of the waste streams and identification of process streams that are candidates for material recovery is first performed automatically by ENVOPExpert using P-graph analysis (Friedler et al, 1994). In this method, ENVOPExpert identifies each material whose presence is either desirable or undesirable in every process output streams. As the presence of ammonia in the purge stream is undesirable, one way of reducing the amount of ammonia in the purge stream is to recover it from the unit that causes its escape to the purge stream. This is done by tracing the path of ammonia in the upstream direction, starting from both the purge stream and the product stream to a unit where the two paths cross. Once the unit is found, waste minimization solutions that focus on preventing valuable material (in this case ammonia) from becoming a waste stream can be suggested using design heuristics (Halim and Srinivasan, 2002b, c). Figure 3 illustrates the P-graph model of ammonia in the purge stream. Each material stream is represented by a circle and an operating unit by a bar. The presence of ammonia in the purge, stream-21, is caused by the splitter unit T-101 and inefficiency in the flash-tank V-101. Two solutions – “use further separation after the splitter to recover ammonia component” and “improve the design and operating conditions of the flash separator” can be proposed for this. The reader is referred to Halim and Srinivasan (2002a, b, c) for the detailed procedure for deriving these solutions.
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5. Quantitative Analysis Once the qualitative solutions have been proposed, they can be quantitatively explored using HYSYS simulator. This is done by suitably manipulating the process variables controlling waste generation. In Figure 3, the split ratio of T-101 can be varied so that
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the ammonia in stream-16 is maximized, while the pressure drop in V-101 can be manipulated to minimize ammonia in stream-22. At the same time, the environmental impact factors and the cost implications can also be evaluated. Table 2 lists the results of this and other changes to the process. As can be seen, there exists a trade-off between plant profitability as represented by the energy requirement and the environmental impact. Also, a decrease in the purge flowrate would greatly reduce the environmental impact but at the expense of higher impurities in the ammonia product stream. Optimization of process variables can also be performed in this framework. We have used the simulated annealing algorithm as implemented in EXCEL-VBA for this purpose. As one example, the optimum feed temperature and cooler energy to maximize the ammonia component in the product stream was calculated. The results showed that maximum ammonia flowrate can be achieved at the lower end of the feed temperature and higher end of cooler load. Table 2. Qualitative and quantitative analysis of ENVOPExpert
Waste minimization solution Base Case Increase energy of compressor K-100 from 488.6 to 500 kW Decrease purge ratio from 0.005 to 0.004 Decrease feed temperature from 200 to 1900C Increase energy of cooler E-101 from 1.063E5 to 1.083E5 kW Increase energy of cooler E-103 from 4.8E4 to 4.86E4 kW
Impact 636.2 636.2 509.0 626.8 605.9 631.5
6. Conclusions The need for waste minimization has challenged the chemical industries to seek new approaches to tackle practical waste minimization problems. We have developed a framework comprising of an expert system and a simulator to automate waste minimization analysis. The framework has been shown to be capable of generating both qualitative and quantitative solutions. Stochastic optimization of process variables that control the waste profile is also made possible. Although the current application of the framework is to HYSYS simulator, it can also be extended to other CAPE-OPEN compliant simulators.
References Cabezas, H., J.C. Bare and S.K. Mallick, 1999, Comp. Chem. Eng. 23, 623. Chen, H. and D.R. Shonnard, 2004, Ind. Chem. Eng. Res. 43(2), 535. Dantus, M.M. and K.A. High, 1999, Comp. Chem. Eng. 23, 1493. Eisenhauer, J. and S. McQueen, 1993, Environmental Considerations in Process Design and Simulation, AIChE, New York. Friedler, F., J.B. Varga and L.T. Fan, 1994, Pollution Prevention via Process and Product Modifications, Eds. M.M. El-Halwagi and D.P. Petrides, AIChE, New York. Halim, I. and R. Srinivasan, 2002a, Env. Sci. Tech. 36(7), 1640. Halim, I. and R. Srinivasan, 2002b, Ind. Eng. Chem. Res. 41(2), 196. Halim, I. and R. Srinivasan, 2002c, Ind. Eng. Chem. Res. 41(2), 208.
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Process Intensification for Systematic Synthesis of New Distillation Systems with Less Than N-1 Columns Ben-Guang Rong* and Ilkka Turunen Department of Chemical Technology, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finland. *Email: [email protected]
Abstract In this paper, a method of process intensification for systematic synthesis of new distillation systems with less than N-1 columns for an n-component mixture is presented. The method is based on the simultaneous thermal coupling and heat integration principles. First, the heat-integrated thermally coupled configurations (HITC) are produced. Then, a prefractionator without any pure products in each of the HITC configurations is incorporated into another column with dividing-wall. This produces the intensified new distillation systems with less than N-1 columns. The intensified new distillation systems have the potential to further significantly reduce the capital investment than the HITCs. The method is illustrated for quaternary distillations and can be extended to any n-component mixture. Keywords: process intensification, process synthesis, distillation system, thermal coupling, heat integration
1. Introduction Process Intensification is defined as a strategy for achieving dramatic reductions in the size of a chemical plant at a given production volume (Ramshaw, 1983). Process intensification has also been claimed to bring other benefits, such as lower capital costs, reduced energy consumption, increased safety and improved product quality (Stankiewicz and Moulijn, 2000). One major approach in process intensification is to reduce the number of equipment units by innovative design of multifunctional equipment which leads to reduced investment costs and significant energy savings. Distillation is the widely used separation method in process industry, whereas it is the largest energy consumer among process units and simultaneously needs a large capital investment. New distillation systems with the potential to significantly reduce both energy consumption and capital investment are desired. For an n-component distillation, the traditional designs of simple column configurations use n-1 columns and 2(n-1) condensers and reboilers. Each column implements one of the n-1 sharp splits. Such simple column configurations have the intrinsic separation inefficiency and suffer from both high energy consumption and large capital investment. It is known that the number of columns and the number of heat exchangers in a distillation system represent not only the final equipment costs but also the installation costs in the final plant construction. The sizes of columns and heat exchangers in a distillation system are directly related to the energy amount consumed for the specified separation which attribute to the energy efficiency of the separation process.
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In this work, we take advantage of simultaneous thermal coupling and heat integration as a process intensification strategy for the systematic synthesis of intensified new distillation systems for a multicomponent distillation. First, the number of condensers and reboilers are reduced by the thermal coupling and heat integration techniques. This produces the heat-integrated thermally coupled configurations (HITC). Then, the prefractionation column without pure product in the HITCs is incorporated into another column with dividing-wall. This produces the intensified new distillation systems with a reduced number of columns than that of the HITCs. There were few works in the literature reported on a few specific distillation systems with dividing walls (Kaibel 1987; Christiansen et al.1997a,b; Lestak and Collins 1997; Agrawal 2001). However, a method of process intensification that can systematically synthesize the possible intensified new distillation systems for an n-component distillation is not available yet.
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Figure 1. Heat-integrated thermally coupled configurations for quaternary mixtures In each of the HITC configurations in Figure 1, thermal coupling and heat integration have been simultaneously used to coordinate the individual splits in their separation sequences. In Figure 1a, three individual splits are coordinated. In each of the parts b-e, g-h, and k in Figure 1, four individual splits are coordinated. In each of the parts f, i-j, lm, o and q in Figure 1, five individual splits are coordinated. Finally, in each of the parts n, p and r in Figure 1, six individual splits are coordinated. Note that in each
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configuration in Figure 1, the number of column sections is twice the number of the individual splits in its separation sequence. In each of the HITC configurations in Figure 1, a column with more than two sections is obtained by combining the individual columns through heat integration. As such, the submixture BC transferring between intermediate locations of two columns is a one-way liquid flow. At the same time, the condensers and reboilers associated with submixtures involving the two extreme volatility components are eliminated, and the two-way vapour and liquid flows are introduced through thermal couplings. In Figure1a, there are two columns with two heat exchangers, while each of the remaining configurations contains three columns (i.e. n-1) with the number of heat exchangers be two, or three or four (i.e. less than 2(n-1)). In each of the configurations in Figure 1, the first column with only two sections is for the first split of the feed mixture, while among the remaining columns, there is at least one column with more than two sections (four or six sections). It is significant to note that, in each configuration in Figure 1, there is at least one column that does not produce any pure products. In other words, its two ends are connected with the other columns by thermal couplings. Such a column without any products serves as the prefractionation to a submixture and is usually called a prefractionator. In the following section, we will show that it is the prefractionator that the intensified new distillation system with less than N-1 columns is achieved for each of the HITC configurations in Figure 1.
3. Intensified New Distillation Systems with Less Than N-1 Columns Before considering the intensified new distillations systems, let us first consider the connections of the prefractionator with a combined column. Originally, when considering only thermal couplings, the two submixtures of the prefractionator can communicate with either one same column or two different columns (Rong et al. 2003). However, in order to achieve intensified new distillation systems with less than N-1 columns, we determine the interconnections of the columns in the HITC configurations in such a way that first a combined column with more than two sections is obtained through heat integration(s) and is selected as the main column for further equipment intensification. Then, a prefractionator with only two sections is connected with the main column. The top end of the prefractionator is thermally linked with the upper part of the main column, while the bottom end of the prefractionator is thermally linked with the lower part of the main column. Notice that each of the HITC configurations in parts a-i, and k in Figure 1 has only one combined column which serves as the only main column, while each of the remaining configurations in Figure 1 has two combined columns of which the one connected with the prefractionator serves as the main column. The simultaneous thermal coupling and heat integration in the HITC configurations in Figure 1 produce the opportunities for equipment integration and intensification. It was practical that a prefractionator could be incorporated into another column to share one same column shell (Wright 1949; Petlyuk et al. 1965; Kaibel 1987; Christiansen et al. 1997a,b). One simple way of such incorporation is through a dividing wall (Wright 1949; Kaibel 1987; Christiansen et al. 1997b). Therefore, by the same principle, the prefractionator in each of the HITC configurations in Figure 1 can be incorporated into the main column as a dividing wall. By doing so, each of the HITC configurations in Figure 1 can achieve an intensified new distillation system with less than N-1 columns, which are shown in Figure 2, respectively. In each of the configurations in Figure 2, there is one intensified column equipment which performs at least three individual splits within the column shell, and it is in this sense that we call it a multifunctional
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equipment. It is interesting to note that there are similar structures among some of the intensified new distillation systems in Figure 2. However, each of the intensified new systems has a distinct separation sequence with a certain number of intended individual splits. The specifications of the intended individual splits can cope with the separation problems with various feed conditions in terms of relative volatilities and feed compositions of the feed components.
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(q) (r) Figure 2. Intensified new distillation systems with less than N-1 columns for quaternary mixtures from Figure 1 It is important to indicate that the dividing wall column for ternary separations has been widely used in many industrial applications, where typically 30% savings on both energy and capital costs were achieved (Kaibel and Schoenmakers 2002; Becker et al. 2001). Moreover, both the steady-state and dynamic performance were confirmed by the practical applications. Therefore, the intensified new distillation systems in Figure 2 are also expected to be promising in industrial applications for quaternary separations.
4. Conclusion A method of process intensification on the basis of simultaneous thermal coupling and heat integration achieves the intensified new distillation systems systematically. The heat-integrated thermally coupled configurations are first generated. Then, the thermally linked prefractionator in the HITCs is incorporated into a combined column with dividing wall. The intensified new distillation systems are expected to have the similar energy savings, but reduced capital investment than that of the HITCs. It is significant to accentuate that it is the synergism of the mass and heat transfers by simultaneous thermal coupling and heat integration that gives the opportunities for equipment integration and intensification. It is the equipment integration and intensification that ultimately produces the intensified new distillation systems with less than N-1 columns for an n-component mixture. The concepts and ideas presented in the method for quaternary mixtures can be extended to any n-component mixtures (nt4).
References R. Agrawal, 2001, Ind. Eng. Chem. Res., 40, 4258. H. Becker, S. Godorr , H. Kreis, J. Vaughan , 2001, Chem. Eng., 108, 68. A.C. Christiansen, S. Skogestad, K. Lien, 1997a, Comput. Chem. Eng. 21, S237. A.C. Christiansen, S. Skogestad, K. Lien, 1997b, IChemE Symp. Series, No.142, p 745. G. Kaibel, 1987, Chem. Eng. Technol. 10, 92. G. Kaibel, H. Schoenmakers, 2002, Computer-Aided Chem. Eng. 14, ESCAPE 12, p.9. F. Lestak, C. Collins, 1997, Chem, Eng. July, 72. F.B. Petlyuk, V.M. Platonov, D.M. Slavinskii, 1965, Int. Chem. Eng., 5, 555. C. Ramshaw, 1983, Chemical Engineer-London, 389, 13-14. B.-G. Rong, A. Kraslawski, I. Turunen, 2003, Ind. Eng. Chem. Res., 42, 1204. A.I. Stankiewicz, J.A. Moulijn, 2000, Chem. Eng. Prog. 96, No 1, 22-34. R.O. Wright, 1949, U.S. Patent 2,471,134.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Mixed-Integer Optimization of Distillation Column Tray Positions in Industrial Practice Ingo Thomasa , Andreas Kr¨onera a
Linde AG, Linde Engineering Division, Dr.-Carl-von-Linde-Str. 6-14, 82049 Hoellriegelskreuth, Germany We present an outer approximation algorithm tailored to the optimization of distillation column feed and side draw tray positions. A new fallback rule based on a generalization of an integer cut for integer parameters is presented. This fallback rule turned out to be a key ingredient to successful application to real-world problems. As an example, a process design optimization of an natural gas treatment plant is presented. Keywords: Mixed Integer Nonlinear Programming, Process Design, Outer Approximation, Tray Optimization 1. Introduction As a leading international engineering and contracting company, the Linde Engineering Division of Linde AG designs and builds turnkey process plants for a wide variety of industrial customers: chemical industry, manufacturers of hydrogen and synthesis gases, natural gas suppliers, air separation companies, and more. For steady state and dynamic simulation and optimization Linde Engineering has deR veloped the simulation tool OPTISIM [2]. In basic engineering it is applied for steady state design calculations and optimization of process plants. Any mathematical solution R algorithm in OPTISIM relies on an equation-based strategy using advanced numerical methods. The ever-growing demand for flexibility and profitability strongly affects both, design and operation of process plants. In the early stage of a plant’s life cycle, design decisions such as fixing distillation column feed and side draw trays, heavily rely on elaborate steady state process simulation and optimization calculations. The optimal design of such separation processes poses a Mixed Integer NonLinear Problem (MINLP) with discrete connection tray numbers and various continuous design parameters. Although numerical mathematics has provided a variety of apparently well suited algorithms, the numerical solution of column MINLPs is still a challenge when it comes to industrial practice. R Different algorithms were implemented in OPTISIM and adapted to cope with practical problems such as instabilities of the underlying distillation column model. As an example a modified version of an outer approximation algorithm is presented. Successful results are demonstrated by the practical example of a natural gas treatment plant.
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The OA/ER method is a general-purpose MINLP scheme [1,4] which generally has the advantage of requiring a relatively small number of NLP solutions, which is by far the most expensive part in our setting. OA/ER has been extended for tray optimization problems by Viswanathan and Grossmann [6,5]. It has, however, the drawback of relying on convexity assumptions for the objective function and the constraints to find global solutions. These restrictions have shown to be a severe problem in real-world problems. By now, the choice of a suitable initial value for the tray variables is of paramount importance to get usable results. In our setting of rather complex process plants we often experience convergence problems or worsening of the objective function when updating the tray variables. Key ingredients to the practical application of the scheme are 1. a fallback mechanism to handle these situations, 2. a limitation on the update of tray variables. 2. Limiting the Integer Update Practical experience as well as theoretical investigation show that the states of a distillation column depend in a highly non-linear way on the tray variables. Hence, the linear approach for objective and constraints that is used in the cutting plane MILP is valid only in a small neighbourhood of the tray variables at the k-th step y k . Therefore, we restrict the integer update to a neighbourhood of the actual point by replacing the integer domain Y of the cutting plane MILP by Y˜ := Y˜ (y k ) := {y ∈ Y : y − y k ≤ DELTA TRAY}, where DELTA TRAY is a problem-dependent constant. 3. The Fallback Method If either the NLP in an outer approximation step turns out to be unsolvable or yields a result larger than the upper bound, a fallback step is triggered. Then, the NLP method is restarted using the second-best integer update, which is computed by re-solving the cutting plane NLP with the additional constraint y˜k+1 = y k+1 .
(INEQ)
The new update is a weaker lower bound to the optimum, but often yields a solvable system of flowsheet equations. Please note that the technique we propose is closely related to the integer cut reported by Duran and Grossmann [1]. It can be regarded as a generalization of this technique for integer variables instead of binary ones, which is quite effective when combined with the integer domain restriction. The catch of this idea is the formulation of inequality (INEQ) – there are only greaterthan/less-than relations and equality constraints in linear programming, but no “not equal” constraints.
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For readability, we omit the major iteration index k + 1. We want to add the condition y˜ = y for a fixed y˜ ∈ Y to the cutting plane MILP. To do this, we represent the (integer type) difference between the new and the old (non-converging, now “forbidden”) integer part of the solution component-wise as a sum of binary variables sl,i , tl,i ∈ {0, 1} (˜ y − y)i =
DELTA TRAY
(sl,i − tl,i ) .
(R)
l=1
To achieve uniqueness of this representation we add the following conditions: At most one component out of the pair sl,i , tl,i is 1. sl,i + tl,i ≤ 1 for all l, i
(U1)
All non-zero entries in the s, t vectors are collected in the leftmost part. sl−1,i ≥ sl,i tl−1,i ≥ tl,i
for all l > 1, i for all l > 1, i
(U2a) (U2b)
We immediately observe: y − y ∗ = 0 is true if and only if TRAY DELTA i
(sl,i + tl,i ) ≥ 1.
(U3)
l=1
This can be seen as follows: If (U3) holds true, then there is an index pair (l∗ , i∗ ) such that either tl∗ ,i∗ = 1 or sl∗ ,i∗ = 1 (not both). By symmetry, it suffices to discuss sl∗ ,i∗ = 1. By (U2a) we have s1,i∗ = 1 and (U1) yields t1,i∗ = 0. Hence (by (U2b)) tl,i∗ = 0 for all l and therefore (˜ y − y)i∗ > 0. Please note that this algorithm potentially introduces a large number of additional constraints in the cutting plane MILP. This number can be considerably reduced in conjunction with the integer step size restriction described above, by using the fact that “forbidden points”y ∈ / Y˜ (y k ) do not need not to be considered by this algorithm. Example. Let DELTA TRAY be 1, and the objective turned out to increase for (y1 , y2 ) = (5, 4). Then, the following constraints are added to the cutting plane MILP: y1 − 5 = s1,1 − t1,1 y2 − 4 = s1,2 − t1,2 s1,1 + t1,1 ≤ 1 s1,2 + t1,2 ≤ 1 s1,1 + t1,1 + s1,2 + t1,2 ≥ 1, s1,1 , s1,2 , t1,1 , t1,1 ∈ {0, 1} This relations could be stated more elegantly using sos-1 variables as introduced by Beale and Tomlin [3], which are by now not available in our optimization environment.
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Feed Gas
Pretreatment
temperature . o
Main Heat Exchanger
flow o
oo oo
C3 Refrigerant pressure , flow C3 Refrigerant pressure , flow
oo
C3 Refrigerant pressure , flow
temperature pressure
o Optimization Parameters
o
tray tray
o
tray
Min. shaft power distance to CO2 freezing
tray o
o
heat o
heat o
pressure
Sales Gas
MTD DTmin
o
Expander Booster
Sales Gas Compression
o o
Demethanizer
o o
product specification
product o specification
2nd side reboiler 1st side reboiler reboiler
o Optimization Constraints o Optimization Objective
heat
NGL product
Figure 1. NGL plant flow scheme
4. Optimization of a C+ 2 recovery process As a real life optimization example we consider a process to separate ethane and heavier components, i.e. natural gas liquids (NGL) or C+ 2 fraction, from natural gas (NG) as sketched in Fig. 1. The lean methane rich sales gas is used as fuel whereas the NGL product serves as feed to ethylene plants. The task is to design a process with a high recovery of NGL at low shaft power for the refrigeration and the recompression. A key success factor of a design is the best integration of process cold from the distillation column (demethanizer) with the heat exchanger as the optimal integration minimizes the shaft power. The objective to be minimized is the compressor shaft power while retaining constraints for heat exchangers and columns. There are 16 real optimization parameters to be adjusted such as heat exchanger outlet temperatures, pressure levels and flow rates of refrigerant and heating rates of reboiler streams. Five integer parameters represent feed and side reboiler tray numbers of the demethanizer. The design is subject to 56 inequality constraints on the delivery pressure, properties of sales gas and NGL product, approach to CO2 solid point, and minimum temperature differences in heat exchangers. The relationship between the optimization parameters and constraints is established by a steady state plant design model which incorporates numerous process details together with auxiliary process sections omitted in Figure 1. The entire flowsheet contains 580 unit models and 460 streams with 27 components posing 18000 equality constraints to the optimization problem. The flowsheet is non-convex and unstable w.r.t. tray positions which leads to several fallback steps as well as inversion of upper and lower bound (see convergence history in
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44 UB LB 43.5
objective function
43
42.5
42
41.5
41 0
2
4
6
8 10 major iteration
12
14
16
18
Figure 2. Convergence history of tray optimization
Fig. 2). It takes about sixteen outer approximations to find the optimal position which is about 2.5% better than the continuous optimum using the initial tray settings. The computation involves the solution of about 100 steady states, and the total computing time on a standard desktop PC (Intel P4 2.8 GHz) is significantly below 1 hour. 5. Conclusion A modified version of the outer approximation/equality relaxation algorithm tailored to the optimization of distillation column feed and side draw tray positions is presented. One challenge of the implementation of this algorithm was to cope with numerical difficulties related to the large scale of the systems of equations that describe complete process plants. Though there is a number of well known MINLP schemes, e.g. OA, GBD, BB, ECP, etc., there are, to the best of the authors’ knowledge, no appropriate and robust implementations available. Hence, the implementation of MINLP schemes requires careful adaption to the hosting process simulation tool and problem setting. Experiences show a considerable benefit regarding the reduction of engineering time in process design as well as operating cost savings. While comparable commercial tools are not available on the market, Linde Engineering is achieving a significant competitive advantage by its proprietary development in the R OPTISIM framework. REFERENCES 1. M.A. Duran and I.E. Grossmann. An outer-approximation algorithm for a class of mixed integer nonlinear programs. Math. Prog., 36:307–339, 1986. 2. E. Eich-S¨ollner, P. Lory, P. Burr, and A. Kr¨oner. Stationary and dynamic flowsheeting
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4. 5.
6.
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in the chemical engineering industry. Surveys on Mathematics for Industry, 7:1–28, 1997. Beale E.M.L. and Tomlin J.A. Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In Lawrence J, editor, Proceedings of the Fifth International Conference on Operational Research, pages 447–54, London, 1970. Tavistock Publications. G.R. Kocis and I.E. Grossmann. Global optimization of nonconvex minlp problems in process synthesis. Ind. Engng. Chem. Res., 27:1407–1421, 1988. Jagadisan Viswanathan and Ignacio E. Grossmann. An alternate minlp model for finding the number of trays required for a specified separation objective. Computers chem. Engineering, 17(9):949–955, 1993. Jagadisan Viswanathan and Ignacio E. Grossmann. Optimal feed locations and number of trays for distillation columns with multiple feeds. Ind. Eng. Chem. Res., 32:2942–2949, 1993.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
A Chemical Process Design Framework Including Different Stages of Environmental, Health and Safety (EHS) Assessment Hirokazu Sugiyamaa, Ulrich Fischera, Masahiko Hiraob, Konrad Hungerbühlera a
Safety & Environmental Technology Group, Institute of Chemical and Bioengineering ETH Zürich, ETH-Hoenggerberg, 8093 Zurich, Switzerland b Department of Chemical System Engineering,The University of Tokyo, 7-3-1 Hono, Bunkyo-ku, 113-8656 Tokyo, Japan
Abstract In this work, we present a novel framework for the design of chemical processes. This framework includes four design stages and aims at a multiobjective decision-making at each stage using suitable environmental, health and safety (EHS) assessment methods in conjunction with technical and economic performance indicators. The different stages that have been defined comprise decisions on the synthesis route, the general process structure, the type of unit operations and the corresponding operating conditions. Suitable methods are selected for each design stage, to cover EHS aspects comprehensively in combination with technical and economic issues. The proposed framework is demonstrated on the design of methyl methacrylate (MMA) processes. By mimicking the framework, several reaction routes for this chemical are eliminated in a step-by-step procedure to obtain the most promising routes together with their optimal process structures and operating conditions. Keywords: chemical processes, stage-by-stage design, integrated approach, multiple objectives, MMA case study
1. Introduction Needs are growing to include environmental, health and safety (EHS) aspects in every decision-making throughout process development. In particular, decisions on reaction path, solvents, unit operations and operating conditions affect the EHS performance of the process significantly. Various methods to reflect such aspects of early stages into decision-makings have been proposed (e.g. Waste Reduction Algorithm by Hilaly and Sikdar, 1994; Environmental Hazard Index by Cave and Edwards, 1997; Inherent Safety Index by Heikkilä, 1999). Selection of these evaluation methods should be done appropriately to cover EHS aspects comprehensively at every design stage. Some authors distinguished different stages of process design, and applied one environmental evaluation method with different amount of available information in a two-step procedure (e.g. Hoffmann et al., 2001; Chen and Shonnard, 2004). We present a novel framework comprising four stages of chemical process design. Each stage is characterized by the available information as a basis for process modeling and assessment and the tasks that have to be solved. For each stage appropriate technical, EHS and economic assessment methods and indicators are selected. The concept is applied to different production processes, i.e. different synthesis, for methyl methacrylate (MMA).
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2. Proposed design framework 2.1. Definition of design stages We defined design stages, major decisions and activities involved at each stage as shown in Table 1. These stages are part of early design stages of grass root design. Product type, candidating synthesis routes and production scale should be determined before the first stage. In design stage Process Chemistry I, synthesis routes are screened on the basis of their ideal status i.e. 100% conversion. More information such as yield, auxiliaries, solvents are included in design stage Process Chemistry II. For selected routes, a process structure is generated and determined at design stage Conceptual Design I. The process structure is transferred to a flowsheet in design stage Conceptual Design II. A process simulator is used to specify unit operations and their equipment sizes. Final outcome of the framework is the optimized flowsheet, which is the input to the detailed engineering stages of design and process implementation. 2.2. Covered aspects and proposed indicators at each defined stage Table 1 includes the covered aspects of the process performance and assigned indicators for multiobjective evaluation at each design stage. 2.2.1. Economic performance Gross profit is used at the first three stages, and is replaced by net present value (NPV) in the last stage. Gross profit at the first stage is the potential maximum of one synthesis route. It is updated by including yield and extra purchase cost of solvent etc. at design stage Process Chemistry II, and further operating cost estimated from the process structure at design stage Conceptual Design I. NPV is suitable in the last stage where the investment cost can be calculated. Table 1: Suitable economic and EHS evaluation methods as well as appropriste technical indicators at early stages of process design. Design stage
Process Chemistry I
Process Chemistry II
Conceptual Design I
Conceptual Design II
Major decisions
Reaction route selection
Reaction route selection
Process structure
Unit operation, sizing
Activities involved
Consideration of ideal reaction performance
Consideration of yield, auxiliary, catalyst, solvent, byproduct
Consideration of recycling structure, separation scheme
Process simulation
Multiobjective evaluation indicators
Economic performance
Supplemental indicator
Gross profit (theoretical max)
Updated gross profit
Gross profit including operating cost
Gate-to-gate environmental Impacts
proxy by MLI
proxy by MLI
e.g. overall energy consumption, emissions
Upstream environmental Impacts
LCA in upstream (e.g. cumulative energy demand)
Updated LCA
Updated LCA
Substance EHS method
Updated EHS
Process EHS method (mass is included)
Updated
e.g. #Reaction steps
Overall yield reaction conditions (e.g. T, P) Technical difficulties (e.g. catalyst activity)
e.g. min #stage in separation
e.g. equipment size
Net present value
Cradle-to-gate LCA
EHS Hazard
Technical aspects
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2.2.2. Gate-to-gate environmental impacts This includes direct emissions or resource use of the process under design. Such environmental impacts are not possible to be quantified in the first two stages where the focus is on the reaction only. Mass Loss Indices (MLI, Heinzle et al., 1998), which are the amout of unwanted mass in the reactor outlet per product amount, can be a suitable proxy for covering this aspect. Later in Conceptual Design I/II, emissions or energy use of the process can be calculated. In the final stage, these direct impacts of the process are merged with upstream impacts, and cradle-to-gate impacts are calculated. 2.2.3. Upstream environmental impacts This aspect covers the environmental damages caused in the raw materials production. Life cycle assessment (LCA) methods are appropriate to cover this aspect, and are used throughout the stages. The inventory data of the upstream processes can be obtained from LCA databases e.g. ecoinvent (Frischknecht et al., 2004). Energy-related impact categories such as cumulative energy demand (CED, e.g. Verein Deutscher Ingenieure, 1997) can be important especially when bulk chemicals are involved. Under limited availability of LCA data which is often the case in specialty chemicals, semiquantitative methods such as ABC assessment (Heinzle et al., 1998) can be useful. 2.2.4. EHS hazard This is the process hazard towards environment, workers’ health and safety. A suitable method is EHS assessment method (Koller et al., 2000). This method, developed for early design stages, provides index scores of a substance in eleven EHS categories. Such index values of substances can be aggregated as EHS score of a process. Methyl methacrylate O
[Z1/L1] CO/CH3OH
[Z2]
Ethylene
HCHO O
[L2] Methylal O
CO/H2O [P1]
[F1]
CH3OH
O
[F 2] HC HO
CO/CH3OH [Q1] CO/CH3OH
[N 1] CO /H /H 2 F
O H3 C
O2
O
H
O
O
[N3] O2
O
O
[N2] H2O F
[A1/E1]
[A2]
OH
O
HO
O
4
NH2 2S
Figure 1:
[B2/C2/F3] O
HO
H
Isobutane
NH2 H2SO4
O2
H H 3O ]C [T2 HCOOCH
OH N
[E3]
3
2] [M
[M1/R1/S1/T1] NH3/O2
[B1/D1] O2
Isobutene
O
H2SO4
[E2]
A to F: Industrial L to N: post industrial P to T: post researched X to Z: currently researched
tert-buthyl alcohol
OH
OH N
HCN
[E 4]
O
Legend: Alphabet = route type Number = step order
CH3OH/O2 [D2]
Propylene
2] [Q
[A 3/ M 3]
[Y1]
Propyne
O
[B3 /C3 /F4/N 4/X2 ]
[P2 ] H CH O
OH
O2
CH 3 OH
O
CO/H2
O
[R2/S2] NH2
O
HCONH2
[R3]
HCN NH3+CO
O
[S3] CH3OH
[C1] O2
[X1]
O2
Various synthesis routes for the production of MMA; adapted from Nagai (2001).
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Throughout the stages EHS score is updated when new substances are introduced e.g. solvents. After design stage Process Chemistry I, process mass, e.g. the process inventory, is included to characterize the magnitude of the hazard. 2.2.5. Technical aspects Technical aspects of the process described by e.g. number of steps, overall yield, reactor conditions etc. are included in the framework. They provide additional information on the technical performance of a process and represent an important basis for decisionmaking.
3. Case study The developed framework was applied to the synthesis of MMA. Possible synthesis routes are shown in Figure 1. Starting from these 17 routes we mimicked the proposed design framework in a step-by-step procedure. 3.1. Design stages Process Chemistry I and II Figure 2 shows selected results of the multiobjective evaluation of these 17 routes at design stage Process Chemistry I. Here, gross profit, MLI, CED and EHS score are selected as indicators and 100% conversion is assumed (compare Table 1). The objectives are to obtain a maximum gross profit and minima for the other indicators, i.e. MLI, CED and EHS score. At this stage the MLI includes only coupled products, which are later updated with other unwanted outlets such as raw materials. For validating the use of CED as a proxy measure, high correlations were observed between scores of CED and other indicators e.g. eco-indicator 99 (Goedkoop and Spriensma, 1999) of chemicals involved in this study. The EHS score of a reaction is the sum of the maximum substance scores in each of the eleven EHS categories. Those synthesis routes that are indicated by bold dots in Figure 2 are Pareto optima with regard to the two objectives in each graph. As Pareto optima in all four dimensions routes C, N, P, Q, X, Y and Z are identified. In the following design stage, Process Chemistry II, the multiobjective evaluation is performed in a similar way. Now more information on the reaction is considered, i.e. reaction yield, auxiliaries, the choice of the catalyst and solvents and possible 2
10
M
MLI [kg/kg-MMA]
A 1
R
T
S B D
0.5
L E
C P Z
F
X
0
0.5
1
1.5
Gross proft [$/kg-MMA]
A M
8
7
X
E R
F
N
6
N
P T S Q B C D Y Z
L
Q Y
0
EHS [Index value]
9
1.5
5 2
0
50
100
150
CED (upstream) [MJ-eq/kg-MMA]
Figure 2. Multiobjective evaluation of MMA synthesis routes as obtained for design stage Process Chemistry I (assuming 100% conversion.
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8
Ammonium Bisulphate
7
Sulphuric Acid
Index value [-]
6
Dimethyl ether
5
Methyl Formate Methyl Methacrylate
4
Methacrylamide 3
Acetone Cyanohydrin
2
Hydrogen Cyanide Acetone
1
Methanol Accumulation Accumulation
Solid Solid Waste wastes
Air mediated Mediated Effects effects
Degradation Degradation
Water Mediated ,mediatedEffects effects
ChronicToxicity toxicity Chronic
Irritation Irritation
Reaction & on/Decomposition decomposition
AcuteToxicity toxicity Acute
Mobility Mobility
Fire & Fire/Explosion explosion
0
Figure 3: Detailed EHS assessment scores of MMA synthesis route A including auxiliaries, catalyst, solvent and byproduct as obtained for design stage Process Chemistry II .
the process alternatives. Considering the results obtained in the first two design stages, several synthesis routes show a high performance and should therefore be selected for further assessment on the basis of a more detailed modeling conducted in the following design stages. Figure 3 shows the EHS assessment result of route A as obtained from information that is available at the design stage Process Chemistry II. Substances which contribute to scores of each EHS category can be identified e.g. hydrogen cyanide and sulphuric acid in acute toxicity. Presentation of this detailed assessment result allows the identification of the problem sources and thus enables the transparency of the framework.
Recycle MAL Isobutene
First O2/air oxidation [C1]
Second oxidation [C2]
MAL, MAA recovery
Waste water MAA
Methanol
Esterification [C3]
MAA recovery
Product MMA
184 CED (cradle-to-gate) [MJ-eq/kg-MMA]
Purge
Recycle inert gas
Coal
182
Heavy oil 180
LNG
Pareto optima
178
176 20
Recycle methanol
Waste water
30
40
50
60
NPV [million $]
Figure 4: Process flow diagram of route C and its multiobjective evaluation including utility options.
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3.2. Design stages Conceptual Design I and II Route C was one of the promising routes found in the previous stages. Therefore we have performed a more detailed process simulation and multiobjective evaluation for this route. Figure 4 shows the process flow diagram of the modeled process together with its multiobjective evaluation using CED and NPV. Here CED covers cradle-to-gate range, namely the upstream processes, i.e. synthesis of raw materials and energy conversion processes, and the chemical production process itself. When different utility options are considered, process alternatives using liquefied natural gas (LNG) or heavy oil are found Pareto optimal in these objectives. These modeling and evaluation procedures have also to be applied to other synthesis routes for the production of MMA that showed a high performance in the design stage Process Chemistry II. On the basis of these results one route can then be selected for further detailed engineering.
4. Conclusions We presented a process design framework that includes different design stages with multiobjective decision-makings using suitable EHS methods along with technical and economic assessments at each stage. In the case study of MMA production, use of the proposed framework enabled the selection of those routes with the highest multiobjective performance. On the basis of these results the best synthesis route can be identified and optimal process structures can be created together with the optimal operating conditions.
Acknowledgement The authors would like to thank Prof. Dr. V. Hoffmann and Ms. E. Antonijuan at ETH Zurich for the valuable contributions to this work. Finantial supports from The Nagai Foundation Tokyo are deeply acknowledged.
References Cave, S. R., Edwards, D. W. (1997). Chemical process route selection based on assessment of inherent environmental hazard. Comput. Chem. Eng. 21, S965-S970. Chen, H., Shonnard, D. R. (2004). Systematic framework for environmentally conscious chemical process design: early and detailed design stages. Ind. Eng. Chem. Res. 43, 535-552. Frischknecht, R., et al. (2004). Overview and methodology, ecoinvent report No. 1, Swiss Center for Life Cycle Inventories, Data v.1.1, Duebendorf, Switzerland. Goedkoop, M. and Spriensma, R.(1999). The Eco-indicator 99: A damage oriented method for life cycle impact assessment, methodology report of PRé Consultants, Netherlands. Heikkilä, A. M. (1999). VTT Publication, Technical Research Centre of Finland, Espoo, Finland. Heinzle, E., et al. (1998). Ecological and economic objective functions for screening in integrated development of fine chemical processes. 1. Flexible and expandable framework using indices. Ind. Eng. Chem. Res. 37, 3395- 3407. Hilaly, AK. Sikdar, SK. (1995). Pollution balance method and the demonstration of its application to minimizing waste in a biochemical process. Ind. Eng. Chem. Res. 34, 20512059. Hoffmann, V. H., Hungerbühler. K., McRae, G.J. (2001). Multiobjective screening and evaluation of chemical process technologies. Ind. Eng. Chem. Res. 40, 4513-4524. Koller, G., Fischer, U., Hungerbühler, K. (2000). Assessing safety, health and environmental impact during early process development. Ind. Eng. Chem. Res. 39, 960-972. Nagai, K. (2001). New developments in the production of methyl methacrylate. Applied Catalysis. 221, 367- 377. Verein Deutscher Ingenieure. (1997). VDI-Richtlinie 4600: Cumulative energy demand, terms, definitions, methods of calculation, Düsseldorf.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Multi-objective reactor network synthesis for industrial mass transfer limited processes Filipe Neves,a Dulce Silva,a Nuno Oliveira,a Fernando Mendes b a
GEPSI-PSE Group, Department of Chemical Engineering - University of Coimbra, Pinhal de Marrocos - Pólo II, 3030-290 Coimbra, Portugal b Quimigal S.A - Química de Portugal, Quinta da Indústria - Beduído, 3860-680 Estarreja, Portugal
Abstract This work describes a methodology for the synthesis of networks of multiphasic reactors, specially suited for processes where complex mecanistic models and a large number of unit types are present. The key concept consists on the decomposition and iteration of the original problem between two different representation levels (corresponding to smaller and less nonlinear problems), to avoid some of the numerical difficulties associated with the original one. This strategy was sucessfully applied to the aniline production phase of Quimigal S.A., allowing a systematic retrofit of this industrial process. Keywords: Reactor networks, multi-objective optimization, triphasic systems.
1. Introduction Various systematic approaches for the synthesis of optimal reactor networks were proposed during the past decade. However, despite the significant contributions, it can be argued that the use of the existing formulations on highly complex practical problems is still hindered by significant restrictions. In [1] the problem of reactor network synthesis is addressed by generating a superstructure where all possible connections between different units are embedded, solving the resulting problem as a MINLP. A different approach, based on the theory of attainable regions, was introduced in [2]. These authors proposed a formulation where no recycle streams are needed, allowing therefore a sequential solution to the synthesis problem. In both of these works, all of the examples refer to homogeneous reaction systems. Because of this, only models of CSTR and plug flow units were considered, with the last ones solved either by the use of orthogonal collocation or approximated by a battery of CSTRs. More recent developments allow also the inclusion of temperature optimization in the synthesis problem, due to the importance of this parameter. Despite the significant number of contributions found in literature, there are actually few references that address multi-phase reaction systems. One of the exceptions is the work presented in [3]. This work deals with two-phase systems, and considers the nonisothermal synthesis problem. However, the examples presented only use two different kind of units (CSTR and plug flow), with the design decisions (besides heat exchange policy) restricted to feed policy and reaction volume. A more recent work [4] elucidates that, for highly complex reaction systems, there are several additional decisions that must be optimized. Accordingly to the authors, the mathematical complexity of the mechanistic models, required to express all the functional relations between state variables and decision variables, is typically very high. Therefore, the resulting models,
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when considered within an optimal network synthesis problem, tend to prevent the use of systematic approaches. The strategy adopted by these authors was to make some a priori decisions, e.g., unit types and partial network topology, decreasing the complexity of the synthesis problem to a level that could be handled. The obvious disadvantage of this approach is that such decisions may hinder, from the start, possible better solutions.
2. Difficulties of industrial multi-phase reactor network optimization The current work addresses the optimization of industrial reactor networks involving complex three-phase units. Besides the required large-scale non-ideal models, there are several aspects that contribute to the complexity of this problem: • Types of reaction units: For example, heterogeneous reactions can theoretically be conducted in different kinds of multi-phase reactors: trickle-bed, fluidized bed, slurry, etc., with more operational differences between them than just the flow pattern. • Unit decision variables: For each type of reaction unit available, there are several operational parameters (and not only volume and feed policy) that must be optimized (e.g., catalyst diameter and load, intensity of agitation). • Presence of additional units: In some cases, streams exchanged between reactors need to go through contactors/separators (also with complex behavior), in order to enhance the feasibility of the overall process. • Inequality constraints: These are needed to characterize the ranges of variation of the operational parameters and also to define the validity zones for the correlations used. When coupled with nonconvex process models, these bounds can introduce disconnected feasible operating regions. 2.1. Objective functions Considering the network synthesis problem in a single context – e.g., economical or conversion oriented – can be insufficient due to conflicting objectives: • Conversion / Reaction volume: When operating near complete conversion, in masstransfer limited processes, a small increase in the reaction yield might require an exponential increase of the reaction volume. • Existing / New technologies: This can be understood as a “fear factor”. Often, industry is only willing to trade current technologies (e.g., reactors type) by new ones (for which no operational experience is available) if the predicted gains are substantially large. • Flexibility / Operability: For the same global reaction volume, a larger number of smaller volume units offers better chances of minimizing production losses due to common plant failures; on the other hand, more complex control rooms, together with higher investment and labor costs are involved. • Complexity / Performance: Very often, the solutions obtained involve a large number of exchange streams that only contribute moderately to the network performance; reducing the number of connections between units can lead to simpler networks, at the cost of only minor performance losses. Hence a multi-objective formulation is more suitable to address the synthesis of practical reaction networks. Two approaches, usually considered for the solution of these problems can be used. A more mathematical approach builds a composite performance index that includes all objectives together with respective weighting factors in a single objective. The second approach, more analytical, tries to elucidate the
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tradeoffs between different conflicting objectives (mainly by graphic studies), leaving the choice of the preferred solution to the decision maker.
3. Proposed Strategy The strategy proposed in this work addresses the synthesis of isothermal networks. The key idea is to decompose the original problem and iterate between two different kinds of reduced subproblems, in order to avoid the numerical difficulties associated with the detailed formulation of the original synthesis problem (Figure 1). This is a common concept in optimization, since avoiding pure equation oriented strategies [5] or using reduced models [6] is sometimes useful, or even strictly necessary, to overcome numerical difficulties of current solvers in process synthesis problems.
Figure 1 – Simplified framework for the developed strategy.
Another important aspect of the developed strategy is that the network synthesis problem is kept as a NLP, avoiding the use of discrete decision variables. This is achieved by introducing additional constraints that drive certain flows to their bounds and smoothing functions to activate/eliminate the unit cost functions. A NLP formulation has the advantage of allowing the use of robust numerical implementations (e.g., IPOPT), that are better suited to handle the severe nonlinearities usually present in the mathematical formulation. At the same time, our personal experience with these problems indicates that adopting an interior point methodology improves, in certain cases, the quality of the optimal solutions found, by reducing the frequency of convergence to local solutions determined by variable bounds. This is similar to the numerical experience described in [7] - note that the structure of reactor network synthesis problems shares many of the difficulties found in the classical pooling problem.
4. Industrial process in study – Case Studies The previous strategy was used to perform optimization studies on the aniline production process, via liquid hydrogenation of nitrobenzene, implemented by
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Quimigal, S.A.. The process is a sum of mass transfer and reaction steps, and can be theoretically carried out in three-phase units – see Figure 2.
Figure 2 – Liquid hydrogenation of Nitrobenzene – microscopic and macroscopic view.
The catalyst used does not support high nitrobenzene concentrations, and, therefore, the first study consisted in identifying the potential benefits in eliminating this restriction. In other words, what would be the new arrangement for the existing units, and in what extent could it increase the network performance. Since a fixed number of reactors must be considered, the objective function used includes only two conflicting goals: the global production rate and the number of interconnecting streams required. The optimal configuration, obtained with the developed strategy, is presented in Figure 3; as can be seen, it involves the opposite arrangement (serial) of the existing one.
Figure 3 – Case Study I: actual (parallel) and optimal (serial) process configurations.
The reason can be found in the kinetic curve – see Figure 2; the maximum nitrobenzene concentration, allowed in the exit network stream, is very low. Working in a parallel
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configuration, all units must therefore fulfill this restriction and present very low reaction rates. In a serial configuration, the first three units can work with greater concentrations for which greater reaction rates are achieved. It is important to emphasize that, since we are dealing with an already existing network, the number of required changes (new interconnecting streams) should be minimal, since their introduction is somehow difficult and represents stopping periods (with production loss). For that reason, in the multi-objective function used, the penalty weight associated to these streams flowrates was large. Nevertheless, performing an additional number of runs, with lower penalty weights, showed that the interconnections increase could not enhance, significantly, the network performance. For the second study, the goal was to determine the number, type and operational parameters of additional units, necessary to allow a duplication of the actual production capacity. Existing units should be used and, contrarily to the previous study, the initial catalyst restriction should also be considered. A superstructure was generated (Kokossis & Floudas, 1990), embedding all units considered (existing and candidate ones). The performance of trickle and fluidized bed reactors was modeled considering batteries of slurry CSTR units, with modified mass transfer coefficients and reaction efficiency factors, accordingly to their respective models. It is our personal experience that this procedure reduces the numerical difficulties observed, when compared with the solution of the original DAE system, by means of orthogonal collocation. The results are shown in Figure 4, where exchange streams that were eliminated, in order to reduce network complexity, are also represented.
Figure 4 – Case Study II: effect of multiple objectives in the final solution.
It should be emphasized that, similar to the observed in the first study, the contribution of these streams was almost negligible – without considering them, the network overall efficiency is only decreased about 1%. Relatively to the type of the selected units, it should be mentioned that trickle-beds were not selected due to both internal and
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external mass transfer problems. The low diffusivity of the liquid mixture, in the solid catalyst, hinders any chances of profiting from the internal particle volume. Therefore, in a presence of a surface reaction, the greater particle diameter, in trickle-bed reactors, is a considerable disadvantage, since it implies larger catalyst loads. Also, fixed-bed configurations decrease the mass transfer at the liquid-solid interface, compared to fluidized and slurry reactors, where the more vigorous mixing is crucial to avoid external diffusional problems. It is important to discuss how the multi-objective functions affect the results obtained. Besides the conflicting goals already discussed (number of interconnections vs. production rates), the use of weighting factors also plays an important role, e.g., in the unit type selection. Fluidized units can be chosen due to an overall better performance, when compared to slurry units, because they require a smaller capital cost and have a simpler mechanical design. Nevertheless, due to the absence of existing operational experience regarding them (since they represent new technology), if the penalty weight given to this attribute is increased, we will observe their substitution by slurry units. Another important result, derived from the use of multiple objectives, is the number of new units selected – minimizing it will conflict with the network operational flexibility. In fact, a fluidized unit with the total volume of the two new ones, represented in Figure 4, would be less expensive. On the other hand, it would also cause greater production losses in a situation of shutdown and restart. With two units, two independent parallel lines can be established, reducing the effects of common operational problems.
5. Conclusions & Future Work The strategy developed managed to overcome typical numerical difficulties that arise when strongly non-linear large-scale models must be considered within a reactor network optimization problem. The decomposition of the original problem and the avoidance of discrete formulations are key aspects that, together with the use of multiobjective functions, allowed the study of a complex industrial three-phase process. In the first case, the results obtained confirm the importance of eliminating a current catalyst restriction; after accomplishing this, and depending on the new catalyst properties (especially in what refers to MNB tolerance), a new network configuration can be readily synthesized in order to accommodate a simple and easy retrofit. When the revamping of the current plant is considered, the results indicate that a different configuration should be used and, therefore, the actual parallel configuration abandoned. In this new network, the units (existing and new ones) can be operated to achieve a more favorable ratio between the production rate and the required reaction volume. The multi-objective functions used showed that the final structure can vary depending on the weights given to certain attributes, with direct influence on the network flexibility and on the type of units to use. Future work will deal with the inclusion of heat integration aspects, with special focus on the energy tradeoffs between the reaction and remaining stages.
References [1] A.C. Kokossis and C.A. Floudas, Chem. Eng. Sci., 45 (1990) 595. [2] A. Lakshmanan and L.T. Biegler, Ind. Eng. Chem. Res., 35 (1996) 1344. [3] V.L. Mehta and A.C. Kokossis, AIChE J., 46 (2000) 2256. [4] R. Diaconescu, R.Z. Tudose and S. Curteanu, Polym. – Plast. Techol. Eng., 41 (2002) 297. [5] D. Alkaya, S. Vasantharajan and L.T. Biegler, Ind. Eng. Chem. Res., 39 (2000) 1731. [6] H. Briesen and W. Marquardt, AIChE J., 50 (2004) 633. [7] M. B. Poku, L.T. Biegler and J.D. Kelly, Ind. Eng. Chem. Res., 43, (2004) 6803.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Synthesis of Separation Systems for Azeotropic Mixtures: Preferred Distillation Region Stanislaw K. Wasylkiewicz Aspen Technology, Inc., 900, 125 - 9th Avenue SE, Calgary, Alberta T2G 0P6, Canada
Abstract An algorithm for automatic generation of sequences of distillation columns and decanters for separation of azeotropic mixtures has been developed where distillation boundaries can be crossed by moving them with pressure change, by exploring curvatures of distillation boundaries or by liquid-liquid splits in decanters. Based on a broad knowledge of distillation regions and distillation boundaries for the separated mixture, open-loop sequences are generated and primary recycles are automatically detected. Then preferred distillation regions are identified and suitable recycle streams are selected. In this paper, we are focused on internal secondary recycles where species present in the sequence feed are introduced as separating agents. This type of recycles can simplify tremendously the whole sequence and reduce significantly the total cost of separation. In this paper, an example based on an industrial case is presented where the internal secondary recycle was automatically calculated during synthesis of column sequences. Keywords: azeotropic distillation, synthesis, recycles
1. Introduction A fundamental problem in synthesis of systems for separation of azeotropic mixtures is a fast and reliable determination of feasible products that can be attained in individual separators for a given feed composition. Contrary to zeotropic mixtures, not all separation schemes generated based on relative volatilities of components are feasible, as well as not all desired specifications can be met. These facts are usually discovered after extensive and lengthy simulation studies if only process simulators are used to solve design problems. This so far the most popular design-by-simulation approach requires several performance simulations for various design parameters selected usually by trial and error. It can be extremely time-consuming to find out that the specified separation is inherently infeasible. An algorithm for automatic generation of sequences of homogeneous as well as heterogeneous distillation columns for separation of azeotropic mixtures has been developed where distillation boundaries can be crossed by moving them with pressure change (Wasylkiewicz, 2004), by exploring curvatures of distillation boundaries or by liquid-liquid splits in decanters (Wasylkiewicz, 2005). In the systematic synthesis of the azeotropic separation schemes, first we generate open-loop sequences by systematic application of split generator to determine feasible products that can be attained in individual separators for specified feed compositions. Then we identify suitable recycling options. From the synthesis point of view, we distinguish two types of recycles: primary recycles and secondary recycles. The main goal of the primary recycles is to reduce the number of separation steps. Recycling should merge units that perform identical tasks. Our algorithm automatically detects primary recycle streams
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and their destinations, and calculates recycle mass balances. A secondary recycle stream is introduced in order to shift the total feed composition and by doing this change the functionality of a separation unit. In an external secondary recycle an external species is introduced as the separating agent. In an internal secondary recycle a species present in the sequence feed is introduced as the separating agent. It can be a pure component produced somewhere in the sequence or any other intermediate stream. In this paper, we are focused on internal secondary recycles. Based on a broad knowledge about distillation regions and boundaries for the separated mixture, a preferred distillation region can be identified and a suitable recycle stream can be selected. This type of recycles can simplify tremendously the whole sequence and reduce significantly the total cost of separation. In practical separation problems, we often do not need to separate the sequence stream to all pure components because quite frequently some of their mixtures can be recycled e.g. to a reactor. In such cases, internal secondary recycles can tremendously simplify the whole sequence and reduce significantly the total cost of separation. In this paper, we present in details an example based on an industrial case where the internal secondary recycles were automatically calculated during synthesis of column sequences.
2. Common Industrial Separation Problem In chemical industry, it is a common problem to identify possible routes of separation of a complex mixture into products (often individual pure components), waste streams and reactor recycles. The mixture quite often contains several dozen components, is highly non-ideal, with numerous azeotropes and distillation boundaries. It can take several months of trial and error to solve this problem using only simulator. Distil (2004) can help to solve this problem much faster, usually in a few days by providing the designer with all information about the system necessary to be able to make the best decisions during the separation sequence synthesis. The real industrial case (Wasylkiewicz, 2001) was solved in a few days. Half of the time was devoted to develop appropriate thermodynamic model by creation of pseudo-components, regression or estimation interaction parameters, checking thermodynamic consistency etc. Then several possible routes to separate twelve-component mixture into individual pure products, waste streams and reactor recycles were identified using automatic column sequencing and split generation features in Distil. Since we can not publish any proprietary information about this industrial case, for the example presented in this paper we selected the fivecomponent mixture (C5) of ethanol, chloroform, methanol, acetone and benzene to show how the internal secondary recycle can be automatically calculated in the synthesis of column sequences. The objective of the example is to identify possible routes of separation of the C5 mixture into the following streams: • Pure ethanol and pure benzene streams (complete recovery). • One or a few streams that can be recycled to the reactor (can not contain ethanol or benzene). In the final five-component mixture of the real industrial case, there was one purecomponent product, another pure component was a waste stream (could not be recycled) and the rest of components should be recycled to the reactor.
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3. Preferred Distillation Region Our systematic synthesis of azeotropic separation schemes is based on the rigorous calculation of distillation regions and distillation boundaries in the mixture. For the specified feed composition, we can precisely determine in which distillation region the feed is and find all feasible products that can be attained in individual separators for a feed belonging to a particular distillation region. If any of the products meets the separation goals we call such a region the preferred distillation region.
Figure 1. Singular points and distillation regions for the C5 mixture at 101 kPa.
At atmospheric pressure, Wilson-Ideal model predicts nine azeotropes in the C5 mixture. All singular points (azeotropes and pure components) and basic distillation regions are shown in Figure 1. One quaternary, two ternary and six binary azeotropes have been found. Singular points are displayed in order of increasing boiling temperatures. There are two unstable nodes, two stable nodes and ten saddles in the mixture. This gives rise to four basic distillation regions. The original feed to the separation system is in distillation region 2. Several feasible splits were calculated by Distil. The most interesting one is the indirect split where the heaviest singular point in the region (benzene, No. 14) can be completely separated as the bottom product from the rest of components (top product). A distillation column, which carries out this separation (Column 1 in Figure 2), fulfils one of the objectives of our synthesis problem - completely recovery of benzene from the original feed. That is why we identify region 2 as the preferred distillation region. Top product from Column 1 (Stream 3 in Figure 4) is practically benzene free. For this four-component (C4) sub mixture four distillation regions have been found and several feasible splits for Stream 3 have been calculated by Distil. Unfortunately, none of them fulfills our next objective in the synthesis - complete recovery of ethanol from the C4 mixture. Stream 3 is not in preferred distillation region. However, there are two other preferred distillation regions that contain ethanol as the stable node and a simple distillation column placed in this region can produce ethanol in the bottoms. To obtain the overall feed composition in one of these regions, we have to add a recycle stream to the Stream 3. This calculation intensive process of selecting proper recycle stream, checking feasibility of the split and arranging another column that produces enough
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recycle stream has been done automatically by Distil. The resulting column sequence is shown in Figure 2 and parameters of all the streams in Figure 4. Mass balances for all three columns and the recycle are shown in a projection of the C5 composition space into 3D space (without benzene) in Figure 3.
Figure 2. Sequence of columns for complete recovery of benzene and ethanol from C5 mixture.
Figure 3. Mass balances for all columns in the sequence and the recycle. Projection of the C5 composition space into 3D space.
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Figure 4. Parameters and compositions of the feed, intermediate streams and products for the sequence of columns for complete recovery of benzene and ethanol from C5 mixture.
4. Algorithm for Calculation of Internal Secondary Recycles As prerequisites, components that must be fully recovered from the original mixture have to be specified (e.g. benzene and ethanol in the presented example). Then during automatic generation of sequences of distillation columns (Distil, 2004; Wasylkiewicz, 2005), program takes one of the products from recently added distillation column, that is not a sequence product, and tries to find a feasible split that can accomplish any from the specified separation goals. If such split was found a new distillation column is added to the sequence (e.g. Column 1 in the presented example). If not (e.g. for Stream S3), program tries to find internal secondary recycle sequence in the following steps: 1. Find distillation region the stream is in at selected pressure. 2. If this is not a preferred distillation region, look if there is another distillation region (the stream is not in) that could be suitable for complete recovery of any component that must be fully recovered, e.g. ethanol from Stream S3. 3. If the preferred distillation region was found, analyze all singular points of this region for possible recycle streams. In the example, ethanol is the heaviest
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4.
5.
component in the preferred distillation region (stable node) and as a result a complete recovery of ethanol can be achieved at column bottom. After the recycle candidate was selected, e.g. pure methanol (Stream S7), calculate minimum recycle ratio to accomplish full recovery of the selected component. First find Stream S6 on composition boundary that will fulfill mass balance around the sequence of Column 2 and Column 3 and provide full recovery of ethanol. Then find Stream S4 that must be in the same distillation region as the stream S5 to make Column 2 feasible. To minimize the recycle ratio the stream S4 is usually chosen close to the distillation boundary but inside the preferred distillation region. Notice also that the overall feed of Column 2 obtained by mixing Stream 7 (recycle stream) with Stream 3 does not need to belong to the preferred distillation region if the distillation boundaries are curved enough. Now look for a pressure, at which a distillation column that separates Stream S4 into Stream S6 and Stream S7 will be feasible by analyzing distillation regions and distillation boundaries for C3 mixture that does not contain the component already completely recovered in Column 2 (ethanol).
Pressure analysis of azeotropes is an important part of Distil (2004). That is why we used different pressure in Column 3 to produce enough methanol for the recycle. At 101.3 kPa Column 3 would not be able to provide appropriate flow rate of Stream 7 as distillation boundary at this pressure is too close to the feed (Stream 4). By changing operating pressure in Column 3 to 1 kPa we moved the distillation boundary away from the methanol vertex what allowed us to obtain higher flow of bottom product (Stream 7). In fact any type of separation could be used instead of distillation Column 3 (e.g. membrane separation) as long as it produces enough methanol for the recycle.
5. Conclusions A new algorithm has been developed for automatic synthesis of sequences of distillation columns with recycle streams even in cases when intermediate streams are on distillation boundaries. Feasible splits for azeotropic mixtures are rigorously and efficiently calculated based on information about all azeotropes in the mixture, distillation regions and distillation boundaries that are generated using adjacency and reachability matrices. The new algorithm facilitates efficient crossing distillation boundaries and quick finding preferred distillation regions for particular separation tasks. It can be used for mixtures of any number of components. This type of internal secondary recycles can significantly simplify the whole sequence and reduce drastically the total cost of separation.
References Distil, 2004, v 6.2 software, Aspen Technology, Inc., http://www.aspentech.com S.K. Wasylkiewicz, 2001, Case Study, Use of Column Sequencing and Split Generation Features in Distil, Calgary, August 2001 S.K. Wasylkiewicz, 2004, Advances in Synthesis of Continuous Separation Sequences for Azeotropic Mixtures, 54th Canadian Chemical Engineering Conference, paper No. 243, Calgary, Canada, October 2004 S.K. Wasylkiewicz, 2005, Crossing Distillation Boundaries in Synthesis of Separation Sequences for Azeotropic Mixtures, AIChE Spring National Meeting, paper No. 83e, Atlanta, GA, USA, April 2005
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Modeling and Designing Powder Mixing Processes Utilizing Compartment Modeling Patricia M. Portillo, Fernando J. Muzzio, Marianthi G. Ierapetritou Department of Chemical and Biochemical Engineering Rutgers University, Piscataway, NJ 08854,USA
Abstract Powder mixing has been the subject of research for a number of years due to its importance in a variety of industrial sectors including pharmaceuticals, food and consumer goods manufacturing. Although a number of different models have been proposed in the literature, most of them are either empirical or require computationally intensive calculations that make them difficult to implement for realistic systems. The aim of this paper is to investigate the characteristics of powder mixing utilizing a simplified framework based on compartment modeling that efficiently and accurately capture the system behavior. A V-blender is used as a case study in order to illustrate the applicability of the proposed approach in characterizing mixing, whereas a continuous mixer is modeled in order to investigate the effect of different control strategies. Keywords: powder mixing, mixing models, compartment modeling, mixing process control design
1. Introduction Many industrial sectors rely heavily on granular mixing to manufacture a number of different products. Pharmaceutical industries are one of the most representative examples where homogeneity is very important to ensure product quality and compliance with strict regulations. In order to improve mixing process design, modeling is of great importance since it is used to improve the mixing performance, reduce manufacturing cost, and ensure product quality. The main difficulty of modeling powder-mixing processes is that granular materials are complex substances that cannot be characterized as liquids or solids (Jaeger and Nagel, 1992). Granular mixing can be described by multiple mixing regimes due to convection, dispersion, and shear (Lacey, 1954). As reviewed by (Fan et al., 1970) a number of publications exist, where powder mixing is modeled in an attempt to improve mixing efficiency and thus reduce production cost and improve product quality. The existing approaches can be categorized in heuristic based methods, approaches that are based on kinetic theory, particle dynamic simulations, and Monte Carlo simulations (Ottino and Khakhar, 2000). The main limitations however of most of the existing methods that restrict their applicability, is that they are either computationally very expensive or they do not include sufficient information of the physical system. As a result, the focus of this paper is to develop an efficient representation of powder mixing process based on compartment modeling.
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2. Compartment Modeling Compartment modeling has mainly been utilized to model liquid mixing in bioreactors, in order to examine the circulation of aerated flow under the influence of different mixers (Cui et al., 1996;Vrábel et al., 1999). In this work, we apply compartment modeling to solid mixers by spatially discretizing the system into a number of subsections that are assumed to be perfectly mixed and contain a stipulated number of particles. Discretizing the time domain, a number of particles are allowed to flow from each compartment to the neighboring ones at each time step. This is referenced as the particle flux, F as shown in Figure 1b. Following the ideas of Fan et al. (1970) that described solid mixing as a random process, the particles selected to leave and enter each compartment are randomly selected following a probability distribution. The number of particles transferred account for the mixing occurring throughout the vessel. The change in the number of particles of species j, in compartment i at time step k is denoted as Δφijk . Thus, the change in each specie j throughout all compartments at every time step must equal zero as dictated by equation (1): w
∑ Δφ i =1
ijk
= 0 (1)
where w are the number of compartments used to represent the entire mixer. The systems studied consist of two groups of particles associated with different physical attributes such as chemical composition, size, and color. For the present analysis, a specific concentration is attributed to each particle group, such that the number of particles within each compartment determines the mixture concentration. In our study, we consider a binary process where one group of particles belong to group 1 with a concentration of 1 and another group of particles belong to group 2 with a concentration of 2. The mixing behavior is characterized using the variance, σ2 as shown in equation (2): 2 n (x − x ) σ2 =∑ i (2) n −1 i =1 where xi is the concentration of sample i; x is the mean of sample concentrations; and n is the number of samples. We also use the relative standard deviation (RSD), defined as the square root of equation (2) (σ).
3. Compartment Comparison
Modeling
and
Discrete
Element
Method
(DEM)
Although DEM has been proven to simulate accurately real mixing processes, one of its main disadvantages is the computational complexity associated with realistic process simulations (Wightman et al., 1998). In this section, a comparison between compartment and DEM modeling is performed to illustrate the effectiveness of compartment modeling in terms of the required computational complexity. A discrete element model analysis was performed by Wightman et al. (1998) examining a horizontal cylindrical vessel undergoing rotational motion. The study consisted of modeling two types of particles, red and blue initially loaded with side-by-side loading, one half of the cylinder filled with red particles and the other half with blue particles. In our study we modeled the horizontal cylindrical vessel using a 16-compartment-model. The compartment modeling approach does not have computational restrictions, as
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Fraction of Red Particles
shown in Figure 1, and as a result the fraction of red particles as a function of axial length was examined at four different time periods. The graph shows that the compartment models capture the particle compositional behavior as a function of axial length with the exception that the compartment simulation required 2,084 CPU seconds on a Sun Sparc 900 MHz Processor 2GB and the DEM simulation performed by Wightman et al. (1998) required about 48 h of CPU time for every 1 s of real time simulated on a Sun Sparc 20.6 Workstation. The significant computational savings serve as a motivating example to implement this model as a predictive tool that can be utilized for on-line production. 1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 1 Time 1
2
3 Time 2
Axial4Length5 Time 3
6
7
Time 4
8 DEM
Figure 1 A compartment model red particle fraction results from a discrete time step 1 through 4 in comparison to a DEM simulation at one time point for a horizontal tumbling cylinder.
4. Case Studies 4.1. Case Study 1: V-blender Compartment modeling can be generally applied to model any mixing process. In this paper, a V-blender is considered as an illustrative case study (Figure 2a). The VBlender rotates around the x-axis from the upright position to the downward position, as described in detail in Brone et al. (1998). Following the experimental observations of Brone et al. (1998) this blender can be modeled considering 5 compartments as shown in Figures 2a, and 2b. Each compartment contains 200,000 particles, for a total of one million particles. In this study compartments V1, V2, and half of V3 have particles pertaining to group 1 and compartments V4, V5, and half of V3 have particles pertaining to group 2. The flux magnitudes of F1 and F4 have significantly larger values to represent higher mixing rates at this mixer location compared to fluxes F2 and F3 that represent the center of the mixer.
F1 V1 (a)
F2 V2
F3 V3
F4 V4
V5
(b) Figure 2a) Discretized V-blender 2b) Compartment model of V-blender
Using this model, we first investigate the effects of sample sizes in characterizing mixing. Since most analytical methods, such as near infrared (NIR) spectrometry, use a small number of particles in each sample (Berntsson et al., 2002), we explore cases involving small sample sizes. In particular the following cases are analyzed: 2 particles
P.M. Portillo et al.
1042
per sample, 4 particles per sample, 8 particles per sample, and 16 particles per sample. The number of samples taken is adjusted to keep a constant number of particles retrieved within one time step. The variance profiles shown in Figure 3a illustrate that sampling an identical system with a smaller number of samples results in a larger variance than if sampled with larger sample sizes. More importantly what should be noticed is that depending on the sample sizes, the variance of the system can be either overestimated or underestimated. This is especially important given that in order to satisfy the existing manufacturing criteria (CGMP’s, 2003), the Relative Standard Deviation (RSD) should be less than 4.0% to be readily passing and less than 6.0% to be marginally passing. Many of the samples in Figure 3a will not satisfy even marginally passing. Since the number of samples is often minimized due to the invasive nature of sampling, it is important to consider the effects of a small number of samples. Thus, the following cases are considered for a constant number of particles: 5 samples of 4,000 particles per sample, 10 samples of 2,000 particles per sample, 20 samples of 1,000 particles per sample, and 40 samples of 500 particles per sample. The histograms evaluated for the time period between 15,000 and 20,000 time steps (Figure 3b) show that a small number of samples are not sufficient to represent the normal distribution. The simulation results emphasizes what is known from experimental experience that a small number of samples may not sufficiently represent the homogeneity of the system as also reported by Fan et al. (1970), and although sampling is invasive limiting the samples can result with erroneous results. Another important consideration of sampling is the sampling location. As pointed out by Allen (1981), the two “golden rules” of powder sampling are that: (1) a powder is sampled only when in motion and (2) a sample is collected uniformly from the entire process stream. Due to physical limitations, samples are not always taken uniformly throughout the vessel. Thus, we examine the variance effects of sampling at different locations. We examine the three different sampling schemes, described as follows: Scheme A retrieves samples from compartment V1 and V3. Scheme B retrieves two samples one at each end of the vessel corresponding to compartments V1 and V5, and scheme C uniformly retrieves samples from each compartment V1 through V5. Following the guidelines of Allen (1981), we consider the most accurate sampling scheme to be the uniform sampling scheme (Scheme C) where samples are retrieved from each compartment. In order to monitor the accuracy among the sampling schemes we consider the following function J: n
J= ∑ k =1
(σ 0 (k) 2 -σ i (k) 2 ) 2 n
(3)
that is defined as the sum of the squared difference between the variance for the uniform sampling scheme (Scheme C), σo, and the variance of the sampling scheme used (i), σi at different time step k; for n time points. Here the uniform sampling scheme (Scheme C) is used as the standard for the variance distribution. The variance difference between the standard and the sampling scheme chosen indicates the variance error. The smaller the variance error, the closer the sampling scheme represents the results of the uniform sampling scheme. The results from this measurement show that scheme A has the largest variance error, 0.6382 and scheme B has a much smaller error values of 0.0349. If the aim is to characterize the variance of the system as accurately as possible, the goal
Modeling and Designing Powder Mixing Processes Utilizing Compartment
1043
is to minimize J and scheme B is favored. Given that there exists a large number of sampling alternatives compartment models offer an effective way of selecting the best sampling location for a specific system. 4500
0.25 10,000 samples, 2 particles per sample 5,000 samples, 4 particles per sample 2,500 samples, 8 particles per sample 1,250 samples, 16 particles per sample
Variance Frequ e
Variance Frequency
0.15 Variance
Variance
0.2
0.1
3500 3000 2500 2000 1500 1000
0.05
500
0
(a) 0
1000
2000
3000
4000
5000 Time St ep
6000
7000
8000
9000
5 samples 10 samples 20 samples 40 samples
4000
10000
(b)
0 0.0001
0.0003
0.0005
Time
0.0007
0.0009
0.0011
0.0013
0.0015
Variance Variance
Figure 3a) Variance as a function of time for extremely small particle samples 3b) Variance frequency histogram for four samples
4.2. Case Study 2: Continuous Mixer The use of continuous mixers increases the control flexibility since the additional parameters involved such as fill level, and recycle stream rate can improve the homogeneity of the mixture. However, the effect of these parameters on granular materials is not well understood. As an attempt towards better understanding of the continuous mixing process, a mixer with a moving blade is modeled in this section using compartment modeling. Four compartments are considered as shown in Figure 4a. We use 1 million particles in our study, and distribute them evenly between the different compartments. Compartments 1 and 3 initially have particles only pertaining to group 1, whereas compartments 2 and 4 have particles pertaining to group 2. We assume that the particle fluxes between each compartment are the same, in order to examine the effects of different control strategies independent of different mixing regimes. y3
y1
y2
1
2
3
4 (a)
1
2
1
2
1
2
3
4
3
4
3
4
(b)
(c)
(d)
Figure 4 a) Compartment model b) Compartment model with 3 recycle stream possibilities c) Optimal recycle stream d) Compartment model valve control
The first control strategy we investigate is a recycle stream since it helps mixing by recirculating the granular material throughout the mixer. In order to optimize the effects of the recycle stream addition, we optimize the recycle stream location. This is achieved by formulating and solving an optimization problem with the objective to minimize the RSD in compartment 4 using binary variables (y1, y2, y3) to represent the addition of recycle stream to compartments 4, 2, and 3. All recycle stream possibilities are shown in Figure 4b. In order to illustrate the effect of the different alternatives, the RSD is plotted for all different schemes in Figure 5a. The optimal solution corresponds to y1=1
P.M. Portillo et al.
1044
and y2=y3=0. The compartment model representing this scheme is illustrated in Figure 4c. The second control scheme considered corresponds to the addition of a valve in the exit stream so that the material is held within a mixer for a finite time, allowing the mixture to mix for a longer period of time. As shown in Figure 5b, the valve control scheme does decrease the initial RSD in comparison to a non-control scenario. 0.35 0.35
No Recycle Recycle Stream 1 Recycle Stream 2 Recycle Stream 3
0.3
.
No Valve Valve Control
0.3 0.25 0.2
0.2
RSD
RSD
0.25
0.15
0.1
0.1
0.05
0
0.15
0.05 0
(a)
-0.05
0
100
200
300
400
500
600
700
800
900
1000
(b) 0
100
200
300
400
500
600
700
800
900
1000
Time
Time Figure 5 RSD versus time a) for all recycle stream possibilities b) for valve control
5. Summary In this paper powder mixing is simulated using compartment modeling. The main advantage of the proposed approach is that the computational cost is low, allowing the simulation of a large number of particles. As shown in the case studies, the proposed approach is efficiently utilized to capture granular mixing for both batch and continuous processes and used to investigate the effects of different control strategies on granular mixing. Work is currently under progress in order to expand the solution approach for different control strategies to optimize mixing performance. A hybrid compartment modeling-discrete element method (DEM) scheme that will allow the detailed modeling of complex geometries is also under development.
References Jaeger H.M, Nagel S., 1992, “Physics of the Granular State”, Science, 256, 20, 1523-1531. Lacey P.M., 1954, “Developments in the theory of particle mixing”, J. Appl. Chem. , 4, 257-268. Fan L.T., Chen S.J., Watson C.A. , 1970,,“Solids Mixing”, Ind. and Eng. Chem., 62,7,53. Ottino J. M., Khakhar D. V., 2000, “Mixing and Segregation of Granular Material”, Ann. Rev. of Fluid Mech., 32, 55-91. Cui Y., Van der lans R., Noorman H., and Luyben K., 1996, “Compartment mixing model for stirred reactors with multiple impellers”, Trans IchemE, 74, 261-271. Vrábel P, Van der lans R., Cui Y., Luyben K., 1999, “Compartment model approach: Mixing in large scale aerated reactors with multiple impellers”, Trans IchemE, 77, 291-302.
Wightman C., Moakher M., Muzzio F. J., 1998, “Simulation of Flow and Mixing of Particles rotating and rocking cylinder”, AICHE, 44, 6,1266-1276.
Brone D., Alexander A., Muzzio F. J., 1998, “Qualitative Characterization of Mixing of Dry Powders in V-Blenders”, AICHE, 44, 271-278. Berntsson O., Danielsson L.-G., Lagerholm B., Folestad S., 2002, “Quantitative in-line monitoring of powder blending by near infrared reflection spectroscopy”, Powd. Tech., 123,185-193. Guidance for Industry, 2003, Powder Blends and Finished Dosage Units – Stratified In-Process Dosage Unit Sampling and Assessment, Pharmaceutical CGMP’s. Allen, T., 1981 Particle Size Measurement, 3rd ed., Chapman & Hall, London.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Design and control of homogeneous and heterogeneous reactive distillation for ethyl acetate process Hao-Yeh Leea, Hsiao-Ping Huanga*, and I-Lung Chienb a
Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan. Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan.
b
Abstract This paper compares conceptual design and control of two reactive distillation (RD) processes for the production of ethyl acetate (EtAc) using homogeneous and heterogeneous catalysts. The homogeneous catalytic RD process has higher capital cost but less energy cost and less concern about the catalyst installation and frequent replacement. The heterogeneous catalytic process has faster reactions in both forward and reverse directions than those of the homogeneous one. Thus, it needs less reactive hold-up and less number of trays to achieve the product specifications. The column composition profiles are with quite different form for these two cases. As for the control of these two catalytic RD processes, they also have quite different dynamic behaviours. From our study, heterogeneous catalytic process has faster closed-loop response and lower steady-state offset in final product purity in the face of throughput and feed composition disturbances than the homogeneous ones. Keywords: Reactive Distillation, Heterogeneous Catalyst, Homogeneous Catalyst, TAC
1. Introduction Esters are of great importance to chemical process industries. Among them, EtAc is an important organic solvent widely used in the production of varnishes, ink, synthetic resins, and adhesive agents. This ester is typically produced from the reaction of acetic acid(HAc) and ethanol(EtOH). In literature, Keyes[1] first reported the study of an ethyl acetate process using a reactive distillation column in combination with a pre-esterification reactor, two recovery columns, and a decanter. Later, the progresses on the steady-state simulation for the RD column were studied by Chang and Seader[2] using homotopy-continuation method, and by Simandl and Svrcek[3] using insideoutside method, and by Alejski and Duprat[4] formulating dynamic mathematical model with experimental validation. Bock et al.[5] then presented an uncatalyzed RD column with excess EtOH and a pressurized recovery column. Vora and Daoutidis[6] studied the operation and control of a single RD column, but the top product is not pure enough. The resulting designs from the above both works aforementioned are not economical, because the outlet streams are out of specifications and need to treat further. Tang et al.[7] studied the process which contains sulfuric acid as catalyst using an RD column with an overhead decanter and a stripping column. Highly pure EtAc product was obtained and all the outlet streams meet product and environmental specifications. Tang et. al. [8] provides generalization for the study of design of reactive distillation with ____________________ *Corresponding author. H. P. Huang, Tel: 886-2-2363-8999; Fax: 886-2-2362-3935., E-mail: [email protected]
1045
H.-Y. Lee et al.
1046
acetic acid as one feed. In that paper for EtAc process, Purolite CT179 as heterogeneous catalyst was used. The objective of this paper is to compare the process design and control dynamics of these two kinds of reaction distillation design with homogenous catalyst in [7] and heterogeneous catalyst in [8].
2. Phase equilibria and reaction kinetics To account for non-ideal vapor-liquid equilibrium and possible vapor-liquidliquid equilibrium for this quaternary system, the NRTL[8] model is used for activity coefficients. Because the processes are to be operated under atmospheric pressure, the vapor phase non-ideality considered is the dimerization of acetic acid as described by the Hayden-O’Conell[9] second virial coefficient model. The NRTL coefficients are given by Tang et al.[7] For the EtAc quaternary system, the models predict three minimum boiling binary azeotrpes and one minimum boiling ternary azeotrope. With the data computed for azeotropic systems encountered in this esterification process, the residual curve map (RCM) diagrams of EtAc-EtOH-H2O is plotted and given in Fig.1. In this diagram, significant liquid-liquid (LL) envelope is observed. The ternary minimum boiling azeotrope lies closely on the edge of LL envelope. In this system, the tie lines slop toward pure water node, consequently, relatively pure water can be recovered from the LL separation in this esterification process. EtAc ( 77.20 oC )
0.8
Decanter Compositions Top and Bottom Compositions of Stripper
0.6
70.37 oC
70.09 oC
0.2
0.4
71.81 oC
78.18 oC 0.2
EtOH ( 78.31 oC )
0.4
0.6
0.8
H2O ( 100.02 oC )
Fig. 1. RCM diagram for EtOH-EtAc-H2O system Table 1 Kinetic equations for homogeneous and heterogeneous catalyst EtAc systems. System (Catalyst) (i) EtAc (sulfuric acid) (ii) EtAc (Purolite CT179) *
Kinetic model r = k1CHAcCEtOH – (k1/Kc)CEtAcCH
2
O
k1 =1000×(4.195Ck + 0.08815) exp(– 6500.1/T ) Kc = 7.558 – 0.012T Pseudo-homogeneous model r = mcat (k1 x1.5 HAc xEtOH − k−1 xEtAc xH 2 O )
k1 = 4.24×10 exp(– 48300/RT ) k-1 = 4.55×105exp(– 66200/RT ) 3
k1 (T=363K)
Keq (T=363K)
1.1568×10-5 [kmol/m3·s]
3.2
4.78×10-4 [kmol/(kgcat·s)]
3.50
R=8.314[kJ/kmol/K], T[K], r[kmol/s], mcat[kgcat], Ci[kmol/m3], xi[mole fraction], Ck=[vol%]. (i) Alejski and Duprat[4], (ii) Hangx et al.[11] The chemical reaction kinetic model with sulfuric acid as homogeneous catalyst is adopted from the paper[4]. The sulfuric acid concentration is assumed to be 0.4 vol%. Since this catalyst concentration is quite low, it can be neglected in the vaporliquid equilibrium calculation. The solid catalyst in use is the acidic ion-exchange resin
Design and Control of Homogeneous and Heterogeneous Reactive Distillation
1047
Purolite CT179 in the pseudo-homogeneous model for EtAc. The reaction rates are expressed in Table 1. Notice that, in the heterogeneous catalyst system, the kinetics is catalyst weight-based. In applying the reaction kinetics to a reactive distillation, it is assumed that the solid catalyst occupies 50% of the tray holdup volume and a catalyst density of 770 kg/m3 is used to convert the volume into catalyst weight (mcat).
3. Steady state design and discussion In a previous study[7][8], It was found that, in this type of process, the reactive section should be extended to the column base of the RD column and, therefore, a much larger holdup is expected in the bottom of the RD column. In this work, the column base holdup is taken to be 10 times of the tray holdup. Based on the similar process flow sheet, the optimal steady-state design is proceeded by systematic procedures. In this study, the feed composition of EtOH is 0.87 mole fraction slightly less than their azeotrope compositions. The feed of acetic acid is 0.95 mole fraction in this system. The specifications include: 50 kmol/hr of EtAc product (99 mole%) in the bottom of the stripper accompanied by less than 0.01 mole% HAc impurity. In the search for the optimal designs for these two cases, all the simulations are carried out using ASPEN PLUS with the RADFRAC module provided with FORTRAN subroutines for the reaction rates. For a system with a given production rate with product specifications, the design steps are: (1) Set the reactants feed ratio to 1 initially (i.e., FR = FHAc /FEtOH= 1). (2) Fix the number of reactive trays (Nrxn). (3) Place the heavy reactant feed (NFHAc) on the top of the reactive zone and introduce the light reactant feed (NFEtOH) on the lowest tray of the reactive zone. (4) Guess the tray numbers in the rectifying section (NR) and the stripping section (NS). (5) Change the organic reflux flow (R) and stripper reboiler duty (QR, S) until the product specifications are met. (6) Go back to (4) and change NR and NS until the total annual cost (TAC) is minimized. (7) Go back to (3) and find the feed locations (NFHAc & NFEtOH) until the TAC is minimized. (8) Go back to (2) and vary Nrxn until the TAC is minimized. (9) Go back to (1) and change the feed ratio (FR) until the TAC is minimized.
The TAC of the following is used to evaluate for the optimal design. TAC = operating cost+(capital cost/payback year)
(1)
Where, the operating cost includes the costs of steam, cooling water, and catalyst, and the capital cost covers the cost of the column, trays, and heat exchangers. The results of the two catalyst systems are given in Table 2. The developed process flow sheets are given in Fig. 2(a) and Fig. 2(b) as well. (a) 1 12 HAc Feed 50.8 kmol/hr 95.0 mol%
Cooling Water
Organic Reflux Decanter
Feed to Stripper 230.59 kmol/hr
10
Steam
Steam
Heat duty = 4300.77 KW
Heat duty = 2058.66 KW
EtOH feed 57.472 kmol/hr 87.0 mol%
10
50.8 kmol/hr 95.0 mol%
EtOH feed 57.472 kmol/hr 87.0 mol%
Cooling Water
Aqueous Product 60.45 kmol/hr
Duty = -881.45 KW
Decanter Feed to Stripper 221.54 kmol/hr
1
HAc = 2.25e-03 mol% EtOH = 5.68 mol% EtAc = 71.48 mol% Water = 22.84 mol%
19 HAc Feed
Product 47.83 kmol/hr HAc = 0.01 mol% EtOH = 0.90 mol% EtAc =99.00 mol% Water = 0.09 mol%
Condenser Duty = -1668.73 KW
390.87 kmol/hr
RD Column
9 55
1 mol/s HAc = 85.5 mol% EtOH = 3.2 mol% EtAc = 3.6 mol% Water = 7.7 mol%
HAc = 1.0072e-03 mol EtOH = 2.17 mol% EtAc = 1.50 mol% Water = 96.33 mol%
Condenser Duty= -4789.92 KW
Organic Reflux
Stripper
54
Bottom recycle
1
HAc = 2.16e-03 mol% EtOH = 5.67 mol% EtAc = 71.52 mol% Water = 22.82 mol%
37
1
Aqueous Product 60.44 kmol/hr
-786 kW
315.57 kmol/hr
RD Column
(b)
-1743.08 kW Condenser
-4084.51 kW Condenser
HAc =1.05e-03 mol% EtOH = 2.18 mol% EtAc = 1.50 mol% Water = 96.32 mol%
Stripper 9
20
Steam
Steam
Heat duty = 5113.40 KW
Heat duty = 1979.01 KW
10
Product 41.61 kmol/hr HAc = 1.00e-2 mol% EtOH = 9.12e-1 mol% EtAc = 99.00 mol% Water = 7.88e-2 mol%
Fig. 2. The process flow sheet of EtAc process (a) homogeneous, and (b) heterogeneous systems
H.-Y. Lee et al.
1048
Comparisons of the results with these two EtAc systems, the configuration of process flow sheet is basically the same except for the homogeneous case needs a recycle stream to circulate the sulfuric acid in the RD column. From Table 2, the homogeneous and heterogeneous systems have almost the same stages of rectifying section and stripper. However, homogeneous system has larger capital cost in the reactive zone because of larger residence time is needed due to lower reaction rate. Although with larger capital cost in homogeneous system, the energy cost of this case is 11% less than the heterogeneous one. Another point is that only very few sulfuric acid is needed, thus the catalyst cost can be neglected in the homogeneous system. As the result in Table 2, the final TAC of homogeneous system is a little less than heterogeneous one. Table 2. The design results of EtAc process with homogeneous and heterogeneous catalysts System
Homogeneous
Heterogeneous
Column configuration
RD
Stripper
RD
Stripper
Total No. of trays including the reboiler
55
10
20
10
No. of trays in reactive section (Nrxn)
44
No. of trays in rectifying Section (NR)
11
Column diameter (m)
3
1.313
2.04
1.37
0.1524
0.0508
0.1016
0.0508
Weir height (m)
11 9
Decanter temperature (°C)
40
40
Total capital cost ($1000)
829.25
700.3
Catalyst cost($1000/year)
0
71.24
Energy cost($1000/year)
519
584.87
1348.25
1356.41
TAC ($1000/year) (50 kmol/hr)
Fig. 3 displays the compositions profiles of these two systems. The column composition profiles are with quite different form in the reactive zone, even though they have the same compositions in the top and bottom streams of the RD column. These two different reaction kinetics lead the composition from opposite way to approach EtOH/EtAc/H2O ternary azeotrope. Different reaction kinetics have significantly effect on the design of EtAc process under the same thermodynamic properties. o
(a)
EtOH (78.31 C) 78.18 C 0.8
Liquid Composition Profile RD Column Stripper Vapor Composition Profile RD Column Stripper
o
o
EtAc (77.20 C)
o
71.81 C
1.0 o
(b)
EtOH (78.31 C) 78.18 C 0.8
o
70.09 C
o
70.37 C
Liquid Composition Profile RD Column Stripper Vapor Composition Profile RD Column Stripper
o
70.09 C o
70.37 C
0.6
XB
XB
0.6
o
EtAc (77.20 C)
o
71.81 C
1.0 o
0.4
0.4
0.2
0.2
0.0 0.0
o
H2O (100.02 C)
0.2
0.4
0.6
XA
0.8
1.0
o
HAc (118.01 C)
0.0 0.0
o
H2O (100.02 C)
0.2
0.4
0.6
XA
0.8
1.0
o
HAc (118.01 C)
Fig. 3. Composition profiles in the EtAc process (a) homogeneous, and (b) heterogeneous systems
4. Dynamic control The non-square relative gain (NRG) of Chang and Yu[12] is used to find the temperature control trays. The NRG (ΛN) is defined as:
Design and Control of Homogeneous and Heterogeneous Reactive Distillation
1049
ΛN= Kp⊗ (K p+)T
(2)
where Kp is the steady-state gain matrix, ⊗ denotes the element-by-element multiplication, the superscript + is the pseudo-inverse, and the superscript T means the transpose. In this work, open-loop tests use FR and QR, S as the manipulating variables to find the steady state gains. The largest row sum of the NRG is selected as the temperature control trays. Fig. 4 shows the row sums for these two systems, thus, the controlled variables for homogeneous system are TRDC,4 and TSTR,7; and for heterogeneous system are TRDC,5 and TSTR,7. The control parings of these two systems are straightforward to use FR to control TRDC and QR, S to control TSTR. 0.6
0.006
0.4
0.004
0.2
0.002
0.0
0.000
-0.2
1
5
9
13
17
21
25
29
33
37
41
45
49
53
2 57
6 10 61 65
-0.002
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RD
1.0
(b) RD column RS(i)
0.008
0.0010
0.8
0.0008
0.6
0.0006
0.4
0.0004
0.2
0.0002 0.0000
0.0 -0.2
Stripper RS(i)
0.010
0.8
Stripper RS(i)
RD column RS(i)
Stripper
RD
1.0
(a)
1
3
5
7
9
11
13
Stage [-]
15
17
19
21 1
23 3
25 5
27 7
29 9
-0.0002
Stage [-]
Fig. 4. NRG and selected temperature control trays (a) homogeneous (b) heterogeneous systems Feed Flow
40
60
80
0
40
60
80
100
20
40
60
80
0
78.0 77.8 40
20
40
60
80
Time (hr)
0.84 20
40
2200 2000 0
5
10
15
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20
40
60
Time (hr)
80
0.84 10
15
QR,S ( KW )
2000
80
1600
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10
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20
0
5
HAc Feed Composition
10
95 5
10
15
20
QR,S ( KW )
80.96 80.92 80.88 80.84 5
10
Time (hr)
15
20
15
60
80
100
20
40
60
80
100
98.70
20
40
60
80
100
0.20 0.16 0.12 0.08 0.04 0.000
20
40
60
80
100
0.12 0.08 0.04 0.00
0
99.1
90 mol%
99.0 98.9 98.8
20
20
99.4 99.2 99.0 98.8 98.6 98.40
0.025 0.020 0.015 0.010 0.005 0.0000
5
10
15
20
5
10
15
20
5
10
15
20
10
15
20
90 mol%
99.02 99.00
0.88
98.98
0.84
2050 2020 1990 1960 1930 19000
40
100 mol%
0.92
0
20
5
10
15
98.960
20
0.020
XB,acid(mol%)
Feed Ratio
97
0
99.04
0.96
99
80.800
100
0.88 5
98.0 0.16
-20%
0.92
0
98.5
Time (hr)
+20%
2400
81
93 0
100
2400
Feed Ratio
O
TRDC( C )
O
TSTR( C )
80
0.96
101 O
60
2600
100
95
79 0
100
0.88
82
TRDC( C )
80
99.0
100 mol%
0.92
0
100
900
60
0.96
100
QR,S ( KW )
78.2
20
XB,acid(mol%)
1200
Feed Ratio
85 75 0
1600
105
O
100
2000
Feed Flow
TSTR( C )
80
2400
HAc Feed Composition
95
0
(b)
60
XB,acid(mol%)
O
40
XB,acid(mol%)
20
O
TRDC( C )
20
99.5
XB,acetate(mol%)
0
78.4 O
0.84
100
QR,S ( KW )
TSTR( C )
77
105
TSTR( C )
0.88
2800
79
XB,acetate(mol%)
20
81
75
100.0
0.92
XB,acetate(mol%)
0
-20%
XB,acetate(mol%)
Feed Ratio
85 75
+20%
0.96
95
O
(a)
TRDC( C )
105
0.015 0.010 0.005
5
10
Time (hr)
15
20
0
5
Time (hr)
Fig. 5. Temperature control responses for disturbances rejection (a) homogeneous (b) heterogeneous systems
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In the dynamic simulation, ±20% throughput changes and ±5 mol% HAc feed composition disturbances are used to evaluate the control performance for these two systems. Fig. 5 shows that the closed-loop responses of these two systems under simple PI temperature control. For the homogeneous system, slower responses and much more oscillatory responses are found in Fig. 5(a). Especially for the negative disturbances which need more than 100 hours to settle and exhibit larger overshoot in the HAc impurity. Also note that the closed-loop behavior of the homogeneous system is quite nonlinear for the temperature control loops. For the heterogeneous system, faster responses and much symmetrical responses can be obtained as shown in Fig. 5(b). The product composition settles in less than 10 hours and much smaller offsets in the EtAc composition can be achieved. From the steady-state design, the TAC of homogeneous system is a little cheaper than heterogeneous system. However from the dynamic responses, heterogeneous system exhibits less overshoot and faster as well as more symmetrical temperature control performance for disturbances rejection than homogeneous system.
5. Conclusion In this study, it is found that the two RD processes have similar flow-sheet configurations. Each includes an RD column, a decanter, and a stripper. Due to the use of sulfuric acid as catalyst with slower reaction rate, the homogeneous catalytic RD process has higher capital cost but less energy cost and less concern about the catalyst installation and frequent replacement. Due to the use of Purolite CT179, the heterogeneous catalytic process has faster reaction rates in both forward and reverse directions than those of the homogeneous one. Thus, it needs less reactive hold-up and less number of trays to achieve the product specifications. As for the control, heterogeneous catalytic process has faster response and lower steady-state offset than the homogeneous ones. Because of faster reaction rate for the heterogeneous system implies less hold-up and thus faster process dynamics. As a result, better control performance can be obtained for the heterogeneous EtAc RD process.
Acknowledgment This work is supported by the Ministry of Economic Affair under grant 94-EC17-A-09-S1-019.
References [1] D. B. Keyes, Ind. Eng. Chem. 24, (1932) 1096-1103. [2] Y. A. Chang and J. D. Seader, Comput. Chem. Eng., 12, (1988) 1243-1255. [3] J. Simandl and W. Y. Svrcek, Comput. Chem. Eng., 15, (1991) 337-348. [4] K. Alejski, and F. Duprat, Chem. Eng. Sci. 51 (1996) 4237-4252. [5] H. Bock, M. Jimoh, and G. Wozny, Chem. Eng. Technol. 20 (1997) 182-191. [6] N. Vora and P. Daoutidis; Ind. Eng. Chem. Res., 40, (2001) 833-849. [7] Y. T. Tang, H. P. Huang, and I-L. Chien., J. Chem. Eng. Japan 36, (2003) 1352. [8] Y.T. Tang, Y.W. Chen, S.B. Hung, H.P. Huang, M.J. Lee and C.C. Yu., AIChE J. 51, (2005) 1683-1699. [9] H. Renon, and J. M. Prausnitz. AIChE J. 14, (1968) 135. [10] J. G. Hayden, and J. P. O'Connell, Ind. Eng. Chem. Process Des. Dev. 14 (1975) 209-216. [11] G. Hangx, G. Kwant, H. Maessen, P. Markusse and I. Urseanu. Technical Report to the European Commission (2001), (http://www.cpi.umist.ac.uk/intint/NonConf_Doc.asp) [12] J. W. Chang, and C. C. Yu,. Chem. Eng. Sci., 45, (1990) 1309.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Decomposition Based Algorithm for the Design and Scheduling of Multipurpose Batch Plants Tânia Pintoa, Ana Paula F. D. Barbósa-Póvoa§b and Augusto Q. Novaisa a
Dep. de Mod. e Sim., INETI, Est. do Paço do Lumiar, 1649-038 Lisboa, Portugal Cent. Est. de Gest., CEG-IST, DEG, IST., Av. Rovisco Pais, 1049-101 Lisboa, Portugal
b
Abstract The optimal design of plants incorporating multiple resources, whose main characteristic is their intrinsic flexibility, it is a complex task, since the design of the plant resources and scheduling should go together. Due to the nature and dimension of these problems, they often result into large Mixed Integer Linear Program formulations (MILP) that come associated with a high computational burden. In order to overcome this difficulty, a decomposition algorithm has been developed, which is described in this paper. It comprises two different levels (a master and a sub-problem) that interact through an iteration procedure that guarantees optimality. The high level consists in a time aggregation model, where the main equipment choices are defined. These choices once established, are transferred and used as input at the lower level, where the optimal scheduling coupled with resources capacities is calculated. The algorithm performance is analysed and improved with the addition of three classes of cuts, while retaining control of the run length of the sub-problem solution. Some examples illustrating the effectiveness of the proposed decomposition approach are presented. Keywords: Design, Decomposition Algorithm, Aggregation, Resource-Task Network.
1. Introduction The design of a multipurpose production facility, in its basic form, involves the selection of the number, type and capacities of the resources involved, as well as the definition of its operability, so as to produce a set of products while guaranteeing a set of pre-defined conditions and optimizing a given objective. Due to the inherent flexibility of the multipurpose resources utilization, where the same resource can be used to perform different tasks, operational scheduling considerations need to be taken into account at the design stage. Thus, in order to guarantee optimal solutions, most design formulations based on mathematical programming approaches have to consider a large number of resources items, out of which the ones which are incorporated into the final plant design are selected. This fact, together with the complexity of the recipes, leads to large MILP problems, often associated with a high computational burden that grows together with the problem dimension. For this reason, the development of efficient tools is required, which still constitutes an open area of research. In recent years, some works addressing the basic design problem were published, where the design of the main equipment and its operability have been considered. BarbosaPóvoa and Pantelides (1997) proposed a decomposition algorithm to solve a ResourceTask-Network (RTN) design problem; Xia and Macchietto (1997) used a simulated §
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annealing approach for the design; van den Heever and Grossmann (1999) applied a logic-based OA within a bi-level decomposition to tackle the design, operation and expansion planning problem; Bernal-Haro et al. (2002a,b) also addressed the multipurpose design problem and used a two stage methodology to solve it, whereby a discrete event simulation was used for the 1st stage and a genetic algorithm for the 2nd. More recently, Heo et al. (2003) dealt with planning and design by means of an MINLP that was solved using a separable programming method based on heuristics. All the above works still point to some difficulties related with the attainment of a solution and to the need for further and more robust solution methods. In this paper the approach proposed by Barbosa-Póvoa and Pantelides (1997) is revisited and applied to a more generic design problem where the plant topology is also considered. A generalization is developed that involves the decomposition into two complementary problems. A master problem, which deals with the selection of the appropriate resource levels, and a sub-problem that attempts to determine the optimal plant schedule and the resources capacities. The master problem is an aggregated design model based on the aggregation techniques proposed by Wilkinson (1996) and extended by Barbosa-Póvoa and Pantelides (1999). The sub-problem is the original RTN design model where some design decisions are fixed. An iteration procedure is implemented between the two problems, with integer and tailor-made cuts being added at each iteration into the master problem (and thus excluding previous design choices). The remaining of this paper is structured as follows. First the definition of the design problem is presented, followed by the description of the decomposition-based algorithm. The applicability of the proposed approach is then illustrated via the solution of some representative examples. The paper concludes with some remarks.
2. Design Problem Given: • The process/ plant description (in RTN terms) including the plant topology; • Resources availability, characteristics and costs; • Time horizon of planning and mode of operation; • Demand over the time horizon (production range) and cost data; Determine: • The optimal plant configuration (i.e. number and type of equipment units and their connections, as well as their sizes); • The optimal process schedule (i.e. timing of all tasks, storage policies, batch sizes, amounts transferred, allocation of tasks and consumption of resources); So as to optimize an index of economic performance of the plant, measured in terms of capital expenditure, operating costs and revenues. Based on the above description a model was developed using the RTN representation. This (Pantelides, 1994) appears as one of the most general and conceptually simpler representations to deal with process/plant problems. Two types of entities are defined: tasks are operations that consume and/or produce a specific set of resources; resources describe all other different entities involved in the process/plant. The model results from a generalization of the work of Barbosa-Póvoa and Pantelides (1997) which was extended to account for the plant topology. A non-periodic plant operating over a given time horizon is considered and a discretization of time is assumed. Mixed storage policies, shared intermediate states, material recycles and multipurpose batch plant equipment with continuous sizes, are allowed.
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3. Decomposition Algorithm The decomposition algorithm involves a master and a sub-problem involved in an iterative procedure. The master problem is an aggregated model, derived from the detailed design formulation, through the application of the temporal aggregation operators proposed by Wilkinson (1996) to the variables of the original model. As a result, the original variables are replaced by a set of aggregated variables and the order of magnitude of the model is reduced. The temporal aggregation involves splitting the planning horizon into smaller portions - Aggregated Time Periods (ATPs). Each ATP involves a set of aggregation operators Am(.) that perform the weighted sum of the corresponding detailed variables over the time intervals spanned by the ATP. The aggregation operator (m≥0, where m is the order of the operator) is defined as follows (Barbosa-Póvoa and Pantelides, 1999):
A m [X ] =
t
∑ g (t − t ' ) X
t ' = t − h +1
(1)
t'
where t is the start of the ATP and g(.) is a given aggregation weighting function. Not all original variables are eliminated through the use of these operators, though. Those variables that represent an activity that starts in one ATP and continues into the next one are not eliminated and remain in the aggregated formulation - linking variables. In conclusion, the aggregated model is consistent with the detailed formulation and appears as a relaxation of it. Its solution leads to the choice of the main equipment as well as of all the plant connections. The sub-problem, as referred above, uses the generalization of the mathematical formulation presented by Barbosa-Póvoa and Pantelides (1997), where the design variables determining the choice of the equipment and connections are assumed to be fixed at the values determined by the master problem. The related capacities remain undefined and will be determined at the sub-problem. By combining the master and the sub-problem, a decomposition approach is developed (Figure 1) where K iterations are considered till convergence is reached. Assuming maximization of production, the master, being a relaxation of the original problem, leads to a valid upper bound (φU) of the optimal solution (φ), while the sub-problem provides a lower bound (φL) of the same solution, since its feasible region is restricted when compared to the original one, on account of variables being fixed based on the results of the master problem. The algorithm converges when the two bounds come within a tolerance (ε). The optimal solution corresponds to the sub-problem solution with the best objective function value. The decomposition algorithm involves the addition of integer cuts at every iteration K. These cut all design assignments(Δr) determined by previous master problems:
∑Δ
r∈U
k
r
− ∑ Δ r ≤ card (U k ) − 1
∀ K=1,…,kmax
(2)
r∈Lk
Where Uk and Lk denotes the subsets of the variables Δr that have taken respectively the values of one or zero at the master solution, iteration K. Additionally to the integer cuts and in order to improve the model solution, tailor-made cuts were also developed. Two sets of these are applied at the master model, namely, cuts I and II, see below. The first is related to the continuous variables and guarantees
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that if a task exists, the associated batches always differ from zero. The second is defined over the design variables and guarantees that if a transfer operation exists it must have an equipment item both at its destination and at its origin. Taylor made cuts I: τk
τk
θ =1
θ =1
ξ kt0 − ∑ ξ k ,t +1−θ − mN kt0 + m ∑ N k ,t +1−θ ≥ 0 ∀ k ∈ Tk , t = h , 2 h ,...H
(3)
τk
ξ kt1 − (τ k + 1) ξ kt0 − ∑ ξ k ,t +1−θ (θ − τ k − 1) − mN kt1 + m (τ k + 1) N kt0 θ =1
τk
− m ∑ N k ,t +1−θ (θ − τ k − 1) ≥ 0 θ =1
(4)
∀ k ∈ Tk , t = h , 2 h , ...H
τk
hξ kt0 − ξ kt1 − ∑ ( h − θ ) ξ k , t +1−θ − m h N kt0 + m N kt1 θ =1
τk
+ m ∑ ( h − θ ) N k , t +1−θ ≥ 0 θ =1
(5)
∀ k ∈ Tk , t = h , 2 h , ... H
Taylor made cuts II:
Δ r∈To ≤ Δ r∈Tt ≤ Δ r∈Td
(6)
∑
(7)
r ∈Td
Δr +
∑
r ∈To
Δ r ≥ Δ r∈ D
4. Examples Four examples are used to illustrate the effectiveness of the proposed decomposition. These are solved in four different situations using the decomposition algorithm (cases 1, 2, 3 and 4) and compared with the detailed approach, which in turn is solved in two different situations: the detailed model and the detailed model with tailor-made cuts II. In the decomposition approach the four cases differ in the use of cuts at the master model: (1) integer cut only; (2) integer cut and tailor-made cuts I; (3) integer cut and tailor-made cuts II; (4) integer cut and both types of tailor-made cuts. Also, in the decomposition approach a control of the solution run length is performed, based on the experience gained with preliminary trials. This considers for the master model a resources availability of 1000 CPU seconds, while for the sub-problem, examples 1, 2 and 3, resources are levelled at 150, 300 and 450 CPUs respectively. Example 4 is run for a maximum resource level of 5000 CPUs due to its large dimension. Example 1- A multipurpose batch plant must be designed at a maximum profit so as to produce [0; 80] ton of products, S5 and S6, from two raw materials, S1 and S2, over a horizon of 24h. Example 2- Same as before, but in order to produce [0;80] ton of products, S5 and S6, [0;50] ton of products S9 and S10, from three raw materials, S1, S2 and S7, over a horizon of 24h. Example 3 - Same as before, but in order to produce [0; 2500] ton of products, S5, S9 and S10, [0; 4000] ton of products S6, from three raw materials, S1, S2 and S7, over a horizon of 120h.
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Example 4 - Same as before, but in order to produce [0; 2500] ton of products, S5, S9 and S10, [0; 4000] ton of products S6, [0; 2142] ton of products S11, from three raw materials, S1, S2 and S7, over 360h. Table 1 sums up the characteristics of the four examples in the detailed and decomposition approaches. For the latter, separate data are shown for the master and the subproblem. The characteristics of the remaining cases explored within each example are omitted since they only differ in the number of cuts introduced. Table 1. Models characteristics. Examples Ex1
Detailed Master Subproblem Detailed Master Subproblem Detailed
Ex2
Ex3
Master Subproblem Detailed Master Subproblem
Ex4
Equ.
Var.
BVar.
Relative Gap %
7112
4918
1896
0.199
1842 7112
5262 4918
1896 1875
0.099
10608
7226
2732
0.1
2681
7728
10608 51639
7226 35436
2732 2700 13464
2748
35951
51639 159790 2841 159790
0.096 3.22
35436 109132
13464 13431
1,86
41189
-
109663 109132
41189 41154
4,439
Equ- equations; Var – variables; BVar – binary variables; - no solution reached;
The solutions performance is shown in tables 2 and 3, respectively for the first three examples and example 4. In the first example (Ex.1) the decomposition approach shows a better performance for all runs in case 4, when compared with the detailed model (2.3 against 3.1: -11.88%) but no solution could be reached in cases 2) and 3). Similarly to Ex.1, in Ex.2 the detailed model with the tailor-made cuts also showed a worst performance. However, case 4 shows an improvement of 92.47% (12.2 against 162.2). In Ex.3, neither case 1 nor case 2 of the decomposition approach, reach a solution for the run length made available to the subproblem or for the iteration upper bound. However, case 4 shows an improvement of 10% (570.1 against 633.1) and 45.68% (570.1 against 1049.6) was obtained when compared to the detailed model with or without tailor-made cuts. Table 2. Performance in CPU seconds, Pentium (R) 4, 3.0GHz, 0.99 GB RAM,.CPLEX 9.0 Ex Det Det + cut Case 150 it. 300 it. 450 it.
1
2.61
3.1
2
162.2
197.5
3
1049.6
633.1
1) 4) 1) 2) 3) 4) 3) 4)
3.3 2.3 23.6 17.4 155.3 12.2 570.1
1 1 1 1 1 1 1
3.3 2.3 23.6 17.4 155.3 12.2 1761 570.1
Det – detailed problem; Dec – decomposition algorithm; - no solution reached; it. – nº of iterations; Det +cut- detailed model with design tailor-made cuts I.
1 1 1 1 1 1 4 1
3.3 2.3 23.6 17.4 155.3 12.2 1761 570.1
1 1 1 1 1 1 4 1
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In Ex.4 (Table 3) no solution was reached in 14400 CPU for the detailed model, which converged in 4171.8 CPU with the tailor-made cuts II. The first case of the decomposition approach showed an improvement of performance of 53%. Table 3. Example 4, Master solution run length is 5000 cpu. Ex Det Det + cut Case 2000 it.
4
--
4171.8
1) 2) 3) 4)
1959.4 6434.4 6811 9183.1
1 1 2 2
--no solution reached in 14400 CPU time.
5. Conclusions A decomposition approach for the detailed design of multipurpose batch plants has been developed. This involves a master problem and a sub-problem within an iterative procedure. The solution of a set of cases was analyzed and a promising solution algorithm performance achieved. The implementation of design and tailor-made cuts is found to improve in general the solution efficiency. With the same aim, a control on the sub-problem solution run length is found to be instrumental, since it promotes model performance, while reducing unproductive time involved in the solution of unfeasible sub-problems. For larger problems the influence of the tailor-made cuts is less evident. Further studies are therefore required to be developed and applied to more general cases. Also some attention will be given to aspects related to the structure of the master problem and the quality and type of information transferred from the master to the subproblem.
Nomenclature Variables: ξkt - batch size of task k at time t
Δ r - amount of resource r
Parameters: h – aggreg. time periods (ATPs) with a pre-defined length. m – small value
Sets: Tt ={k: set of all transfer tasks} D = {r: set of all equipment resources} To= {k: set of all tasks requiring an equipment ahead, r} Td= {k: set of all tasks requiring a sink equipment, r} Tk = {k: set of all tasks operating in an equipment resource}
Acknowledgment The authors gratefully acknowledge the financial support FCT, grant SFRH/17728/2004.
References Barbosa-Póvoa, A.P. et al. (1997). Comput. Chem. Engng., 21S, S703-S708. Barbosa Póvoa, A. P. F. D. et al. (1999). Comp. Chem. Engng, 23, S7-S10. Bernal-Haro, L., Azzaro-Pantel, C., Pibouleau, L., and Domenech, S. (2002a). Ind. Eng. Chem. Res.41(23), 5727-5742. Bernal-Haro, L., Azzaro-Pantel, C., Pibouleau, L., and Domenech, S. (2002b). Ind. Eng. Chem. Re,41(23), 5743-5758 Pantelides, C.C. (1994), In proceedings of FOCAPO, 253. van den Heever, S. A., and Grossmann, I. E. (1999),Comp. Chem. Engng 23(8), 1075-1095. Heo, S. K., Lee, K. H.., Lee, I. B., and Park, J. H. (2003). Ind. Eng. Chem. Res., 42(4), 836-846. Wilkinson, S.J. (1996). PhD Thesis University of London. Xia, Q. S., and Macchietto, S. (1997). Comp. Chem. Engng , 21, S697-S702.
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A knowledge-based approach for accident information retrieval Mikiya Suzukia , Rafael Batresa∗ , Tetsuo Fuchinob , Yukiyasu Shimada c , Paul W. Chungd a Toyohashi b Tokyo
University of Technology, Toyohashi 441-8580, Japan
Institute of Technology, Tokyo 152-8552, Japan
c National
Institute of Industrial Safety, Tokyo 204-0024, Japan
d Loughborough
University, Loughborough, LE11 3TU, England
Abstract There is an enormous amount of information available on past accidents in the form of incident reports available as documents and managed by databases. Engineers who perform safety analysis can benefit from this information. However, incident reports and accident databases are written the form of textual natural language descriptions. This paper presents a knowledgebased approach to extract knowledge directly from such sources while reducing the number of mismatches. Keywords: accident databases, causality, reasoning, ontologies 1. Introduction Safety plays a very important role throughout the life cycle of a chemical plant. To ensure safety and minimize later plant changes, safety is evaluated during process and plant design stages. Safety evaluation typically involves the analysis of possible abnormal situations that may occur in the plant. This task is typically carried out by experts that use their knowledge and experience to relate the process or plant with potential problems. For example, HAZOP is carried out by defining intended characteristics of the process and plant, and then proposing deviations from the intent so as to identify causes of deviations, and consequences (process drifts, equipment malfunctions, failures and operation errors). On the other hand, there is an enormous amount of information available on past accidents in the form of incident reports available as documents and managed by databases. Engineers who perform safety analysis can benefit from this information. However, incident reports and accident databases are written the form of textual natural language descriptions. Extracting knowledge directly from such sources has a considerable number of mismatches. False positives are reported when a word has the same spelling but a different meaning such as the word tank. Also, false negatives are obtained because existing accident representations lack the ability to deal with types and subtypes of things. For example, a query for finding accidents that resulted in explosions may not show reports containing the word BLEVE (a kind of explosion involving vapors from boiling liquids). ∗
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To overcome the limitations of text-based descriptions, we propose an ontology that provides generic and accident related definitions in a form that is processable by a number of software reasoners. The development of the ontology uses an upper ontology based on ISO 15926 that defines general-purpose classes and relations and acts as a foundation for more specific domains. Consequently, accident knowledge bases can be integrated with other tools such as those related to process, plant and equipment design. Inherently safer design can also benefit from this approach as the knowledge base can provide clues on what can go wrong with some of the alternatives and on how to eliminate the hazard. Causality information contained in the accident reports is represented by means of causality networks that are composed of activities and events. Causality networks can be built graphically and then converted to the OWL ontology language that can be used directly in a number of inference software packages. OWL is an ontology language for the Web that provides modeling constructs to represent knowledge with a formal semantics. OWL inherits features of frames such as basic constructs for defining classes and instances. 2. Accident databases Chung et al., identified five organizational hierarchies of information. They represented the equipment involved in the incident, the operational activity at the time of the incident, the chemicals involved, the cause of the accident and its primary consequences. The definition of the five main classes and their hierarchies constitutes the first step in building an accident ontology. We have encoded the hierarchies in OWL so that queries between classes and subclasses can be carried out with OWL-compatible inference engines. For example, the class BLEVE is defined as a subclass of the class explosion pressure release. Therefore, a query made in terms of eexplosion pressure releasef will retrieve those reports documenting a BLEVE along with reports documenting other kinds of explosions and pressure release phenomena. The subClassOf relation makes the representation of hierarchies possible. However, there are many other relations that can be defined which add semantic content to the reports. The key advantage of ontologies is about all the things that can be expressed by using those relations. It is because of the use of relations and the axiomatic expressions of an ontology that an inference engine can distinguish between things. In order to understand what relations are applicable to a given class, it is necessary to define the meaning of the members of that class. Upper ontologies define generic classes and relations that constitute the basis for doing such task. In this research, ISO 15926 is used as an upper ontology that defines a domainindependent ontology that provides a common set of classes and relations [1]. According to ISO 15926, things such as explosions and equipment are classified as possible individuals, in other words things that exist in space and time from the beginning they are created to their end. Certainly, explosions and equipment are different things. An explosion brings about change. Equipment participates in such change. In addition, the ontology supports physical quantities and units of measure. This and other features are detailed in the ontology. 3. Causality The remaining two hierarchies require more analysis. Accident scenarios include one field for causes and one field for consequences. The causes typically are equipment malfunctions or human errors, while the consequences involve hazards. However, a consequence can be a cause of other consequences so rigid classification of things as causes or consequences is unrealistic. For example, explosion can be consequence for one accident but it could be a cause for another. It is an instance of the explosion that can be a cause or a consequence, not the class. The knowledge
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domain
ending range
range
possible_individual
beginning
participation is_a
range
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domain
activity
event Class Relation
range
cause_of_event
domain
Figure 1. Causality related classes and relations
representation about causes and consequences leads us to the concept of causality which has been discussed in philosophy since ancient times and more recently in Artificial Intelligence [2]. Shoham lists properties of causation some of which are listed here: 1. 2. 3. 4.
Causation is antisymmetric. A cannot cause B if B is the cause of A Causation is antireflexive. A cannot cause itself. Causes cannot succed their effects in time. A(s) causes A(t) ⇒ s ≺ t Entities participating in the causal relation have a temporal dimension. For example, explosions, runaway reactions, mixing operations, all have a beginning and an ending.
Domotor adds the property of transivity: If A causes B and B is the cause of C, then A is also the cause of C. But what is the nature of causes and effects? Property 4 has the answer. Participants in a causal relation exist in time and may be temporally bounded. This observation is in tandem with the definition of activity and event in ISO 15926. Activities in ISO 15926 are defined as possible individuals that bring about change by causing an event. Also, as shown in Figure 1, activities can have temporal boundings because activity is a subclass of possible individual which is bounded by beginning and ending events. In other words, an instance of event is a possible individual with zero extent in time that marks the beginning or the ending of a possible individual. 4. Causality networks In order to create and visualize causality information, we introduce the concept of causality network which is based on ISO 15926. A causality network is defined as a tuple N = A, E, P, R where, • • • •
A is a set of nodes representing activities, E is a set of nodes representing events, P is a set of nodes representing participating entities, R is a set of arcs containing the subsets of causal relations S, a subset of life-cycle relations T , and the subset of participation relations V . • S ⊆ (A × E), T ⊆ (E × A), V ⊆ (P × A) The causal relations in S are represented by instances of cause of event. A cause of event is a relationship that indicates that the caused (event) is caused by the causer (activity). The set of life-cycle relations T is composed of the beginning and ending relations. A beginning is a temporal bounding that marks the temporal start of a possible individual. An ending is a temporal
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bounding that marks the end of a possible individual. The subset of participation relations is made of instances of the participation relation. A participation is a composition of individual that indicates that a possible individual is a participant in an activity. Currently, causality networks are built manually but automatic approaches are under study. 4.1. Example 1 The following is a fragment of textual description of an incident as found in the IChemE accident database. Record.: 129451998 An explosion occurred at a chemical plant injuring nine workers and releasing chemicals into the surrounding area. The explosion occurred due to a runaway chemical reaction in a 2000-gallon kettle being used to produce dye. The causal information to be represented is from the sentence The explosion occurred due to a runaway chemical reaction in a 2000-gallon kettle. Figure 3 shows the causality network for this sentence. ev1 Comment: Pump failure
Condenser
Cooling Water
p3
tp_p3 beginning
Temporal part of Pump P3
vacuum trap
spatio_ temporal_part
Pump P3
beginning
Kettle
act1 Description: Pressure increases in the line P3−T1
T1
T1 relative_location
2000−gallon kettle
relative_location
2000−gallon kettle
LP Steam P3 50% HA (ions: 40−60 ppb)
Vacuum pump
Heater Cooling Water
P1 Feed pump
cause_of_event ev2 Comment: Temperature starts to increase
Product tank beginning
P2
Figure 2. HA Flowsheet
act1 Description: Runaway Chemical Reaction
act2 Description: Bubble temperature increases
po1 containment_of individual
2000−gallon kettle
cause_of_event
ev3 Comment: Onset temperature is reached
cause_of_event
activities
beginning
act3 Description: Runaway Chemical Reaction
T1 relative_location
2000−gallon kettle
participation
physical_object ev1 Comment: Start of the explosion
T1
cause_of_event
HA (50%)
ev4 Comment: Start of the explosion
event beginning
act2 Description: Explosion
Figure 3. Causality diagram
beginning
act4 Description: Explosion
Figure 4. Causality diagram for the HA accident
5. Retrieval of accident information The classes and relations presented in the previous sections are implemented in the form of an ontology encoded in the OWL language. The knolwedge represented in OWL and based on the
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ontology can be processed by a number of inference engines which enables not only information searching (in the database sense) but also new knowledge extraction. New knowledge extraction is possible because of the inference mechanisms such as modus ponens, forward chaining and backward chaining along with logical expressions containing axioms such as transitive, irreflexive and symmetric axioms. 6. The explosion at Gunma At 18:00 hrs on June 10, 2000, a distillation unit processing a solution of hydroxylamine (HA) exploded at a plant operated by Nisshin Chemical Co. located in Gunma prefecture about 70 miles north of Tokyo. The accident databases that were consulted during the writing of this paper had little information on the possible causes of this particular incident. The information that follows was obtained from a detailed report carried out by an accident investigation team [4]. The explosion caused 4 fatalities, injured 4 persons in the plant and 54 persons in the surrounding area and caused significant damage to the plant. Prior to the explosion, the plant had been shut down for 5 hours to replace oil in a vacuum pump. The explosion occurred about 30 minutes after startup. The production facility involved two phases. The first phase consisted of the production of 34 wt% acqueous solution of hydroxylamine sulfate. The second phase involved the steps shown in Figure 5: neutralization with sodium hydroxide, distillation and redistillation. N eutralization
HA(34%)H2 SO4 −−−−−−−−−→
Product
←−−−
HA(34%)
+
Distillation
−−−−−−−→ Redistillation
HA(50%) ⏐ ⏐
(1)
+
50%HA(M < 1ppb) ←−−−−−−−− 50%HA(M : 40 − 60ppb)
Figure 5. HA Production steps
The explosion is believed to have occurred at the redistillation section. In the redistillation section, the 50 wt% HA solution is further concentrated and ion concentration is reduced from 1ppb to 40-60 ppb. The feed is heated by steam in a double-pipe heat exchanger and flows up to a distillation equipment that is operated at low pressure by means of a vacuum pump as shown in Figure 2. The accident report describes a possible scenario in which the explosion was initiated by an abnormal increase of temperature at the redistillation equipment. The temperature increase was possibly caused by a failure at the vacuum pump. The causalitiy network of this scenario is shown in Figure 4. The failure of the vacuum pump caused an increase of pressure that possibly elevated the boiling point of the HA solution, and caused an increase of the concentration inside the redistillation equipment. Concentrations in the redistillation system may have reached levels of more than 85 wt% that can result in decomposition of hydroxhylamine solution followed by explosion (85%) [3]. The decomposition of hydroxylamine is highly exothermic with a heat release similar to that of TNT. The decomposition reaction is given below according to [6]. 31.2NH2 OH −→ 12.2NH3 + 7N2 + 0.2H2 + 28.6H2 O + 2.4N2 O + 0.4NO ΔH = −29.6kcal/kg(2)
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The instances involved in the accident scenario can be loaded into an OWL compatible reasoner. We have developed an end-front for JTP (Java Theorem Prover) which is a reasoning system that can derive inferences from OWL files. JTP translates each OWL statement into a KIF sentence of the form (PropertyValue Value Predicate Subject Object). Then it simplifies those KIF sentences using a series of axioms that define OWL semantics. OWL statements are finally converted to the form (Predicate Subject Object). Queries are formulated in a format similar to KIF, where variables are preceded by a question mark. Reasoning on physical quantities is possible using a reasoner that is plugged in JTP for logical comparisons (e.g. less than) and arithmetic relations. Following are some of the queries that can be made: 1. What were the causes of explosions? (and (ecm:cause_of_event ?event ?explosion) (ecm:beginning ?event ?activity)) 2. What were the causes of explosions involving hydroxylamine? (and (ecm:cause_of_event ?event ?explosion) (ecm:beginning ?event ?activity) (ecm:participation ?HA ?activity) (rdf:type ?HA hydroxylamine)) 7. Conclusions This paper introduced a knowledge-based approach for representing accident information. Specifically, it examined the concept of causality networks as a model to integrate causality information with objects involved in the accident. The causality networks and knowledge of substances and equipment are encoded in OWL so as to enable reasoning and knowledge retrieval. To this end, an upper ontology based on ISO 15926 is used as a common view of the world. For accidents such as the Gunma explosion where specific causes could not be determined, the cause-consequence format implemented in current accident databases results inadequate. For designers to be able to reuse accident information one accident should be related to one or more scenarios. This paper introduced a knowledge-based approach that describes the scenario causality and participating physical objects. Future work is planned to integrate some of the classes in the accident database with the ontology. More research is needed to determine the extent to which causality information can be extracted from existing text-based databases. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
R. Batres, Proceedings of ESCAPE-5 (2005) N. V. Findler and T. Bickmore, Applied Artificial Intelligence 10 (1996) 455-487 L. Long, Process Safety Process 23 (2004) 114-120 Masamitsu Tamura. Accident Report on the Hydroxylamine Explosion in a Gunma Chemical Plant. March, 2001 Y. Shoham. Reasoning about Change, MIT Press, Cambridge, Massachusetts, 1988 L. O. Cisneros. Adiabatic Calorimetric Studies of Hydroxylamine Compounds. PhD Thesis, Texas A&M University, 2002 S. Scholes, Discuss. Faraday Soc. No. 50 (1970) 222. O.V. Mazurin and E.A. Porai-Koshits (eds.), Phase Separation in Glass, North-Holland, Amsterdam, 1984. Y. Dimitriev and E. Kashchieva, J. Mater. Sci. 10 (1975) 1419. D.L. Eaton, Porous Glass Support Material, US Patent No. 3 904 422 (1975).
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Feasibility study of batch reactive distillation in hybrid columns Csaba Steger1, Endre Rev1, Zsolt Fonyo1 , Michel Meyer2, Zoltan Lelkes1 1 2
BUTE, Muegyetem rkp. 3., 1521, Budapest; Hungary LGC, ENSIACET, BP 1301, 5 rue Paulin Talabot, 31106, Toulouse, France
Abstract Reactive distillation (RD) is an effective way to integrate reaction and distillation. Since no general methodology to perform the conceptual design of RD is known it is not a widespread process in the industry. A general feasibility study is presented in this paper to study the feasibility of batch RD with fast equilibrium limited reaction. Its applicability is presented on the example of ethyl acetate production. Several configurations is found infeasible, but a complex configuration is found able to produce the desired products. Keywords: batch, reactive, distillation, hybrid configuration
1. Introduction Reactive distillation (RD) integrates two processes, reaction and separation, in the same operation unit. Recovery and selectivity of the equilibrium limited reactions can be increased by applying RD, and the capital cost may be decreased [1]. RD is a widely studied process but its application is limited because of the absence of a reliable and general conceptual design method. Articles are published about fully reactive columns [2], coupled systems [3, 4] and about hybrid configurations [5] (distillation column both with reactive and non-reactive sections). Till now the complex configurations have been investigated for continuous processes with stage to stage model only [5]. Batch reactive distillation in complex configurations has not been studied. Eight possible arrangements of batch reactive distillation in a batch rectifier are presented in Figure 1. The reboiler is reactive in each case, and at most one feed is allowed. In the case of a middle vessel column, the apparatus can be considered as an agregate of a batch rectifier and a batch stripper. If the same assumptions are taken into account in the batch stripper, then 64 variants of the middle vessel column are generated. Investigation of all of these variants one by one takes a lot of time. Therefore our aim was to create a general feasibility methodology, which can treat as much combinations of the batch reactive configurations as possible.
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Figure 1 – Reactive batch distillation configurations with reactive boiler
2. Feasibility study The feasibility study always contains simplifying assumptions so that the results can be obtained much faster than with rigorous models. Our methodology applies fast, equilibrium limited, reactions in the liquid phase with zero reaction enthalpy; it neglects the liquid holdup on the stages and the effect of the applied catalyst on the vapor-liquid equilibrium. The process is modelled through quasi steady states. Separation is called feasible if the specified product(s) can be withdrawn, starting from the charge composition in the vessel. The column profiles inform us about the concentrations along the column; they must be investigated together with the still path (evolution of the vessel composition in time). 2.1. Model equations The feasibility methodology incorporates solution of differential equations describing the concentration profiles in the non-reactive sections (1), in the reactive sections (2), and determining the reactive still path (3). Integration of the differential equations (1-2) determine the extractive and non-extractive sections as well, the only difference is the existence of the member containing the feed; F is equal to zero in the case of nonextractive sections.
dx j dh ~ dl j dh
=∓
(
)
V y j (x j ) − y *j (x ) L
( ()
where
)
y j (x j ) =
()
L D F x j + xD , j − z j V V V
~ ~ ~ ~ ~ = ± v~j l j − v~j* (x ) where v~j l j = l j + d j − f j
d (U S ⋅ x S ) ~ ~ ~ = −d − b + f dt
(2)
(3)
νj νj ~ lj = l j − l ref , v~ j = v j − v ref , etc. ν ref
(1)
ν ref
(4)
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Equation (1) (published by Lelkes et al; [6]) and equation (2) include the driving force of the process, the difference between the actual and the equilibrium concentrations in the vapor phase at the given height of the unit. Equations (4) are the transformation relations, which are similar to the transformation published by Doherty et al. [7], and provides a two dimensional mapping of the reactive space. By integrating equations (1-2), the concentration profiles can be predicted in each section of a column. If the studied configuration consists of different sections, their crossing point must also be investigated. 2.2. Feasibility condition in fully reactive and non-reactive configurations In a fully non-reactive column a non-reactive rectifying and a non-reactive extractive sections can cross each other; the column profile is affected only by the applied feed. The feasibility is already studied in these systems (see e. g. [6]). The rectifying and extractive profiles have to cross each other to constitute a feasible column profile inside the column. This is the situation in the case of a fully reactive column, as well. To obtain a feasible reactive composition profile with entrainer feeding, the reactive rectifying and reactive extractive column profiles must cross each other. 2.3. Feasibility condition for hybrid configurations A sudden change may happen in the concentration profile between two sections if one is reactive and the other is not. The transformation (equation 4) describes the concentration jump caused by the reaction [7], thus the crossing condition can be studied in the transformed space. Both the reactive and non-reactive composition profiles must be transformed. If the transformed profiles cross each other, the concerning non-transformed profiles are feasible. Consequently, composition profiles are predicted in each column sections with the specified products. The calculated profiles are visualized in transformed and/or in nontransformed space. To decide whether a configuration is feasible or not the relations between the predicted column profiles must be investigated. The recovery of a component can be estimated by investigating the feasible column profiles and the still path together.
3. Production of ethyl acetate with reactive distillation This section presents the application of the feasibility study for the production of ethyl acetate with esterification: CH3COOH + C2H5OH Ù C2H5OOCH3 + H2O
(5)
The equilibrium limited reaction (5) is characterized with the chemical equilibrium constant K=3.94 [8]. 3.1. Investigation of the non-reactive sections Figure 2 presents the coupled system (reactive boiler and non-reactive sections), and the residue curves of the studied quaternary system. It contains four binary and one ternary azeotropes. Both products (EtOAc, H2O) are saddle points. Thus, they cannot be produced with infinite number of theoretical stages. The rectifying part produces the ternary azeotrope as distillate (unstable node, UN), and the stripping part produces the acetic acid as bottom product (stable node, SN). The shaded area in Figure 2 represents the feasible area of non-reactive rectifying and stripping profiles with the product specification x > 0.95 and finite number of
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theoretical stages. There is no common part of these areas; they do not intersect the reactive space; thus, neither the rectifying nor the stripping part can provide the specified products. AcOH (SN)
EtOAc (S)
AcOH (SN)
S S
EtOH (S)
UN S
AcOH (SN)
H2O (S)
Figure 2 – a) Residue curves map, feasible stripping and rectifying areas with finite number of stages b) Middle vessel configuration with non-reactive sections Table 1 – Coordinates of the 4th stable node of the extractive profiles F/V
EtOH
AcOH
EtOAc
H20
0.1
0.1891
0.0966
0.5526
0.1616
0.2
0.1563
0.1827
0.5337
0.1273
0.5
0.0319
0.4137
0.4923
0.0622
0.6
0.0111
0.4880
0.4652
0.0357
0.75
0.0024
0.5915
0.4021
0.0040
1
0.0007
0.7274
0.2718
0.0001
The ternary stable nodes of the non-reactive extractive profiles are also shown in Figure 2 as dots, with different feed flow rates. The coordinates of the fourth, quaternary, stable node are given in Table 1. This stable node cannot be drawn in two dimensional space; that is why it is presented with coordinates. All the extractive profiles starting from an inner point of the composition terahedron ends in this fourth stable node. Three of the four stable nodes, including the one presented in the table, are situated in the shaded area if the F/V ratio is above 0.75; thus, the upper column part can produce nearly pure ethyl acetate with continuous acetic acid feeding. 3.2. Investigation of the rective sections The reactive residue curves are shown in Figure 3. The reactive space contains only two binary azeotropes (EtOH/H2O and EtOAc/EtOH); the other ones are reacted away. Both products (EtOAc, H2O) remain saddle points; they cannot be produced with infinite number of theoretical stages. The rectifying part produces the EtOAc/EtOH
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azeotrope as distillate (UN); the stripping part produces acetic acid or EtOH as bottom product (SN). However, neither the upper nor the lower column part can produce the desired products with infinite number of stages, the stripper part is able to produce nearly pure water with finite number of stages. The feasible region of the reactive stripping section is shaded in Figure 3. H 2O (S)
AcOH (SN)
EtOAc (S)
UN
EtOH (SN)
Figure 3 –a) Reactive residue curves map and feasible area of the reactive stripping section with finite number of theoretical stages b) Feasible middle vessel configuration
The middle vessel configuration is considered as a combination of a batch rectifier and a batch stripper. A non-reactive rectifier is able to produce pure EtOAc with finite number of theoretical stages in the rectifying section from any point of the composition space. A reactive stripping section is able to produce pure water with finite number of stages from a significant area of the reactive space. Thus the combination of the two parts (Figure 3b) is a feasible middle vessel column configuration to produce the desired products if the still composition is located in the shaded area in Figure 3a.
4. Summary A general feasibility study is created to investigate the feasibility of batch reactive distillation processes in middle vessel column. The presented methodology can deal with fully reactive, fully non-reactive and complex column sections with a reactive vessel. The feasibility of producing ethyl acetate is presented. A complex column with non-reactive rectifying and extractive sections and with a reactive stripping section is found feasible.
Acknowledgements Hungarian OTKA F046282, OTKA T037191, and French Government Grant.
Captions b D d F f
bottom product component flowrate distillate flowrate distillate component flowrate feed flowrate feed component flowrate
C. Steger et al.
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dimensionless height liquid component flowrate molar liquid flowrate time molar holdup in the still molar vapor flowrate vapor component flowrate liquid composition vapor composition vapor composition in equilibrium with x feed composition
Subscripts D j ref S
distillate jth component reference component still (middle) vessel
Superscript ~
transformed variable
Greek letter ν
stoechiometric coefficient
References 1. 2. 3. 4. 5. 6. 7. 8.
Sundmacher, K.; Kienle, A. (eds.): Reactive Distillation, Wiley-VCH, Weinheim, 2002 Mujtaba, I. M.; Macchietto, S., Ind. Eng. Chem. Res., 36, (1997), 2287 Guo Z. et al, AIChE J. 49, (2003), 3161 Guo, Z.; Lee, J. W, AIChE J., 50, (2004), 1484 Dragomir, R. M.; Jobson, M., Chem. Eng. Sci., 60, (2005), 4377 Lelkes, Z. et al, AIChE J., 44, (1998), 810 Doherty M. F., Buzad G., Icheme Symposium Series, 128 (1992), A51 Lee, J. H.; Dudukovic, M. P, Comp. Chem. Eng., 23, (1998), 159
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Chapter 1
Heat integration between processes: Integrated structure using stage-wise model Anita Kovač Kralja, Peter Glaviča a
Facultyof Chemistry and Chemical Engineering, University of Maribor, Smetanova 17; Maribor, Slovenia
1. Abstract Integration between processes can reduce energy usage and emissions. Simultaneous method of heat integration between processes has been modified by using a stage-wise mathematical programming model. The model can be used to design heat exchanger networks (HENs) exploiting heat exchange between processes easily and well enough in short time. The method is including streams of different processes which are being heated or cooled with a utility only. The existing heaters and coolers can be left unchanged in original processes or can be used for integrating heat between processes and hot and cold utility can be saved. The proposed method is useful for simultaneous integration between complex industrial processes, applicable to new design or existing process. It can apply efficient transfer of heat between processes by using pinch analysis, too, adding new alternatives to the superstructure. The approach has been illustrated by integrating three existing processes and a retrofitted one using the stage-wise model. The objective was to maximize the annual additional profit of integration between processes. The simultaneous integration of the three processes has forecasted the additional profit of 177 kEUR/a. Keywords: modification, stage-wise model, MINLP model, industrial plant
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2. Introduction Integration between processes can be performed either by pinch analysis or by mixed integer nonlinear programming, MINLP. Rodera and Bagajewicz developed mathematical models for direct and indirect integration in the special case of two plants (1999). They have extended the previous work for a set of more plants (2000) by using mathematical programming (MILP). However, the system of equations is difficult to converge for simultaneous process optimization and heat integration when processes are complex and energy intensive because the number of variables increases with the number of combinations. In this paper we have modified the heat integration between processes (Kovač Kralj et al., 2005) in step 3 by using the stage-wise model. 3. Stage-wise model for heat integration between processes The simultaneous integration between processes enables energy saving. Alternatives suggested by pinch analysis (step 2) of the heat transfer between several internally nonintegrated and integrated (step 1) processes can be included. The fraction of the maximum possible integration between processes can be calculated, too. The parameters of individual process operations are not changed. In this paper the simultaneous method of heat integration between processes in step 3 was modified by using a stage-wise model (Yee and Grossmann, 1990), because it was the simplest and fastest mathematical optimization method available. The heat exchanger network between processes can be optimized using a mixed integer nonlinear programming (MINLP) method which is based on the stage-wise model (heat integration intervals between hot and cold process streams) using superstructural representation. The stage-wise model can be extended for heat integration between different processes P (P = P1, P2, …). An example of a superstructure involving two hot and two cold streams of two different processes is shown in Fig. 1. In it streams of different processes which are being heated or cooled with a utility only are included. Existing coolers zci and heaters zhj can be disposed of by fixed areas (shown shaded). The existing heaters and coolers can be left unchanged in original processes or can be used for heat integration between processes saving the hot and cold utility. Streams of different processes are divided into two sets, hot processes streams (HP) set for hot streams, represented by index i and cold processes streams (CP) set for cold streams, represented by index j. Within each stage of the superstructure, potential exchange between any pair of hot and cold streams of different processes can occur. In each stage k, the corresponding processes stream is split to be directed to an exchanger for a potential match between each hot stream and each cold stream. Index k is used to denote the superstructure − stages of a set of temperature locations (ST). The binary variable zijk denotes the match ij in
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stage k. The outlets of the exchangers are isothermally mixed defining the stream for the next stage. The outlet temperatures of each stage are treated as variables in the optimization model. The number of stages in the superstructure can be set, for instance equal to the number of integration processes P. Utilities in existing coolers zci and heaters zhj are located at both ends of the superstructure. Indices of the existing hot utility (HU) and cold utility (CU) correspond to the heating and cooling utilities in the existing processes. Heat flow rates, inlet and outlet tempeatures are treated as fixed. The existing hot stream H1 of process P1 can be cooled by the existing utility or can be integrated with cold streams C1 and/or C2 of another process P2. Note that the derivation of the superstructure does not require the identification of the pinch point or the partitioning into subnetworks. The network does not involve any stream splitting. using the existing heaters
all combinations of matches between processes zijk = zheijk + zcoijk zijk = zheijk + zcoijk S tage 1 S tage 2
zhj H1-C1
zci
H1-C1 C1 from P2
H1 from P1 heater
using the existing coolers
H1-C2
H1-C2
H2-C1
H2-C1
cooler
C2 from P2
H2 from P1 heater temperature location k=1
H2-C2
H2-C2 temperature location k=2
cooler temperature location k=3
Fig. 1: Two-stage HEN superstructure for heat integration between processes. The stage-wise model for heat integration between processes can include additional equations. The existing coolers zci or heaters zhj can be used in place of zijk for possible heat integration. Therefore, the binary variables zijk have to be divided between the binary variables zheijk or zcoijk which denote existing heaters or coolers in the match ij of stage k: i ∈ HP, j ∈ CP, k ∈ ST (1) zijk = zheijk + zcoijk but only one unit is chosen: i ∈ HP, j ∈ CP, k ∈ ST (2) zheijk + zcoijk ≤ 1 The existing coolers zci can be left at the original place or can be displaced to zijk for integration between processes, but only one exchanger decision is chosen:
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zci + ∑ zcoijk j,k
≤1
i ∈ HP
(3)
The existing heaters zhj can be left at the original place or can be displaced to zijk match for integration between processes, but only one exchanger decision is chosen: (4) zhj + ∑ zheijk ≤ 1 j ∈ HP i,k
The stage-wise model was extended by including additional equations (Kovač Kralj and Glavič, 2005) of heat balances for the heat exchangers performing heat integration between processes: i ∈ HP, j ∈ CP, k ∈ ST (5) Aijk = $ ijk /(Kij LMTDij) Equation 5 presents the calculated heat exchanger area of match (i, j) in stage k (Aijk). The areas can be calculated using the following variables: $ ijk − heat flow rate of match ij in stage k, Kij − overall coeficient of heat transfer between streams i and j, LMTDij − driving force of match ij in stage k using Chen’s approximation (Yee and Grossmann;1990). The existing areas of coolers (Acoexijk) and heaters (Aheexijk) in matches, available for heat integration between processes can be reused and they can be extended by additional areas (Aaijk): (6) Aaijk ≥ Aijk – (Acoexijk zcoijk + Aheexijk zheijk) i ∈ HP, j ∈ CP, k ∈ ST The HEN model of heat integration between processes can include additonal equations of heat integration between hot i and cold j stream only one time in stage k, because of the additional control: i ∈ HP, j ∈ CP (7) ∑zijk ≤ 1 k
Equation 8 can be used to control that only one exchanger in one stage exists: i ∈ HP, k ∈ ST (8) ∑zijk ≤ 1 j
The objective function of heat integration between processes is to maximize additional annual profit (Ctot) which sums up annual savings of hot (Csteam) and cold (CCW) utility and subtracts the annual depreciation (Di = Cfix + Cvar Aexp). The annual depreciation sums up the additional areas (Aaijk), displacement of heat exchanger (Cdis) and buying of new piping (Cpip). The objective function is considering the pay back multiplier (r = 0,216). It searches for the best heat integrated matches between processes: Ctot = ∑(Csteam zheijk $ ijk + CCW zcoijk $ ijk − Cvar Aaijk exp r − Cdis zijk r ijk
− Cpip zijk r) i ∈ HP, j ∈ CP, k ∈ ST (9) The HEN model of heat integration between processes can be extended by using additional binary variables which denote new heat exchanger. The problem is more exact but it is difficult to solve. The binary variables of heat exchanger match ij in stage k (zijk) can be divided between the binary variables zheijk or zcoijk or znijk which denote using existing heaters or coolers or buying new heat exchangers:
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zijk = zheijk + zcoijk + znijk i ∈ HP, j ∈ CP, k ∈ ST but only one unit is chosen: i ∈ HP, j ∈ CP, k ∈ ST zheijk + zcoijk + znijk ≤ 1 A new objective function can be defined now: Ctot = ∑(Csteam zheijk $ ijk + CCW zcoijk $ ijk − Cvar Aaijk exp r − Cpip zijk r
(10) (11)
ijk
− Cdis zijk r − (Cfix znijk + Cv Aijk exp) r) i ∈ HP, j ∈ CP, k ∈ ST (12) Selection of heat integration between the streams can be included into the HEN model of heat integration between processes. In the HEN, hot stream H1 or H2 can be selected, for example: i ∈ HP, k ∈ ST (13) ∑ (zH1jk + zH2jk) ≤ 1 j
In the HEN, cold stream C1 or C2 can be selected, for example: j ∈ CP, k ∈ ST ∑ (ziC1k + ziC2k) ≤ 1 i
(14)
4. Case stady The approach will be illustrated by integrating three existing processes using the stage-wise model (Table 1). The processes are operating within one location and, therefore, heat integration between them can be used (Kovač Kralj et al., 2005). Table 1: Hot and cold streams of processes. Stream Existig solvent plant E402 E204 E202 E404 Existing formalin plant E6 E5 E10 Existing methanol plant AC1 Retrofitted methanol plant Synthesis gas; SG
Ts/oC
Tt/oC
I/kW
119,7 69,2 149,5 152,1
40,0 114,6 164,0 160,0
143,46 81,72 158,05 188,02
15,0 53,3 94,0
54,0 55,1 94,5
74,10 139,32 4142,00
128,0
65,0
7780,50
377,0
290,0
2862,30
Heat integration between an existing and retrofitted methanol processes was carried out including all the equations from 1 to 12, including the selection between existing and retrofitted process (eq. 13 and Table 2). The fraction of the maximum possible integration between existing processes is 0,40. The fraction of the maximum possible integration between the existing and the
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retrofitted processes is 0,402. The optimal matches selected have included the integration between: • retrofitted stream SG and solvent stream E202 (158,05 kW), • retrofitted stream SG and solvent stream E404 (188,02 kW), • retrofitted stream SG and formalin stream E10 (2516,23 kW) and • solvent stream E402 and formalin stream E6 (74,10 kW). This heat integration can save 2936,4 kW of the hot utility and 74, 1 kW of the cold utility by using the existing heat exchanger of streams: E202, E404 and E6, and a new heat exchanger for match SG and E202. Additional profit is 177 kEUR/a. Table 2: Cost data for example processes. Installed costs of heat exchanger: Di = Cfix + Cv A exp Energy costs: - cost of 8 bar steam (C8): - cost of 5 bar steam (C5): - cost of cooling water (CCW): Cost of displacing one heat exchanger (Cdis): Cost of insulation piping (Cpip):
(60 200 + 4 690 ⋅ A0,83) EUR
2,4 2,1 0,3 1 660,0 1 970,0
EUR/(GJ) EUR/(GJ) EUR/(GJ) EUR EUR
5. Conclusions We have applied the simultaneous integration method to transfer heat flow between the processes using the stage-wise model. Simultaneous integration between processes can be performed using the MINLP algorithm to solve the stage wise model. 6. References Bagajewicz, M. J. and Rodera, H. (2000). Energy savings in the total site. Heat integration across many plants. Comput. chem. Engng 24, 1237−1242. Kovač Kralj A., Glavič P. and Kravanja Z. (2005). Heat integration between processes: Integrated structure and MINLP model. Comput. chem. Engng 29, 1699−1711. Kovač Kralj A. and Glavič P. (2005). Optimization by stage-wise model for complex industrial heat exchanger network. Proceedings of ESCAPE − 15, Barcelona, 343−348. Rodera, H. and Bagajewicz, M. J. (1999). Targeting procedures for energy savings by heat integration across plants. American Institute of Chemical Engineering Journal 45, 1721−1742. Yee T. F. and Grossmann I. E. (1990). Simultaneous optimization models for heat integration II. Heat exchanger network synthesis, Comput. chem. Engng 14/10, 1165−1184.
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Generic Model Framework for the Synthesis of Structured Reactive Separation Processes Guido Sand, Markus Tylko, Sabine Barkmann, Gerhard Schembecker, Sebastian Engell Universität Dortmund, Department of Biochemical and Chemical Engineering, 44221 Dortmund, Germany
Abstract A synthesis procedure for structured reactive separation processes based upon a six layer generic model framework is presented. It allows for a systematic determination and optimization of structured arrangements of reaction, separation and integrated functionalities. Its applicability to reactive distillation, reactive adsorption and reactive extraction is demonstrated by examples. Keywords: process synthesis, reactive separation
1. Synthesis of structured reactive separation processes The integration of reaction and separation processes is known to potentially increase the overall process efficiency since thermodynamic and chemical boundaries may be overcome. However, the coupling of reaction and separation reduces the degrees of freedom for process synthesis such that structured arrangements of reaction, separation and integrated (e.g. reaction combined with separation) functionalities (see Fig. 1) may be advantageous. The complex interactions of reaction and separation governed by incomplete information at early stages make the systematic synthesis of structured processes a demanding task. In general, process synthesis is addressed by an iterative multi-layer procedure applying specific methods on the various layers. However, the generalization of synthesis procedures for structured processes and the integration of the synthesis layers are still open issues [1, 2]. In this contribution, a synthesis procedure based upon a framework of consistent models with increasing level of detail is presented (Sec. 2). The procedure is applied to two examples from reactive distillation (Sec. 3), and its generalized applicability to reactive adsorption and reactive extraction is sketched (Sec. 4). For a detailed exposition the reader is referred to [3]. Reaction functionality Separation functionality Integrated functionality
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Fig. 1. Structured arrangements of reaction and separation (from left to right: homogeneous, heterogeneous, partially integrated, and sequential distribution of functionalities)
2. Generic model framework A process synthesis task is specified by a given substance system (including raw materials and products) and its objectives. The synthesis procedure is based upon the model framework shown in Table 1. It applies the concept of generalized building blocks comprising one or two phases which provide either reaction functionality (one reactive phase, e.g. in a CSTR-type reactor), separation functionality (two transport phases, e.g. on a tray of a distillation column) or integrated functionality (reactive phase and transport phase, e.g. on a tray of a reactive distillation column). In practice, these building blocks are aggregated in order to increase the efficiency of the functionalities (e.g. reactor cascades, distillation columns or reactive distillation columns). In the proposed generic model framework, the level of aggregation is decreased from top to bottom. The procedure starts with the selection of a phase system, it proceeds with the determination of a coarse structure of aggregated building blocks and ends up with a fine structure of disaggregated building blocks. Necessary thermodynamic and kinetic data are assumed to be available on each layer. On the first layer, only the reaction phase is considered. Based on the analysis of the reaction kinetics, incentives for an integrated transport phase are identified. Accordingly, possible transport phases are evaluated with respect to their separation effect and operating conditions, resulting in a favorable integrated process (layer two). On the third layer, the integrated functionality as provided by aggregated building blocks comprising a reactive phase and a transport phase is studied under simplifying assumptions: In each of the building blocks chemical and phase equilibrium is assumed and the aggregate is provided with a single feed only. Under the aforementioned conditions, all concentration trajectories lie within a reduced dimensional reaction space. A simulation based analysis of the reaction space in transformed coordinates [4] results in product regions which can be reached in principle. If the objectives cannot be met by concentrations from within the reactive product regions, the consideration is broadened onto the entire component space (layer four). Additional reaction and separation functionalities as provided by aggregated building blocks with two transport phases or a single reaction phase in addition to multiple feeds are considered. They are evaluated with respect to their potential to manipulate the concentration trajectory appropriately. On layer five, the simplifying assumptions are dropped such that the preliminary process structure from layer four is mapped into a superstructure. The building blocks are disaggregated and their existence, thei r functionalities and the distribution of feed streams become degrees of freedom. The optimum structure is determined by MINLP techniques subject to material and energy balances in addition to geometric and the operating conditions of the regarded equipment with respect to an economic objective. The resulting optimized conceptual process design can be extended further based upon a sensitivity analysis of the optimal solution (layer six). Table 1. Generic model framework
Layer 1 Reaction phase 2 Integrated phase 3 Integrated functionality
Method Reaction kinetics analysis Phase equilibrium analysis Reaction space analysis
Result Incentives Integrated process Product regions
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4 Combined functionalities 5 Process superstructure 6 Optimized design
Component space analysis MINLP optimization Sensitivity analysis
Process structure Process design Extensions
3. Reactive distillation 3.1. Production of methyl tertiary butyl ether In this example, the objective is to produce high-purity methyl tertiary butyl ether (MTBE) and to separate high-purity n-butane from methanol (MeOH) and a mixture of isobutene (IB) and n-butane. Layer 1: The reaction in liquid phase is equilibrium limited. Neither MTBE nor nbutane can be obtained with high-purity, thus they have to be separated from the reaction phase. Significant production rates of MTBE are obtained at temperatures between 50°C and 110°C. Layer 2: For separation, a liquid transport phase (leading to reactive extraction) is unsuitable due to the small miscibility gap and the tie lines being orthogonal to the MTBE concentration gradient. A vapor phase (leading to reactive distillation) is promising as the reactive distillation lines are next to n-butane. The range of boiling points at 8 bar (60°C-136°C) is overlapping with the abovementioned reaction conditions. The creation of a transport phase by energy supply is preferred to the creation of a second phase by auxiliary components, which leads to recycle streams and additional separation steps. Consequently, adsorption, absorption and membrane processes are not considered. Layer 3: The reaction space exhibits four reactive distillation areas (RDA), two of which of technical relevance (Fig. 2). The feed concentration should be in RDA1 to shift the product region towards the n-butane vertex. The product region is bordered by the material balance lines, the reactive distillation line through the total feed concentration and the reactive distillation boundary. Nevertheless, MTBE is not in the product region. Layer 4: A separation functionality adjacent to the high-boiling edge of the integrated functionality is necessary to purify MTBE. A distillation process is a feasible choice as MTBE is a stable node in distillation area DA 2 (Fig. 3). As a prerequisite, the concentration at the interface between the integrated and the separating functionality has to be in DA2. This can be achieved either by reaction or by positioning the low-boiling IB/n-butane feed accordingly. Furthermore, the concentration has to be in RDA1 only at the low boiling edge of the integrated functionality, which can be achieved by positioning the feed of the relatively high-boiler MeOH accordingly. As their operation windows overlap, both functionalities can be implemented in one apparatus.
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Layer 5: The superstructure of the reactive distillation column comprised 60 optional trays with optional MeOH and IB/n-butane feeds and optional reactive holdup [5]. The reaction was modeled as kinetically controlled, and non-ideal separation was modeled
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by the gas phase Murphree efficiency. The existence of reaction functionality is controlled by the continuously variable reactive holdup, and the existence of separating functionality is controlled by a binary switch of the efficiency (zero or a finite value). The geometry was constrained by a maximum height of the reactive holdup and maximum and minimum vapor velocities. The economic cost function comprised annualized investment costs plus annual operating costs. The solution obtained by classical MINLP solvers [6] can slightly be improved by global techniques (profit of 1,050·103 € p.a. vs. 1,051·103 € p.a.). The best basic process design obtained is shown in Fig. 4. Layer 6: An analysis exhibited that the lower vapor velocity constraints on the lower trays and the holdup constraints were binding. In order to decrease the vapor load and to increase the reactive holdup, the superstructure was extended by an external reactor with heat exchanger. The solution obtained by the classical solver could by significantly improved by the global one (1,080·103 € p.a. vs. 1,102·103 € p.a.); the optimized extended design is shown in Fig. 5 [7].
xH2O= 99.6%
3.2. Production of methyl acetate In the second example, the objective is to produce high-purity methyl acetate (MeAc) from methanol (MeOH) and acetic acid (HAc) with water as by-product. Layers 1-4 of the synthesis procedure lead to an aggregated process structure comprising an integrated functionality augmented by a separation functionality adjacent to its low-boiling edge. The low-boiler MeOH is proposed to be fed next to the high-boiling edge of the integrated functionality in order to extend the product area to the water vertex. The high-boiler HAc is proposed to be fed next to the low boiling edge of the separation functionality in order to break the azeotropes MeAc/MeOH and MeAc/water. The disaggregated cost-optimized design differs significantly wrt. two aspects from the preliminary structure (see Fig. 6): Firstly, a separating functionality is employed adjacent to the high-boiling edge of the integrated functionality in order to purify water. Secondly, the HAc feed is positioned close to the MeOH feed. Despite this unpredicted position, the HAc concentration on the upper trays is sufficiently high to break the azeotropes. Fig. 6. Basic process design (MeAc)
4. Generalized applicability 4.1. Reactive adsorption In the third example, the objective is to produce high-purity fructose from a glucose/fructose-mixture. In order to overcome the chemical equilibrium in the liquid phase, a solid transport phase is proposed (reactive chromatography). In the one dimensional component space the reaction space collapses to a point, such no separation
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effect is gained from a homogenously distributed functionality. A de-integrated structure (similar to Fig. 1) is proposed, which is realized as a Hashimoto-type simulated moving bed reactor [8]. 4.2. Reactive extraction In the forth example, the objective is to produce high-purity mono-telomere (MT) from butadiene (But) and ethylene glycol (EG) in aqueous phase, along with a minimal loss of palladium catalyst. The employment of a sole reaction phase is prohibited by three side reactions. To increase the selectivity, a liquid organic transport phase is proposed to remove MT from the reaction phase and provide the reaction phase with But in low concentrations. The proposed structure comprises three stages (plus flash) in a crosscounter-current flow pattern (Fig. 7). The interstage feeding of butadiene causes a high difference in polarities between the liquid phases in stage 3 such that MT is separated and catalyst is regained. The EG-feed position causes a low polarity difference in stage 2 such that the reaction rate is increased. A more detailed description of the examples in sections 4.1 and 4.2 can be found in [3]. Butadiene recycle But Feed
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Fig. 7. De-integrated cross-counter-current process design
5. Conclusions A process synthesis procedure for structured reactive separation processes based upon a general modeling framework was presented. Four examples, regarding different phase systems, showed that the general procedure and concepts can successfully be applied to various process types. In none of the cases, a single integrated functionality was sufficient to reach the process design objective. Furthermore, it was shown, that the detailed optimization based upon disaggregated building blocks may lead results which differ significantly from the rough considerations based upon aggregates of building blocks. Nevertheless, while the simulation techniques used on the aggregated model layers three and four are well established, robust MINLP-techniques as used on the disaggregated layers five and six are currently available for reactive distillation only. The transfer of these results to reactive extraction is subject to ongoing research.
6. Acknowledgements The financial support from the Deutsche Forschungsgemeinschaft under grant EN 152/33-4 as part of the research unit "Integrated Reaction and Separation Processes" and from the graduate school of "Production Engineering and Logistics" at Universität Dortmund is gratefully acknowledged.
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References 1. G. Schembecker and S. Tlatlik, Chem. Eng. Process., 42 (2003) 179. 2. C. Almeida-Rivera, P. Swinkels and J. Grievink, Comput. Chem. Eng. 28 (2004) 1997. 3. H. Schmidt-Traub and A.Górak (eds.), Process Intensification by Integrated Reaction and Separation Operations, Springer, Berlin, 2006, chapter 2 (to appear). 4. S. Ung and M. Doherty, Ind. Eng. Chem. Res. 34 (1995) 3195. 5. A. Barbosa-Povoa and H. Matos (eds.), European Symposium on Computer-Aided Chemical Engineering-14, Elsevier, Amsterdam, 2004, 493-498. 6. C. Floudas and R. Agrawal (eds.), Sixth International Conference on Foundations of Computer-Aided Process Design, CACHE, Austin, 2004, 319-322. 7. P. Jansens, A. Stankiewicz and A. Green (eds.), Sustainable (Bio)Chemical Process Technology, BHR Group, Bedfordshire, 2005, 317-324. 8. K. Hashimoto, S. Adachi, H. Noujima and Y. Ueda, Biotechnol. Bioeng., 25 (1983) 2371.
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Knowledge Extraction during the Design of Activated Sludge Systems Xavier Floresa, Manel Pocha, Ignasi Rodríguez-Rodaa, Laureano Jiménezb and René Bañares-Alcántarac a
Laboratori d’Enginyeria Química i Ambiental, Universitat de Girona, Campus Montilivi s/n, 17071 Girona, Spain. b Departament d’Enginyeria Química, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain. c Department of Engineering Science, University of Oxford, Parks Roads, OX1 3PJ Oxford, United Kingdom.
Abstract This paper presents a support module that extracts and reuses the knowledge generated during the sensitivity analyses for design of activated sludge plants. This support module, consists of three steps: 1) Data handling and knowledge extraction, where all the data generated by numerical simulators during the sensitivity analyses is processed in order to extract qualitative knowledge, 2) Knowledge codification, where the extracted knowledge is codified as IF-THEN rules, and 3) Knowledge reutilization, where numerical and qualitative methods are integrated to guidethe designer through the design process. The usefulness of the support module is demonstrated in a case study, where the bioreactor of a denitrifying activated sludge plant is retrofitted to achieve simultaneous carbon, nitrogen and phosphorous removal. Keywords: Activated sludge, Process design, Nutrient removal, Knowledge extraction, Multicriteria analysis.
1. Introduction. The design of activated slu dge systems is particularly complex because it includes biological multiphase reaction, solid-liquid separation and several types of recycles and purges (Metcalf & Eddy, 2003). Moreover, activated sludge systems must take simultaneously into account a variety of objectives with different relevance; as a result, the evaluation process becomes a multicriteria problem (Flores et al., 2005). Nowadays, the design of activated sludge systems is supported by the use of numerical simulators (Henze et al, 2002) and by (iterative) sensitivity analyses near a base case to account for the inherent uncertainty of the design variables. Large amounts of data are generated during these analyses which, conveniently processed, could be transformed into knowledge and used by the designer during the rest of the design process. The objective of this paper is to present a design support module that extracts and reuses the knowledge generated during the sensitivity analysis process.
2. Methodology. First, a base case is defined. The relative desirability of the base case and of any alternative case can be evaluated through a set of different evaluation criteria [X = {X1,...,Xy}] that measure the satisfaction of several design objectives [OBJ = {OBJ1,...,OBJx}].
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The importance of each objective is set by its weight [w = {w1,...,wx}] which reflects the preferences of the stakeholders. Weightse ardistributed among the evaluation criteria [X] and normalized to 1 in order to avoid unbiased comparisons
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Thus, a design can be measured by a score profile: [A = (X1,…, Xy)]. Then, value functions of the form [v (Xj)] map the score profile of the proposed option [v(A) =( v(X1),…, v( Xy))] into a normalized value from 0 to 1, where the 0 and 1 values are associated with the worst (Xj*) and the best (Xj*) situation, whilst a mathematical function is used to evaluate the intermediate situations. Finally, the weighted sum is computed according to equation 1, using the normalized criterion v (Xj) and its corresponding weight (wj). s(A) = w 1 ·v(X 1 ) + ... + w j ·v(X j ) + ... + w y ·v(X y ) =
y
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The score in the weighted sum indicates e degree th of satisfaction of an alternative design with respect to the set of design objectives [OBJ]. A value of 1 for v(A) would denote that design objectives are fully satisfied. A set of analyses using numerical simulators is then performed to test the sensitivity of the base case design with respect to the variables that rule the design [V ={Vi, …, Vz}]. In these sensitivity analyses, evaluation criteria [X] are recalculated varying the value of the selected variables [V] by a percentage (± 5-10 %) with respect to the base case. All the data generated during these analyses is processed by the proposed design support module. This support module extracts and reuses the knowledge generated during the sensitivity analyses thorough a three step systematic procedure: In Step 1, all the data is handled and qualitative knowledge is extracted by the proposed design support module. The objective of the first step is to classify the relationship between the effect on the evaluation criteria [v(X)] and the analyzed variables [V] in four different categories: (a) directly proportional, (b) indirectly proportional, (c) constant and (d) no monotonic when the value of the analyzed is moved. In Step 2, thisknowledge is codified as IF-THEN rules. This set of rules [R = (R1,1,…,Rz,y)] conforms a knowledge base that contains the information from the analysis performed at step 1. For example, IF Vk is increased THEN Xj decreases. Next, these rules are used as the input of an inference engine that keeps a record of design decisions together with the influence of design variables on the behaviour of the evaluation criteria. Finally, in Step 3, the codified qualitative knowledge extracted from the detailed models can be reused guiding the designer through the design process. Thus, the designer can be supported by the system suggesting which actions [Ac] should be implemented to improve the design base case in terms of a specific objective OBJi or criteria Xj. Additionally, the system also informs the designer which criteria worsen in the event of the application of the proposed actions.
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3. Case study. 3.1. Plant Redesign and Sensitivity Analysis. This paper is illustrated with a case study, where a bioreactor of the benchmark denitrifying activated sludge plant (Copp 2003) is retrofitted to achieve simultaneous carbon, nitrogen and phosphorus removal. The plant has a modified Ludzack-Ettinger configuration with five reactors in series (tanks 1 and 2 are anoxic with a total volume of 2000 m3, while tanks 3, 4 and 5 are aerobic with a total volume of 4000 m3) linked with an internal recirculation from the 3rd aerobic tank to the 1st anoxic tank, a 10-layer secondary settling tank (with a total volume of 6000 m3) and two PI control loops. The first loop (OD) controls the dissolved oxygen in the 3rd aerobic tank by the manipulation of the aeration flow, and the second loop (NO) controls the nitrate in the 2nd anoxic tank by manipulating the internal recycle flowrate. For this new plant (A), five variables (z = 5) were selected for the sensitivity analyses: anaerobic volume (V1), waste flow (V2), recycle flow (V3), OD set-point (V4) and NO set-point (V5). All these variables were varied a maximum of ± 10 % considering 4 points above and 4 points below the base case (i.e. every ± 2.5 %). We studied fifteen criteria (y = 14) to measure the accomplishment of the four redesign objectives (x = 4): minimize environmental impact (OBJ1), minimize economic costs (OBJ2), maximize technical reliability (OBJ3) and comply with the limits fixed by the 91/271/ECC Directive (OBJ4). In this case study we assume equal importance for all four redesign objectives, wi= 0.25. Thus, for OBJ1, criterion X1 (global cleaning efficiency) is proposed (Gernaey and Jorgensen, 2004). Construction costs (X2, U.S EPA, 1982) and operation costs (X3, Vanrolleghem and Gillot, 2002) are used for OBJ2. Plant robustness (X4) and flexibility (X5), nitrate control performance (X6) and sensitivity to separation problems (X7-1= foaming risk; X7-2 = bulking risk and X7-3 = rising risk) measure the satisfaction of OBJ3. A detailed description of X4 and X5 can be found in Flores et al., (2005) while X6 and X7 are reported in Copp (2003) and Comas et al., (2005) respectively. Finally, X8-X12 reflect the percentage of time that the concentration of the pollutant exceeds the legal limits or time plant in violation (TIV) for TSS, COD, BOD5, TN and TP (Copp 2003). X2 is calculated by model based estimation with the CAPDET model (U.S EPA, 1982) and the rest of criteria through dynamic simulation. The IWA Activated Sludge Model # 2d was chosen as the biological process model (Henze et al. 2002) and the doubleexponential settling velocity function of Takács et al. (1991) was chosen as a fair representation of the settling process Once the criteria are quantified a score profile [A] for the proposed redesign is obtained. Hence, the score of the base case is [A = (82.03 %, 7.7·105 €, 9.1·105 €·year-1, 14.70, 10.27, 0.78 (gN/m3)2·d, 0 %, 61.46 %, 31.10 %; 0 %, 0 %, 0 %, 90.18 %, 29.32 %).] The score profile [A] is normalized from 0 to 1 using value functions: v (A) = (0.82, 0.38, 0.91, 0.74, 0.51, 0.22, 1.00, 0.39, 0.69, 1.00, 1.00, 1.00,0.10, 0.71). As an example, criterion X1 has the value function v(X1) = 0.01·X1 associating the best (1) and the worst (0) situation with 100% and 0 % respectively. Finally, the weighted sum (eq. 1) for the base case is calculated, with a result of 0.69. The fifteen evaluation criteria are recalculated every 2.5 % for the five variables analyzed. Figure 1 shows an example of two evaluation criteria (X11 and X12) during the sensitivity analysis of variable V1.
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Figure 1. Evolution of TIV for the sensitivity analysis of V1. (a) TN (X11); (b) TP (X12.). 3.2. Knowledge Extraction and Reutilization Step 1. All the data generated in the previous analyses is used for the extraction of qualitative knowledge. The objective is to find the correlations between the five analyzed variables with the fourteen criteria. Trends in the evaluation criteria are computed and then classified in the four categories summarized in Table 1. Step 2. Once all this knowledge is extracted from the data is codified in form of IFTHEN rules. The set of rules [R= (R1,1,…R5-12)] conforms a knowledge base containing the relationship between 5 variables and 15 evaluation criteria. Finally, these rules are used as input to the inference motor (e.g. CLIPS). As example, the syntaxis of a couple of rules extracted from Table 1 are shown:. eases THEN TIV for P (X 12) decreases. R1-12 = IF V1 (anaerobic volume) incr R5-6 = IF V5 (NO setpoint) increases THEN control perfomance (X6) increases Table 1. Evolution of TIV for the sensitivity analysis of V1. (a) TN (X11); (b) TP (X12.).
V1 V2 V3 V4 V5
Indirectly proportional v(X6), v(X7-3), v(X12) v(X7-1), v(X7-2), v(X7-3), v(X12) v(X3), v(X4), v(X5), v(X6), v(X11) v(X4), v(X5), v(X6), v(X11)
Directly proportional v(X2), v(X4), v(X5), v(X7-2), v(X11) v(X1), v(X3), v(X5), v(X6), v(X11) v(X7-2), v(X7-3), v(X12) v(X7-2),v(X7-3), v(X12)
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Constant v(X7-1), v(X8), v(X9), v(X10) v(X2), v(X8), v(X9), v(X10) v(X2), v(X7-1), v(X8), v(X9), v(X10) v(X2), v(X7-1), v(X8), v(X9), v(X10) v(X2), v(X7-1), v(X8), v(X7-2), v(X9), v(X10)
Non monotonic v(X1), v(X3) v(X4) v(X1) v(X1) , v(X3) -
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Step 3. In this section is shown, how the rules codified in Step 2 can be reused to identify the most suitable design variables to improve the achievement of a particular criterion (Xj). The proposed scenario proposes a tight compliance in the limits fixed by the European Directive concerning nitrogen concentration (X11). This scenario is relevant because for the base case 90.18 % of the time there is a violation of the nitrogen, while the design removes all other pollutants (TSS, COD, BOD and TP)(see Figure 1a for more details). To analyze any scenario, the system can search automatically all the design variables that are implied to improve this new requirement. At the same time, the system highlights the negative effects of the proposed actions in other criteria. The following examples clarify the situation. If criterion X11 (TIV for nitrogen) has to be improved, the positive effects of the proposed actions had to be balanced with the negative associated. The proposed actions are summarised in Table 2. Table 2. Proposed action to improve criteria X11 and adverse effects of this modification Action 1 Ac1 Ac2 Ac3 Ac4 Ac5
Description and adverse effects Criterion X11 can be improved decreasing V1; but When V1-1 is decreased, X6, X7-3 and X12 decrease Criterion X11 can be improved decreasing V2, but When V1-2 is decreased, X7-1, X7-2 , X7-3 and X12 decrease Criterion X11, can be improved increasing V3 but When V1-3 is increased X7-2, X7-3 and X12 decreases Criterion X11, can be improved increasing V4, but When V1-4 is increased X7-2, X7-3 and X12 decreases Criterion X11, can be improved increasing V5 but When V1-5 is increased X3 , X6 and X12 decreases.
Then, numerical and qualitative methods are integrated to evaluate the convenience of the proposed actions in different ways. For example, ordering the actions according to the highest decrease in X11 (without considering the rest of criteria), the recommended actions would be arranged as follows: Ac3 (-12.38%), Ac1 (-8.09%), Ac2 (-7.59%), Ac4 (-1.98%) and finally Ac5 (-0.66%). Another approach is evaluating the actions considering their overall effect on the redesign objectives [OBJ], selecting only those actions that suppose a net gain in the weighted sum (eq1). Thus, from the above mentioned actions, only two (Ac2 and Ac3) are recommended, as they achieve a gain in the objective function of 0.66% and 0.08% (i.e. lower time plant in violation). Another approach could be screening of actions that suppose a decrease in the effect of a certain criteria Xj , or a combination of several criteria simultaneously. As in many other tools, the inference system indicates which actions should be revisited, but the final decision remains on the designer. A positive side effect of this procedure is that it keeps a record of the design process in form of rules. On the one hand, because enables the understanding of the whole design process and the designer is conscious of both weak and strong points of any possible action. On the other, because improves the communication both among designers and stakeholders, such as present and future inspectors. Furthermore, plant modifications are made easier in the event of changes in legislation, technologies or plant capacity.
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4. Conclusions This paper presents a design support module that extracts and reuses the knowledge generated in the sensitivity analyses carried out during the design of activated sludge plants. The significance of this approach resides in the advantages of integrating numerical and qualitative methods for decision making. Detailed numerical models are used to recalculate the evaluation criteria around the base case, from which qualitative knowledge is extracted and used in turn, to guide the designer thorough the design process. As a result, it is possible to: (a) reduce the cognitive load on the designer and enhance his/her understanding of the process; (b) improve decision making, e.g. decreasing the number of design iterations, by the concurrent manipulation of multiple criteria –more than a dozen in the case study- highlighting both strong and weak points of the proposed actions and (c) reuse the knowledge (geographically and temporally) that otherwise would be tacit and only usable by a single designer.
References Copp J.B. 2003. Respirometry in Control of the Activated Sludge Process: Benchmarking Control Strategies. IWA Scientific and technical Report No.11. Comas J, Rodríguez-Roda I., Poch M., Gernaey K., Rosen C. and Jeppsson U. 2005. Extension of the IWA/COST simulation benchmark to include expert reasoning for system performance evaluation. 2nd IWA Conference on Instrumentation, Control and Automation for Water and Wastewater Treatment and Transport System (ICA). Gernaey, K.V. and Jørgensen, S.B. 2004. Benchmarking Combined Biological Phosphorus and Nitrogen Removal Wastewater Treatment Processes. Control Eng. Pract., 12, 357-373. Flores X., Bonmatí A., Poch M., Bañares-Alcántara R., and Rodríguez-Roda I. 2005. Selection of the Activated Sludge Configuration during the Conceptual Design of Activated Sludge Plants. Ind. Eng. Chem. Res. 44, 3556. Henze M., Gujer W., Mino T., and van Loosdrecht M.C.M. 2002. Activated Sludge Models ASM1, ASM2, ASM2d and ASM3. IWA Scientific and Technical Report No. 9 IWA. London Metcalf & Eddy. 2003. Wastewater Engineering: Treatment, Disposal and Reuse. McGraw-Hill: New York. U.S. EPA - Office of Water Program Operations. 1982. Process Design and Cost Estimating Algorithms for the Computer Assisted Procedure for Design and Evaluation of Wastewater Treatment Systems (CAPDET), Ed. Harris R.W., Cullinane M.J.Jr. and Sun P.T., Vicksburg, Mississippi (EPA ref: PB82-190455). Takács, I., Patry, G.G. and Nolasco, D. 1991. A dynamic model of the clari fi cation thickening process. Wat. Res., 25(10), 1263-1271. Vanrolleghem, P. and Gillot, S. 2002. Robustness and Economic Measures as Control Benchmark Performance Criteria. Wat. Sci. Tech., 45(4/5), 117-126.
Acknowledgements The authors gratefully acknowledge financial support from Spanish “Ministerio de Ciencia y Tecnología” (DPI2003-09392-C02-01). The authors also wish to thank Krist V. Gernaey (CAPEC, Technical University of Denmark) for providing the ASM2d benchmark MatLab/Simulink© code to carry out the simulations and sensitivity analyses.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Pharmaceutical process development applying automated laboratory reactors Tibor Chován,a Imre Markovits,b Béla Farkas,b Kálmán Nagy,b Lajos Nagy,a Károly Nyíri,b Ferenc Szeifert,a a
University of Veszprém, POB 158, Veszprém, H-8200, Hungary EGIS Pharmaceuticals Ltd., Keresztúri út 30-38., H-1106, Hungary
b
Abstract Application of automated laboratory reactor systems and suitable model-based approaches can speed up significantly the process development of active pharmaceutical ingredients. The paper presents the design and functions of such a reactor system including the recipe-oriented control solutions. The system performance was enhanced by implementing a model-based algorithm for estimating the process heat flow and therefore allowing the application of reaction calorimetric approaches. The application is demonstrated on two pharmaceutical process development problems. In the first, the calorimetric function was used for safety risk evaluation of a chemical reduction step. In the second, optimal process parameters of a crystallization step were determined. These examples underline that pharmaceutical and fine chemical industries can benefit from the application automated reactors and appropriate model-based methods. Keywords: reaction calorimetry, process development, crystallization, risk evaluation, pharmaceutical industry.
1. Introduction Majority of pharmaceutical and fine chemical processes are accomplished in batch or fed-batch operations. Process development generally means scaling up the batch process in a few steps. The cost and time of the development can be significantly reduced by applying efficient and reliable laboratory and pilot systems. These requirements can only be achieved by well instrumented and accurately controlled, still flexible systems. The information gathered by these systems at different scales can provide a sound basis for the later application of simulation-based approaches which opens lots of new possibilities in process development. The typical unit of pharmaceutical and fine chemical processes is an autoclave with a stirrer, a heating-cooling jacket, feeding systems and a condensator with collectors. This unit inherently has reasonable flexibility since it can execute lots of different operations (like thermal operations, reactions, crystallization, extraction, distillation, etc.). This study presents a laboratory scale solution and its application. Main objectives of the use of laboratory scale equipments are to collect kinetic information of the processes taking place, to optimize the laboratory procedures experimentally and to provide efficient recipes for the further development steps.
2. Automated laboratory reactor systems At laboratory scale the main tool is the automated laboratory reactor system (ALRS) which allows conducting well defined and reproducible experiments, optimizing the process experimentally and, with suitable analytics, determining kinetic information.
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Besides fulfilling the above requirements, the ALRS developed here is prepared to execute the same process operations as those of the pilot and plant scale, particularly: • reactions: different reactions under predefined conditions, • heating and cooling operations: to given temperature or following a time program, • feeding: one or more components at given flow rates, • boiling operations: reflux and distillation, • crystallization: different methods (cooling, evaporation, dilution), • extraction: component removal (mixing and feeding). 2.1. The design and control of the reactor system The system was designed with a modular structure assuring that new equipments and instrumentation can be easily implemented. The structure of the reactor system is given in Fig. 1 in form of an overview screen. The precise and reproducible experimental procedures are assured by an extensive instrumentation and a PC-based control subsystem. This subsystem can also be implemented on existing, manually operated laboratory reactors. The main features of the control system are: • collection of all measured and calculated data as well as process events, • export of collected data for later evaluation, • control of experimental procedures based on recipes (separate recipe editor), • control of independent instrumentation (pumps, stirrer, thermostat, etc.), • precise reactor temperature control by model-based algorithm, • several operator screens for monitoring and control of the experiments.
Figure 1. Overview screen of the reactor system
2.2. Reaction calorimetry Reaction calorimetry provides a very general method to evaluate the heat effects of different processes taking place in the reactor and to follow the general progress of processing steps. Special laboratory instruments, the so called reaction calorimeters, allow precise measurements of heat effects, however their implementation and operation is quite expensive. In many cases a well designed and instrumented reactor system complemented with suitable evaluation software can provide adequate information for the analysis of the heat effects [1]. Calorimetric functions of the automated laboratory reactor system can be derived from the following heat and component balances of the reactor:
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dT1 = F1 c p (Tin,1 − T1 ) + F2 c p (Tin ,2 − T1 ) + dt T + T3 + α F( 2 − T1 ) − (α F ) v (T1 − Te ) + Q 2
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The first two terms r.h.s. of the unsteady-state heat balance are the heat flows carried by the two feeds, the 3rd is the heat transfer between the jacket and the reactor, the 4th is the heat loss, while the 5th is the gross heat flow corresponding to the processes (e.g. reactions, crystallization, etc.) taking place in the reactor. The calorimeter function involves determining the heat flow, Q (t ) . First the model parameters ( c p ,α F , (α F ) v ) are estimated by evaluating data collected in some experiments. Then the Q (t ) heat flow can be calculated on-line based on the measurements of F1 , F2 (or maybe m ) as well as the (accurate) measurements of temperatures involved in the model. Based on the measured and estimated variables of the model, Q (t ) can be calculated applying different methods. Difficulties might emerge calculating the differential term, dT1 / dt from measured data. To avoid this problem, a technique based on an inverse formation was elaborated to estimate the Q (t ) heat flow. This calculation was available in the model-based control algorithm of reactor temperature almost as a “side product”, therefore in spite of its apparent complexity it can be employed at low “cost”. measured variables estimated parameters
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The inverting was solved using the method known from the Globally Linearizing Control (GLC) [2] as shown in Fig. 2. Since the relative order of the nonlinear model, Eq. (1) is one, the following linear model can be defined between the temperature described by the model and the measured reactor temperature ( T1,m ):
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This way Eq. (3) provides the process heat flow, Q (t ) . Modifying parameter τ , the errors caused by the measurement noise can be appropriately reduced. The accuracy of the method was tested by a standard heat flow generated by an electric heater.
3. Industrial application of ALRS The following two application examples are taken from real-life process development problems of EGIS Pharmaceuticals Ltd. These examples are aimed at demonstrating the broad range of possible applications and the benefits expected. 3.1. Study of a reduction step of a pharmaceutical intermediate 3.1.1. Safety risk evaluation of reaction Safety risk evaluation is an important issue before scaling-up the process. The questions of interest are the rate of the heat evolution and whether it carries any risk regarding the scale-up. The observed temperature profile and the heat flow curve of the examined reaction (a reduction of a carbon-carbon double bond with sodium borohydride using cobalt containing catalyst) can be seen in Fig. 3. The three peaks correspond to the three heat evolving steps: preparation, reaction, and cleavage. The main observations are: The heat flow curve shows sensitively the exothermic reaction steps. The reaction heat for each peak ( Q prep , Q react , Qcleav ) can be
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calculated and their ratio is Q prep : Q react : Qcleav = 1 : 20 : 30 . During the cleavage the prompt heat evolution is able to warm up the reaction mixture by 15°C. In the worst case, if all heat is released at the same time, this is still not enough to make the reaction mixture boil. Therefore the conclusion is that there is only a minor safety risk. On the other hand, it is proved, that ALRS is useful to get a quick, still sufficiently accurate estimation on the safety risk, enabling to speed up the process development. 3.1.2. Heat flow curve and the conversion The objective was to reveal whether the heat flow analysis can reliably indicate the reaction rate. Integrating the heat flow curve of the reduction allows the estimation of the conversion. For comparison, a periodic sampling and a subsequent off-line analysis by high-pressure liquid chromatography (HPLC) was performed to determine conversion.
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It can be concluded that the agreement between the two methods is reasonably good at the end of the reaction; therefore the heat flow measurement can serve as an online analytic tool determining the end of reaction as shown on Fig. 4. 3.2. Crystallization process development of an active pharmaceutical ingredient (API) In the second example the optimization of a crystallization step of an API was aimed at. The objective was to determine the parameters or parameter sets which have significant influence on the particle size distribution (PSD) of the crystalline product. Two crystallization techniques were examined: crystallization by cooling and by dilution. 3.2.1. Cooling crystallization A typical crystallization heat flow curve is shown on Fig. 5. The single peak refers to the crystallization, and features of the peak convey important information for the process development. The start of the peak indicates the start of crystallization; this feature made it possible to determine the width of the metastable zone and revealed the effect of stirrer speed. The metastable zone turned out to be quite broad supporting the application of seeding. The effect of stirrer speed during crystallization is rather interesting. The application of high stirrer speed caused a slight decrease in the median crystal size of the API. The plausible explanation is the attrition effect of the stirrer, but this was not justified by the experiments. Instead, the heat flow analysis revealed that in this case the crystallization started at higher temperature, which means, that the stirrer speed has larger effect on the crystal nucleation than on crystal growth. For the comparison of PSD of the API produced by different methods the median crystal size was used. The particle size distribution obtained and the experimental conditions are presented on Fig. 6 and 7. 100 Treactor
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The experiments showed, that large crystals were produced by seeding at a temperature as high as possible, cooling with a low cooling rate (cooling rate C) and applying the suspension reheating-cooling method (D, E), on the other hand small crystals formation is favored using very fast cooling rate (A) and high stirrer speed. 80
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3.2.2. Crystallization by antisolvent addition (dilution) The particle size distribution was studied during the crystallization of the API by antisolvent addition. The parameters to alter were the feeding rate of the antisolvent, the intensity of mixing, the temperature of antisolvent addition, the combined cooling and salting-out method. It was found that the most sensitive parameter on PSD is the feeding rate of the antisolvent, which is inversely proportional to PSD. The median crystal size was between 25-75 μm varying the parameters of crystallization.
4. Conclusions The automated laboratory reactor system presented in the paper provides great flexibility in process development through its modular design and advanced control solutions. The algorithm implemented to estimate the process heat effects allows the application of reaction calorimetric approaches. The two real-life application examples, one from chemical and one from the physico-chemical development, illustrate well how ALRS can be used in the pharmaceutical process development, and justify that it is an essential tool for speeding up this type R&D activity.
References [1] R.N. Landau, 1996, Thermochimica Acta, 289, 101. [2] J. Madar, J. Abonyi, F. Szeifert, 2005, Artificial Intelligence, 18, 341.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Addressing the design of chemical supply chains under demand uncertainty Gonzalo Guillén, Fernando D. Mele, Antonio Espuña and Luis Puigjaner Universitat Politècnica de Catalunya, Chemical Engineering Department, E.T.S.E.I.B., Diagonal 647, E-08028, Barcelona, Spain
Abstract This work addresses the design of chemical supply chains (SCs) with multi-product, multi-echelon distribution networks under demand uncertainty. Given a design horizon consisting of several time periods in which product demands materialize, the objective is to select the design that maximizes the expected profit. To tackle this problem, it is derived a multi-stage stochastic formulation wherein the design decisions are made irrespective of the realization of the uncertain parameters. A special decomposition technique is introduced aiming at the overcoming of the numerical difficulties associated with this formulation. At the upper level of this strategy (master problem), a genetic algorithm GA is implemented for managing the design problem and proposing potential SC structures. At the inner level (slave problem), a two-stage stochastic SC scheduling model is computed within a rolling horizon mode in order to provide an estimate of the expected profit achieved by each SC design under the uncertain environment. Computational results indicate that the proposed approach performs better than the rigorous formulation for a wide range of CPU times fixed to the solver, i.e. our algorithm leads to better solutions in terms of optimality gap than the multistage stochastic MILP formulation for the same CPU times. Keywords: supply chain design, optimization under uncertainty, genetic algorithms, stochastic programming.
1. Introduction Supply Chain Management (SCM) aims to integrate plants with their suppliers and customers so that they can be managed as a single entity and to coordinate all input/output flows (of materials, information and funds) so that products are produced and distributed in the right quantities, to the right locations, and at the right time. The SCM problem may be considered at different levels depending on the strategic, tactical and operational variables involved in the decision-making. The complexity of a SC is increased by the high degree of uncertainty brought about by external factors, such as continuously changing market conditions, customer expectations, and internal parameters, such as product yields, qualities and processing times. This work focuses on the strategic level of the SCM problem and addresses the design under uncertainty of a SC. In simple terms, the design of a SC involves the determination of the optimal combination, in terms of some predefined criteria, of suppliers, producers, and distributors that are able to provide the right quantity of products and services to the right locations and at the right time. Compared to the approaches that have been presented in the literature to date, our approach focuses on developing a model and solution strategy for a broader problem that deals with the design of chemical SCs with a non-fixed structure in terms of the
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echelons of the network, and taking also into account the detailed configuration of its nodes and the uncertainty under which the SC operates. A scenario-based multi-stage stochastic programming model with binary variables in stages beyond the first one, which are given by the scheduling constraints applied to estimate the capacity of the batch plants embedded in the network, is developed to tackle this problem and a decomposition strategy is also presented for overcoming the numerical difficulties associated with such mathematical formulation. The main contributions of this work are the insertion of scheduling constraints at the design stage of the SC and the development of a novel decomposition technique to solve the underlying large-scale stochastic MILP formulation associated with this problem.
2. Problem statement Production-distribution networks that can be described through the state-task-network (STN) representation (Kondili et al., 1993) are considered. Materials are manufactured in batch plants and stored in warehouses prior to being sent to final markets where they become available to customers. The following data is assumed to be known in advance: • Set of raw materials, intermediate and final products • Set of production recipes and prices of final products. • Superstructure of the network, i.e. number of available nodes and their discrete capacities. • Cost functions associated with the equipment units, the consumption of raw materials and utilities, the lost demand due to inadequate production and holding inventory. The operation of the SC under demand uncertainty for the whole time horizon is as follows (see Figures 1 and 2). Sales of products are executed at the beginning of each of the periods of time in which the time horizon is divided. The demand associated with each of these periods can not be perfectly forecasted and its uncertainty is represented by a set of scenarios with given probability of occurrence. Decisions regarding scheduling tasks are made in stages as information arrives and uncertainty unveils over time. Therefore, while the scheduling decisions must be taken at the beginning of each period, that is, prior to the realization of the uncertainty, sales are computed once the random events take place at the end of the period, i.e. beginning of the next one. Thus, the problem contemplated involves a sequence of decisions, i.e. schedules, purchases of raw materials and sales of final products, that react to outcomes, i.e. demand realization, that evolve over time. Unlike other works in the literature, we assume that some of the demand can actually be left unsatisfied because of limited production capacity.
Figure 1. Problem statement.
Figure 2. Scenario tree.
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3. Multi-stage stochastic programming approach The design problem previously presented can be formulated as a (|K|+1)-stage stochastic MILP, whose solution involves the computation of the scheduling decisions to be implemented at each node of the scenario tree. Three types of constraints are considered within this formulation, the assignment, the mass balance and the capacity constraints. The mathematical formulation derived is based on the STN representation developed by Kondili et al. (1993) which implies the discretization of each time period k into |T| scheduling intervals of lower length. The mass balance equations have been modified to properly model the transport of materials among the SC entities. The basic assignment constraint has been taken from the work of Shah et al. (1993), in which the authors reformulated the original assignment constraint used by Kondili et al. (1993). Scheduling decisions are assumed to be taken within each period of time and are linked to adjacent periods of the scenario tree through the initial amounts of materials. Hence, the mass balances are enforced via the following constraints: (1) (2) (3) That is, Eq. (1) states that for every scenario mk the initial mount of state s (Sst0kmk) plus the amount purchased (Purchskmk) must equal the hold-up (Sstkmk) plus the amount consumed (BIistkmk) and plus the sales (Salesskmk). This equation assumes that purchases and sales of products take place in the first time interval of each period of time in which the time horizon is divided. Eq. (2) represents the mass balance for time intervals beyond the first one. Eq. (3) forces the sales to be lower or equal than the demand (Demskmk). Here we assume that some of the demand can be left unsatisfied because of limited production capacity. Three types of equipments are considered in the formulation, namely production equipments, single storage tanks and warehouses. Three binary variables, Xjdj, Yldj and Zndn, are defined in order to represent the existence of an equipment of discrete size d. These variables take the value of 1 if the equipment being represented is of discrete size d, and 0 otherwise. (4) (5) Eq. (4) forces all the binary variables representing the starting of the tasks performed in equipment j (Witkmk) to be equal to 0 if the equipment j is not selected and remain inactive otherwise. Moreover, Eq. (5) limits the batch sizes according to the capacities of the equipments. In this equation, µi is the size factor of task I, Bitkmk is the batch size of task i started in time interval t of time period k in scenario mk, and ESizejdj represents the discrete size d of equipment j. (6)
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Eq. (6) expresses the limits for the storage tanks. The amount of state s stored in storage tank l in any time interval and scenario is bounded by the capacity of the corresponding storage tank. Here, STSizeldl represents the discrete size d of storage tank l. (7) In this work, the warehouses are modelled as common storage tanks, i.e. as storage tanks which can store simultaneously more than one product. Eq. (7) expresses the limits for the warehouses. The total amount of states s stored in a warehouse n in any time interval and scenario is bounded by its capacity. Here, WSizendn represents the discrete size d of warehouse n. (8) (9) The presented model accounts for the maximization of the total expected profit, as it is stated in Eq. (8). Revenues are obtained through sales of final products, while costs are due to the underproduction, i.e. leaving part of the demand unsatisfied, the purchases of raw materials, the consumption of utilities and holding inventories. The investment cost is given by Eq. (9), and is computed as the sum of the cost of the production equipments (ECostjdj) plus the cost of the single storage tanks (STCostldl) plus the cost of the warehouses (WCostndn).
Figure 3. Decomposition strategy.
Figure 4. Evaluation of the designs alternatives.
4. Decomposition strategy The decomposition strategy is as follows. At the strategic level, the SC structure is considered. This structure concerns the capacities of the warehouses and the capacity of the plants, which is given by the number and size of their production equipments and storage tanks. A the lower level, the tactical and operational decisions, namely the quantities manufactured at the factories, transported among nodes and the sales, are computed by applying a two-stage stochastic SC scheduling model which is based on the structure of the network being assessed. Demand uncertainty is accommodated within this framework by running the scheduling model in the so-called rolling horizon
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mode (Reklaitis, 1982). The overall algorithm is depicted in Figure 3. At the upper level (master problem), a GA generates potential design alternatives. At the inner level (slave problem), the operational variables associated with each design candidate are computed under the uncertain environment by running the corresponding two-stage stochastic SC scheduling model in a rolling horizon mode (see Figure 4). This computation provides the fitness functions of the individuals belonging to each population generated by the GA.
Figure 5. STN representation: batch plants.
Figure 6. STN representation: warehouses and retailers.
5. Computational results A numerical example is next presented to illustrate the performance of our algorithm. The outer loop of the proposed strategy (GA) has been implemented in MATLAB® taking advantage of the GA toolbox available in version 7.0 of the software. The inner loop, which involves the computation of the SC scheduling formulation within a rolling horizon mode, has been implemented in GAMS (Brooke et al., 1988) and has been solved with the MIP solver of CPLEX 7.0 on a AMD Athlon 3000 computer. The interfacing between both programs is carried out by means of the software library developed by Ferris (1988).
Figure 7. Superstructure.
Figure 8. Computational results.
Here we investigate the retrofit under uncertainty of a SC comprising several plants, warehouses and retailers placed in different locations. The STN representation of the
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network under study is given in Figures 5 and 6, while the superstructure of the embedded plants is depicted in Figure 7. These plants are assumed to be multi-purpose batch chemical plants, whose structure is adapted from the case study proposed by Kondili et al. (1993). We assume a time horizon of 72 time units, which is divided into three equally long time periods of 24 scheduling intervals of 1 time unit. 4 possible events in each period of time, i.e. 4 possible values for the demand, are considered. This leads to 64 scenarios for the whole time horizon. The computational results associated with both approaches are given in Table 1. In this case, the best solution computed by the multi-stage stochastic formulation in 50,000 seconds is equal to 12,478,622 m.u., still within 35.2 % of the best relaxed solution (in our case 19,269,000 m.u.) computed by the branch and bound when the time limit of 50,000 seconds was exceeded. The GA computes a solution equal to 16,574,806 m.u. for the same CPU time, which is within 14.0 % of the best relaxed solution computed by the branch and bound. In Figure 8, the best solutions computed by the GA and the rigorous model have been plotted against the computation time. Let us note that the curve corresponding to the GA lies entirely above that of the rigorous model. This means that for CPU times below 50,000 seconds, the GA performs better than the multi-stage stochastic programming model, i.e. it provides better solutions than the rigorous model for the same CPU times. Table 1. Computational results.
6. Conclusions In this work, a multi-stage stochastic formulation has been derived to address the design of chemical supply chains under demand uncertainty. To overcome the numerical difficulties associated with the resulting large-scale stochastic MILP, a special numerical technique based on decomposition has been presented. This decomposition strategy combines GAs and mathematical programming tools. The main advantages of the proposed approach have been highlighted through a case study where the comparison with the rigorous multi-stage stochastic formulation has been carried out. Results indicate that the proposed strategy provides better solutions than those computed by the multi-stage stochastic formulation for the same CPU times.
References A. Brooke, D. Kendrick, A. Meeraus, 1988, GAMS-A User's Guide. M. C. Ferris, 1998, Interfacing optimization and visualization software. Computer Sciences Department, University of Wisconsin, Madison, Wisconsin. E. Kondili, C. C. Pantelides, R. Sargent, 1993, A General Algorithm for Short-Term Scheduling of Batch Operations I. MILP Formulation, Computers and Chemical Engineering, 17 (2), 211229. The MathWorks. MATLAB 7.0. User’s Manual. The Math Works Inc., 2004. G. V. Reklaitis, 1982, Review of scheduling of process operations, AIChE Symposium Series, 78 (214), 119-133. N. Shah, C. C. Pantelides, R. Sargent, 1993, A General Algorithm for Short-Term Scheduling of Batch Operations-II. Computational Issues, Computers and Chemical Engineering, 17, 229244.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Development of a mixture design methodology for problems with incomplete information. Application to PVC heat stabiliser design. Ulisses G. da Cruz, Galo A. C. Le Roux Departamento de Engenharia Química, Escola Politécnica da Universidade de São Paulo, Av. Prof. Luciano Gualberto, trav. 3, 380, São Paulo, SP, 05508-900, Brasil
Abstract The objective of this work is to optimize the formulations of heat stabilizers for PVC (Poly Vinyl Chloride) by means of mixture design techniques extended to deal with problems with incomplete information. The mixture design problem consists in minimizing the cost of the formulation while obtaining a product that satisfies technical and market specifications. In the case studied, the formulations are mixtures of 8 to 10 components chosen out from 16 basic compounds. Each of the compounds confers different characteristics to the final formulation, and there can be interactions between them. The mixture characteristics of interest are obtained from assays performed in a standard torque rheometer for different stabilizer formulations. Data from a database, in which many different test and commercial formulations are recorded, were used in order to build models that correlate the properties with the compositions. As the number of data points that would be needed in order to obtain the necessary information for a complete description of the mixture would be prohibitive, mixture models were built using Principal Components Regression (PCR). The mixture design problem can be expressed as Linear and Nonlinear depending on the type of model used. In order to avoid extrapolation problems that would result from the poor availability of information, additional equations that confines the solution to the subset where information is available (the space spanned by the main principal components retained for correlation) are added to the problem. This is an extension to the standard mixture design problem. Solutions obtained are compared to validation experiments which shows that the methodology is very promising for industrial design problems. Keywords: mixture design; principal components regression; multivariate calibration; heat stabilizer.
1. Introduction A methodology combining optimisation and mixture design is presented and applied to the design of PVC heat stabilisers. PVC is one of the most important commercial thermoplastics but, because of its low thermal stability, it cannot be processed without the usage of a heat stabiliser [6]. The stabiliser is prepared in accordance to the conditions PVC will be submitted to (while being processed) and to the final usage it is suited for, which can vary largely [9]. The updating and adjustment in stabiliser formulations in order to satisfy market necessities shall follow its dynamics, with a minimum delay. Reaching a product with desired characteristics with a minimum cost must be obtained as fast as possible.
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Stabilisers are mixtures usually containing 8 to 10 components. Each component confers specific characteristics to the final product and there are usually synergistic interactions between them. In order to state the mixture design problem mathematically, mixture models shall be able to describe the relationship between the system properties and the mixture composition [1]. In this work, design problems are stated as Linear Programming problems (LP) or Nonlinear Programming problems (NLP) depending on the order of the polynomial mixture model. As the number of data points that would be needed in order to obtain the information necessary for a complete description of the mixture would be prohibitive, it is proposed to build mixture models based on Principal Components Regression. Data extracted from a database in which development test and commercial formulations are stored are used for building the models. As the number of data points is limited and the availability of information is poor, extrapolation difficulties would be an issue. As an original contribution, it is proposed for the linear problem that additional equations that confine the solution to the subset where information is available, be added to the problem, in order to avoid the extrapolation problems,. In order to validate the methodology, solutions obtained are compared to validation experiments.
2. Multivariate Calibration And Optimization 2.1. Multivariate Calibration The objective of multivariate calibration techniques is to represent an important quantity of information contained in the data by a reduced set of variables, without loss of the relevant information. The main motivation for data compression is that a great number of variables present redundant information that lead to conditioning problems [7,10]. In this work, Principal Components Regression is used in order to build models for mixtures based on historical data with incomplete information. 2.1.1. Principal Components Regression (PCR) Principal Components Regression (PCR) is one of the most traditional multivariate calibration techniques [10]. Instead of using system properties (concentrations, temperatures, etc …) as independent variables, they are replaced by the magnitude of their projection in a reduced set of Principal Components [7]. 2.1.2. Information Principal Components Regression provides a decomposition of the space spanned by the independent variables. The Principal Components (PCs) represent a basis set that concentrates the availability of information. In order to avoid extrapolation problems, the solution is confined to the space spanned by the Principal Components. 2.2. Optimization Problems of mixture design [3,4,6] arising from linear mixture model can be expressed as LP. LP problems are very well known in literature and extremely performing techniques are nowadays available which can solve problems with thousands of variables[2,5]. The quadratic mixture model is expressed as an NLP problem.
3. Application 3.1. Introduction The data set consists of 155 stabilizers formulations obtained during years of product development, in which 16 different components were mixed in different proportions. Formulations are mostly mixtures of 8 to 10 components. A more deep
Mixture Design Methodology for Problems with Incomplete Information
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description of the formulations will not be presented here because of industrial proprietary reasons. Formulations are grouped in a 155x16 matrix, called X, each line of which corresponds to a different formulation. The dependent variables, that it is aimed to predict in order to describe the properties of the stabilizer, are based on torque rheometer assay results. Results were obtained in a torque rheometer Rheomix 600p model, from Haake. The rheometer has a heated chamber of 60 cm3 with two rotors. The PVC compound contains 2% of heat stabilizer. During the operation of the rotors, the torque applied to the mixture of the molten mass is measured, and a rheology curve (torque versus time) is supplied, as shown in figure 1. The procedure simulates the behavior of the compound when extruded. Three different points of the rheological curve were strategically chosen in order to summarize its behavior. This leads to four different parameters for each curve. These four parameters condense the main characteristics of the curves. These are grouped into a 155x4 vector, called Y.
Figure 1 – Some standard torque rheometer curves for four different stabilizers
3.2. Numerical Tools Matlab version 6.5 (Mathworks Inc.) was chosen as the framework for the solution. PLS Toolbox version 3.0 (Eigenvector Vector Inc.) was employed in order to build PCR correlation models. In this work we will restrict the analysis to linear and quadratic mixture models. 3.2.1. PCR Correlation Model 16
Linear model: yˆ =
∑a x
(1)
i i
i =1
X and Y variables were preprocessed. X variables were centered between -1 and +1 and Y variables were centred at their mean. 14 PC’s were chosen, based on Cross Validation analysis based on minimum the Root Mean Square Error of Cross Validation criterion, where the “leave-one-out” (PRESS) methodology was applied. 16
Quadratic Model: yˆ =
16
16
i< j
j
∑ a x + ∑∑ a x x i i
i =1
ij i j
(2)
X and Y variables were preprocessed as for linear models. Combinations xixj were added to the model, and the regressor matrix is (155x136) elements. 13 PC’s were chosen based on Cross Validation analysis The R2 statistic for both the models is presented in the table 1.
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Table 1. R2 values for the four independent variables. Variables 2
Linear model - R
2
Quadratic model – R
Y1
Y2
Y3
Y4
0,73
0,85
0,77
0,64
0,75
0,86
0,77
0,67
3.3. Optimization Two different approaches are used in order to solve the linear mixture design problem. In the first one, called base case (BC), the regression model is taken as is, that is, no modeling incertitude is considered in predictions. In order to limit extrapolation problems an approach called XI, in which the solution is confined to the space spanned by the Principal Components is proposed. For the nonlinear problem only the BC approach was used. 3.3.1. Linear Programming Base Case (BC) The Linear problem generated in order to solve the base case is stated as follows: Aeq x = beq min f T x subject to: lb ≤ x ≤ ub where f, x, beq, lb and ub are vectors and Aeq is a matrix. ⎡ x1 ⎤ ⎢x ⎥ (3) cost function: f T x = [ p1 p 2 … p16 ]× ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ x16 ⎦ where the pi are the costs of the individual components of the stabilisers. The equality constraints are given by: ⎡(ma − mi )1 ⎢ a11 ⎢ Aeq x = beq ⇒ ⎢ a21 ⎢ a 31 ⎢ ⎢⎣ a41
⎡ ⎡ x1 ⎤ ⎢2 − ⎥ ⎢x ⎥ ⎢ ⎥ ⎢ 2⎥ ⎢ ⎥ • ⎢ x3 ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎦ ⎣ x16 ⎦ ⎢ ⎢ ⎣
(ma − mi )2
(ma − mi )16 ⎤
a12
a116
a22 a32
a216 a316
a42
a416
⎤
16
∑ (ma − mi )⎥⎥ 1
Y1tg Y2tg Y3tg Y4tg
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(4)
where ma is the maximum value the of variable x; mi is the minimum value of the variable x. The first equality constraint expresses the fact that it is a mixture problem, that is, that the sum of the original variables must be one. The other four constraints express the target for the parameters of the rheology curve of the mixture. The aij coefficients are the parameters obtained from the PCR regression. Ytg are the four target values for the dependent variables. 3.3.2. XI approach In addition to the mixture constraint, we introduce the restriction that vector x must not belong to the space where the information was not considered by the calibration model. In order to express this, we impose the vector to be orthogonal to the PC's not used by the PCR model.
Mixture Design Methodology for Problems with Incomplete Information ⎡(ma − mi )1 (ma − mi )2
⎢ ⎢ ⎢ ⎢ Aeq x = beq ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎢
⎡ ⎢2 − ⎡ x1 ⎤ ⎢ ⎥ ⎢x ⎥ ⎥ ⎢ 2⎥ ⎢ ⎥ ⎢ x3 ⎥ ⎢ ⎥ ⎢x ⎥ ⎢ ⎥•⎢ 4 ⎥= ⎢ ⎥ ⎢ x5 ⎥ ⎢ ⎥ ⎢ x6 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢x ⎥ ⎢ ⎦⎥ ⎣ 16 ⎦ ⎢ ⎢⎣
(ma − mi )16 ⎤
a11
a12
a116
a 21 a 31
a 22 a 32
a 216 a 316
a 41
a 42
a 416
l m1
l m2
l m16
l n1
l n2
l n16
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16
∑ (ma − mi )⎥⎥ 1
Y1 tg
tg
Y2 Y3 tg
Y4 tg 0m 0n
⎥ ⎥ ⎥ ⎥ ⎥ (5) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦
Where the elements lij in Aeq are the coordinates of the PCs excluded from the PCR regression. This implies that the optimization takes place in the space orthogonal to the PCs that had not been used in the regression. 3.4. Nonlinear Programming Base Case The Nonlinear problem generated in order to solve the base case is as follows ceq(x ) = 0 min f (x ) subject to: Aeq x = beq lb ≤ x ≤ ub where x, beq, lb and ub are vectors and Aeq is a matrix, ceq(x) is a function that returns a vector, and f(x) is a function that returns a scalar. The cost function is the same as (3). The nonlinear equality constraints are the models for the prediction of the properties of the mixtures which must be equal to a given target The restriction of the sum the 16 components, is a linear equality constraint.
3.5. Results In order to validate the methodology, it was applied to an example taken from a commercial stabiliser. Using the commercial stabiliser properties as targets, the optimisation led to formulations with smaller costs than the original product (4,03). The results are presented in table 2. Table 2. Results and experimental errors of the linear and quadratic models for example. Target BC Linear XI Linear BC Nonlinear Properties Y1 Y2 Y3 Y4 RMSEP Cost of the formulation
5,6 17,9 16,3 16,0
Value 5,7 20,2 18,3 15,4
4,03
2,14
Error 0,1 2,3 2,0 -0,6 1,54
Value 5,5 17,9 17,9 16,8 2,33
Error -0,1 0,0 1,6 0,8 0,89
Value 5,7 19,2 18,3 17,0
Error 0,1 1,3 2,0 0,9 1,28
2,53
4. Conclusions The XI method improves the quality of the predictions. The nonlinear programming solution leads to a result close to the LP-BC. The rheometer curves in figure 2 show that
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the formulations obtained for the example are close to the target (black curve). The methodology thus provides solutions that are close to the target, with reduced costs. In addition, as only historical data are used for the construction of the models, no additional development costs are implied by the methodology. The methodology proposed is simple and leads to feasible solutions, thus simplifying the process of design of formulations, with acceptable commercial results. 34,00 32,00 30,00 28,00 26,00 24,00
Torque (Nm)
22,00
Objective [Nm]
20,00
BC Linear [Nm]
18,00 16,00
XI Linear [Nm]
14,00
BC Nonlinear [Nm]
12,00 10,00 8,00 6,00 4,00 2,00 0,00 0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
5,5
6,0
6,5
7,0
7,5
8,0
8,5
9,0
9,5
10,0
Time (min)
Figure 2 – Torque rheometer curves as validated for the commercial example.
References 1. ACHENIE, L. E. K., KARUNANITHI, A. T., GANI, R., 2005 - A new decomposition-based computer-aided molecular/mixture design methodology for design of optimal solvents and solvent mixtures, Ind. Eng. Chem. Res. 44, 4785-4797. 2. BIEGLER, L. T., GROSSMANN, I. E., WESTERBERG, A. W., 1997 - Systematic Methods of Chemical Process Design, Printice Hall, New Jersey. 3. CAFAGGI, S. et al., 2003 - An example of application of mixture design with constraints to a pharmaceutical formulation, Chemometr. Intell. Lab. Syst. 65, 139-147. 4. CORNELL, JOHN A., 1990 - Experiments with Mixtures – 2nd edition; John Wiley & Sons, USA. 5. EDGAR, T. F., HIMMELBLAU, D. M., 1988 - Optimization of Chemical Process – 4th edition; McGraw-Hill, USA. 6. GACHTER, R.; MÜLLER, H., 1993 - Plastics Additives Handbook – 4th edition; Hanser Publishers – Cincinnati. 7. MARTENS, H., NAES, T., 1989 - Multivariate Calibration, John Wiley & Sons, Great Britain. 8. NETO, B. B., SCARMINIO, I. S., BRUNS, R. E., 2002 - Como Fazer Experimentos – 2ª edição, Editora Unicamp, Campinas. 9. NUNES, L. R., JR. R., ORMANJI W., 2002 - Tecnologia do PVC, Braskem – ProEditores Associados, São Paulo. 10.WISE, B. M., et al., 2003 - Manual PLS_Toolbox 3.0, Eigenvector, Manson WA.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
PRINCIPLES FOR CHEMICAL PRODUCTS DESIGN Luis A. Cisternasa, Edelmira D. Gálvezb a
Department of Chemical Engineering, University of Antofagasta, Casilla 170, Antofagasta, Chile b Department of Metallurgical Engineering, Northern Catholic University, Casilla 1280, Antofagasta, Chile
Abstract In this work it is suggested to develop a theory for chemical products that defines the bases and serves to guide and put in order the development of knowledge in the area. A nature representation of chemical products is carried out based on the simple observation that a chemical product is a system consisted of different chemical substances, and that it is manufactured for one or more purposes, in other words, it is formed by components, an organization and an environment. This representation is used to identify different kinds of chemical products design problems. Keywords: chemical product, product design, product theory.
1. Introduction Chemical products (ChP) have been present more in the industry than in the academy, and that is why the used language and the available knowledge have served more to the industrial sector and the market. In a recent work it has been set out the necessity to develop a theory to chemical products like the one developed for chemical processes [1]. This statement is based on the observation that the engineering of chemical processes, and the design of chemical processes in particular, have been developed due to the existence of a theory, it means a series of principles and /or laws which allow to explain, classify and communicate the chemical processes phenomena. The flow sheet (graphic language), unitary operations and the principles (thermodynamics, kinetics, control) make up the main elements of this theory. In the case of chemical products this development does not exist, reason for what a language for its representation is not identified, for example in a kind of mathematic or graphic representation, there is not a clear definition neither of products nor for the governing behavior principles. This lack limits the development of chemical products, because there are not solid bases to build on its development. Notice that in spite of some attempts, there is not a procedure yet that allows the design of all kind of chemical processes. The efforts have been focused on the development of tools for the design of any specific process or on the answer of a particular design problem (e.g. energy integration). That is why; it is natural to think that in the case of chemical products, where there is less knowledge, we have to develop tools to solve particular problems. In this case it could be useful to classify the products and the existing design problems. In this work our previous manuscript [1], which is a description of chemical products nature, is extended with the purpose of developing the bases for the development of chemical products engineering and to identify the problems of ChP design.
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2. Chemical Product Design A chemical product, ChP, is defined as a system made up of different chemical substances, which is manufactured for one or more purposes [1]. This definition recognizes that a ChP is a system, it means that a product is made up of arranged parts in a specific form. On the other hand, it recognizes the existence of one or more uses of the product, and that is why a product is not totally defined if there is not a specification of the environment in which it interacts. So, mathematically we can represent a chemical product, Q, in this way: Q = C ∪O ∪ M
(1)
Here, C represents the components, O the organization and M the environment. The performance or the properties of the ChP depend on C, O and M. The components may be represented as:
{
}
C = C j | i = 1,2,...m
(2)
Where Cj is a chemical product that is a component of the main chemical product, m is the number of components which make up the main ChP. Then a product can be broken down into sub components and so on until identifying the primary components (which are single products). Every time we work with an entity without distinguishing components, we identify the single products, in other words, when its performance or properties depend on itself as a whole [1]. This representation allows the description of the components of a ChP like a tree in which its base is made up only for primary components. A ChP made up of several components will be identified as a compound product, whereas a ChP that is made up from just one primary component will be identified as a single product. The organization of a ChP is a description, it could be qualitative, quantitative or both, on how the different components are included. It includes a description of the physical form about every component (e.g. size, shape), and the structure, it means of how the different components are interconnected. Then the organization is made up of organized collections in a tree, where in its top it is found a description of the ChP as a whole, and in its bottom a description of each one of the primary components. So, the organization can be represented in this way: O = {O k | k = 1,2,...n}
(3)
Where Ok is one of the n descriptions that make up the product organization. As every component of the product has a specific organization, the organization of a product includes organizations of each of its components. The environment M, is made up of different niches [1] which interact with the product to fulfill its function. A niche is an environment section in which a particular property of the product interacts. The presence of different niches is not only due to the fact that a product carries out different functions in a particular use, but also the product in its
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life changes the niche. It is to say that different product-environment interactions occur during the processing, storage, the use and disuse of a product. So: M = {N l | l = 1,2,...q}
(4)
Where Nl is one of the q niches that made up the M environment of the ChP Q. The performances or properties, P, of a ChP are a function of the components, its organization and the environment, in other words: P = f (Q) = f (C , O, M )
(5)
When comparing the Eqs. (1) and (5) it is clear that the chemical product definition is given by the components definition, its organization and environment, and not by its properties, that is to say several chemical products can have the same properties. On the other hand, P is a set of properties, like P = {Pi | i = 1,2,...r}
(6)
Where Pi is each one of the properties or performances that characterize to the ChP. In some cases, some or all the properties can only be function of the primary components or only of primary components and its organization. The representation of chemical products through the Eq. (1) may have different applications. For example, it can be useful for the classification of chemical products. Usually the classification of ChP is based on its prices or uses. Both, prices and uses meet the market needs and the interest of companies, this kind of classification has been, in some cases, ambiguous. Based on the Eq. (1), Cisternas [1] has proposed a classification based on the number of components, the organization (structure) and its relation with the environment (see Fig. 1). Number of components Compound 4 products
6
8
5
Single products
Whitout complement
Properties are components properties
1
3
Some key properties are properties as a whole Structure
With complement 2
7
Environment Interaction
Figure 1. Classification of Chemical Products[1]
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Another application may be related to the definition of a new area in chemical engineering: chemical products engineering (CPE): CPE is a service function of chemical engineering that involves a selection of the product to be produced, determination of the functions, environment, components and structures of the chemical product. Area that can be subdivided into two considering that the organization and the environment are fundamental aspects of the ChP: a) Structured chemical product engineering, which can be defined as a branch of CPE concerned with the composition, arrangement and internal structure of chemical products, its relationship with the process and the interaction with the environment. b) Functional Chemical Product Engineering, which can be defined as the aspect of CPE that is concerned with the product objectives, environment, and functions rather than with its components and organization.
3. Applications to the Design Different authors have identified several stages in the design of chemical products [2,3]. In those works, clearly, like in the processes design, synthesis and analysis activities are present. In the synthesis stage different knowledge or tools are needed to identify a potential chemical product, Q, based on the specifications of the wanted product S, that is to say: Q = K (S )
(7)
Where K represents the knowledge used in the synthesis stage and S represents the ChP specifications. Different kinds of knowledge are used in the synthesis stage, which includes heuristics, experience and the designer’s creativity. By its side, the S specifications are a function of the market requirements and also of the experience of those who define them. In general a product specification can be represented by a set of specifications. S = {S l | l = 1,2,...s}
(8)
By its own, the analysis stage includes an evaluation of performance or properties of potential chemical product given by the Eq. (5). So, this process is recursive, where the objective is that product properties, P, are equal, or the closest possible to the product specifications S, that is to say: f (C , O, M ) = K ( S )
(9)
In the solution to the problem represented by the Eq. (9), all of them, C, O, M and S are variable, that is the reason for what this problem rests with an ill structured problem. It is necessary to clarify that the function f(C, O, M) forces to do different design activities like modeling, experimentation, prototype, and so on. Due to the problem to solve with the Eq. (9) is complex, it is recommended to develop particular strategies for different specific problems that can be identified in the Eq. (9).
Principles for Chemical Products Design
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For example, the design of fertilizers can be seen as make up of several steps (see Fig. 2): a) generate the desired specifications of the product, S. This means to specify properties related with product manipulation, NH4+ decomposition reduction, and liberation control of the active component [4]. These specifications depend on the use that will be given to the product and in the market requirements. b) use the available knowledge in the synthesis stage to design a product, that is to say to define their components, C, their organization, O, and their environment M. For our example it means to define the nutrients to include in the fertilizers and the additives, define the form the additives are incorporated in the fertilizer (its structures) and the environment characteristics like roots properties, soil characteristics, climate (temperature, humidity, kind of rains (acid or not)), nutrient demand of plants, etc.[5] c) once defined C,O and M is necessary to analyze if the synthesized product responds appropriately to the product specifications. These steps are carried out several times until achieving that the synthesized product properties reach the desired product specifications. Experimentation will be necessary to evaluate physical properties (as the viscosity), liberation profiles, etc. d) finally, the evaluation criteria should be revised by the light of the characteristics of the prototype product, if the criteria change new product specifications or refinement can be generated. From the fertilizer behavior, through their life cycle, new specifications can be necessary because several factors, as the segregation level, chemical stability, physical characteristics, etc.
4. Structured chemical product Design If the environment, through its niches, has a minor role in the design process, or if it is wanted to focus the design problem on the organization, we can say that the problem of the design rests with a structural design problem of a chemical product. The Eq. (9) can be re-written in this way: f (C , O) = K ( S )
(10)
Where the problem consists on determining the components, and the organization of the product with the purpose of getting the desired specifications S or properties. As the design and the analysis stage progress, the specifications may suffer modifications, whether because new specifications are added or because the specifications are defined again as the knowledge increases over the desired product. It is necessary to indicate that in this kind of problem, there is not a lack of environment or niches, but they are not part of the design problem variables, in other words, they are fixed. The molecular design with a specific set of properties can be considered as a special case of design, where the problem to solve is: f (C ) = K ( S )
(11)
And C is made up of just one component.
5. Functional chemical product Design If the design problem is focused on the search of strategies to replace products, insert products or to find new uses for the already known products, then the design problem
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Evaluation criteria generator
S
Chemical product synthesis
Q
Chemical product analysis
P Appraiser of criteria
S≠P
Figure 2. Chemical product design loop
can be called as functional design of chemical products. The Eq. (9) can be re-written as:
f (M ) = K (S )
(12)
Where the problem consists on determining the environment niches, with the purpose of achieving the searched objectives. Once the niches are defined, the problem can be considered as structural design problem again (Eq.10), where it is necessary rearranging the components and/or the structure as to adapt it to the new environment.
6. Conclusions If a chemical product is considered as made up of components, an organization and an environment, then it is possible to represent it for the union set that represents each of those elements. On the other hand, if it is considered that a product design consists in identifying the components, the organization and the environment that meet efficiently to the desired specifications of a ChP, then it is possible to identify that the problem is recursive and that different design sub problems can be identified from this definition. The last idea allows us to study particular strategies for each of these ChP design problems.
References [1] L.A. Cisternas, Nature of Chemical Products, in K.M. Ng, R. Gani, and K. Dam-Johansen, ed. “Chemical Product Design: Towards a Perspective through Case Studies,” Elsevier, to appear in 2006. [2] E.L. Cussler & G.D. Moggridge, 2001, Chemical product design, Cambridge University press, Cambridge, United Kingdon. [3] M. Hill, 2004, product and process design for structurated products, AIChE J., 50(8), 16561661. [4] U. Bröckel & C. Hahn, 2004, Product design of solid fertilizers, Chem. Eng. Res. & Design, 82(A11), 1453-1457. [5] J.E. Sheehy, P.L. Mitchell, G.J.D. Kirk, A.B. Ferrer, 2005, Can smarter nitrogen fertilizers be designed? Matching nitrogen suply to crop requirements at high yields using a simple model, Field Crops Research, 94, 54-66.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Mathematical development for scaling-up of molecular distillators: strategy and test with recovering carotenoids from palm oil Batistella, C.B.; Moraes, E.B.; Maciel Filho, R. and Wolf-Maciel, M. R. Separation Process Development Laboratory (LDPS). Chemical Engineering School , State University of Campinas, (UNICAMP), CP 6066, ZIP code 13081-970, CampinasSP, Brazil. Email: [email protected]
Abstract Studies for recovering carotenoids from palm oil were developed in the Separation Process Development Laboratory (LDPS-FEQ-UNICAMP) through molecular distillation process (modeling, simulation and experiments) (Batistella and Maciel, 1998; Batistella et al., 2002). All the studies, however, have involved laboratory scale distillators, but due to the success of the developments, a lot of industrial applications are becoming possible. So, aiming to design molecular distillators with industrial dimensions, it is important to develop a methodology looking for an easy and fast form for scaling-up, which is the aim of this work. The procedure makes use of the knowledge obtained from the simulation and operational strategy developed for small scale equipment. Moreover, the nonideality condition of the vapor phase was considered on the developments of the scale-up (Batistella et al., 2000), since this an important macrobebaviour to be considered. The obtained results show that the developed methodology is robust. Keywords: Molecular Distillation, Scale-Up, Carotenoids, Short Path Distillation
1. Introduction Scaling-up art in process system design is a stage of great importance in the development of high added value products and also in a shorter time. This is even true for molecular distillators, since phenomena at molecular scale are taking place in the system. Molecular distillation, also known as short path distillation, is extremally sensitive to the process variables, in such way that scale-up analyses must take into consideration the most influencing design and operating parameters in order to carry out suitably the task from laboratory data. On the other hand, the performance is drastically affected, increasing operating costs and decreasing the separating efficiency. Additionally, in molecular distillation, material thermal decomposition can occur. In fact, molecular distillators are specially designed process to deal with such constraint. When a scale-up study is developed based on analysis of dimensionless parameters, this fact is usually not fully taken into account. So, the residence time of the material inside the equipment is an important variable to be considered in the development of a scaleup procedure as well as in the operating strategy, preserving the thermally sensitivity products. 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), the one of reference (1) and the equipment to be scaled-up (2), since these parameters have a
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significant impact on the dominant process phenomena ( Batistella et al., 2000a). Considering the parameters studied in the scale-up procedure, such as feed flow rate and distillate flow rate, the results were good enough, indicating that the used methodology represented very well the molecular distillation phenomena, when comparing the simulated and predicted data, even with scale fifty times higher.
2. Scale-up Conditions In scale-up studies for molecular distillators, the following conditions in the periphery of the evaporator (Bhandarkar and Ferron, 1988) are considered:
W1 = W2
(1)
S1 = S 2
(2)
where W and S are the mean speed and the thickness of the liquid film in the periphery of the evaporator, respectively. The residence time is approximately the same for both distillators, and the thickness of the liquid film is different from zero in any point of the evaporator, avoiding risk of thermal decomposition (thickness zero is related to infinite time). In this work, it was considered molecular centrifugal still for recovering carotenoids from palm oil as study case.
3. Centrifugal Molecular Distillator The mean velocity of the liquid on the evaporator, schematically shown in Figure 1, is given by ( Batistella and Maciel, 1996):
W=
S 2Ω2 x sen 2 φ 3μ
(3)
where Ω represents the rotor angular speed, x is the distance from the center of the evaporator , μ is the kinematics viscosity of the liquid phase on the evaporator and φ is the angle of the half cone.
Figure 1. Scheme of the centrifugal molecular distillator and its coordinate system.
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The equation of the thickness of the liquid film on the evaporator is given by the equation (Batistella, 1996): x ⎡⎛ ⎞⎤ ⎛ n ⎞ ⎢ ⎜⎜ mo − ∑ ⎜ ∑ Ei ⎟πΔx sen φ (2 x + Δx) ⎟⎟ ⎥ x = x o ⎝ i =1 ⎠ ⎢ ⎠⎥ S = ⎢⎝ ⎥ πρΩ 2 x sen 2 φ ⎢ (2 x sen φ − S cos φ ) ⎥ ⎢⎣ ⎥⎦ 3μ
1 3
(4)
where ρ is the density of the liquid phase, Ei is the evaporation flow rate and m the rate of liquid on the evaporator ( the subscript (0) means initial, i means component i and n means the total number of components ). Taking into account the same angle for both size evaporators, and the same liquid system, the thickness of the liquid film is proportional to: 1
⎛ m ⎞3 S ∝⎜ 2 2 ⎟ ⎝Ω L ⎠
(5)
where L is the total length of the evaporator (x=L). The angle of half cone of evaporator is near to π, so that it may be demonstrated that: 2x sen φ >> S cos φ . Here, the proportionality constant is function of the properties of the distilled materials as well as of the temperature. The flows in the liquid film in both evaporators are the same. Considering that both distillators present the same temperature profiles (same thermal conditions), it can also be said that the distillate flow rates are expected to be the same (strong temperature function). Finally, considering the equations 1, 2, 3, 4, 5, and the evaporation area (π*L2): D1 D2 = L21 L22
(6)
In order to apply the proposed procedure, it is necessary to know only the dimension of the larger scale distillator, what makes the proposed scale up procedure very attractive. Then, the speed of the rotor and the feed flow rate can be determined. The distillate flow rate can be estimated by equation (6). The simulation is made considering the predicted feed flow rate and the speed of the rotor (simulated distillate flow rate). The carotenes concentrations are also presented. To carry out these calculations, the simulator DISMOL, developed by Batistella and Maciel (1996), was used. In the ideal case, the distillation rate is equal to the evaporation rate. By definition, the evaporator rate is the liquid film surface evaporation, while the distillation rate is the final distillate product that leaves the molecular distillator. The equation of the distillation rate for the ideal vapor phase consideration (Ei) is given by Langmuir´s equation (Langmuir, 1913): 1
⎛ M W i ⎞2 Ei = CiS Pi sat ⎜ ⎟ ⎝ 2πRg Ts ⎠
(7)
where Ei is the evaporation rate (kg/m2.s), Pisat is the vapor pressure of component i, MWi is the molecular weight, Rg is the gas constant, TS is the surface temperature of the liquid film and CiS is the mole fraction of the liquid in the film surface.
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This equation considers the phenomena of the liquid phase, but it does not take into consideration conditions of the vapor phase. This equation is valid when: the system pressure is smaller than 10-3 mmHg; the distance between the evaporator and the condenser is smaller than the mean free path of the molecules; the condenser temperature is lower than the evaporator temperature; and the evaporator and the condenser are plane and parallel surfaces (Batistella et al., 2000). Such conditions are not usually found in the practice and intermolecular collisions in the vapor phase are considerable and, therefore, the distillation is a nonequilibrium process. Then, the Langmuir´s equation does not represent the vapor phase realistically. With a more rigorous modeling of the vapor phase (Batistella et al., 2000), the simulator DISMOL will allow to evaluate the behaviors of the liquid and the vapor phases for the molecular distillator. This will be useful for studying nonideal systems. The behavior of the molecules in the vapor phase under high vacuum can be described by the Boltzmann's equation (Ferron, 1986):
⎛ ∂f ⎞ n u zi ⎜ i ⎟ = ∑ J ij ⎝ ∂z ⎠ j=1
(8)
This equation refers to a planar, one-dimensional and steady-state flow, where i represent the species of the vapor mixture, fi is the distribution function and uzi is the molecular velocity of component i in direction z. Jij represents the collision integral for the interaction between species i and j. The solution of the collision integral is complex (Borgnakke and Larsen, 1975). It is important to consider that evaporators and condensers of molecular distillators present complex geometries. To avoid the complex solution of the Boltzmann's equation, Bird (1970) has applied the direct simulation Monte Carlo method (DSMC) in problems involving the dynamics of rarefied gases. He has proved that this method can be directly related to the Boltzmann's equation and that it is entirely consistent. He also showed that the results obtained by this method constitute the solution of the Boltzmann's equation. In this work, it was also considered the Monte Carlo method for the vapor phase modeling (non ideality of the vapor phase).
4. Results The results were obtained from carotenoids recovery from palm oil system. This study was initially evaluated in laboratory level, which results were applied in the developed methodology. Table 1 shows an appreciable agreement between the predicted (equation 6) and simulated (DISMOL) values of the distillate flow rate. The concentration of carotenes was determined through Dismol, where the input feed flow rate value was obtained through the scale-up technique for each distillator. Also, it was verified a concentration of carotenes of about 30 times (starting with 600 ppm). These results show that the procedure developed for scaling-up is good in predicting operating conditions for larger distillators. It can be noted that for a distillator processing 27.55 kg/h, 44 times larger than the reference distillator, the deviation of the prediction of the distillate flow rate compared to the result of the simulation is just 2%. For intermediate scale-up dimensions, the deviation is still smaller.
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Table 1. Results of the Scale up for the Centrifugal Molecular Distillator for carotenoids recovery. Diameter of the Rotor (cm)
Speed of the Rotor (rpm)
7.5
1300
10.0 20.0 30.0 40.0 50.0
1126 796 650 563 503
Feed Flow Rate (kg/h) Reference 0.63 Scaled-Up 1.12 4.43 9.94 17.64 27.55
Distillate Flow Rate (kg/h): Predicted Simulated (equation 6) (DISMOL)
Carotenes Concentration (ppm)
---
0.62
18,460
1.10 4.39 9.89 17.58 27.47
1.09 4.35 9.76 17.31 27.03
19,550 18,630 18,440 19,070 18,300
On the other hand, elimination curves for the system palm oil, esters and carotenes, can be seen in Figure 2. Dismol simulator was used in this study, considering the reference still (diameter of the Rotor = 7.5 cm). The pre-treatment, as well as the compositions of the obtained products, were already mentioned previously in Batistella and Maciel (1998). In Figure 2, the elimination curves of the modified components (esters) of the palm oil in a centrifugal molecular distillator are shown. 45
40000
EthylOleate Linoleate Stearate
40
Elimination Curves, % -Vapor Phase-
45000
Carotene -Simulated-
35
Carotene -Experimental-
35000 30000
30
25000
25 Diglycerides
20 15
20000
Monoglycerides
Ethyl Palmitate
10
15000
Triglycerides 10000
5
Carotene Concentration, ppm -Liquid Phase-
50
5000
0
0
110
120
130
140
150
160
170
180
190
200
210
220
230
Temperature, °C Figure 2. Elimination Curves from a Centrifugal Molecular Distillation considering the palm oil .
It is possible to see clearly that each component presents different elimination curves, mainly in cases where the molecular weights are considerably different, as well as their vapor pressures. The first component to be separated is ethyl palmitate, which presents the lowest molecular weight, reaching a maximum of elimination at 175°C. However, the components ethyl estearate, ethyl oleate and ethyl linoleate present the profiles very similar with maximum elimination points at 187 °C. This fact is due to their close values of molecular weight and vapor pressure. The elimination curve of the triglycerides presents a band considerably wider, comparatively to the other components, indicating low vaporization enthalpy at 0.001 mmHg (Batistella et al., 2003). For comparative analysis, Figure 2 presents the profile of simulated and experimental concentration data of carotene in the liquid phase. It is possible to verify
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that the carotene concentration reached values above 60 times the value of the original concentration (from 600 ppm to close to 40000 ppm) for both simulated and experimental profiles. This has occurred due to the almost total elimination of ethyl esters and mono and di-glycerides, remaining basically triglycerides, fact that is easily observed in Figure 2 for high temperatures (from 205 to 210°C). In this range, practically, all ethyl esters (representing more than 92% of the mixture) were already eliminated. The remaining mono-diglycerides and, mainly, triglycerides form the mixture (in the liquid phase) rich in carotenes. For larger operating temperatures, the benefit would be reduced because the smaller quantity of mono and diglycerides to be removed. Moreover, considerable thermal decomposition of carotenes occurs, showing that temperatures of 200-210 °C would be the operational limit for the considered system in a centrifugal distillator.
4. Concluding Remarks In this work, it was proposed and tested an easy to use scaling up procedure for centrifugal molecular distillator. The obtained results show that the developed methodology is robust and, thus, it can be used to proceed with scale up studies. It is, also, worthwhile mentioning that the software to carry out these studies is available, which can be considered as another contribution of this work, since it can be used for other systems.
Acknowledgements The authors are grateful to FAPESP and CNPq for the financial support.
References Batistella, C.B; Moraes, E.B. and Wolf Maciel, M.R., 2003. “Experimental and Simulated Elimination Curves For recovering Carotenoids From Vegetal Oil Through Molecular Distillation Process”. Chemical Engineering Transactions, 3, 569-574. Batistella, C.B.; Moraes, E.B.; Wolf-Maciel, M.R.; Maciel Filho, R., 2002. “Molecular Distillation Process for Recovering Biodiesel and Carotenoids from Palm Oil”. Applied Biochemistry and Biotechnology, vol 98, 1149-1159. Batistella, C. B.; Wolf-Maciel, M. R. ; Maciel Filho, R., 2000. “Rigorous modeling and simulation of molecular distillators: development of a simulator under conditions of non ideality of the vapor phase”. Computers and Chemical Engineering, vol 24, 1309-1315 Batistella, C. B.; Moraes, E.B.; Wolf-Maciel, M. R. ; Maciel Filho, R. 2000a. “Strategy and Mathematical Development for Scale-up of Molecular Distillators for Recovering Carotenoids from Palm Oil”. Computer-Aided Chemical Engineering, 8, 505-510. Batistella, C.B. and Maciel, M.R.W, 1998, Recovery of Carotenoids from Palm Oil by Molecular Distillation. Computers & Chemical Engineering, 22, S53-S60. Batistella, C.B. and Maciel, M.R.W., 1996, “Modeling, Simulation and Analysis of Molecular Distillators: Centrifugal and Falling Film”. Computers Chemical Engineering, vol. 20, Suppl., pp. S19-S24. Bhandarkar, M. and Ferron, J. R., 1988, Transport Process in Thin Liquid Films during HighVacuum Distillation. Ind. Eng. Chem. Res.,27, 1016 – 1024. Batistella, C.B., 1996. Modelagem e simulação de destiladores moleculares de filme descendente e centrifugo. Master Thesis-UNICAMP, Campinas-SP-Brazil. Bird, G.A ., 1970, Direct Simulation and the Boltzmann Equation. The Physics of Fluids, 13, 2676-2681. Borgnakke, C. and Larsen, P.S., 1975. Statistical Collision Model for Monte Carlo Simulation of Polyatomic Gas Mixture., J. Comput. Phys., 18, 405-420. Ferron, J.R., 1986, Evaporation and Condensation of Mixture under Rarefied Conditions. Ind. Eng. Che. Fund.,25,594-602. Langmuir, I., 1913. The vapor pressure of metallic tungsten. Phys. Rev. Ser. 2, 2, 329-342.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Case Study on Design of Regulatory Policies for Sustainable Emission Reduction Andres Malcolm, Libin Zhang and Andreas A. Linninger Laboratory for Product and Process Design, University of Illinois at Chicago, Chicago, 60607, USA
Abstract Current environmental regulations typically follow a command-and-control approach mandating the installation of state-of-the-art abatement technology and hard emission thresholds. However, this type of regulation does not encourage process improvement and technological research to cut down emissions below compliance levels. On the other hand, market-based environmental regulatory models stimulate continued improvement of cleaner manufacturing practices by creating economic incentives for sustained emission reduction. This paper aims at furnishing regulators and manufacturers with a tool to assess the impact of future regulatory scenarios. In order to assess the impact of different regulations, this paper proposes a holistic model for commodity chemicals, pharmaceutical and specialties manufacturing operations including standard recycle and treatment options. This proposed work employs realistic chemical engineering models of pollution abatement operations in order to assess the feasibility of a treatment option, estimate its cost and expected emissions. Furthermore, this work introduces rigorous mathematical optimization techniques for designing regional emission reduction efforts at reasonable cost to manufacturers. This approach will offer plant managers a systematic tool to ascertain expected compliance cost of new environmental regulations and regulators a systematic methodology to design regulations considering manufacturers' budgets. Keywords: Pollution Prevention, Emission Trading, Design of Environmental Regulations, Multi-Period Optimization, Sustainable Manufacturing.
1. Introduction Regulators like the Environmental Protection Agency (EPA) use different regulatory tools to ensure socially acceptable emission levels caused by polluting industries. This work analyzes cost and environmental efficacy of different types of environmental regulations to reduce and control pollution. The paper will compare three different types of regulations for lowering air emissions from industry: Command-and-Control, Environmental Tax and Emission Trading. There are many different operations for transforming hazardous waste into environmentally benign material. The Combinatorial Process Synthesis (CPS) automatically produces a set of recycle treatment options for converting a given effluent stream into one or more benign residuals. A complete description of the CPS is available elsewhere (Chakraborty et al., 2002, 2004). This work uses a modified CPS considering a group of plants to analyze the impact of different regulatory scenarios on a whole region. In order to obtain the optimal plant strategies, the CPS needs the plant inventory, the business plan and the waste forecast as inputs, as depicted in Figure 1. In this paper we wish to study the impact of different regulations on reducing the manufacturers’ emission levels. We choose CO2 as pollutant, because the global interest
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in reducing greenhouse gases. The methodology is flexible so that it can be applied to other pollutants or groups of pollutants (Malcolm et al., 2006). Combinatorial Process Synthesis for Regional Emission Control
Regulatory Models
WASTE MANAGEMENT SCENARIO, W Waste forecast for each plant (Amount, Composition)
PLANT PRODUCTION FORECAST
Plant 1 Plant 4
1. SUPERSTRUCTURE GENERATION
Plant 2
DIAGNOSIS Generation of Treatment Goals
Plant 3
PRESELECTION Treatment Step Identification
TREATMENT DATABASE, T (Recycle, Distill, Extract, etc.)
POLLUTANTS TRESHOLDS, R Dictated by occupational health and safety standards
EXECUTION Simulate Residue Estimate Cost Possible Technological Options for Recycle and treatment
2. SUPERSTRUCTURE OPTIMIZATION PLANT MODELS Actual inventory of equipment types and capacities, dmax
NETWORK OF TREATMENT PLAN
REGULATORY FORECASTS A- Command and Control B- Emission Tax C- Emission Trading
TREATMENT PLAN OPTIMIZATION FOR ALL PLANTS (MILP)
PLANT BUDGETS Maximum investments, Imax
REGION-WIDE OPTIMAL TREATMENT POLICY
Figure 1– Methodology for assessing regional emissions and cost for a change in environmental regulations.
2. Methodology This section aims at predicting the compliance cost and emission reduction induced by three different types of environmental regulation in a region consisting of the four plants. It is assumed that CO2 needs to be reduced by 15% within a period of five years. The first section estimates the compliance cost for this region under a command-andcontrol type of regulation. The second example estimates the impact of a similar regulatory change guided by pollution tax. The third problem assesses cost for air pollution reduction using a cap-and-trade scenario. It will be interesting to asses whether the market-based model really leads to the desired air emission reduction at minimal cost to manufacturers as suggested by economic theory (Milliman and Prince, 1989; Tietenberg, 1985, 1996). Problem (1)-(4) is a mixed-integer-linear program to find the optimal industry's waste management strategies under different environmental regulations. This formulation implies a multi-objective problem according to each plant's cost minimization goals p with weight, γ . The objectives in (1) minimize the total net present cost, NPC, factoring the net present operating cost and the annualized capital investment which accounts for the purchase price of each equipment type, Ce, and the number of units bought in the period, Δn (t ) . Constraint (3) performs facility allocation of each recycle p and treatment tasks x j (t ) into a corresponding plant equipment, e. p
e
Case Study on Design of Regulatory Policies for Sustainable Emission Reduction
min NPC =
p p x j , Δne
∑γ ∀p
p
⎛ N ⎞ n ⎜ ∑ ∑ (1 + r ) -t ⋅ C p ( t ) ⋅ x p (t ) + ∑ (1 + r ) -t ∑ C ⋅ Δn p (t ) ⎟ j e e
⎟
t =0 ⎜ j t =0 ∀e∈I e CapitalCost ⎝ ⎠ OperatingCost
1121
(1)
s.t. p
p
p
ne (t ) = ne (t - 1) + Δne (t ) p
map ( x j (t ), e ) ;
∑x
p (t ) ⋅ j
p
∀x jp ≠ 0 p
E j (t ) ≤ emax (t)
Equipment Investment
(2)
Facility Allocation
(3)
Environmental regulations
(4)
where p ∈ { Plants} , e ∈ {equipments} , r : Interest Rate
The optimal solution fixes binary decision variables, xp(t), reflecting different choices of recycle or treatment steps in each plant during the planning period (n = 10) and an economic horizon of N=20 years. The solution allocates each unit operation to a specific piece of equipment, e, and optimally places investment decisions to augment the plant capacity, C or acquire equipment for new recycle or treatment operations. Capital investments for new plant equipment, e, at site, p, are represented by integer increments, Δn (t ) . Eq. (4) models different environmental regulations; in this case the command and control threshold is presented. Due to space limitations only the results will be presented in this work. The complete mathematical and more details are available elsewhere (Malcolm et al., 2006). p
e
3. Command-and-Control
Invesments (k$)
Command-and-control regulations prescribe hard limits on total emissions for each site. The total emissions emanating from a site must therefore remain below the regulated emission threshold. The regulatory forecast of the command-and-control foresees the 15% reduction in CO2 emissions. Accordingly, the regulation establishes hard bounds on the maximum permissible discharges for each plant. The computational analysis synthesizes plant operating schemes which comply with the hard CO2 reduction goal. This is a challenge since the industry’s average production is assumed to grow at a rate of 3% per year. The corporations have to invest in new treatment technologies in order to reduce pollution, while at the same time expand production. Figure 2 shows that plants one, three and four have to purchase new separation equipment to meet the new CO2 standard. Most investments are 1,400 necessary immediately after the Plant 1 Plant 2 Plant 3 Plant 4 enactment of the lower CO2 limit. These 1,200 technological investments include 1,000 distillation columns to recover solvents, 800 thus eliminating CO2 emissions 600 associated with waste incineration. 400 Plant-2 equipped with the best initial 200 plant infrastructure can afford to delay 0 investments by initially shifting from 0 1 2 3 4 5 6 7 8 9 10 Year waste incineration to more solvent Figure 2. Yearly investments under recovery with available in-house command-and-control regulation with the equipment. This site maintains in-house corresponding equipment types.
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solvent recycling until year nine. At that point it has to acquire more distillation columns to handle its increase in organic wastes. Plant-3 needs to reduce waste incineration in favor of more environmentally friendly solvent reuse. The change to waste recycling requires substantial capital investment in year five and seven.
4. Environmental Tax The environmental tax model tries to improve company's pollution prevention behavior by charging a tax for emissions. The specific regulatory design question is as follows: Which pollution tax level leads to 15% emission reduction in the region in the next five years? We propose to solve this problem in two steps. First, the tax level is set at the current average marginal abatement cost. The marginal abatement costs were estimated by repeated treatment cost simulations with small increments in the waste loads; the base tax level was set accordingly. Then the base tax level of 0.03$/Ton was increased until the total emissions of the region were compliant. Thus, the necessary tax increase was of 0.07 $/Ton CO2. The increased tax burden forces plant-1 and plant-4 to invest in newer, cleaner technology. The other two plants prefer to pay the higher tax, while using existing equipment capacity to minimize their air discharges. This analysis demonstrates that environmental tax succeeds in inducing the desired emission reduction. However, the ideal tax level requires a good estimate of the manufacturers' "hidden" marginal abatement costs which may be difficult for a regulator to assess. In addition, necessary iterations for determining the tax levels for achieving a desired emission targets causes regulatory uncertainty for the industry. Environmental Tax may be therefore hard to implement politically.
5. Cap and Trade Regulatory Framework
Buyers S ellers
Tons of CO2 Tradaed
Finally, we wish to examine a cap-and-trade regulatory model. It introduces two new adjustable parameters: the calling price for permits titles and the total permits cap (Tietenberg, 1985,1996). In this work, we will demonstrate how the regulator can adjust these parameters to achieve desired levels of emission reduction. The cap limits the total emissions in a region, thus a limited distribution of emission permits guarantees a 15% pollution reduction. Under the assumption of utility maximization governing the decision for each manufacturer, the mathematical program of (1) – (4) predicts the compliance cost and projected emissions for the entire region. Initially, it is beneficial for plant-1 to buy emission permits in order to defer investments into cleaner technology. Therefore, it purchases pollution credits on the emission market. Permits are available since the expansion of existing solvent recovery capacity in plants 2 and 3 reduces CO2 output creating free titles for trading. In year 4, plant-1 finally has to invest in additional recovery technology. Its Plant 1 Plant 2 900 subsequent low CO2 discharge creates a Plant 3 Plant 4 700 return on investment from the sale of 500 surplus permits. These free permits 300 No trading provide plants 3 and 4 with the Permitted 100 flexibility to optimally time necessary -100 upgrades on their sites. The trading 0 1 2 3 4 5 6 7 8 9 10 activity among the polluters is shown in -300 Figure 3 according to which plants 3 -500 and 4 are net buyers and plants 1 and 2 -700 are net sellers. Eventually plants 3 and 4 Year make their major investment decisions Figure 3. Emission trading activity under in year eight and nine respectively as cap-and-trade with minimum cost.
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Investments (k$)
depicted in Figure 4. Plant-1 emits 1,400 Plant 1 Plant 2 more than initially by buying credits Plant 3 Plant 4 1,200 from the other plants with improved 1,000 pollution prevention technologies (e.g. 800 plant-3). After plant-1 invests in the 600 new recovery technology, its 400 emissions are reduced drastically. On 200 the other hand, plant-3 with the better 0 initial infrastructure, in early periods 0 1 2 3 4 5 6 7 8 9 10 Year recycles solvents using their solvent recovery free capacity lowering its Figure 4. Yearly investments under capinitial emissions. In year 8 due to its and-trade with symbolic representation of growing production, plant-3 has to solvent recovery. invest in expanding its solvent recovery facilities lowering its emissions. Although the plants can buy and sell emission credits, the total cap is never violated and the regional emission levels are reduced at minimum cost.
6. Discussion
Total Net Present Cost (k$)
Total Net Present cost (k$)
Economical Impact. Figure 5 shows that the cap-and-trade strategy achieves similar pollution levels than a command-and-control scenario by spending almost 10% less ($360,000 less). The environmental tax model is twice as expensive for the industry. With taxation, an arbitrarily high 7,000 charge is added to the production cost in order to achieve desired emission 6,000 reduction. More taxation may lead to 5,000 prohibitively high production costs for the whole region thus distorting the 4,000 competitiveness of the manufacturers. 104 kTon 102 kTon 3,000 110 kTon This problem is usually solved by tax of CO of CO of CO Emitted Emitted Emitted return policies, but since this an 2,000 Command Cap and Trade Tax arbitrary measure, it was not Control considered in this work. A detailed Figure 5. Total regional net present cost analysis of the individual plants and total CO2 emitted under the different annualized expenditures for emission regulatory scenarios. control is depicted in Figure 6. The Command Control 3,600 comparison shows that the total Cap and Trade 3,200 annualized costs are always lower in Environmental Tax 2,800 emission markets not only for the 2,400 region, but also for each company 2,000 1,600 individually. This property seems to 1,200 indicate that emission trading creates 800 the smallest business interference 400 caused by the regulatory change. In 0 Plant 1 Plant 2 Plant 3 Plant 4 contrast, command-and-control as well as taxation creates arbitrary Figure 6. Plant’s net present total annualized losers and winners. cost of manufacturing under the different regulatory scenarios. 2
2
2
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7. Conclusions The article proposes a systematic framework based on detailed unit operation models to accurately predict treatment options, emissions and the optimal investments policies of a region under different regulatory scenarios. The approach shows the successful combination of chemical engineering knowledge with simplified market models to predict the economic and environmental impact of regulatory changes. This work can assist regulators in designing the environmental policies that satisfy environmental objectives with consideration of financial burdens to manufacturers. The quantitative analysis reveals that the flexibility of the emission-trading model benefits plant managers in three aspects. First, they can optimally time their investments decisions in accordance with their production plans and available cash flow. Secondly, they may earn ongoing benefits from technological improvements by selling surplus permits. Finally, desired air emission reduction targets for a whole region were more tightly satisfied under the cap-and-trade regulation. Consequently, the market driven approach gives the regulator a very effective control instrument to adjust desirable levels of tolerable pollution. In conjunction, these three advantages may encourage the introduction of pollution prevention efforts in industry, currently missing in the prevailing command-and-control environmental regulations. This paper has considered a simplified market model using a multi-objective utility maximization assumption to predict corporate decisions. In the future, an improved model could consider price flexibility using a game theoretic approach for demand and supply of emission titles. Another interesting feature to be included in future work is the consideration of uncertainty in the market and price forecasts.
8. Acknowledgments Financial support from NSF Grant DMI-0328134 and the Environmental Manufacturing Management (EvMM) fellowship from the UIC Institute for Environmental Science and Policy are gratefully acknowledged.
References Chakraborty A. and Linninger A. A.; "Plant-Wide Waste Management. 1. Synthesis and MultiObjective Design", Industrial and Engineering Chemistry Research, 41 (18), pp 4591 - 4604, 2002. Chakraborty A. and Malcolm A., Colberg, R. D., Linninger A. A., "Optimal Waste Reduction and Investment Planning under Uncertainty", Computers and Chemical Engineering, vol. 28, pp. 1145 – 1156, 2004. Linninger, A. A. and Malcolm, A.; "Pollution Prevention in Batch Processes". Book chapter for "Batch Processes" edited by E. Korovessi, DuPont and A. Linninger, UIC, CRC Press, 2005. Malcolm, A.; Zhang, L.; and Linninger, A. A.; "Design of Environmental Regulatory Policies for Sustainable Emission Reduction", accepted for publication in AIChE Journal, 2006. Milliman SR and Prince R. Firm Incentives to Promote Technological Change in Pollution Control. Journal of Environmental Economics and Management; 17:247-265, 1989. Tietenberg T. Emission Trading: an exercise in reforming pollution policy. Recourses for the future;, Washington, DC, 1985. Tietenberg T. Environmental and Natural Resource Economics (4th edition), New York, NY: Harper Collins College Publishers, 1996.
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A decomposition/reconstruction algorithmic procedure for computer aided case based reasoning – implementation in biodegradation Fragiskos A. Batzias Laboratory of Simulation on Industrial Processes, Dept of Industrial Management and Technology, University of Piraeus, Karaoli & Dimitriou 80, 185 32 Piraeus, Greece
Abstract This work deals with the design/development/implementation of a flexible algorithmic procedure for Case Based Reasoning (CBR), capable to be used in Computer Aided Process Engineering (CAPE) problems of process/product investigation/design either as a main technique or as a supplementary tool in searching for an acceptable solution. Its main characteristics are (i) decomposition of each case in constituent unit cases and storing within the same Knowledge Base (but in a different domain) with the corresponding whole cases, (ii) hierarchical and combinatorial searching within both domains, for finding similar whole/unit cases to compare with the case under investigation, (iii) application of fuzzy multicriteria analysis and fuzzy logic for decision making on best solution choice, after reconstruction of the whole case, (iv) dynamic behaviour by taking into account knowledge acquired within the same search session. An implementation concerning the pathway identification of pollutant RDX (hexahydro-1,3,5-trinitro-1,3,5-triazine) degradation is presented. Keywords: biodegradation, biosensors, case based reasoning, decomposition.
1. Introduction Case-Based Reasoning (CBR) is usually formalized, for enhancing algorithmic approaches, as a four-step (named 4R) process, involving the following stages: retrieve, reuse, revise, retain. The Knowledge Base (KB) that supports CBR is continually enriched by storing new cases but the CBR mechanism is actually a batch process, as this kind of reasoning is activated only when the target problem appears and demands a solution. By considering the KB as part of the CBR system, we might say that this system delays generalization of its stored cases until specific problem solving time – a strategy of lazy generalization, in contrast to Rule-Based Reasoning (RBR), when generalizations are made from the set of cases already stored before the target problem is even known. In RBR, some kind of analysis/decomposition is made in order to find common characteristics among the resulting parts and derive the corresponding rules (partonomy/taxonomy function, widely applied in any ontological approach). In CBR, the target problem or probe case is compared for matching with each whole case already stored. This is one of the weak points in the entire procedure, as the expert (system or/and human) is called to understand/discriminate/isolate not simply some common elements between the probe P and the already stored case J but the common element(s) that is(are) critical for solving the problem by similarity; as a matter of fact, this is an identification/interpretation issue in the sense of second order cybernetics. The approach adopted herein to make CBR more effective when dealing with pathway identification of pollutants degradation is decomposition of the whole cases
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into parts (partonomy function) if the conventional technique of matching the whole cases fails; then, reconstruction is performed on the basis of the closest neighbouring parts to form the corresponding wholes upon which final test for matching is made. In this way, CBR comes closer to RBR, where decomposition facilitates the derivation/testing of rules, and formal logic comes closer to content, since the algorithmic decomposition into parts follows the physicochemical pathway of degradation into intermediate and final products.
2. Methodology The methodological framework designed/developed in the form of an algorithmic procedure to solve the problem described in introduction consists of the following stages (for their interconnection, see Fig. 1). 1. Description of the probe case under consideration. 2. Determination of the problem to be solved and/or the partial targets to be achieved. 3. Identification of parameters / variables corresponding to significant attributes and setting of specifications for a solution to be accepted. 4. Design/synthesis/adoption of criteria for selecting external Information Libraries containing similar whole cases. 5. Retrieval of the next most similar whole cases and temporal storing of them. 6. Fuzzy multicriteria ranking of retrieved whole cases by means of a modified Delphi method described in [1]. 7. Usage of the first whole case (from the set of ranked cases which remain unexamined) in successive trials, without and with revisions, to solve the problem initially determined. 8. Evaluation and characterisation (based on taxonomic/partonomic criteria) of the solution for KB updating. 9. Incorporation of the information acquired (due to unsuccessful trials) into the KB. 10. Decomposition of the probe case into discrete partial/unit cases. 11. Selection of the critical pathway probe case (for the solution of the problem set in stage 2) to be examined and reformulation of stage 3 accordingly. 12. Decomposition of the similar whole cases already retrieved into partial/unit cases corresponding to those obtained from the probe case. 13. Setting up of the necessary recomposition mechanisms/interfaces/protocols. 14. Design/synthesis/adoption of criteria for selecting external Information Bases of similar partial/unit cases. 15. Retrieval of the next most similar partial/unit cases and temporal storing of them. 16. Fuzzy multicriteria ranking of retrieved partial/unit cases (including those decomposed in stage 12) by means of a modified Delphi method. 17. Usage of the first partial/unit case (from the set of ranked cases which remain unexamined) in successive trials, without and with revisions, to achieve satisfactory matching. 18. Reconstruction of the (modelled) whole case proposed as the most similar one. 19. Incorporation of the information acquired (due to unsuccessful trials) into the KB. 20. Design/development of an expert system on the grounds of RBS. 21. KB operations (enrichment, information processing consultation). 22. Intelligent agent performance for knowledge acquisition from external Bases according to an operational interface described in [2]. P. Is the problem solved satisfactorily? Q. Is there any other ranked whole case unexamined?
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Is the probe case decomposable? Is the increase of the number of parameters/variables and/or the loosing of the specifications set initially in stage 3 feasible/acceptable? T. Is the matching achieved in accordance with the specifications? U. Is the problem solved satisfactorily? V. Is there any other ranked partial/unit case unexamined? W. Is loosing of the specifications set initially in stage 3 and formulated in stage 11 feasible/acceptable? X. Is there an adequate amount of data/information formulating a complete set of rules to solve the same initial problem? The criteria used for selecting the critical probe pathway in stage 11 are: extent and completeness; f1; uniqueness of intermediates, f2; reliability of experimental evidence, f3; plausibility/rationality, f4; uncertainty, f5. The criteria used is stages 6 and 16 for ranking whole and partial cases, respectively, are: similarity to probe, without any revision/modification in order to improve matching, h1; empirical support by induction, h2; theoretical support by deduction, h3; simplicity (Occam’s razor) as a measure of fulsifiability, after Popper, h4. The multicriteria evaluation method adopted is a fuzzy modification of classical PROMETHEE for allowing partial compensation,
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indifference (aIb), incomparability (aRb), weak preference (aQb) and strict preference (aPb) between alternatives a and b.
3. Implementation and Results An implementation of the methodology described above is presented subsequently in brief, where biodegradation of the cyclic nitramine explosives (commonly used in conventional munitions and various military applications) is examined as regards the mineralisation pathways followed through their intermediates to final dead end products; i.e., the probe case under examination and the problem to be solved (stages 1 and 2, respectively) considered together is enhancing or weakening the hypothesis on an explosive’s biodegradation pathway by simulating its key metabolites with the intermediates of another more established/identified biodegradation pathway of the same explosive. From the point of view of scientific methodology, this hypothesis resembles a Lakatosian Research Programme, which can be proved to be progressive or degenerating while under investigation until final identification in depth. The importance of the studied biodegradation lies in the fact that activities associated with manufacturing of nitramine explosives, training, waste disposal, and closures of military bases have resulted in significant soil and ground water contamination with toxic products, thus necessitating their neutralization which can be done mostly by microbial degradation. The identification of pathway intermediates permits the design/development/application of relatively cheap tailor-made biosensors for (i) following the fate of these explosives in Nature and (ii) assessing the results of measures taken to prevent soil pollution or accelerate biodegradation (e.g., by providing supplementary nutrients or/and more efficient microbial species, possibly modified genetically to consume the specific explosive/contaminant more effectively). The input/output summary screenshot in Fig. 2, extracted from a specimen run of the computer program we have developed, is a significant node of the network of the relevant activities performed according to the main stages of the decomposition/reconstruction algorithmic procedure depicted in Fig. 1. In the first column the most important cyclic nitramine explosives 2,4,6,8,10,12-Hexanitro2,4,6,8,10,12-hexaazaisowirtzitane, octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine, hexahydro-1-nitroso-3,5-dinitro-1,3,5-triazine, hexa-hydro-1,3,5-trinitro-1,3,5-triazine are presented under the common/market abbreviated names CL-20, HMX, MNX, RDX, respectively, which are more recognisable and suitable for keyword searching by the intelligent agent of stage 22. The last explosive (RDX) has been selected for investigating its biodecomposition pathway which is presented in the second column, under the form of molecular formulae of the intermediates most likely to participate in the chain starting with the RDX-formula and ending with the intermediate suggested by performing stage 11; the complete chain, without the end products, is as follows (RSS stands for Rhodococcus sp. Strain DN22 [3]): C3H6N6O6 (RDX)
+RSS - (NO2
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The third column gives, as L-R triangular fuzzy numbers, the results of multicriteria evaluation of pathway fragments, considered as alternatives; the fourth column shows the results in crisp numbers after defuzzification by applying the centroid method (the Tseng and Klein method can also be used for higher resolution [4]); the last
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Fig. 2. Screenshot presented the INPUT-OUTPUT SUMMARY, when the probe pathway used for CBR is the fragment or partial chain C3H6N6O6 – C2H5N3O3 (first and last chemical term, respectively).
column shows the end products of biodegradation. The ‘critical’ (possibly a key metabolite if it actually appears in the pathway – consequently its presence in other RDX decomposition pathways may enhance the hypothesis of its existence) intermediate is the hypothesised substance C2H5N3O3 (ranked first in preference order, with a score of 82.4 expressed as a crisp number) and the matching results obtained via CBR appear under the OUTPUT heading: alkaline hydrolysis and photodegradation are suggested as the closest processes that give C2H5N3O3 during the corresponding RDX decomposition routes. The button functions are as follows; Taxonomy: gives all stored chains (in alphanumeric code) that contain the probe chain; Partonomy: gives the complete pathway (after a query by the user for a specific code number) which the probe chain is part of (italics used for the semantic operators functioning in the ontology); Sequence: gives all stored alternative chains with starting and ending compound identical to those of the probe chain; Upstream: gives all stored alternative chains leading to the ending compound of the probe chain; Downstream: gives all stored alternative chains involving the starting compound of the probe chain in their pathway towards full decomposition (mineralisation); Matching: presents the files with the sets of all the matching results obtained in stages 7 and 17; Results: presents the files with the sets of all the ranking results obtained in stages 6 and 16; Conditions: gives details on the conditions required for the realization of the processes in the OUTPUT section, where the results of CBR appear, ranked in decreasing order of matching. A ‘Documentation’ file, containing the corresponding ‘Reference’, is linked with ‘Conditions’; e.g. [5, 6] can be found within the sub-files corresponding to the first two processes of ‘Output’, Alkaline Hydrolysis and Photodegradation of RDX, respectively.
4. Discussion and Concluding Remarks Some pathways stored in the KB may include hypothetical intermediates, whose existence depends on ‘specific assumptions’. Evidently, the fuzzy grades assigned, according to criterion f2, to such an alternative pathway in stages 6 and 16 should be at a minimum but the grades according to the rest criteria may be higher enough to rank this
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alternative first. In such a case the ‘specific assumptions’ will appear by activating the file (through the corresponding button, see Fig. 2) of ‘Conditions’. If these assumptions are not realistic for the probe case, the user can either select the alternative ranked next or rerun the fuzzy multicriteria module, the second disjunctive to be preferred if sensitivity analysis is going to follow for providing information about the robustness of the solution. On the other hand, successful matching with pathways containing hypothetical or real (but not easily detectable) critical intermediates could contribute to research intensification for the isolation/identification of metabolites of explosives degradation [7]. Finally, the economic criterion should be also taken into account when CBR is used within large Knowledge Bases, as we have shown recently [8]. In conclusion, the approach adopted herein offers some help in the design of computer aided CBR procedures, especially when dealing with pathway identification of pollutants degradation, bringing this method closer to RBR and formal logic closer to empirical content, since the suggested algorithmic decomposition into parts follows the physicochemical pathways of degradation; on the other hand, the processing of fragments obtained by decomposition demands computational power and combinatorial techniques of a significantly advanced level in comparison with the conventional CBR techniques where the analyst’s mind, having to deal with a lower number of whole cases, takes over.
Acknowledgments This work was performed within the framework of Pythagoras II EU-GR Research Programme (Section: Environment) for the design/development/ implementation of bioindicators/sensors. Support provided by the Research Centre of the Piraeus University and contribution of an anonymous referee to clarifying certain points of this presentation are also kindly acknowledged.
References [1] A. Batzias and F. Batzias, 2003, Fuzzy fault tree analysis as a mechanism for technical support to small/medium electroplaters on a quasi online/real-time basis, Proc. IEEE Intern. Conf. on Industrial Technology, Maribor, Slovenia, 36-41. [2] F. Batzias and E. Markoulaki, 2002. Restructuring the keywords interface to enhance CAPE knowledge via an intelligent agent. Computer-Aided Chem. Eng., 10, 829-834. [3] D. Fournier, A. halasz, J. Spain, P. Fiurasek, J. Hawari, 2002, Determination of the key metabolites during biodegradation of hexahydro-1,3,5-trinitro-1,3,5 triazine with Rhodococcus sp. Strain DN22, App. Environ. Microbiol., 68, 166-172. [4] T.Y. Tseng and C.M. Klein, 1989, New algorithm for the ranking procedure in fuzzy decisionmaking, IEEE Trans. Systems, Man and Cybernetics, 19, 1289-1296. [5] V.K. Balakrishnan, A. Halasz, J. Hawari, 2003, Alkaline hydrolysis of the cyclic nitramine explosives, RDX, HMX, and CL-20: New insights into the degardation pathways obtained by the observation of novel intermediates, Environ. Sci. Technol., 37, 1838-1843. [6] J. Hawari, A. Halsz, G. Groom, S. Deschamps, L. Paquet, C. Beaulieu, A. Corriveau, 2002, Photodegradation of RDX in aqueous solution: A mechanistic probe for biodegradation with Rhodococcus sp., Environ. Sci. Technol., 36, 5117-5123. [7] M.C. Rodriguez, M.R. Monti, C.E. Argarana, G.A. Rivas, Enzymatic biosensor for the electrochemical detection of 2,4-dinitrotoluene biodegradation derivatives, Talanta, 68, 16711676. [8] F.A. Batzias, Optimisation of measuring equipment network for preventing photochemical pollutin episodes – a model case-based reasoning approach under EN 14181, Proc. of the 7th International Conference on Emissions Monitoring, Paris, February 2006, 355-365.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
ENVIRONMENTALLY CONSCIOUS DESIGN OF ETHANOL FED FUEL CELL SYSTEM Liliana Hernández, Viatcheslav Kafarov Center for Simulation and Control of Process, Industrial University of Santander. A.A. 678, Bucaramanga, Colombia
Abstract In this work, a clean technology for electricity production from renewable sources is proposed. For this aim, an integration of bioethanol steam reforming and a fuel cell system (SOFC) was developed by computer aided design using HYSYS®. For process integration, was taking into account that steam reforming of bioethanol is an endothermic process and this reaction improves its conversion with fed steam excess. Moreover, typical SOFC operational conditions were used. An integrated flowsheet was developed using heat and mass integration of several process streams achieving high energy efficiency. Additionally, a discussion about other integration schemes is presented. Keywords: computer aided design, SOFC, bioethanol steam reforming, heat integration, mass integration.
1. Introduction Currently the hydrogen is considered as the new “energetic vector” because of its advantages related to possibility its production from renewable sources with an important positive environmental impact and its high mass energetic density (Zabalza, et. al. 2005). Hydrogen can be obtained from biomass or its products as bioethanol which is produced by fermentation. Hydrogen from bioethanol can be produced by partial oxidation, autothermal reforming or steam reforming. The last, has the disadvantage that involve an endothermic reaction but with this process best conversions to hydrogen can be obtained. Electricity production from bioethanol and its steam reforming to hydrogen for fuel cells is a clean technology which offers high energy efficiency and zero emissions pollutants. Solid oxide fuel cells (SOFCs) are one of the most attracting kinds of fuel cells, for its advantages as high efficiency (near to 60%), high rate in reaction kinetics and high quality heat, which can be used as source for heat integration with endothermic bioethanol steam reforming. The SOFC – based power plants fuelled by bioethanol have been studied for recent 3 years. The most of these researches are focused on SOFC direct ethanol fed because the high operational temperatures of these fuel cells which can let the reforming of this fuel inside the device (Gupta, et. al., 2005, Douvatzides, et. al., 2004, Assabumrungrat, et. al., 2004), but the slow diffusion of ethanol on electrodes and the coke deposition at those temperatures (1000°C) are some challenges that must be solved. Other kind of these power plants is which the bioethanol reforming take place on external reformer at SOFC. In this case, the reformer operation can be improved to get more hydrogen quantities and the reforming can be operated at moderated temperatures to avoid the coke deposition on the catalyst. The state of the art about this process is on developing and we can make mention of one work only of Douvartzides, et. al. (2003),
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although at the present on Europe are going ahead several research projects as BioH2 and many others. In Douvartzides work, energy – exergy analysis and optimization of operational conditions was made, using an afterburner to develop the heat integration of this process. In this work, other alternative for this process integration is discussed. Typical operational conditions for SOFCs are used and, the favorable conditions for bioethanol steam reforming as moderate temperatures and steam excess on the fed stream was taking into account. The research is a part of joint international project “Bioethanol fed fuel cells” CYTED IV.21 with the participation of eight European and Latinamerican countries in a frame of program “Biomass as source of energy and chemical products”.
2. Bioethanol as Hydrogen Storage Bioethanol as hydrogen has several advantages such as hydrogen carrier for fuel cells, because of it is easy to store, handle and transport in a safe way due to its lower toxicity and volatility. Moreover, bioethanol is a chemical substance which can storage hydrogen at greater than its liquid density at atmospheric pressure, it has a hydrogen volume density more than other organic compounds as ethane, propane and methanol. Additionally, bioethanol as storage medium has a total density near to 1 g/cm3 because of this it is among the most promising hydrogen storage fuel cell substances (see Figure 1).
Figure 1. The storage density of hydrogen plotted as a function of the hydrogen mass fraction. Crabtree, 2005
3. Modelling of Bioethanol Steam Reforming The most of research made about bioethanol steam reforming is addressed to develop catalyst for this process and to study its behaviour, but only few works have been reported about the development of reaction mechanisms and kinetics models. For catalyst development, different metals with several supports of metallic oxides have been probed among them: Rhodium: (Diagne, et. al., 2004) (Diagne, et. al., 2002); Platinum: (Freni, et. al., 2003); Nickel: (Laborde, et. al., 2004 a) (Sun, et. al., 2005) (Sun, et. al., 2004) (Athanasios, et. al., 2002) (Athanasios, et. al., 2003) (Freni, et. al., 2003); Cobalt: (Batista, et. al., 2004) (Llorca, et. al., 2003 b); Gold: (Freni, et. al., 2003); Palladium: (Goula, 2004) (Freni, et. al., 2003) and Ruthenium: (Dimitris, 2003). Also, have been studied metallic oxides on metallic oxides as Copper oxide (Nishiguchi, et. al., 2005) and bimetallic catalyst as Cu-Ni/Al2O3 (Laborde, et. al., 2004 b) and Cu-Ni/SiO2 (Fierro, et. al., 2002).
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For feed ratio of ethanol/water have been probed experimentally ratios from 1:6 (Laborde, et. al., 2004 a) to 1:13 (Llorca, et. al., 2003 a), but for ratios superior at 1:8 was not registred an improving of conversion and selectivity for bioethanol steam reforming (Diagne, et. al., 2004). In relation to the discernment of reaction mechanisms for steam reforming of bioethanol, have been found only few such as for Nickel catalyst supported on La2O3 (Athanasios, et. al., 2004), Rhodiun on TiO2 (Raskó, 2004), Nickel on Al2O3 (Laborde, et. al., 2004 b) and Cobalt on ZrO2 (Benito, et. al., 2005). About kinetic models only have been found one complete work for Ni/Al2O3 catalyst (Therdthianwong, 2001) and this model was correlated for only one temperature of 400°C. In this work, was considered that bioethanol is vaporized and reformed by mixing to steam in a packed bed reactor which contains a Ni-based catalyst where its overall reaction can be represented by the irreversible process between one ethanol molecule and three water molecules according to the reaction o C 2 H 5 OH + 3H 2 O → 6 H 2 + 2CO2 ΔH Rx = 1.704 x10 5 kJ / kgmole
(1)
Bioethanol steam reforming take place for temperatures more than 250°C. In this work the reformer is operated to 400°C with a non elemental kinetic, Therdthianwong, 2001 (see Equation 2). 2.52 − rC2 H 5OH = 280075 PEtOH PH72O
Where − rC2 H 5OH is the rate of consumption of bioethanol [mol/g-cat h] and P is the pressure [atm.]. Reformer was simulated by a one dimensional pseudohomogeneous model for packed bed reactors considering a plug flow for gas phase and calculating the pressure variation throughout reactor using traditional Ergun equation.
4. Modelling of Solid Oxide Fuel Cells (SOFC) and Co-generation Solid oxide fuel cells with operating temperature of >800°C promotes rapid kinetics by no precious materials, and produces high quality by-product heat for cogeneration. For SOFC modelling on recent years, it has been developed complex models 3D using finite elements (Khaleel, et. al., 2004), CFD using commercial software (Autissier, et. al., 2004) (Lockett, et. al., 2004), finite volume analysis (Campanari and Iora, 2004) and thermo-electrochemical models (Petruzzi, et. al., 2003) to calculate profile temperatures, currents, electrical and thermal power density and gases concentrations. However the knowledge about steam reforming is limited and to avoid the introduction of information noise for joint process was selected uncomplicated SOFC model based on HYSYS®. For process integration was using SOFC with typical operational conditions: T=1000°C, fuel utilization: 85% and oxidant utilization: 25%.
5. Simulation Results: Process Description, Design and Integration 5.1. Process Description The system to produce electricity from bioethanol by steam reforming and fuel cells (SOFC), can be described as follow. The bioethanol is vaporized and fed at a packed bed reactor which contains a Ni-based catalyst. Additionally, water is vaporized and fed at reactor too, and the molar ratio between bioethanol and steam fed was 1:6 to improve the reaction conversion to 99%. The reaction products are fed to SOFC at 485°C after a heating with a hot flow from SOFC. This stream has a volume composition on hydrogen of 53% and taking into account that a typical fuel stream for SOFC has a 67% of hydrogen (Allied-Signal, 1992), is assumed that the SOFC performance no is affected considerably. An air stream is fed to SOFC is also preheated to 485°C to improve the
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fuel cell performance. Finally, the SOFC stream products are used to heat the steam reformer and after that, they are split on four streams to vaporize the bioethanol, the water, to preheat the air and to feed a turbine to produce more energy
Name Temperature(°C) Pressure (kPa) Molar flow (kgmole/h) Mass flow (kg/h) Name Temperature(°C) Pressure (kPa) Molar flow (kgmole/h) Mass flow (kg/h)
Ethanol 25 101,325 58,068 2675,187 304 1000,012 110,3312 3055,737 87710,21
Water 25 101,32 348,34 6275,3 400 400 118,6 629,83 8950,5
Air 25 101,3 3198 92268 401 485 117,2 629,8 8951
100 25 131 58,068 2675,2 404 985,58 108,95 3685,6 101218
103 400 120,7 58,07 2675 500 903,3 105,5 3686 1E+05
200 25 131 348,3 6275 603 82,75 101,4 165,9 4555
203 400 120,7 348,3 6275 703 64,73 101,4 1548 42512
300 303 45,939 485 121,36 117,2 3198,2 3198 92268 92268 803 901 108,14 895 101,37 101,3 751,85 1220 20649 33503
Figure 2. Electricity from bioethanol by steam reforming and fuel cell (SOFC) flowsheet 5.2. Process Design For process design following criteria was used. Environmental criteria: the process has zero emissions of SOx (raw materials does not contain sulphur components), NOx (electrochemical process without combustion) and CO2 produced on bioethanol reforming it is consumed for the biomass growth. Heat integration criteria: the high heat content of SOFC product stream which has a temperature of 1000°C, provides the necessary duty for the endothermic reaction of bioethanol reformer and the preheating requirements of streams fed to process such as bioethanol, water and air. The process was designed to produce 1MW by SOFC on steady state conditions join with an additional cogeneration by gas turbine. As row materials were used liquid bioethanol, and water to feed the reformer and to produce hydrogen as fuel, and air to feed the SOFC as oxidant. This design was computer aided for HYSYS® software. The process flowsheet is showed on Figure 2. For SOFC simulation in HYSYS, was used a subflowsheet, where the fuel cell was simulated as a chemical reactor by using the typical reactions that take place on SOFC (Fuel Cells Handbook, 2000). 5.3. Process Integration For process integration, a Pinch analysis to hot and cool process streams was made, and the splitting of streams was used to satisfy the process requirements. A heat exchanger network for each stream to preheat was designed. Minimum heat exchangers topology were calculated taking into account the phase changes involved on this stage for bioethanol and water streams. To find out this topology, an equilibrium point between the heat transfer efficiency and the heat exchangers number was found. The best process
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flowsheet found is showed on figure 2 and described as follows. The SOFC product streams were mixed and used to heat the steam reformer up to reaction temperature 400°C. The hot stream leaves the reformer at 903°C then; it is split on four streams to vaporize and preheat bioethanol, water, to preheat air and to feed a gas turbine. For each of three heating flows, a network of three heat exchangers was designed to heat cold streams, with the purpose to improve the heat transfer between the streams and avoid the energy losses for thermal shock due to big streams temperature differences. For the design of each heat exchangers network was taking into account, that the process cold streams had a temperature of 25°C and the hot streams was exhausted thermally and its out temperatures was between 87°C and 116°C. The process designed is a net energy exporter because, it produces 1MW from SOFC and 100 kW from turbine K-100, and process energy requirements are only for two pumps to move the bioethanol and water which consume 63.35 W and 116.2 W respectively and one compressor to fed air for SOFC which consume 543.5 kW. Finally this process can produce net 556.32 kW.
6. Conclusions The environmentally conscious design proposed is a base to develop clean technologies to produce electricity by using renewable sources such as bioethanol. A new proposal of a clean technology to produce electricity from bioethanol by steam reforming and SOFC using are designed. This alternative can produce net 556.32 kW and a burner to heat the steam reformer no is required. The heat process integration was made using the hot streams that leaves the SOFC and optimizing the heat transfer process using an appropriate heat exchangers network design.
Acknowledgements We thank the international project CYTED IV.21: “Bioethanol fed fuel cells” and at the Colombian Institute for Science and Technology Development “Francisco José de Caldas” – COLCIENCIAS and Iberoamerican Cooperation on Science and Technology for the Development for the financial support of this research.
References Allied-Signal Aerospace Company (1992). In: Fuel Cells Handbook. EG&G Services Parsons, Inc. Science Applications International Corporation. U.S. Department of Energy. Office of Fossil Energy. National Energy Technology Laboratory. 5th edition. Morgantown. 2000. Assabumrungrat, S. et. al. (2004). Thermodynamic analysis for a solid oxide fuel cell with direct internal reforming fueled by ethanol. Chemical Engineering Science. 59 (24) 6015-6020. Athanasios, N. et. al. (2002). Production of hydrogen for fuel cells by reformation of biomassderived ethanol. Catalysis Today. 75.145–155. Athanasios, N. et. al. (2003). Steam reforming of biomass-derived ethanol for the production of hydrogen for fuel cell applications. chemcomm communication. Athanasios, N. et. al. (2004). Reaction network of steam reforming of ethanol over Ni-based catalysts. Journal of Catalysis. 225. 439–452. Autissier, N. et. al., (2004). CFD simulation tool for solid oxide fuel cells. Journal of Power Sources. 131. 313–319. Batista, M. et. al. (2004). High efficiency steam reforming of ethanol by cobalt-based catalysts. Journal of Power Sources. 134. 27–32. Crabtree, G. (2005). The hydrogen economy. www.physicstoday.org/vol-57/iss-12/p39.html Benito, M. et al. Bio-ethanol steam reforming: Insights on the mechanism for hydrogen production. Journal of Power Sources. Article in press. Campanari, S. and Iora, P. (2004). Definition and sensitivity analysis of a finite volume SOFC model for a tubular cell geometry. Journal of Power Sources. 132. 113–126.
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Diagne, C., et. al. (2002). Hydrogen production by ethanol reforming over Rh/CeO2–ZrO2 catalysts. Catalysis Communications. 3. 565–571. Diagne, C. et. al. (2004). Efficient hydrogen production by ethanol reforming over Rh catalysts. Effect of addition of Zr on CeO2 for the oxidation of CO to CO2. C. R. Chimie. 7. 61–622 Dimitris, K. (2003). Production of hydrogen for fuel cells by steam reforming of ethanol over supported noble metal catalysts. Applied Catalysis B: Environmental. 43. 345–354. Douvartzides, S. et. al. (2004). Electricity from ethanol fed SOFCs: the expectations for sustainable development and technological benefits. International Journal of Hydrogen Energy. 29 (4) 375-379. Douvartzides, S., Coutelieris, F. and Tsiakaras P. (2003). On the systhematic optimization of ethanol fed SOFC-based electricity generating systems in terms of energy and exergy. Journal of Power Sources. 114. 203-212. Fierro, V. et. al. (2002). Oxidative reforming of biomass derived ethanol for hydrogen production in fuel cell applications. Catalysis Today. 75. 141–144. Freni, S. et. al. (2003). Production of hydrogen for MC fuel cell by steam reforming of ethanol over MgO supported Ni and Co catalysts. Catalysis Communications. 4. 259–268. Fuel Cells Handbook. EG&G Services Parsons, Inc. Science Applications International Corporation. U.S. Department of Energy. Office of Fossil Energy. National Energy Technology Laboratory. Fifth edition. Morgantown. 2000. Gaurav, G. Comparison of conversion and deposit formation of ethanol and butane under SOFC conditions. Journal of Power Sources, Article In Press. Goula, M. (2004). Hydrogen production by ethanol steam reforming over a commercial Pd/ Al2O3 catalyst. Applied Catalysis B: Environmental. 49.135–144. Khaleel, M., et. al. (2004). A finite element analysis modeling tool for solid oxide fuel cell development: coupled electrochemistry, thermal and flow analysis in MARC®. Journal of Power Sources. 130. 136–148. Laborde, M. et. al. (2004 a). Bio-ethanol steam reforming on Ni/Al2O3 catalyst. Chemical Engineering Journal. 98. 61–68. Laborde, M. et. al. (2004 b). Hydrogen production via catalytic gasifcation of ethanol. A mechanism proposal over copper–nickel catalysts. International Journal of Hydrogen Energy. 29. 67 – 71. Llorca, J., et. al. (2003 a). CO-free hydrogen from steam-reforming of bioethanol over ZnOsupported cobalt catalysts Effect of the metallic precursor. Applied Catalysis B: Environmental. 43. 355–369. Llorca, J., et. al. (2003 b). In situ magnetic characterisation of supported cobalt catalysts under steam-reforming of ethanol. Applied Catalysis A: General. 243. 261–269. Lockett, M., Simmons, M. and Kendall, K. (2004). CFD to predict temperature profile for scale up of micro-tubular SOFC stacks. Journal of Power Sources. 131. 243–246. Nishiguchi, T., et. al. (2005). Catalytic steam reforming of ethanol to produce hydrogen and acetone. Applied Catalysis A: General. 279. 273–27. Petruzzi, L., Cocchi, S. and Fineschi, F. (2003). A global thermo-electrochemical model for SOFC systems design and engineering. Journal of Power Sources. 118. 96–107. Raskó, J. (2004). Surface species and gas phase products in steam reforming of ethanol on TiO2 and Rh/TiO2. Applied Catalysis A: General. 269. 13–25. Sun, J. et. al. (2004). Hydrogen from steam reforming of ethanol in low and middle temperature range for fuel cell application. International Journal of Hydrogen Energy. 29. 1075 – 1081. Sun, J. et. al. (2005). H2 from steam reforming of ethanol at low temperature over Ni/Y2O3,Ni/La2O3 and Ni/Al2O3 catalysts for fuel-cell application. International Journal of Hydrogen Energy. 30. 437 – 445. Therdthianwong, A., Sakulkoakiet, T. and Therdthianwong, S. (2001). Hydrogen production by catalytic ethanol steam reforming. Science Asia. 27. 193-198. Zabalza, I, Valero, A. and Scarpellini, S. Hidrógeno y pilas de combustible: Estado de la técnica y posibilidades en Aragón. Fundación para el desarrollo de las nuevas tecnologías del hidrógeno en Aragón. Zaragoza. 2005.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
Morphogenesis of polyolefin particles in polymerization reactors Blanka Horackova, Juraj Kosek ICT Prague, Department of Chemical Engineering, 166 28 Prague 6, Czech Republic
Abstract The evolution of the spatially 2D or 3D morphology of porous polymer particles during their growth in the catalytic polymerization of olefins is addressed by the concept of discrete element method (DEM). The polyolefin particle is discretized into number of micro-elements with visco-elastic interactions acting among individual micro-elements. The evolution of this agglomerate of micro-elements is employed in the prediction of particle morphology and in the mapping from the parametric space of the architecture of catalyst particles and reactor conditions into the space of final particle morphologies. First-principles based models of particle morphogenesis can be employed to avoid unwanted phenomena in industrial reactors, e.g., the disintegration of growing polymer particles into fines or the fouling of particles at reactor walls. Keywords: morphogenesis, discrete element method, catalytic polymerization, polyolefin particles, structure-property relationship.
1. Introduction The polymerization of olefins on heterogeneous catalysts is the main source of polyolefins. The fundamentals of these polymerization processes on the meso-scale level describing the evolution of particle morphology and the causes of frequent problems in industrial reactors are still not completely understood [1-3]. The polymer particle grows from the initial catalyst particle of the typical diameter 20 to 50 μm that consists of the porous support and the active catalyst system immobilized on its surface. In the early stage of particle growth the pores of the catalyst support are filled with the polymer that causes the gradual fragmentation of the support into fragments [4]. However, these fragments remain agglomerated by the polymer during the particle growth. The diameter of the final polymer particle is 0.2 to 3.0 mm. We describe the evolution of the spatially 2D or 3D structure of porous polyolefin particles formed in gas-dispersion reactors as the diagenesis of the agglomerate of micro-elements with binary and ternary visco-elastic interactions acting among individual micro-elements. In difference from our previous work [5] we consider several different types of micro-elements and a more general description of monomer transport in the particle, hence a broader class of problems can be addressed. The rate of the growth of each particular micro-element depends on the local catalyst activity and on the local concentration of the monomer. The DEM concept was successfully applied to industrially relevant problems, for example: (i) the formation of large pores (macrocavities) in polymer particles, and (ii) the disintegration of growing polymer particles into fines [6]. The control of the morphology of the nascent polymer particle is important because the particle morphology affects the polymerization rate, the molecular architecture of the produced polymer and the down-stream processing of porous polymer powder. This contribution also investigates the structure-property relationships in hetero-phase polymer particles.
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B. Horackova and J. Kosek
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2. Discrete element model of polyolefin particle Discrete element method has been adopted for the modeling of the polymerization process in the fluidized bed reactor on the level of a single growing particle. The polyolefin particle is represented by the agglomerate of spherical micro-elements of different types (e.g., porous catalyst support, catalyst support with immobilized active sites, polymer phases). The i-th micro-element is described by its position vector xi = (xi, yi, zi), by its velocity vector vi = (vxi, vyi, vzi), by its radius ri and by the average concentration of monomer at the micro-element ci. The discrete element model is illustrated in Figure 1 on the example of the particle consisting from two different polymer phases, as is the case of impact polypropylene consisting from the hard homoand soft co-polymer phase).
B
ry ina
ns tio c a er int
η
M
ax we ll m od E el
ti as El
E
ηM Ternary interactions
ηK
Three-parametric model
Figure 1. Polyolefin particle consisting from two different types of micro-elements (displayed in different colors) represented by the discrete element model. The movement of the i-th micro-element is governed by the second Newton’s law d ( mi v i ) / dt = Fij + Fijk , dx i / dt = v i , (1)
∑ j
∑ j ,k
where t is the time, mi is the mass of the i-th micro-element, Fij is the force by which the j-th micro-element acts on the i-th micro-element and Fijk is the force representing ternary interactions among micro-elements i, j and k. The growth of the mass of the i-th micro-element mi is approximated by the simple polymerization kinetics dmi / dt = 4π ri2 ρ i (dri / dt ) = k p mcat ,i ci ,
(2)
where ρi is the density of the i-th microelement, kp is the polymerization rate constant and mcat,i is the mass of catalyst present in the i-th micro-element. In order to reduce the computation time we assume the pseudo-stationary state in the balance of monomer, i.e., the molar flux of monomer to the i-th micro-element from its neighbouring microelements Nij is equal to its consumption 0 = −k p mcat,i ci / M M + N ij , (3)
∑ j
where MM is the molar mass of monomer. The transport of monomer in our model proceeds by the diffusion between connected micro-elements, Nij = Aij Dij (cj – ci) / dij ,
Morphogenesis of Polyolefin Particles in Polymerization Reactors
1139
where dij is the distance and Aij is the area of connected micro-elements. The binary diffusion coefficient Dij depends on the type of connected micro-elements i and j. The gas phase is discretized into “invisible” micro-elements with mass transport resistance but with no force interactions. Binary and ternary visco-elastic interactions acting among micro-elements are considered. Binary force interactions represent the resistance against the push and pull of two micro-elements and ternary interactions represent the resistance against the change of the bonding angle in the system of three connected micro-elements, cf. Figure 1. The character of polymer materials lies between the limiting concepts of the elastic spring described by the Hook’s law and the fluid with a linear viscous behavior, which is formally represented by the viscous dashpot. The force interactions between micro-elements are approximated by models formed by combinations of viscous and elastic components, cf. Figure 1. These models provide constitutive equations for force interactions Fij and Fijk required in eq (1). Force interactions depend not only on the current position of micro-elements, but also on the history of force interactions. The Maxwell model consisting of spring and dashpot connected in series describes well the relaxation of the stress [7] and the characteristic time of the stress relaxation is τR. During the simulation of the particle morphogenesis we have to update the connectivity of micro-elements. Two micro-elements i and j become connected when they touch each other, i.e., their distance dij becomes dij ≤ u0,ij , where u0,ij is the equilibrium distance between these micro-elements. Two elements become disconnected if the strain eij exceeds the maximum value emax. The triplet connection between microelements A–V–B is initiated when both the connection A–V and the connection B–V are formed and it disappears if either one of the connections A–V or B–V is disconnected or the deviation between the actual angle α = ∠ AVB and the initial angle α0 exceeds the limiting value αmax .
3. Evolution of the morphology of polyolefin particles Figure 2 shows the example of the morphogenesis of the growing polyolefin particle. The uniform distribution of catalyst activity among the micro-elements is considered. Monomer mass transport limitation causes the uneven growth of micro-elements and the stress is gradually built up in the particle and leads to the formation of fractures [8] and to the loss of the initially circular shape of the particle.
catalyst fragmentation
catalyst
t=0s
t=0.3s t=5s
Figure 2. Evolution of the polyolefin particle morphology during the polymerization. Polymer yield rate AY = 100 kgpol/(gcath) and initial Thiele number Θ0=8.9. Systematic mappings from the parametric space of the architecture of the porous catalyst particle, reactor temperature and visco-elastic properties of the polymer phase
B. Horackova and J. Kosek
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into the parametric space of final particle morphologies have been conducted in [6]. Additional parametric studies are provided in Figures 3 and 4. The formation of various morphological features from the initially spherical particle with the uniform distribution of catalyst activity is demonstrated in Figure 3. Monomer mass transport limitation is the reason of uneven growth of micro-elements and can result in the formation of particles with cracks and macro-cavities [9,10]. The bending modulus G of ternary interactions characterizes the resistance against the change of bonding angle of microelements and is assumed to be related to Young modulus E by G = 0.2E. The initial value of the Thiele modulus Θ0 describes the monomer transport resistance in the particle. The monomer transport resistance reduces as the particle grows because the concentration of the catalyst per unit volume of the particle decreases [5]. No ternary interactions AY = 10 kg pol /(g cat h) Θ0 = 2.8
AY = 100 kg pol /(g cat h) Θ0 = 8.9
Ternary interactions, G=0.2E AY = 10 kg pol /(g cat h) Θ0 = 2.8
AY = 100 kg pol /(g cat h)
Θ0 = 8.9
Figure 3. The effects of the diffusion resistance and of ternary force interactions on the resulting morphology of the polyolefin particle at t=10 s. In the case of the negligible monomer transport limitation (corresponding to small value Θ0), the resulting polyolefin particle is relatively compact and replicates well the initial particle shape. When the characteristic rate of particle growth and the related rate of stress accumulation are faster than the rate of stress decay by the Maxwell model, the particle can eventually disintegrate into fines. The disintegration into fines can happen also at low reactor temperatures, when the polymer is brittle and characteristic times of stress relaxation are long.
Elastic
Maxwell τ R = 1s
non-uniformity of catalyst distribution
Figure 4. The effects of the non-uniform distribution of catalyst activity and of the stress relaxation on the resulting morphology of polyolefin particle.
Morphogenesis of Polyolefin Particles in Polymerization Reactors
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Figure 4 demonstrates the effects of the non-uniform distribution of catalyst activity among individual micro-elements and of the stress relaxation on the resulting particle morphology. No monomer mass transport resistance is considered. The first column results from the uniform distribution of catalyst activity among all microelements. The second column results from the consideration of two values of polymer yield rate AY randomly distributed among micro-elements within the polymer particle. And the third column displays the morphologies resulting from the non-uniform distribution of catalyst activities given by 10X/β, where X is the random number with normal distribution and β=0.5. The relaxation of the stress in the growing particle (e.g., by the Maxwell model with τR = 1 s) reduces the porosity of the polymer particle [11].
4. Structure – property relationship for impact polypropylene The discrete element approach can be used in the prediction of the impact resistance of hetero-phase polymers, e.g., of impact polypropylene (iPP) particles consisting from two polymer phases. The skeleton of the particle is formed by “hard” grains of isotactic PP and the particle contains also “soft” ethylene-propylene copolymer. The aim of the hetero-phase structure is to absorb the mechanical energy during the impact by the rubbery copolymer phase. The impact behavior of homopolymer PP particle and of four different particles of iPP with the same content of rubber phase (37.5 wt.%) but with different morphology is demonstrated in Figure 5. The first row in Figure 5 shows the morphology of the particle and its relative position to the wall. The dark color represents the micro-elements of semi-crystalline homopolymer, while the light color represents the micro-elements of rubbery copolymer phase. The 2nd and 3rd row display the resulting structure of the particle after its collision with the wall in dependence on the initial particle velocity. For large initial velocities we can observe the formation of the crack in the more brittle isotactic homopolymer phase. At the velocity v = 20 m/s the homopolymer particle is broken. At the same velocity the particles of iPP with large rubbery domains are badly damaged and the iPP particles with small domains and the random distribution of the rubber phase survived the collision with a little damage.
t=0 s
t=5×10-5s v=10 m/s
t=5×10-5s v=20 m/s
Figure 5. The effect of the structure of four iPP particles and one homopolymer PP particle on the impact resistance of the particle.
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5. Conclusions The model of polymer particle represented by the agglomerate of micro-elements with binary and ternary force interactions acting among individual micro-elements has the capability to describe the evolution of polymer particles resulting in various morphologies. The model is capable of predicting industrially important cases of the formation of macrocavities, the hollow particles, the formation of fines, and the poor replication of the shape of the original catalyst particle. DEM representation of the particle can be also used for the mapping of the structure–property relationships of impact polypropylene particles and for the dynamic simulation of the fragmentation of catalyst carriers [12]. The framework of DEM model can be applied for prediction of the spatially 3D structure of materials resulting from the force interactions of the constituents of their structure [13]. We believe that this type of the model – after the incorporation of additional features, such as electrostatic force fields – is capable of addressing the problems of colloid aggregation, sol-gel precipitation, and other diagenetic processes in mesostructured materials. Acknowledgments. The support from the Czech Grant Agency (project 104/03/H141) and from the Ministry of Education (MSM 6046137306) is acknowledged.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Debling J.A., Ray W.H. J. Appl. Polym. Sci. 81, 3085-3106 (2001). Llinas J.R., Selo J.L.: WO Patent No. 01/66610 (2001). Kittilsen P., McKenna T.F.: J. Appl. Polym. Sci.82, 1047-1060 (2001). Estenoz D.A., Chiovetta M.G.: J. Appl. Polym. Sci. 81, 285-311 (2001). Grof Z., Kosek J., Marek M.: AIChE J. 51, 2048-2067 (2005). Grof Z., Kosek J., Marek M.: Ind. Eng. Chem. Res. 44, 2389-2404 (2005). Young R.J., Lovell P.A.: Introduction to Polymers, Chapman & Hall, London (1995). Kitilsen O., Swendsen H., McKenna T.F.: AIChE J. 49, 1495-1507 (2003). Han-Adebekun G.C., Hamba M., Ray W.H.: J. Polym. Sci. Part A: Polym. Chem. 35, 2063-2074 (1997). Pater J.T.M., Weickert G, van Swaaij W.P.M.: J. Appl. Polym. Sci. 87, 1421-1435 (2003). Naik S.D., Ray W.H.: J. Appl. Polym. Sci. 79, 2565-2579 (2001). Kosek J., Grof Z., Horáčková B.: Particle growth in olefin polymerization, in 8th International Workshop on Polymer Reaction Engineering, DECHEMA Monographs, Vol. 138, pp. 141-150, Wiley-VCH (2004). Kosek J., Stepanek F., Marek M.: Modeling of transport and transformation processes in porous and multiphase bodies, in Advances in Chemical Engineering, Vol. 30 „Multiscale Analysis“, edited by Marin G.B., Elsevier (2005).
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Modeling and design of a biochemical process for NOx removal C. S. Bildeaa, M.L. Oudshoorna, C. Picioreanua, A.C. Dimianb a
Delft University of Technology, Julianalaan 136, 2628BL Delft, The Netherlands University of Amsterdam, Nieuwe Achtergracht 166, 1018WV Amsterdam, The Netherlands
b
Abstract Modeling and design of an industrial scale biochemical process for removal of NOx from flue gases (BioDeNOx) are presented. The process is based on the absorption of NOx in Fe(II)EDTA solution followed by the reduction to nitrogen in a biochemical reactor. Rate-based models of the absorption and reaction units are developed, taking into account the kinetics of chemical and biochemical reactions and the rate of gasliquid mass transfer. We demonstrate that the process is feasible at industrial scale. A spray column is more efficient as absorber than a bubble column since it minimizes the deactivation of Fe(II)EDTA by concurrent oxidation. In achieving high yield in NOx reduction, the regeneration and recycle of the Fe(II)EDTA complex is determinant. Keywords: flue gas treatment, BioDeNOx, modeling, design
1. Introduction Flue gas from energy plants contains nitrogen oxides that contribute to both acid rain and global warming. Existing processes for removal of NOx from flue gas are chemically-based and require high temperature and expensive catalysts. Environmental biotechnology, where micro-organisms function as a catalyst, provides an alternative to these processes (Jin et al. 2005). However, the low solubility of NOx in water hinders the transport over the membrane into the cell and thus slows-down the whole process. [T1]The reduction of nitric oxide (NO) in aqueous solutions of Fe(II)EDTA is one of the core processes in BioDeNOx, an integrated physicochemical and biological technique for NOx removal from industrial flue gases (Van der Maas et al., 2003). The BioDeNOx process, presented in Figure 1, takes advantage of the formation of a complex between NO and Fe(II)EDTA that increases the solubility of NO in water following the reaction[T2]: Fe(II)EDTA2- + NO R Fe(II)EDTA-NO2-
(1)
This reaction can be performed in an absorber, which may be a spray or bubble column. In a second step, the absorbed NO is converted into N2 in a biochemical reactor using a mixed microbial population in which Bacillus azotoformans is the main denitrifying bacteria, and ethanol as electron donor (Kumaraswamy et al., 2005). The reaction is: 6 Fe(II)EDTA-NO2- + C2H5OH → 6 Fe(II)EDTA2- + 3N2 + 2H2O + 2 CO2
(2)
However, reaction (1) is diminished by the oxidation of Fe(II)EDTA complex with oxygen dissolved in the liquid phase: 4 Fe(II)EDTA2- + O2 + 4 H+ → 4 Fe(III)EDTA- + 2 H2O
(3)
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Therefore, Fe(II) must be regenerated, for example by means of Fe(III) reducing bacteria Deferribacteres using ethanol as electron donor (Kumaraswamy et al., 2005): 12 Fe(III)EDTA- + C2H5OH + 5 H2O→ 12 Fe(II)EDTA2- + 2H2O + 2 CO2 + 12H+
(4)
The biochemical processes of denitrification and Fe(III) reduction have been experimentally investigated in a batch reactor (van der Maas et al. 2004). The feasibility of the integrated absorption – bioreaction process was demonstrated on a laboratoryscale setup (van der Maas et al., 2005). In this contribution, we present modeling and design studies of an industrial scale BioDeNOx process. Rigorous rate-based models of the absorption and reaction units are developed, taking into account the kinetics of the chemical and biochemical reactions and the rate of gas-liquid mass transfer. After transforming in dimensionless form, the mathematical model is solved numerically and used to design the process. The critical design and operating parameters are identified by sensitivity studies. Our contribution demonstrates that bringing together chemistry, microbiology and engineering results in a feasible and economically-advantageous process for removal of NOx from flue gases. Decontaminated gas
Fe(II)EDTA2N2, CO2
ξ=1
Microbiological conversion unit
Absorber
ξ=0 Flue gas containing NOx
substrate (C2H5OH)
bleed
Fe(II)EDTA-NO2Fe(III)EDTA-
Figure 1. Biochemical process for NOx removal
2. The mathematical model 2.1. Film region In the absorption unit, the mass transfer between gas and liquid phases is described by the traditional film theory. The resistance to mass transfer is concentrated in two thin films adjacent to the gas-liquid interface. Within the two films, the mass transfer occurs only by steady state molecular diffusion. The chemical reactions take place in the liquid film. The film model variables are spatial coordinate x, and concentrations ci,, i = NO, O2, FeE, FeENO and Fe(III) (where E denotes EDTA). The dimensionless parameters of the model are Hatta numbers (HaNO = 125.87, HaO2 = 0.12), gas-film mass transfer coefficients (κNO= 0.0145, κO2= 0.0204), Henry coefficients (hNO = 53200, hO2 = 80800), liquid phase diffusion coefficients (βFeE= 0.0132, βO2=0.006, βFeENO= 0.0132) and equilibrium constant of reaction (1) (keq= 4130). The reaction kinetics is taken from Demmink et al. (2000), Zang and van Eldik (1990) and Wubs and Beenackers (1993). Mass transfer coefficients are calculated using well-known correlations. Inlet concentrations of NO and FeE are used as references for gas and liquid dimensionless variables, respectively. The differential equations describing the concentration profiles in the liquid film are:
Modeling and Design of a Biochemical Process for NOx Removal d 2 cNO dx 2 d 2 cO2 dx 2 d 2 cFeE dx 2
2 = HaNO cNO cFeE −
2 2 = HaO2 cO2 cFeE
2 = HaNO β FeE cNO cFeE −
d 2 cFeENO dx 2
d 2 cFe(III) dx
2
2 HaNO cFeENO keq
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(5)
(6) 2 HaNO 2 2 β FeE cFeENO + 4HaO2 β O2 cO2 cFeE keq
2 = − HaNO β FeENO cNO cFeE +
2 HaNO β FeENO cFeENO keq
2 2 = −4 HaO2 β O2 cO2 cFeE
(7)
(8)
(9)
The following boundary conditions represent the flux continuity for the absorbed species, and the non-volatility condition for Fe complexes. dcNO |x = 0 = −κ NO (φNO − hNO cNO dx dcO2 |x = 0 = −κ O2 (φO2 − hO2 cO 2 dx
x =1
x =1
)
)
dcFe( III ) dcFeE dc |x = 0 = FeENO |x = 0 = |x = 0 = 0 dx dx dx
(10)
(11)
(12)
The boundary conditions (14) represent the concentration continuity at the interface between liquid film and liquid bulk: L L L L L cNO |x =1 = cNO ; cO2 |x =1 = cO2 ; cFeE |x =1 = cFeE ; cFeENO |x =1 = cFeENO cFe(III) |x =1 = cFe(III) (14)
2.2. Absorption column The concentration profiles along the absorption column result from the integration of equations (15) to (20) , representing mole balance for bulk gas and bulk liquid derived in the assumption of plug-flow. The absorber model variables are axial coordinate ξ and concentrations φi (gas phase) and ciL (liquid phase). The model parameters are gas-film and liquid-film mass transfer coefficients corrected by the interfacial area,(γNO=1.3·10-7, γO =1.32·10-7, ωNO=33.07, ωO =23.9), Damköhler number (Da = 12·108), and ratio of rate constants (Γ = 1.1·10-6). The dimensionless numbers correspond to a column of diameter D=2.4 m and height, H = 8 m, treating 2.8 m3/s flue gas. dφNO i ) = − Da γ NO (φNO − hNO cNO dξ
(15)
C.S. Bildea et al.
1146 dφO2 i ) = − Daγ O2 (φO2 − hNO cO2 dξ L dcNO dc = ωNO NO dξ dx
L dcO2 dc = ωO2 O2 dξ dx
L L + Da(cNO − cFeE x =1
dξ
(17)
L L 2 cFeE + DaΓcO2
+ DacNO cFeE − x =1
L dcFeENO ω dcFeENO = NO dξ β FeENO dx
= ωFe(III)
L cFeENO ) keq
(18)
x =1
L dcFeE ω dc = NO FeE dξ β FeE dx
L dcFe(III)
(16)
dcFe(III) dx
Da 2 cFeENO + 4 DaΓcO2 cFeE keq
− Da (cNO cFeE − x =1
cFeENO ) keq
2 − DaΓcO2 cFeE
(19)
(20)
(21)
x =1
Boundary conditions (23) represent the NO and O2 concentrations at absorber gas inlet (ξ=0). Inlet concentration of the Fe species in liquid (ξ=1) are derived from the bioreactor equations (24) to (26). L φNO |ξ =0 = 1 ; φO2 |ξ = 0 = 400 ; cNO |ξ =1 = 0 ; cOL2 |ξ =1 = 0
(23)
2.3. Bioreactor The biochemical conversion unit is modeled as a CSTR. The kinetics of reactions (2) and (4) and the reactor size are lumped into the Damköhler numbers Da1 and Da2, respectively. L L L cFeE (ξ = 1) + Da1cFeENO (ξ = 0 ) + Da2 cFeL ( III ) (ξ = 0 ) − cFeE (ξ = 0 ) = 0
(24)
L L L cFeENO (ξ = 1) − Da1cFeENO (ξ = 0 ) − cFeENO (ξ = 0 ) = 0
(25)
L L L cFe ( III ) ( ξ = 1) − Da2 cFe ( III ) ( ξ = 0 ) − cFe ( III ) ( ξ = 0 ) = 0
(26)
The reactor Damköhler numbers have been defined assuming first order kinetics with respect to Fe species, with bacteria activity and concentration incorporated in the rate constants. In equations (24) to (26), ξ = 0 and ξ = 1 indicate reactor outlet (absorber inlet) and reactor inlet (absorber outles) values.
3. Results and discussion The solution of the model equations is found by discretizing the spatial coordinates x and ξ by finite differences and solving the resulting set of non-linear algebraic equations using the FORTRAN solver NLEQ1 from ZIB library (http://www.zib.de/Software).
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Figure 2 shows concentration profiles along the absorption column, for the case of complete regeneration of the Fe(II)EDTA complex in the biochemical reactor. The profiles demonstrate that NO is removed with high efficiency despite its low concentration in the incoming stream. However, a large amount of oxygen is also absorbed, leading to significant oxidation of Fe(II)EDTA. 400
0.8
200 0.4
NO
c FeE, c Fe3+
0.6
100
0.2
0.010
FeE
0.8
300
φ O2
φ NO
1.0
O2
0.008
0.6
0.006
0.4
0.004
Fe
0.2
3+
c FeENO
1.0
0.002
FeENO 0.0
0 0.0
0.2
0.4
0.6
0.8
0.0
1.0
0.000 0.0
0.2
0.4
ξ / [-]
0.6
0.8
1.0
ξ / [-]
Figure 2. Bulk gas and liquid concentration profiles along the absorption column. Da =12·108; Da1 = Da2 = 106
Concentration profiles in the film near the gas-liquid interface, at the top and bottom of the column, are shown in Figure 3. Reaction 1 is fast and NO is completely converted in the liquid film. Reaction 3 is very slow, and oxidation of Fe(II) to Fe(III) takes place in the bulk liquid. 1.2
0.3
c
c
0.4
0.6 0.4
107c FeENO
2
10 c FeENO
0.5
0.8
0.2
c FeE
0.6
c FeE
1.0
102c O2
102c O2
0.2 0.1
7
10 c NO
0.0
4
10 c NO
0.0
0.0
0.2
0.4
0.6
x
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 3. Concentration profiles in the liquid film, at the top and bottom of the absorption column. Da =12·108; Da1 = Da2 = 106
From Figures 2 and 3, we conclude that removal of NO by absorption in Fe(II)EDTA solution is feasible at industrial scale. For the main reaction, there are no kinetic limitations. Because the secondary reaction takes place in the bulk liquid, large gas/liquid ratios are preferred and spray columns are recommended. The sensitivity of the process performance with respect to the size of the absorption column and the performance of the regeneration steps were analyzed. Figure 4 presents the NO concentration in the effluent gas versus the size of the absorption column (Da number) for different values of the bioreactor Damköhler numbers, Da1 and Da2. As expected, a larger column leads to improved NO removal. However, the overall process performance is not restricted by the size of absorption column, but by the efficiency of the regeneration step. The efficiency of the process remains high, for Da1 =Da2 ≈ 100. This sets a target for the bioreactor design (volume, bacteria concentration) and operating conditions.
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(1, 102)
0.1
0.1
c NO,out
c NO,out
1
(102, 1) 0.01 2
0.1
1
0.01
Da 2 = 102
2
(Da 1, Da 2) = (10 , 10 ) 0.001
0.001 0
2
4
6
Da / 108
8
10
12
0.1
1
10
100
1000
10000
Da 1
Figure 4. Effect of the size of the absorption column (Da) and biochemical reaction kinetics (Da1, Da2) on the process performance
4. Conclusions 1. Model simulations indicate that removal of NO from flue gases by absorption in solution of Fe(II)EDTA followed by biochemical conversion to N2 in the presence of ethanol is feasible at industrial scale. 2. The absorption of NO is fast and controlled by the interfacial area. In contrast, the absorption of oxygen, which degrades the yield, takes place in the bulk liquid. Therefore, a spray column is recommended as absorption unit. 3. The recycled absorbent flow rate does not influence the performance of the process, and therefore it may be set according to hydrodynamic constraints. 4. The regeneration of Fe(II)EDTA complex in the microbiological reactor is of major importance. Damköhler numbers of magnitude of 100 for both reactions may be taken as basis for the reactor design.
References J.F. Demmink, I.C.F van Gils and A.C.M. Beenackers, 1997, Absorption of nitric oxide into aqueous solutions of ferrous chelates accompanied by instantaneous reaction, Ind. Eng. Chem. Research, 36, 4914-4927. Y. Jin, M. C. Veiga and C. Kennes, 2005, Bioprocesses for removal of nitrogen oxides from polluted air, J. Chem. Tech. Biotech., 80, 483-494. R. Kumaraswamy, G. Muyzer, J.G. Kuenen and M.C.M. van Loosdrecht, 2004, Biological removal of NOx from flue gas, Water Sci. Tech. 50(6), 9-15. R. Kumaraswamy, U. van Dongen, J.G. Kuenen, W. Abma, M.C.M. van Loosdrecht and G. Muyzer, 2005, Characterization of microbial communities removing nitrogen oxides from flue gas: the BioDeNOx process, 2005, Appl. Env. Microbiology, 71(10), 6345-6352. P. van der Maas, T. van de Sandt, B. Klapwijk and P. Lens, 2003, Biological reduction of nitric oxide in aqueous Fe(II)EDTA solutions, Biotechnol. Prog., 19(4), 1323 -1328. P. van der Maas, L. Harmsen, S. Weelink, B. Klapwijk and P. Lens, 2004, Denitrification in aqueous FeEDTA solutions, J..Chem. Tech. Biotech., 79, 835-841. P. van der Maas, P. van den Bosch, B. Klapwijk and P. Lems, 2005, NOx removal from flue gas by an integrated physicochemical absorption and biological denitrification process, Biotech.Bioeng., 90(4), 433-441. V. Zang, R. van Eldik, 1990, Kinetics and mechanism of the autoxidation of iron(II) induced through chelation by ethylenediaminetetraacetate and related ligands, Inorg. Chemistry, 29, 1705-1711. J.H. Wubs and A.C.M. Beenackers, 1993, Kinetics of the oxidation of ferrous chelates of EDTA and HEDTA in aqueous solution, Ind. Eng. Chem. Research, 32, 2580-2594.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Multicriteria Design of Separation Sequences by Including HSE Criteria and Uncertainty Krisztina Cziner, Mimi Hassim and Markku Hurme Helsinki University of Technology, Department of Chemical Technology, Laboratory of Plant Design, P.O. Box 6100, Espoo, FIN-02015 HUT, Finland.
Abstract In this paper optimal separation sequence selection problem under uncertainty of feed compositions is solved by the combination of Analytic Hierarchy Process, genetic algorithm and stochastic simulation. The criteria used for multicriteria process evaluation are cost, environmental aspects, safety and occupational health. Keywords: Multicriteria decision making, Analytic Hierarchy Process, genetic algorithm, stochastic simulation, separation sequence synthesis
1. INTRODUCTION Major decisions affecting the entire process lifecycle are done during early stages of process design. The criteria used are not any more only economy but health, safety and environmental (HSE) aspects are also taken into consideration. These criteria cannot be measured with same metrics and combined directly into the same objective function therefore a multicriteria analysis is required. There are many methods on multicriteria decision making available, even not many methods are practical. The Analytic Hierarchy Process (AHP) is a systematic approach for structuring decision problems hierarchically (Saaty, 1990). The AHP method deals with selections between process alternatives, which have to be generated by the user. In this paper the combination of the Analytic Hierarchy Process with genetic algorithms (GA) and stochastic simulation is presented. The method allows uncertainty aspects of design variables to be considered in multicriteria design of separation sequences. The work extends the earlier study (Cziner and Hurme, 2003), where AHP and GA were combined for process synthesis, which was based on the criteria weightings derived from the strategy and values of the company.
2. STOCHASTIC SIMULATION COMBINED WITH GA AND AHP The method presented consists of five levels: 1. AHP deriving of criteria weightings for the fitness (objective) function 2. Stochastic variation of selected input variables to describe design uncertainty 3. Genetic optimization of sequences 4. Distillation calculation 5. Evaluation of process alternatives by using the AHP derived weightings The Analytic Hierarchy Process is a weighted scoring method, typically used for structuring the decisions hierarchically. It is used for deriving the weights for the multicriteria objective function (fitness), which is used for the evaluation of alternative
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process alternatives. The decision criteria are assigned weights according to their relative importance to the decision maker. A genetic algorithm provides a fast method to search for optimal solution to discrete problems such as the process synthesis problem. Selection of optimization methods is discussed by Cziner et al. (2005). In real plant operation the variables such as feed concentrations are frequently not constant but they vary with time. Uncertainty is taken into account by stochastic simulation of the design variables such as feed concentrations. Stochastic simulation utilizes random numbers for creating probability distributions for the selected input variables.
3. CRITERIA SELECTION Because many companies are nowadays committed to sustainability, more indicators are needed for the evaluations. There are three groups of criteria, that should be taken into consideration in engineering decision making; 1) economic, 2) technical and 3) sustainability (EHS) aspects. These groups can be further divided into subcriteria (Tuomaala et al., 2001). This criteria set is claimed to be complete for all engineering decision making situations. The criteria are measured with indicators. The selection of indicators depends on the design stage, when the evaluation is made.
4. CASE STUDY Distillation is the major separation method is chemical in petrochemical industry. Therefore a process synthesis problem on separating hydrocarbons was selected. The separation methods are ordinary distillation (method 1) and extractive distillation with two different solvents acetonitrile (ACN) (method 2) and N-methyl-2-pyrrolidone (NMP) (method 3) shown in Table 1. In the case study the feed concentration variation effects on the optimal separation sequence are analyzed. The feed stream concentration of 1-butene is varied between 10 and 20 mol% and 2-butenes between 23 and 35 mol%. The concentration of propane and pentane are constant and the concentration of n- butane is the rest. The stochastic simulation is done so that the model combined with optimization algorithm is run hundreds of times with stochastically distributed values. Table 1. Base case feed compositions, their variation, adjacent relative volatilities and modified volatilities Feed concentration, mol%
Original volatilities
Modified volatilities
α1
α2
α3
α1
α2
α3
the rest
2.57
3.82
6.22
2.57
2.05
2.10
14.75
10 - 20
0.89
1.47
1.43
0.89
1.22
1.19
4 2-butenes
27.58
23 - 35
1.22
1.19
1.35
1.22
1.09
1.18
5 n-pentane
5.90
5.90
2.17
1.85
2.60
2.17
1.45
1.80
Component
Base case
Variation
1 Propane
1.47
1.47
2 n- butane
50.29
3 1-butene
For the genetic algorithm the structure of the process is presented as a string of integer pairs. The first integer represents the separation type. The second represents the light key component number in the separation. In the case study there are components
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1,2,3,4,5 (see Table 1) and separation methods 1,2,3 (normal distillation and extractive distillations with ACN and NMP, respectively). If the column uses ordinary distillation (method 1) to separate components 1,2,3 and 4,5, the code is 13, because the light key in separation is component 3). The whole sequence is represented by a string such as [13 32 11 14] representing the four separation steps. 4.1. Economical Evaluation The economic aspects have been taken into account by calculating the sum of the vapor flows in the columns and estimating the cost of the solvent. Column vapor flow can be used for approximating both capital and operating costs (Porter and Momoh, 1991). Since extractive distillation is known to be more expensive operation than ordinary distillation, their relative volatilities have been scaled by method of Souders (1964) to present the true economic feasibility (see Table 1). 4.2. Safety Considerations Safety can be divided into inherent and add-on safety. Only inherent safety can be considered in conceptual design. Heikkilä et al. (1996) have discussed the indicators for inherent safety. In this process synthesis case the main safety variable is the inventory of flammable chemicals. The process alternative that handles less material (i.e. has lower inventory of chemicals) is inherently safer. The inventory is proportional to the volume, whereas the vapor flow is proportional to cross sectional area of the column. Therefore the inventory was approximated with V1.5, where V is the column vapor flow rate. SEQUENCE SELECTION
COST
VAPOR FLOW
ALT 1
SAFETY
SOLVENT
INVENTORY
ENVIRONMENT
LC50
ALT 2
GW
HEALTH
OEL
NFPA
HE
ALT N
Fig.1 Criteria hierarchy for the separation sequencing case. 4.3. Environmental Aspects Environmental aspects can usually be measured by waste amounts. In this case, however, there are no continuous waste streams present, only fugitive emissions and waste water due to rain fall and maintenance. Therefore the properties of chemicals have been used as indicators. Environmental lethal concentration (LC50) values for Fathead minnow were used for ecotoxicology in water environment. For VOC emissions the global warming potential (GWP) was used as an indicator.
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Occupational Health Hazards
Solvent
LC50
OEL(mg/m3)
HE
NFPAmodif.
Hydrocarbons
6-74
1800
7
2
N-methyl-2-pyrrolidone
1072
400
5
2
Acetonitrile
1640
68
3
1
4.4. Occupational Health Aspects The inherent occupational health hazards can be evaluated from the properties of chemicals present in the process. The main toxicity difference between the process alternatives is due to the extractive solvents NMP and ACN, because the hydrocarbons are present in all process concepts anyway. For health aspects three indicators were used based on the work of Hassim and Edwards (2006); 1) The Occupational Exposure Limit (OEL) acts as an indicator for chronic toxicity effects by indicating the level of concentration that may not be exceeded. The lower the OEL value, the more hazardous the chemical is. 2) The Occupational Health and Safety Association (OSHA) Health Effect (HE) values represent the ability to cause typical occupational diseases. 3) National Fire Protection Agency values for health hazard (NFPA) represent the acute toxicity potential. To ensure that the indicator scales are consistent, i.e a low value indicates more severe situations, NFPA values were modified as presented in Table 2; NFPAmodif = 4 - NFPA.
5. SEPARATION SEQUENCING RESULTS The objective functions used were: to minimize vapour flow and inventory, use the process which is the best in LC50, GWP, OEL, NFPA and HE criteria, in addition apply the cheapest solvent type which is possible. The optimal sequence depends both on the weighting of criteria and the stochastic variation of the feed concentrations. 65
Cost 50, NMP
60 55
Cost 90, ACN Cost 50, ACN
Occurence
50 45 40
Optimal sequences: 1- [22 11 14 13] 2- [22 11 13 14] 3- [32 11 13 14] 4- [32 11 14 13]
Cost 10, ACN
35
Cost 10, NMP
30 25
Cost 90, NMP
20 15 10 0
1
2
3
4
5
6
Sequence number
Figure 2. Distribution of optimum sequences as function of cost weighting the relative solvent costs with base case feed concentrations.
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Figure 2 presents, how the selection of optimum separation sequence depends on the weightings of the criteria. [Cost90, ACN] means: high weighting on cost; ACN is cheaper than NMP. Cost has either high (90) or low (10) weighting compared to the environmental and health criteria. Also relative solvent price affects. It can be seen that the ACN based separations are favored only when the weighting of cost is high and the ACN is cheaper than NMP. (The case on left in Fig.2). This is because ACN has more HSE related problems than NMP. In every case there are at least two optimal sequences depending on the feed concentrations. The effect of feed concentration variation on the optimum separation sequence is presented in Figure 3 for case [Cost 90, NMP], in which cost has high weighting and NMP is cheaper than ACN. In this case the selection between sequences [32 11 14 13] and [32 11 13 14] depends on the concentration of 1-butene in feed. Concentrations below 15% lead mainly to the first sequence as seen in Figure 3. The optimal areas however overlap, meaning there are also other affecting variables.
100
[32 11 13 14]
[32 11 14 13]
90 Occurence
80 70 60 50 40 30 20 10 0 10
11
12
13
14
15
16
17
18
19
20
1-butene feed concentration [mol %]
Figure 3. The effect of 1-butene feed concentration on distribution of optimal sequences for the case [Cost90, NMP] i.e. high weighting on cost; NMP cheaper than ACN. Solid line represents sequence [32 11 14 13] slashed line [32 11 13 14].
6. CONCLUSIONS Since conceptual process design is the most critical stages throughout the whole process life cycle, it should be guided by company’s policy. The priorities for process synthesis can be derived by using AHP and implemented as criteria weightings in separation sequence optimization. The design conditions, such as feed concentrations, are rarely constants but they vary. Therefore also the effect of variation should be considered in separation process synthesis. This can be accomplished by the method presented, which combines genetic algorithm with stochastic simulation.
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REFERENCES Cziner, K., Hurme, M., 2003, Process Evaluation and Synthesis by Analytic Hierarchy Process Combined with Genetic Optimization, Comp. Aid. Chem. Eng. 15B, Elsevier, Amsterdam 778-783. Cziner, K., Virkki-Hatakka, T., Hurme, M., Turunen, I., 2005, Evaluative Approach for Process Development. Chem. Engin. and Technol., 28, 1490-1499. Hassim, M., Edwards, D.W., 2006, Development of a methodology for assessing inherent occupational health hazards, Process safety and environmental protection, TransIChemE Part B (accepted). Heikkilä A.- M., Hurme M. and M. Järveläinen, 1996, Safety considerations in process synthesis, Comp. Chem. Eng., 20, S115-S120. Porter, K. E. and Momoh, S. O., 1991, Finding the optimum sequence of distillation columns - an equation to replace the rules of thumb (heuristics). Chem. Eng. J., 46, 97. Saaty, T. L., 1990, How to make a decision: The analytic hierarchy process, European Journal of Operational Research, 48, 9. Souders, M., 1964, The countercurrent separation processes Chem. Eng. Prog., 60, 75-82. Tuomaala, M., Hurme, M., Hippinen, I., Ahtila, P., Turunen, I., 2001, Analysis of process performance in process integration, Proceedings of 6th World Congress of Chemical Engineering, Melbourne.
16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.
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Optimal Scheduling of Tests for New Product Development Hong-Rok Sona, Soon-Ki Heoa, In-Beum Leea a
Department of Chemical Engineering, Pohang University of Science and Technology, San 31 Hyoja-Dong, Pohang, Kyungbuk, 790-784, Korea
Abstract Many industries typically use screening process to discover new products. In these industries, many tasks involved in producing a new product are regulatory requirements, such as environmental and safety tests. For testing tasks in a new product development process, two kinds of mathematical models which considered scenario-based methods and assumptions of outsourcing and retest are proposed. Throughout the application study, the proposed model allows realistic problems to be solved with reasonable computational effort. Keywords: New Product Development, Scheduling, Resource Constraint, Retest
1. Introduction Many industries typically use screening process to discover new products. For examples agricultural chemical and pharmaceutical companies usually use the screening processes. In these industries, many tasks involved in producing a new product are regulatory requirements, such as environmental and safety tests. The failure of a single required test may prevent a potential product from reaching the marketplace and therefore it must be explicitly included in the model. There are uncertainties in the costs, probabilities of success, durations of the tasks, income and so on.
2. Pre-Studied Models Schmidt and Grossmann (1996) provide 7 models for schedule the test tasks in new product development. This paper reviews only M1 and M2 and employs the same notations of Schmidt and Grossmann (1996). Model M1 is a nonlinear, convex, disjunctive formulation of the problem. The index k represents one of the Nk possible scenarios, where each scenario k occurs with probability Pk (and Σ Pk = 1 ). Let the indices i and j represent tasks in stochastic k
scenario k. Let cik be the cost of task i in scenario k, let r be the discount factor, using continuously compounded interest. Let yij be a binary variable that is 1 if there is a precedence constraint that task i must finish before task j can begin and 0 otherwise. Let tk be a nonnegative, continuous variable representing the overall project completion time in scenario k. Let sik be a nonnegative, continuous variable representing the starting time of task i in scenario k. Model M1: Nonlinear, convex, disjunctive
min ∑ Pk (∑ cik e wik + ∑ f kmu km ) k
i
m
(1)
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sik + d ik ≤ t k
∀i, k ,
wik = −rsik + ∑ ln( p jk ) y ji
bm − tk + ukm ≥ 0
∀i, k ,
j ≠i
⎡ yij ∧ ¬y ji ⎢ ⎢⎣ sik + d ik ≤ s jk ,
∀k , m
Lik = − rU ik + ∑ ln( p jk )
∀i, k
(2, 3) (4, 5)
j ≠i
⎤ ⎡ y ji ∧ ¬yij ⎤ ∀(i, j ) (i < j ) (6) ⎥∨⎢ ⎥ ∨ ¬yij ∧ ¬yji ∀k ⎥⎦ ⎢⎣ s jk + d jk ≤ sik , ∀k ⎥⎦ (7, 8) ∀ (i , j ) ∈ A , yij + y ji ≤ 1 ∀(i, j ) i < j
[
yij = 1, y ji = 0
yij + y jl + yli ≤ 2
]
∀ (i , j , l ) i < j < l
y ji + yil + ylj ≤ 2
(9)
∀(i, j , l ) i < j < l
(10)
0 ≤ sik ≤ U ik , Lik ≤ wik ≤ 0 , 0 ≤ t k , 0 ≤ ukm , yij ∈ {0, 1} More details of nomenclatures are explained in Schmidt and Grossmann (1996). The problem is difficult to solve in this form. It would be easier if the objective function were linear. The nonlinearity of the objective function of M1 can be eliminated, since w
the cost term e ik involves a single variable. Using the separable programming method, the exponential is approximated by n-1 piecewise linear segments between the grid points aikn. The standard λ formulation is used in eqs (11)-(14), which is given in many textbooks.
min ∑ Pk (∑ cik ∑ (e aikn λikn ) + ∑ f km u km )
(11)
∑a
(12)
k
i
n
m
λ = −rsik + ∑ ln( p jk ) y ji
ikn ikn
n
∑λ
ikn
∀i,k
j
=1
∀i, k ,
λikn ≥ 0
∀i, k
(13, 14)
n
The nonlinear equation (1) and (4) in M1 can be replaced by the piecewise linear form of eqs (11)-(14). Model M2: Piecewise linear, disjunctive Objective eq (11) Subject to eqs (2), (3), (6)-(10), (12), (13) 0 ≤ sik ≤ U ik , 0 ≤ λikn ≤ 1 , 0 ≤ tk , 0 ≤ ukm , yij ∈ {0, 1}
3. Proposed Model 3.1. Resource-Constrained New Model The number of tests which have to be passed is much larger than the number of teams which can conduct the test until releasing a new product onto the market. This case could be handled by resource constrained problem. To add this constraint, the ‘time slot’ is introduced in the proposed model. One team has some slots and one test occupies one of them. The number of slots in each team equals to the number of all test. Each slot is represented as a number and smaller number means the higher order of precedence. A new binary variable Xist is introduced to represent a numerical expression of the resource constraint and the time slot. i means test, s means time slot and t means team. Each test must be conducted only one slot of a certain team.
Optimal Scheduling of Tests for New Product Development
∑X
ist
=1
1157
∀i
(15)
s ,t
It is possible that a certain test must be conducted by a certain team. B can be assumed a set of those constraints.
∑X
=1
ist
(i, t ) ∈ B
(16)
s
Binary variable yij shows the context between tests conducted in same team.
(1 − yij ) ≤ M (2 − X ist − X js 't )
∀s < s' , i ≠ j , t
(17)
Model NM1: Piecewise linear, disjunctive, resource constrained Objective eq (11) Subject to eqs (2), (3), (6)-(10), (12), (13), (15)-(17) 0 ≤ sik ≤ U ik , 0 ≤ λikn ≤ 1 , 0 ≤ tk , 0 ≤ ukm , yij ∈ {0, 1} , X ist ∈ {0, 1}, M = 100 3.2. Outsourcing Constraned Model The aim of outsourcing is to conquer the inadequate number of team and reduce the test period. But in the most case, it takes more cost to outsource. To solve the problem, outsourcing regards one of the internal team in the proposed model.
min ∑ Pk (∑ cik ∑ (e aikn λikn ) + ∑ c'ik ∑ (e aikn λ 'ikn ) + ∑ f kmukm ) k
i
n
i
n
(18)
m
The parameter cik is the cost of task i conducted by internal team, and c’ik is the cost of task i conducted by outsourcing. Using λikn and λ 'ikn , it should be taken one cost of different costs in the object function.
∑λ
ikn
n
= 1 − ∑ X ist ' ,
λ 'ikn ≥ 0
s
∑ λ' n
ikn
= ∑ X ist '
t ' = outsourcing team, ∀i, k
(19, 20)
s
∀i, k
(21)
Model NM2 can solve the problem which considered resource-constraints and outsourcing constraints by replacing the objective function and constraints in NM1. Model NM2: Piecewise linear, disjunctive, resource constrained, outsourcing Objective eq (18) Subject to (2), (3), (6)-(10), (12), (13), (15)-(17), (19), (20) 0 ≤ sik ≤ U ik , 0 ≤ λikn ≤ 1 , 0 ≤ λ 'ikn ≤ 1 , 0 ≤ tk , 0 ≤ ukm , yij ∈ {0, 1}, X ist ∈ {0, 1} , 3.3. Retest-Constrained New Model In case of the failure of the high probability of success and inexpensive test, the test can be conducted once more. The probability of success and the cost of reconducted tests are the same as the original test. In this case, the cost of reconducted tests must be considered in the objective function. The expected cost of the test is
expectedcost = p4 × (c1 + p1c2 + p1 p2c4 + p1 p2c3 ) + (1 − p4 ) × (c1 + p1c2 + p1 p2c4 + p1 p2c4 + p1 p2 p4c3 )
(22)
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First of all, the retest regards an imaginary test. The retest must be conducted immediately after the failure of original test.
(original test, retest) ∈ A
(23)
yot ,i = yrt ,i , yi ,ot = yi ,rt
∀i ot = original test, rt = retest
(24)
To add retest constraints, some variables in M2 are changed. Let the index z represents the retest. z=1 when the retest is conducted and z=2 when the retest is not conducted. sik, tk, ukm and Uk become sikz, tkz, ukmz and Ukz. The probability of success pik is the same as the probability of test of the next conducted task j and the probability of failure (1– pik) is the same as the probability of test of the reconducted task i’. pik becomes p’ikz and p’ikz would be
p'ikz = pik ∀(i, k , z ) i is not concerned with retest
(25)
p 'ikz = 1
∀k z = 1, i is an original test
(26)
p'i 'kz = pik
∀k z = 1, i' is a retest of i
(27)
p'ikz = 1
∀k z = 2, i is an original test
(28)
The constraints in M2 are changed with changed variables and parameters. Equation (2) in model M2 becomes
sikz + d ik ≤ t kz
⎧1, then i = 1,...,5 ∀k , z = ⎨ ⎩2, then i = 1,...,4
(29)
The upper and lower bound on sikz are given by
U ikz = ∑ d jk j ≠i
0 ≤ sikz ≤ U ikz
⎧1, then i = 1,...,5 ∀i, k , z = ⎨ ⎩2, then i = 1,...,4 ∀i, k , z
(30) (31)
Equations (3), (6), (12)-(14) in model M2 become
bm − tkz + ukmz ≥ 0
∀k , m, z
(32)
⎧1, then i = 1,...,5 i≠ j ∀j , k , z = ⎨ ⎩2, then i = 1,...,4 ⎧1, then i = 1,...,5 z=⎨ ⎩2, then i = 1,...,4
sikz + d ik ≤ s jkz + M kz (1 − yij )
(33)
M kz = ∑ d ik
(34)
i
∑a
iknz
n
∑λ
iknz
λiknz = −rsikz + ∑ ln( p ' jkz ) y ji j
=1
∀i, k , z ,
λiknz ≥ 0
⎧1, then i = 1,...,5 ∀i,k , z = ⎨ ⎩2, then i = 1,...,4 ∀i, k , z
(35) (36, 37)
n
The objective function is also changed with changed variables and parameters.
⎧(1 − pi ) min ∑ ez ∑ Pkz (∑ cik ∑ (e aiknz λiknz ) + ∑ f kmu kmz ) , ez = ⎨ z k i( z) n m ⎩ pi
if z = 1 if z = 2
(38)
Optimal Scheduling of Tests for New Product Development
1159
These changes yield model NM3: Model NM3: Piecewise linear, Big M, Retest-constrained Objective eq (38) Subject to eqs (7)-(10), (23)-(29), (32), (33), (35), (36) 0 ≤ sikz ≤ U ikz , 0 ≤ λiknz ≤ 1 , 0 ≤ t kz , 0 ≤ ukmz , yij ∈ {0, 1}
4. Example of New Models Table 1 gives the data for an example. Table 1. Example Data task
pi
ci
di
{12, 13, 14}
P(d ) = {0.2, 0.6, 0.2}
1
0.807
75,500
2
0.775
105,500
{4, 5, 6, 9, 10, 11} P(d ) = {0.3, 0.4, 0.1,0.075, 0.1, 0.025}
3
0.889
222,700
{4, 5, 7}
P( d ) = {0.3, 0.5, 0.2}
4
0.900
285,000
{6, 8, 10}
P (d ) = {0.25, 0.5, 0.25}
In the example 1, there are 2 teams and 4 tasks. The optimal schedule is in Figure 1. Team 1 conducts task 1, 3 and team 2 conducts task 2, 4.
Fig. 1 Optimal Schedule and histogram of Example 1
In the example 2, there are 1 internal team and 1 outsourcing. The optimal schedule is in Figure 2. Task 1 is conducted by outsourcing
Fig. 2 Optimal schedule and
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In the example 3, it is assumed that task 4 can be retested. The optimal schedule is in Figure 3. The histogram shows the distribution of objective values
Fig. 3 Optimal schedule and histogram of Example 3
5. Conclusion In conclusion, this research focuses on a scheduling of tests in new product development. For testing tasks in new product development process, two kinds of mathematical models which considered scenario-based methods and assumptions of outsourcing and retest are proposed. Throughout the application study, the proposed model allows realistic problems to be solved with reasonable computational effort.
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