Computer Aided Process and Product Engineering (CAPE)

CAPE Edited by Luis Puigianer and Georges Heyen

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ISBN 3-527-30831-8

Kai Sundmacher, Achim Kienle (Eds.)

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CAPE Computer Aided Process and Product Engineering Edited by Luis Puigjaner and Ceorges Heyen

WILEYVCH

WILEY-VCH Verlag CmbH & Co. KGaA

The Editors

Professor Dr. Luis Puigjaner Universitat Politechnica de Catalunya Chemical Engineering Department ESTEIB, Av. Diagonal 647 08028 Barcelona Spain Professor Dr. Georges Heyen Laboratoire d’Analyse et Synthese des Systemes Chimiques Universitk de Liege Sart Tilman B6A 4000 Liege Belgium

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the British Library. Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Typesetting Mittemeger & Partner, Plankstadt Printing betz-druck GmbH, Darmstadt Binding Litges & Dopf GmbH, Heppenheim Cover Design 4t Mattes + Traut, Werbeagentur

GmbH, Darmstadt

Printed in the Federal Republic of Germany Printed on acid-free paper ISBN-13

978-3-527-30804-0

ISBN-lo 3-527-30804-0

I"

Table of Contents Preface X I I I Foreword X X I List of Contributors XXV

Volume 1 1

Introduction 1

Section 1 Computer-aided Modeling and Simulation 1

Large-ScaleAlgebraic Systems

15

Cuido 5uzzi Ferraris and Davide Manca

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 2

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Introduction 15 Convergence Tests 17 Substitution Methods 20 Gradient Method (Steepest Descent) 20 Newton's Method 21 Modified Newton's Methods 24 Quasi-Newton Methods 27 Large and Sparse Systems 28 Stop Criteria 30 Bounds, Constraints, and Discontinuities 30 Continuation Methods 31 Distributed Dynamic Models and Computational Fluid Dynamics 35 Young-il Lim and Sten 5ayJmrgensen Introduction 35 Partial Differential Equations 35

Method of Lines 40 Fully Discretized Method 58 Advanced Numerical Methods 68 Applications 75 Process Model and Computational Fluid Dynamics 98 Discussion and Conclusion 102

Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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Molecular Modeling for Physical Property Prediction Vincent Gerbaud and XavierJoulia

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Introduction 107 What is Molecular Modeling? 108 Statistical Thermodynamic Background 112 Numerical Sampling Techniques 175 Interaction Energy 121 Running the Simulations 124 Applications 125 Conclusions 132

4

Modeling Frameworks o f Complex Separation Systems 137 Michael C. Ceorgiadis, Eustathios 5. Kikkinides, and Margaritis Kostoglou

4.1 4.2

Introduction 137 A Modeling Framework for Adsorption-Diffusion-based Gas Separation Processes 138 Modeling of PSA Processes in gPROMS 148 Efficient Modeling of Crystallization Processes 149 Modeling of Grinding Processes 1GO Concluding Remarks 1GG

4.3 4.4 4.5 4.6 5

Model Tuning, Discrimination, and Verification Katalin M. Hangos and Rozalia Lakner

5.1 5.2 5.3

Introduction 171 The Components and Structure of Process Models Model Discrimination: Model Comparison and Model Transformations 174 Model Tuning 179 Model Verification 183

5.4 5.5

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Multiscale Process Modeling 189 Ian T: Cameron, Gordon D. Ingram, and Katalin M. Hangos

6.1 6.2 6.3 6.4 6.5

Introduction 189 Multiscale Nature of Process and Product Engineering 189 Modeling in Multiscale Systems 193 Multiscale Model Integration and Solution 203 Future Challenges 218

7

Towards Understandingthe Role and Function o f Regulatory Networks in Microorganisms 223 Krist V. Gernaey, Morten Lind, and Sten BayJmrgensen

7.1 7.2 7.3 7.4 7.5 7.6

Introduction 223 Central Dogma of Biology 228 Complexity of Regulatory Networks 229 Methods for Mapping the Complexity of Regulatory Networks 236 Towards Understanding the Complexity of Microbial Systems 247 Discussion and Conclusions 259

Section z Computer-aided Process and Product Design

1

Synthesis of Separation Processes 269 Petros Proios, Michael C. Georgiadis, and €$ratios

1.1 1.2 1.3 1.4 1.5

Introduction 269 Synthesis of Simple Distillation Column Sequences 272 Synthesis of Heat-integrated Distillation Column Sequences 279 Synthesis of Complex Distillation Column Sequences 285 Conclusions 292

2

Process Intensification 297 Patrick Linke, Antonis Kokossis, and Albert0 A h - A r g a e z

2.1 2.2 2.3 2.4

Introduction 297 Process Intensification Technologies 299 Computer-Aided Methods for Process Intensification 303 Concluding Remarks 324

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Computer-aided Integration of Utility Systems Franqois Marechal and Boris Kalituentzef

3.1 3.2 3.3 3.4 3.5

Introduction 327 Methodology for Designing Integrated Utility Systems 330 The Energy Conversion Technologies Database 334 Graphical Representations 339 Solving the Energy Conversion Problem Using Mathematical Programming 349 Solving Multiperiod Problems 367 Example 369 Conclusions 379

3.6 3.7 3.8

N. Pistikopoulos

327

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Equipment and Process Design 383 1. David, L. Bogle, and B. Eric Ydstie

4.1 4.2 4.3 4.4 4.5 4.6

Introduction 384 The Structure of Process Models 384 Model Development 390 Computer-aided Process Modeling and Design Tools 390 Introduction to the Case Studies 393 Conclusions 416

5

Product Development 419 Andrzej Kraslawski

5.1 5.2 5.3 5.4

Background 419 Definition Phase 424 Product Design 431 Summary 439

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Section 3 Computer-aided Process Operation 1

Resource Planning 447 Michael C. Georgiadis and Panagiotis Tsiakis

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Introduction 447 Planning in the Process Industries 448 Planning for New Product Development 460 Tactical Planning 462 Resource Planning in the Power Market and Construction Projects 465 Solution Approaches to the Planning Problem 469 Software Tools €or the Resource Planning Problem 472 Conclusions 474

2

Production Scheduling 481 Nilay Shah

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

Introduction 481 The Single-Site Production Scheduling Problem 483 HeuristicslMetaheuristics: Specific Processes 487 Heuristics/Metaheuristics: General Processes 488 Mathematical Programming: Specific Processes 489 Mathematical Programming: Multipurpose Plants 493 Hybrid Solution Approaches 500 Combined Scheduling and Process Operation 501 Uncertainty in Planning and Scheduling 502 Industrial Applications of Planning and Scheduling 506 New Application Domains 508 Conclusions and Future Challenges 509

3

Process Monitoring and Data Reconciliation 517 Ceorges Heyen and Boris Kalitventzef

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Introduction 517 Introductory Concepts for Validation of Plant Data Formulation 520 Software Solution 527 Integration in the Process Decision Chain 527 Optimal Design of Measurement System 528 An Example 534 Conclusions 538

4

Model-based Control 541 Sebastian Engell, Cregor Fernholz, Weihua Cao, and Abdelaziz Toumi

4.1 4.2 4.3

Introduction 541 NMPC Applied to a Semibatch Reactive Distillation Process 543 Control of Batch Chromatography Using Online Model-based Optimization 552

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4.4 4.5 4.6

Control by Measurement-based Online Optimization 556 Nonlinear Model-based Control of a Reactive Simulated Moving Bed (SMB) Process 565 Conclusions 572

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Real Time Optimization 577 Vivek Dua, John D. Perkins, and Efstratios N. Pistikopoulos

5.1

Introduction 577 Parametric Programming 578 Parametric Control 581 Hybrid Systems 584 Concluding Remarks 589

5.2 5.3 5.4

5.5 6

6.1 6.2 6.3 6.4 6.5 6.6

Batch and Hybrid Processes 591 Luis Puigjaner andlavier Romero

Introduction 591

The Flexible Recipe Concept 597 The Flexible Recipe Model 601

Flexible Recipe Model for Recipe Initialization 602 Flexible Recipe Model for Recipe Correction 610 Final Considerations 617 621

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Supply Chain Management and Optimization Lazaros C. Papageorgiou

7.1 7.2 7.3 7.4 7.5 7.6 7.7

Introduction 621 Key Features of Supply Chain Management 623 Supply Chain Design and Planning 624 Analysis of Supply Chain Policies 630 Multienterprise Supply Chains 635 Software Tools for Supply Chain Management 637 Future Challenges 639

Section 4 Computer-integratedApproaches in CAPE 1

Integrated Chemical Product-Process Design: CAPE Perspectives Rafqul Can;

1.1 1.2 1.3 1.4 1.5

Introduction 647 Design Problem Formulations 648 Issues and Needs 654 Framework for Integrated Approach 658 Conclusion 663

2

Modeling in the Process Life Cycle 667 Ian T: Cameron and Robert 6.Newell

2.1 2.2

Cradle-to-the-GraveProcess and Product Engineering 667 Industrial Practice and Demands in Life-Cycle Modeling 675

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2.3 2.4

Applications of Modeling in the Process Life Cycle: Some Case Studies 681 Challenges in Modeling Through the Life Cycle 689 695

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Integration in Supply Chain Management Luis Puigjaner and Antonio Espuiia

3.1 3.2 3.3 3.4 3.5 3.6 3.7

Introduction 695 Current State of Supply Chain Management Integration 697 Agent-based Supply Chain Management Systems 702 Environmental Module 707 Financial Module 71 1 Multiagent Architecture Implementation and Demonstration 718 Concluding Remarks 727

4

Databases in the Field o f Thermophysical Properties in Chemical Engineering 731 Richard Sass

4.1 4.2

Introduction 731 Overview of the Thermophysical Properties Needed for CAPE Calculations 732 Sources of Thermophysical Data 733 Examples of Databases for Thermophysical Properties 733 Special Case and New Challenge: Data of Electrolyte Solutions 740 Examples of Databases with Properties of Electrolyte Solutions 741 A Glance at the Future of the Properties Databases 744

4.3 4.4 4.5 4.6 4.7 5

Emergent Standards 747 Jean-Pierre Belaud and Bertrand Braunschweig

5.1 5.2 5.3 5.4

Introduction 747 Current CAPE Standards 751 Emergent Information Technology Standards 755 Conclusion (Economic, Organizational, Technical, QA) 765

Section 5 Applications 1

Integrated Computer-aided Methods and Tools as Educational Modules 773 Rafiqul Cani andJens Abildskov

1.1 1.2 1.3

Introduction 773 Integrated Approach to CAPE Educational Modules 776 Conclusion 797

1.4

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Data Validation: a Technology for Intelligent Manufacturing Boris Kalitventzefi Ceorges H eyen, and Miguel Mateus

2.1 2.2 2.3 2.4 2.5 2.6

Introduction 799 Basic Aspects of Validation: Data Reconciliation 799 Specific Assets of Information Validation 806 Advanced Features of Validation Technology 815 Applications 821 Conclusion 826

3

Facing Uncertainty in Demand by Cost-effective Manufacturing Flexibility 827 Petra Heijnen andjohan Grievink

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Introduction 827 The Production Planning Problem 829 Mathematical Description of the Planning Problem 830 Modeling the Profit of the Production Planning 832 Modeling the Objective Functions 836 Solving the Optimization Problem 840 Sensitivity Analysis of the Optimization 845 Implementation of the Optimization of the Production Planning Conclusions and Final Remarks 851 Authors’ Index 855 Subject Index 857

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1 Preface Computer Aided Process and Product Engineering (CAPE): Its Pivotal Role for the Future of Chemical and Process Engineering

Chemical and related industries are at the heart of the great number of scientific and technological challenges involving computer-aided processes and product engineering. Chemical and related industries including process industries such as petroleum, pharmaceutical and health, agriculture and food, environment, textile, iron and steel, bitumous, building materials, glass, surfactants, cosmetics and perfume, and electronics are evolving considerably due to unprecedented market demands and constraints stemming from public concern over environmental and safety issues. To respond to these demands, the following challenges faced by these process industries involve complex systems, both at the process-scale and at the productscale: 1. Processes are no longer selected on a basis of economic exploitation alone. Rather, compensation resulting from the increased selectivity and savings linked to the process itself is sought after. Innovative processes for the production of commodity and intermediate products need to be researched where patents usually do not concern the products but the processes. The problem becomes more and more complex as factors such as safety, health, environment aspects including nonpolluting technologies, reduction of raw materials and energy losses, and productlby-product recyclability are considered. The industry, with large plants, must supply bulk products in large volumes and the customer will buy a process that is nonpolluting and perfectly safe, requiring computer-aided process engineering (CAPE). 2. New specialities, active material chemistry, and related industries involve the chemistry/biology interface of agriculture, food, and health industries. Similarly, it involves upgrading and conversion of petroleum feedstock and intermediates, conversion of coal-derived chemicals or synthesis gas into fuels, hydrocarbons or oxygenates.This progression from traditional chemistry is driven by the new market objectives where sales and competitiveness are dominated by the end-use properties of a product as well as its quality. It is important to underline that today, 60 % of all products sold by chemical companies are crystalline,polymer, or ComputerAided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

amorphous solids. These complex and structured materials have a clearly defined physical shape in order to meet the designed and the desired quality standards. This also applies to plastics, ceramics, soft solids, paste-like products, and emulsions. New developments require increasingly specialized materials, active compounds, and special effects chemicals. The chemicals are much more complex in terms of molecular structure than traditional, industrial chemicals. Control of the end-use property (size, shape, color, aesthetics, chemical and biological stability, degradability, therapeutic activity, solubility, touch, handling, cohesion, rugosity, taste, succulence, sensory properties, etc.), expertise in the design of the process, continual adjustments to meet changing demands, and speed to react to market conditions are the dominant elements. For these specialities and active materials the client buys the product that is the most efficient and first on the market. He will have to pay high prices and expect a large benefit from these short life-time and high-margin products, requiring most often computer-aided process and product engineering. The triplet molecular processes-product-processengineering (3PE) approach requires the tools of CAPE. Today, chemical and process engineering are concerned with understanding and developing systematic procedures for the design and optimal operation of process systems, ranging from nano and microsystems to industrial-scale continuous and batch processes: this is illustrated by the chemical supply chain concept. In the supply chain, it should be emphasized that product quality is determined at the micro and nanolevel and that a product with a desired property must be investigated for both structure and function. A comprehension of the structure-property relationship at the molecular (e.g., surface physics and chemistry) and microscopic level is required. The key to success is to obtain the desired end-use property of a product, and thus control product quality by controlling complexity in the microstructure formation. This will help to make the leap from the nanolevel to the process level. Moreover, most chemical and biological processes are nonlinear, belonging to the socalled complex systems for which multiscale structure is common nature. Therefore, an integrated system approach for a multidisciplinary and multiscale modeling of complex, simultaneous, and often coupled momentum, heat and mass transfer processes is required: 0

0

Different time scales (10-15-108 s) are used from femto and picoseconds for the motion of atoms in a molecule during a chemical reaction, nanoseconds for molecular vibrations, hours for operating industrial processes, and centuries for the destruction of pollutants in the environment. Different length scales (10-'-lO6m) are used from nanoscale for molecular kinetic processes; microscale for bubbles, droplets, particles, and eddies: mesoscale for unit operations dealing with reactors, columns and exchangers: macroscale for production units; and megascale for environment and dispersion of emissions (see the following figure):

Preface

Therefore, organizing scales and complexity levels in process engineering is necessary in order to understand and describe the events at the nano and microscales, and to better convert molecules into useful products at the process scales. This multiscale approach is now also encountered in biotechnology, bioprocesses, and product engineering, to manufacture products and to better understand and control biological tools such as enzymes and micro-organisms. In such cases, it is necessary to organize the levels of increasing complexity from the gene with known properties and structures, up to the product-process relation, by modeling coupled mechanisms and processes at different length scales: the nanoscale is used for molecular and genomic processes and metabolic transformations; pic0 and microscales are used for enzyme and integrated enzymatic systems, and biocatalyst and active aggregates; mesoscale is used for bioreactors, exchangers, separators; and macro and megascales are used for production units and interactions with the biosphere. Thus, organizing levels of complexity at different length scales, associated with an integrated approach to phenomena and simultaneous and coupled processes, are the heart of the new view of biochemical engineering (see next figure). Indeed this capability offers the opportunity to apply genetic-level controls to make better biocatalysts, novel products, or developing new drugs, new therapies, and biomimetic devices. Understanding an enzyme at the molecular level means that it may be tailored to produce a particular end-product. Also, the ability to think across length scales makes chemical engineers particularly well poised to elucidate the mechanistic understanding of molecular and cell biology and its large-scale manifestation, i.e., decoding communications between cells in the immune systems.

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These examples are at the center of the new view of chemical and process engineering: organizing levels of complexity, by translating molecular processes into phenomenologicalmacroscopic laws to create and control the required end-use properties and functionality of products manufactured by a continuous or batch process. I have defined this approach as the triplet molecular processes-product-processengineering (3PE): an integrated system approach of complex pluridisciplinarynonlinear and nonequilibrium processes and phenomena occurring on different length and time scales, involving a strong multidisciplinary collaboration between physicists, chemists, biologists, mathematicians, computer-aided specialists, and instrumentation specialists. Today’s tools are wide-ranging for the success of chemical and process engineering for modeling, complex systems at different scales encountered in the process and product engineering. It’s possible to understand and describe events on the nano and microscale in order to convert molecules into useful products on process and unit scales thanks to significant simultaneous breakthroughs in three areas: molecular modeling (both theory and computer simulation), scientific instrumentation and noninvasive measurement techniques, and powerful computational tools and capabilities for information collection and processing. At the nanoscale, molecular modeling assists in maintaining better control of surface states of catalysts and activators, obtaining increased selectivity and facilitating asymmetrical synthesis, e.g., chiral technologies. Molecular modeling also assists in explaining the relationship between structure and activity at the molecular scale in order to control crystallisation,coating and agglomeration kinetics.

Preface

At the microscale, computational chemistry is very useful for understanding complex media and all systems whose properties are controlled by rheology and interfacial phenomena. At the meso and macroscales, computer fluid dynamics (CFD) is required for scaling up new equipment, or for the design of new operation modes for existing equipment such as reversed flow, cyclic processes, and unsteady operations. It is especially useful when rendering multifunctional processes with higher yields in chemical or biological reactions coupled with separation or heat transfer. It also provides a considerable economic benefit. At the production unit and multiproduct plant scale, dynamic simulation and computer tools for simulation of entire processes are needed more and more. These tools analyze the operating conditions of each piece of equipment in order to simulate the whole process in terms of time and energy costs. New performances (product quality and final cost) resulting from any change due to a blocking step or a bottleneck in the supply chain will be predicted in a few seconds. It is clear that such computer simulations enable the design of individual steps, the structure of the whole process at the megascale, and place individual processes in the overall context of production, emphasizing the role and the place of computer assistance in process and product engineering. The previous considerations on the necessary multidisciplinary and multiscale integrated approach for managing complex systems encountered by chemical and related process industries in order to meet market demands led to the proposal of four main parallel objectives involving the tools of CAPE. The first objective concerns a total multiscale control of the process to increase selectivity and productivity by the nanotailoring of materials. The nanotailoring can be produced with controlled structure, or by supplying the process with a local “informed” flux of energy and materials, or by increasing information transfer in the reverse direction, from process to man, requiring close computer control, relevant models, and arrays of local sensors and actuators. The second objective concerns the process intensification by the design of novel equipment based on scientific principles, new operating modes, and new methods of production. Process intensification with multifunctional equipment that couples or uncouples elementary processes (transfer-reaction-separation),involving the reduction in the number of equipment units leads to reduced investment costs and significant energy recovery or savings. Cost reduction between 10% and 20 % are obtained by optimizing the process. But the use of such hybrid technologies is limited by the resulting problems with control and simulation leading to interesting but challenging problems in dynamic modeling, design, operation and strong nonlinear control. Also, process intensification using microengineering and microtechnology will be used more and more for high-throughput and formulation screening. Indeed microengineered reactors have some unique characterictics that create the potential for high-performance chemicals and information processing on complex systems. Moreover, scale-up to production by replication of microreactor units used in the laboratory eliminates costly redesign and pilot plant experiments, thus accelerating the transfer from laboratory to commercial-scaleproduction.

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The third objective concerns the extension of chemical engineering methodology to product-focusseddesign and engineering in using the multiscale modeling of the above-mentionedapproach, 3PE. Indeed to be able to design and control the product quality of structured materials, and make the leap from the nanolevel to the process level, chemical and process scientists and engineers face many challenges in fundamental concepts (structure-activityrelationships on molecular level, interfacial phenomena, adhesive forces, molecular modeling, equilibria, kinetics, and product characterization techniques); in product design (nucleation growth, internal structure, stabilization, additive);in process integration (simulation and design tools based on population balance); and in process control (sensors and dynamic models). It should be underlined that much progress has been made in product-oriented engineering and in process control using the scientific methods of chemical engineering. The methods include examination of thermodynamic equilibrium states, analysis of transport processes and kinetics when they are separate and linked by means of models with or without the help of molecular simulation, and by means of computer tools of simulation, modeling and extrapolation at different scales for the whole supply chain up to the laboratory-scale. But how can operations be scaled up from laboratory to plant? Will the same product be obtained and will its properties be preserved?What is the role of the equipment design in determining product properties? This leads to the fourth main objective, which is to implement the multiscale application of computational chemical engineering modeling and simulation to reallife situations from the molecular scale to the overall production scale in order to understand how phenomena at a smaller length scale relate to properties and behavior at a longer length scale. The long-term challenge is to combine the thermodynamics and physics of local structure-forming processes like network formation, phase separation, agglomeration, nucleation, crystallization, sintering, etc., with multiphase computer fluid dynamics. Indeed through the interplay of molecular theory, simulation and experimentation measurements a better quantitative understanding of structure-property relations evolves, which, when coupled with macroscopic chemical engineering science, form the basis for new materials and process design (CAPE). Turning to the macroscopic scale, dynamic process modeling and process synthesis are also increasingly developed. Moreover, integration and opening of modeling and event-driven simulation environments in response to the current demand for diverse and more complex models in process engineering is currently taking a more important place. The aim is to promote the adoption of a standard of communication between simulation systems at any time and length-scale level (thermodynamic, unit operations, numerical utilities for dynamic, static, batch simulations, fluid dynamics, process synthesis, energy integration, process control) in order to simulate processes and allow the customers to integrate the information from any simulator into another one. Thus expanding and developing interface specification standards to ensure interoperabilityof CAPE software components justifies the creation of a standardization body (CAPE-OPENLaboratories Network, CO-LAN) to maintain and disseminate the software standards in the CAPE domain that have been developed in several international projects.

Preface

The CAPE Present Book a Vade Mecum in Process Systems Engineering

It is clear and I have shown that chemical and process industries have to overcome challenges linked to the complexity.They need to master phenomena in order to produce products “first on the market” with “zero pollution, zero accident, and zero defects” processes. Therefore, never before have enterprises invested so much in information processing and computer-controlled production, which proved in many cases that it was capable of reducing the costs and greatly increasing the flexibility than any other technology in past decades. Moreover, information and communication technology offered a great number of standardized but also specific possibilities of applications and solutions as never before. Therefore, a strategy aiming at the strengthening of competitiveness of production should obviously incorporate CAPE as a guideline for the reunion of the flexible production, the technical and administrative data processing, and the complete penetration of the enterprise activities with date processing. Within the European Federation of Chemical Engineering, the CAPE Working Party has been very active in this area since the end of the 19GOs, as shown by the success of the ESCAPE series of symposia. The activity of the Working Party is also shown by the publication of this book, which aims to present and review the state-ofthe-art and latest developments in process systems engineering. Its contents illustrates the modem day multidisciplinary and multiscale integrated approach for the integrated product-process design and decision support to the complex management of the entire enterprise. It also highlights the use of information technology tools in the development of new products and processes, in the optimal operation of complex and/or new equipment found in the chemical and process industries, and in the complex management of the supply chain. Actually, this book, based on the competences of scientists and engineers confronted with industrial practice, is a reference tool. Its ensuing and clear objectives in the topic of process systems engineering are: 0

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the necessary multidisciplinary bases required for understanding and modeling the phenomena occurring at the different scales from molecular range up to the global supply chain and extended enterprise; the experimental and knowledge-based methods and tools available to assist in the conception of new processes and products design, and in the design of plants able to manufacture the products in a competitive and sustainable way; the presentation of needed advances to fight ever-increasing complexity involved within the product-process life cycle; some tutorial examples and cases studies aiming to the state-of-the-artcomputeraided tools.

The theoretical and practical aspects of the computer-aided process engineering COVered in this book, involving computer-aided modeling and simulation, computeraided process and product design, computer-aided process operation, and computer integrated approaches in CAPE should find use in libraries and research facilities and make a direct impact in the chemical and related process industries.

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This book judiciously titled “Computer Aided Process and Product Engineering CAPE” is a vade mecum in process systems engineering. It is a valuable and indispensable reference to the scientific and industrial community and should be useful for engineers, scientists, professors and students engaged in the CAPE topic. Bravo and many congratulations to our collegues Prof. Puigjaner and Prof. Heyen for this publication and to the many authors involved in the CAPE Working Party of the European Federation of Chemical Engineering that have made this vade mecum a reality. Prof. Dr. Ing. Jean-Claude Charpentier President of the European Federation of Chemical Engineering

Foreword The European Working Party on Computer Aided Process Engineering has been an important and highly effective stimulus for and promoter of research and educational advances in process systems engineering for over 40 years. The 1991 redirection of the Working Party from the broad and all inclusive scope of embracing "the use of computers in chemical engineering", which was its theme for some thirty years, to its present emphasis on product and process engineering has had important consequences. It has reenergized the organization, sharpened its focus and promoted higher levels of technical achievement. Indeed over the past decade, the ESCAPE series of annual conferences sponsored by the Working Party has become a vital world forum for disseminating, discussing and analyzing progress in state-ofthe-art methodologies to support product and process design, development, operation and management. This volume represents a well-directed effort by the Working Party to capture the current status of developments in this field and thus to give that field its current definition. To be sure, the volume is an ambitious undertaking because the process systems engineering field has expanded enormously from its traditional primary focus on the design, control, and operations of continuously operated commodity chemical processes and its secondary concern with the design and control of batch unit operations. That expansion has included methodologies to support product design and development, increases in both scope and complexity of the processing systems under consideration, and approaches to quantify the risks resulting from technical and market uncertainties and incorporate risk-reward trade-offs in design and operational decisions. The scope of systems encompassed by process systems engineering now ranges from the molecular, biomolecular and nanoscale to the enterprise-wide arena. The levels of complexity include self-assemblyprocesses at the nanoscale, self regulating processes at the cellular level, the combination of mechanical, electrical and surface energetic phenomena in heterogeneous particulate systems, the interplay between thermodynamic, reaction and transport phenomena in integrated reactionlseparation operations, and even the decentralized and semiautonomous interactions of customers, suppliers, partners, competitors and government regulators at the enterprise level. Certainly, these developments have been greatly facilitated by remarkable advances in computing and information technologies. However, at least as important has been the expanded scope of the models that underpin design and operational decisions as well as key advances in the tools for creating, analyzing, solving and maintaining these models over the life cycle of the associate productlproComputer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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cess. The models of interest are now defined not just in terms of the traditional algebraic and differential equations, but also include systems of partial differential and integral equations, graphs/networks, logical relations/conditions, hybrids of logical conditionslrelations and continuous equations, and even object-orientedrepresentations of information, decision and work flows. Has this volume succeeded in addressing the expanded role of models, the thrusts in product design and development, the much enlarged scope and complexity of applications, and the innovative approaches to addressing uncertainty and risk? While it is impossible to address the full scope of these developments within the limited pages of a single volume, the editors and authors, all active contributors of the Working Party, have indeed done remarkably well in capturing and highlighting many of the most important developments. In Section 1, we find coverage of fundamental issues such as the development of modeling frameworks, model parameter estimation and verification methods, approaches to treatment of multiscale models, as well as numerical methods for solution of algebraic, differential and partial differential systems. The applications to particulate-basedprocesses such as crystallization, grinding, and granulation are of continuing special interest. Computational fluid dynamics and molecular modeling tools, which have become integrated into the process systems engineering toolbox are reviewed and the state of methods for the modeling and analysis of microorganisms are presented. The computational biology domain is receiving a high level of attention by the systems engineering community and will certainly receive even more extensive coverage in future reviews. The second section, which principally treats process design, reviews current developments in overall process synthesis as well as synthesis of reaction, separation and utility subsystems. The area of process intensification, which seeks to capture the potential synergies from exploiting the complex interactions of reaction-separationphenomena, is noted, discussed and recognized as an important direction for further research in process systems engineering. The third section on process operations covers important developments in the well-established hnctional levels of the process operations hierarchy: monitoring and data reconciliation, model based control, real-time optimization, scheduling, planning, and supply chain management. Additionally, issues related to the operation of flexible batch plants are reviewed. Key to progress in developments in scheduling, planning, supply chain and flexible batch plant operations have been advances in the formulation and solution of large-scale mixed integer optimization problems. The importance of these and need for continuing advances cannot be overemphasized. The fourth section treats three key integration issues as well as two supporting technology developments. While the basic features of product design are reviewed in Section 3, the progress in meeting the challenges of integrating product and process design are addressed in Section 4.As noted in that chapter, to date most of the progress has been in applications such as structured or formulated products in which the linkage between product and process is very close but hrther developments are on the horizon. The modeling technology required to support the product/process lifecycle raised in one of the chapters is a key issue facing many industry sectors. This

Foreword

issue is not yet as intensively addressed in the process systems community as it should be, given its importance in capturing product/process knowledge and managing corporate risk. The chapter on integrated supply chain management at roots deals with strategic and tactical enterprise-wide decisions. Uncertainty and risk are critical components of such decisions requiring more attention and intense future study. The sections on physical property estimation and databases, as well as open standards for CAPE software, discuss components that comprise an essential infrastructure for CAPE developments. The importance of tools for physical property prediction/estimation is evident in domains such as pharmaceutical products, in which the absence of such predictive tools has significantly retarded CAPE efforts in product and process design. The volume concludes with several enlightening case studies spanning the technologies reviewed in the preceding sections that are well chosen to make these technologies, their strengths and weaknesses more concrete. Given the limitations of a single volume, there necessarily are additional topics that will in the future require more intensive review and discussion. These include process intensification research at the micro and even nanoscales. While there have been research on microscale process design and control, the essential complexity of the phenomena that occurs at micro and nanoscales makes work in this area both challenging and of potential high impact. There has been progress in rigorous treatment of the full range of external and internal parameter uncertainties and promising computational methods for generating risk-reward frontiers that deserves notice, including the integration of discrete event simulation and discrete optimization methods. Algorithms and strategies for addressing multistage stochastic decision problems and incorporating the full valuation of the decision flexibility in multistage decision frameworks are receiving increased attention. Finally, large-scale optimization methods for attacking enterprise level decision problems of industrial scope are emerging and will become even more prominent in the near future. In summary, this volume is remarkably thorough in capturing the current state of development of the process systems engineering field and representing its broad scope. It will serve this field well in stimulating further research and in encouraging students to learn and contribute to a vital and growing body of knowledge that has important applications in broad sectors of the chemical, petrochemical, specialty, pharmaceuticals, materials, electronics and consumer products industries. The editors and authors are to be congratulated for a job well done. G. V. Reklaitis

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List of Contributors lens Abildskov

Bertrand Braunschweig

CAPEC Department of Chemical Engineering Splltofts Plads, Building 229 2800 Kgs. Lyngby Denmark

Institut Franiaise du Petrole Division Technologie, Informatique et Mathematiques Appliquees 1 and 4 avenue de Bois Preau 92852 Rued Malmaison CCdex France

Albert0 Alva-Argaez

Hankyong National University Department Chemical Engineering Kyonggi-do Anseong 456-749 Korea

/an T. Cameron

The University of Queensland School of Engineering Division of Chemical Engineering St Lucia QLD 4072 Australia

Jean-Pierre Belaud

National Polytechnic Institution of Higher Learning at Toulouse INPT Department of Process Systems Engineering 118, route de Narbonne 31077 Toulouse Cedex 4 France

Vivek Dua

University College London Department of Chemical Engineering Centre for Process Systems Engineering Torrington Place London WClE 7JE UK

1. David L. Bogle

University College London Centre for Process Systems Engineering Department of Chemical Engineering Torrington Place London WClE 7JE UK

Sebastian Engell

University of Dortmund Department of Biochemical and Chemical Engineering Process Control Laboratory Emil-Figge-Str.70 44221 Dortmund Germany

Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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Antonio Esputia

Universitat PolitPcnica de Catalunya Chemical Engineering Department ESTEIB, Av. Diagonal 647 08028 Barcelona Spain Cregor Fernhob

Process Systems Enterprise Limited Merlostrasse 12 50668 Koln Germany Cuido Buzzi Ferraris

Politecnico di Milano CMIC Department Piazza Leonard0 da Vinci, 32 20133 Milano Italy Rafiqul Gani

CAPEC Department of Chemical Engineering Sdtofts Plads, Building 229 2800 Kgs. Lyngby Denmark Weihua Cao

GE Global Research Center Real Tirne/Power Controls Laboratory 1800 Cailun Road Zhangjiang High-tech Park, Pudong New Area 201203 Shanghai P. R. China Michael C. Ceorgiadis

Imperial College London Centre for Process Systems Department of Chemical Engineering, Engineering Roderic Hill Building South Kensington Campus London SW7 2AZ UK

Vincent Cerbaud

Laboratoire de Genie Chimique BP 1301 5 rue Paulin Talabot 31106 Toulouse, Cedex 1 France Krist Y. Cernaey

Technical University of Denmark Department of Chemical Engineering Sdtofts Plads, Building 229 2800 Kgs. Lyngby Denmark lohan Crievink

Delft University of Technology Faculty of Applied Sciences Department of Chemcial Technology Julianalaan 136 2628 BL Delft The Netherlands Katalin M. Hangos

Hungarian Academy of Sciences Computer and Automation Research Institute Process Control Research Group Kende u. 11-13, PO Box 63 1518 Budapest Hungary Petra Heijnen

Delft University of Technology Department of Technology, Policy and Management PO Box 5015 2600 GA Delft The Netherlands Ceorges Heyen

Laboratoire dAnalyse et SythPse des SystPmes Chimiques UniversitC de LiPge Alee de la Chimie 3-BG 4000 LiPge Belgium

List ofcontributors

Gordon D. lngram

Margaritis Kostoglou

CSIRO Land and Water Private Bag 5 Wembley WA 6913 Australia

Aristotle University of Thessaloniki Department of Chemical Technology, School of Chemistry Box 116 54124 Thessaloniki Greece

Sten Bay jirgensen

Technical University of Denmark Department of Chemical Engineering Sdtofts Plads, Building 229 2800 Kgs. Lyngby Denmark Xavier joulia ENSIACET -

Laboratoire de GCnie Chimique 117 Route de Narbonne 310077 Toulouse, Cedex 4 France

Andrey Kraslawski

Lappeenranta University of Technology Department of Chemical Technology PO Box 20 53851 Lappeenranta Finland Rozalia Lakner

Department of Computer Science Pannon University Egyetem Street 10, PO Box 158 8200 Veszprem Hungary

Boris Kalitventzd

BELSIM s.a. Rue Georges Berotte 29A 4470 Saint-Georges-sur-Meuse Belgium Eustathois S. Kikkinides

University of West Macedoinia School of Engineering and Management of Energy Resources Sialvera and Bakola Street 50100 Kozani Greece Antonis Kokossis

University of Surrey Centre for Process and Information Systems Engieering Guildford, Surrey, GU2 7XH UK

Young4 Lim

Natural Resources Canada CETC - Varennes Energy Technology and Programs Sector PO Box 27043 Calgary, Alberta T3L 2Y1 Canada Morton Lind

Technical University of Denmark 0rsted DTU, Automation Elektrovej, Building 326 2800 Kgs. Lyngby Denmark Patrick Linke

University of Surrey Centre for Process and Information Systems Engineering Guildford, Surrey, GU2 7XH UK

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Davide Manca

Politecnico di Milano CMIC Department Piazza Leonard0 da Vinci, 32 20133 Milano Italy Francois Marechal

Ecole Polytechnique FCderale de Lausanne Industrial Energy System Laboratory LENI-ISE-STI-EPFL Station 9 1015 Lausanne Switzerland Miguel Mateus

Belsim s.a. Rue Georges Berotte 29A 4470 Saint-Georges-sur-Meuse Belgium Robert 6. Newel1

Daesim Technologies Pty Ltd PO Box 309 Toowong QLD 4066 Australia Lazaros C. Papageorgiou

University College London Centre for Process Systems Engineering Department of Chemical Engineering Torrington Place London WC1E 7JE UK John D. Perkins

University of Manchester Institute of Science and Technology Sackville Street, PO Box 88 Manchester MGO IQD UK

Efstratios N. Pistikopoulos

Imperial College London Department of Chemical Engineering Centre for Process Systems Engineering Roderic Hill Building South Kensington Campus London SW7 2AZ UK Petros Proios

Imperial College London Department of Chemical Engineering Centre for Process Systems Engineering Roderic Hill Building South Kensington Campus London SW7 2AZ UK Luis Puigjaner

Universitat PolitScnica de Catalunya Chemical Engineering Department ESTEIB, Av. Diagonal 647 08028 Barcelona Spain Javier Romero

Universitat PolitPcnica de Catalunya Chemical Engineering Department ESTEIB, Av. Diagonal 647 08028 Barcelona Spain Richard Sass

DECHEMA e.V. Department of Information Systems and Databases Theodor-Heuss-Allee25 GO486 Frankfurt-am-Main Germany

Nilay Shah

Panagiotis Tsiakis

Imperial College London Department of Chemical Engineering Centre for Process Systems Engineering Roderic Hill Building South Kensington Campus London SW7 2AZ UK

Process Systems Enterprise Ltd. Bridge Studios, 107a Hammersmith Bridge Road London WG 9DA UK

Abdelaziz Toumi

Bayer Technology Services GmbH PMT-AMS-APC,Bld. E41 51368 Leverkusen Germany

B. Erik Ydstie

Carnegie Mellon University Department of Chemical Engineering Pittsburgh 5000 Forbes Avenue Pennsylvania 15213-3890 USA

Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

I’

Introduction Since 1991 the working party (WP) of the European Federation of Chemical Engineering with the title “The Use of Computers in Chemical Engineering” adopted new term of reference and a new title-the Working Party on Computer Aided Process Engineering. This decision followed an internal debate on issues involving concepts like computer integrated manufacturing (CIM), computer-aided process operations (CAPE) and computer-aided process design (CAPD).The consensus reached in the new title naturally embraced computer-aided process operations as a subset of CIM, which is focused on the application of computing technology to integrate and facilitate the key technical decision processes which arise in chemical manufacture. Thus, CAPE focuses on algorithms, procedures and frameworks, which automate the operating/ design decisions for those functions which are automatable and support those operating decisions for which human intervention is necessary or desirable (Pekny et al. 1991). This re-structuring of the WP reflected the actual trends in the process industries. Never before had the enterprises so much invested in information processing. Computer-controlled production proved in many cases that it was capable of reducing the costs and increasing the flexibility far more greatly then any other technology in the past decades. Information and communication technology offered plenty of standardized but also specific possibilities of applications and solutions as never before. Therefore, a management strategy aiming at the strengthening of the competitiveness of production should obviously incorporate CAPE as a guideline for the reunion of the flexible production, the technical and administrative data processing and the complete penetration of all enterprise activities with data processing (Westkamper 1992). During this last decade these trends have consolidated and expanded. Thus, CAPE is a network with separate functions which are linked to each other and/or integrated by using common data and information. CAPE systems are used in the development and design, operations planning and production equipment planning departments. Otherwise, order processing is carried out by process control systems (materials resource planning). Both systems deliver data to an information system, which is the basis for the operation of the production with its flexible and control systems. Moreover, as the system boundaries have expanded, CAPE contribution and opportunities are present in the integrated product-process design and decision support to the complex management of the entire enterprise. Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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I Introduction

This book aims to present and review the state of the art and latest developments in process systems engineering. It seeks to highlight the use of information technology tools in the development of new products and processes, in the optimal operation of complex equipment found in the process industries, and in the complex management of the whole business. The book is intended as a valuable reference to the scientific and industrial community, and should contribute to the progress in computer-aided process and product engineering. This work should be also useful for teachers of postgraduate courses in these areas. Following this introduction, the book consists of 27 chapters organized into five major sections: Section 1 Section 2 Section 3 Section 4 Section 5

Computer-aided Modeling and Simulation, Computer-aided Process and Product Design, Computer-aided Process Operation, Computer-integrated Approaches in CAPE, Applications

Section 1 presents a review on actual trends and shows new advances in mathematical modeling and digital simulation techniques that permit practitioners to solve the complex scenario that describes real engineering systems. Basic techniques needed to develop and solve models are reviewed this extends from applied mathematics to model validation and tuning, model checking and initialization, and to the estimation of physical properties. In Chapter 1 steady state process simulation is introduced, which involves largescale algebraic systems. The most known solving algorithms are first presented. Alternative methods of the quasi-Newtonfamily are then described and some issues such as convergence, ill-conditioned and singular Jacobian matrices are also discussed. Then, it focuses on large and sparse algebraic systems, how to work with bounds, constraints and discontinuities as well as how to deal with the stopping criteria to be adopted. Finally, a short introduction to continuation methods is provided. Chapter 2 deals with distributed dynamic models, that is, partial differential equations (PDEs) or partial differential algebraic equations (PDAEs) incorporating convection, diffusion, reaction and/or thermodynamic property terms. Numerical methods for solving PDEs are reviewed in three sections treating first semidiscretized (method of lines) and fully discretized methods before discussing adaptive and moving mesh methods. Some applications are discussed, such as preparative chromatography, fured-bed reactors, slurry bubble columns, crystallizers and microbial cultivation processes. Finally, an approach for combining computational fluid dynamic (CFD)technology with process simulation is discussed. Process and product design studies rely on good knowledge of materials behavior and physical properties. Chapter 3 deals with estimation methods based on molecular modeling, describing the behavior of atomic and molecular systems subject to energetic interactions. It allows running numerical experiments and as any experiment, it can provide not only accurate physicochemical data but also increase the knowledge on the system studied. Molecular modeling concepts are presented so as

1 Introduction

to demystify them and stress their interests for chemical engineers. Mdtiscale approach including molecular modeling is not illustrated due to restricted space. Rather, routine examples on the use of several molecular techniques suitable to get accurate vapor-liquid equilibrium data when no data is available are provided. Chapter 4 presents a critical review of modeling frameworks for complex processing systems with emphasis not only on the models themselves but also on specialized techniques for the efficient solution of these models. More specifically, due to their increased industrial interest a general modeling framework for adsorptiondiffusion-based gas separation processes is presented in detail with focus on pressure-swing adsorption and membrane-based processes for gas separations. For subsequent sections of this chapter, a critical review of models and specialized solution techniques for crystallization and grinding processes is made. Finally, concluding remarks are drawn up and future research challenges are discussed. Process models are of increasing size and complexity, therefore the methods and tools for their tuning, discrimination and verification are of great importance. The widespread use of process models for design, simulation and optimization requires the proper documentation, re-use and retrofit of already existing models, which also need the above techniques. Thus, Chapter 5 deals with computer-aided approaches and methods of model tuning, discrimination and verification that are based on a formal structured description of process models. Besides the length and time scales, a detail scale could also be considered which seeks to develop models with varying degrees of fidelity in relation to the real world phenomena. This is the subject of Chapter 6, which presents coverage of multiscale modeling through a discussion of the origins of such phenomena in process and product engineering, as well as discussing the approaches to the modeling of systems that seek to capture multiscale phenomena. The chapter discusses the development of the partial models that make up the multiscale model particularly focusing on the characteristics of those models. The issue of partial model integration is also developed through the use of integrating frameworks. Those frameworks are analyzed to understand the important implications of model coupling and computational behavior. Throughout the chapter reference is made to granulation processing that helps to illustrate the concepts and challenges in this important area. Finally, Chapter 7 of Section 1 addresses one of the current challenges facing the modeling community: the description of regulatory networks in micro-organisms. Micro-organisms constitute examples of entire autonomous chemical plants, which are able to produce and reproduce despite shortage of raw materials and energy supplies. Understanding the intracellular regulatory networks of micro-organisms is important to process systems engineering for several reasons: microbial systems still constitute relatively simple biological systems, the study and understanding of which may enable better understanding of higher biological systems such as human beings. Furthermore, microbial systems are used, often following genetic manipulation, to produce relatively complex organic molecules in an energy efficient manner. Understanding how to couple the microbial regulatory functions and the higher level process and production control functions is a prerequisite for process engineering.

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7 Introduction

Section 2 of this book brings together process engineering and product design. One major use of models is the development and improvement of processes and products. This is a multidisciplinary approach, and it requires consideration of many aspects of the behavior of a production plant: equipment design, steady state and dynamic operation of integrated processes, raw materials and energy usage, economy, health and safety. Section 2 reviews the methods currently available to integrate knowledge from different disciplines, and presents tools available to assist in the conception of new products, and in the design of plants able to manufacture them in a competitive and sustainable way. Section 2 starts with a comprehensive review of the process separation synthesis problem with emphasis on complex distillation systems (Chapter 1). First, a critical overview of the synthesis of simple column sequences is presented with emphasis on the novel generalized modular representation framework developed at Imperial College London. Then, the synthesis problem of heat-integrated distillation trains is thoughtfully reviewed. Current state-of-the-art methodologies and algorithmic frameworks for the synthesis of complex distillation sequencing are also critically discussed. The term process intensification is associated mainly with more efficient and compact processing equipment that can potentially replace large and inefficient units commonly used in chemical processing but also includes methodologies, such as process synthesis methods, that enable the systematic development of efficient processing units. Chapter 2 provides an overview over current process intensification technologies and presents a number of recently developed systematic computeraided process intensification methods and tools. They enable the systematic screening and scoping of large numbers of alternative processing options and can identify novel options of phenomena exploitation that may lead to higher efficiencies. Such tools provide the basis for systematic approaches to novelty in process intensification and have the potential to identify processing options, which can easily be missed in design activities that rely on intuition and past experiences. Industrial processes require the use of energy, other utilities such as water and solvents, and produce wastes that need to be treated. The system performances rely not only on the efficiency of the process but also on the quality of its integration considering the energy conversion technologies, the possible combined heat and power production, the water usage and the waste treatment techniques. Chapter 3 presents computer-aided methods for solving the optimal integration of utility systems. Graphical representations support the engineer’s creativity; they are used to define the characteristics of a utility system, to analyze the potential of combined heat and power production and to analyze the quality of subsystems integration. From the requirement analysis, a utility system superstructure can be developed, to be later optimized using mathematical programming techniques. Several formulations are presented and discussed in order to integrate the different types of utility subsystems (e.g., combined heat and power, heat pumps and refrigeration) and optimize their integration with the processes. The problem of the water circuit integration can be addressed using similar concepts.

1 Introduction

The computational basis for equipment and process design in the chemical manufacturing industries is introduced in Chapter 4. Problems encountered are discussed through the use of case studies that range from modeling, simulation and optimization of existing and proposed processes. The work described in the case studies represents recent developments and trends in the industry in the area of Computer Aided Process Engineering (CAPE).The applications focus on the use of optimization techniques for obtaining optimal designs and better approaches for controlling the processes close to or at the optimal point of operation. Designs use rigorous models based on thermodynamics, conservation laws and accurate models of transport and fluid flow, with particular emphasis on dynamic behavior and uncertainty in market conditions. Product development and design starts to be a third paradigm in chemical engineering. This emerging field of research undergoes a phase of defining its scope and methods as well as the generalization of the existing industrial experience. Chapter 5 introduces the main phases of product development and the classes of the applicable methods. The special attention is given to the definition phase: methods for translation of the consumer requirements into the product parameters and approaches to generation of the product ideas. Also given is an introduction to the experimental and knowledge-based methods and tools for product design. Finally, the challenges that face CAPE in the field of product development and design are presented. Section 3 reviews the current problems facing process operations. It presents the state of relevant methods and technology, and needed advances to combat everincreasing complexity. The scope covers resource planning and production scheduling, extending to the analysis of supply chain. Process monitoring and measurement validation are described, as being preliminary steps for real-time process optimization and model-based process control. This section starts with a comprehensive review of state-of-the-art models, algorithms, methodologies and tools for the resource planning problem covering a wide range of manufacturing activities (Chapter 1). First, the long-range planning problem in the process industries is considered including a detailed critical discussion on the effect of uncertainty, the planning of refinery operations and offshore oilfields, the campaign planning problem and the integration of scheduling and planning. Then, the planning problem for new product development in pharmaceutical industries is discussed in some detail. Next, the tactical planning problem is briefly presented followed by a description of the resource planning problem in the power market and construction projects. Recent computational solution approaches to the planning problem are reviewed, while available software tools are outlined in the penultimate section of this chapter. Finally, concluding remarks are drawn and future challenges in this area are proposed. The complex problem of what to produce and where and how to produce it best is considered through an integrated hierarchical approach. Chapter 2 deals with production scheduling, focussing on the single site problem. Problem solution using heuristics is described, before presenting solution methods based on mathematical programming. Hybrid solutions are also mentioned, as well as the combined solution of scheduling and optimal operation. The state of the art for industrial applica-

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tions is described, before concluding with new application domains and future challenges. Measurements are needed to monitor process efficiency and equipment condition, but also to take care that operating conditions remain within acceptable ranges to ensure good product quality, avoid equipment failure and any hazardous conditions. However, measurements are never error free. Model-based data reconciliation techniques allow the detection and correction of random experimental errors, taking profit of redundant measurements. Chapter 3 deals with process monitoring and online estimation of performance indicators. This includes also fault detection capability, and is required as part of a model-based control system, since model tuning should be based on validated plant data. The design of effective redundant sensor network is also addressed. Operating a real plant at its optimal design conditions does not guarantee optimal operation. Some plant-model mismatch cannot be avoided, nor the effect of disturbances, this is why some sort of feedback control is needed. Chapter 4 deals with model-based control, i.e., the use of rigorous process models for feedback control by model-based on-line optimization. Several implementations with increasing level of complexity are discussed. Some plant inputs can be fixed by an off-line optimization while other inputs are controlled to keep some key process parameters on target. When nonlinear process models are available this leads to nonlinear model predictive control (NMPC)where the future values of the controlled variables are predicted over a finite horizon (the prediction horizon) using the model, and the future inputs are optimized over a certain horizon (the control horizon). As an extension of this concept, feedback control can be combined with model adaptation and re-optimization. Such a control scheme is presented for the example of batch chromatographic separations, including experimental results. Structural plant-model mismatch is a major problem also addressed. A solution is the use of optimization strategies that incorporate feedback directly; this idea is presented in detail and the application to batch chromatography is used to demonstrate its potential. To conclude, the problem of controlling quasicontinuous chromatographic separations is formulated as an on-line optimization problem where the measured outputs have to meet the constraints on the product purities but the optimization goal is not tracking of a pre-computed trajectory but optimal process operation. With the increasing fundamental understanding of the underlying physicochemical phenomena of various processes and strict environmental, safety and energy consumption constraints the need for efficient real time optimization tools has reached unprecedented levels. A better understanding of the processes is leading to highfidelity but complex mathematical models that can not always be solved efficiently in real time. The computation of the best operating or control strategy, given these models, is further complicated by the presence of constraints on control variables. In Chapter 5 a parametric programming approach is presented, which moves real time computational effort off-line. This is achieved by a priori computing the optimal control Variables as a set of explicit functions of the current state of the plant, where

7 Introduction

these functions are valid in certain polyhedral regions in the space of the state variables. This reduces real-time optimization to simple function evaluations. Actually, many production facilities constitute large hybrid systems, making it necessary to consider the continuous-discrete interactions taking place within an appropriate framework for plant and process simulation and optimization. The next chapter of Section 3 (Chapter 6) briefly discusses existing modeling frameworks for discrete/hybrid production systems embodying different approaches. A very recent framework for process recipe initialization that integrates a recipe model into the batch plant-wide model is introduced. The on-line and off-line recipe adaptation from real-time plant information is presented. Finally, a model-based integrated advisory system is described. This system gives on-line advice to operators on how to react in case of process disturbances. This way, an enhanced overall process flexibility and productivity is achieved. Application of this promising approach is illustrated through examples of increasing complexity. Process operation management and business competitiveness cannot be understood without considering supply chain activities. The main aim of Chapter 7 is to provide a comprehensive review of recent work on supply chain management and optimization mainly focused on the process industry. The first part describes the key decisions and performance metrics required for efficient supply chain management. The second part critically reviews research work on enhancing the decision-making for the development of the optimal infrastructure (assets and network) and planning. Next, different frameworks are presented, which capture the dynamic behavior of the supply chains by establishing efficient replenishment inventory management strategies. Finally, available software tools for supply chain management are outlined and future research needs for the process systems engineering community are identified. Section 4 focuses on recent developments aiming at the integration of different components in the CAPE world that offer a different degree of practical implementation. Supporting databases and a presentation of the necessary emergent standards in the CAPE domain are also included here. As chemical product design involves different disciplines, different types of data and tools, different solution strategies, etc., the need for a framework for integrated chemical product-process design becomes a subject of paramount importance. Moreover, there are chemical products where the reliability of the manufactured chemical product is more important than the cost of manufacture, while there are those where the cost of manufacture of the product is at least as important as the reliability of the product. Thus, product-centred process design is also very important. Identifying a feasible chemical product, however, is not enough; it needs to be produced through a sustainable process. The objective of Chapter 1 of Section 4 is first to define the general integrated chemical product-process design problem, then to identify the important issues and needs with respect to their solution and finally to illustrate through examples, the challenges and opportunities for CAPE/PSE methods and tools. Integrated product-process design where modeling and supply chain issues play an important role is also highlighted. Chapter 2 deals with the important issues of where, why and how models of various types are used throughout the life of an industrial or manufacturing process. The

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chapter does not deal specifically with the modeling of the life cycle process but concentrates on the use of models to address a plethora of important issues that arise during the many stages of a process' life, from the cradle to the grave. In this chapter, the life cycle concept is first discussed in relation to a "cradle to the grave" viewpoint, and then in subsequent sections consideration is made to specific issues related to the modeling goals and realizations. Some important issues are discussed which surround model development, reuse, integration, model documentation and archiving. Consideration is also made to the future needs of such modeling approaches and the important implications of life cycle modeling for corporations. Throughout this chapter the authors refer to several specific industrial case studies that help illustrate the importance of modeling throughout the life cycle as well as the challenges of doing so. What is evident in the following sections of this chapter is that there is a huge range of modeling used to help answer vital sociotechnical questions through the life cycle of the process or product. It is important to appreciate that process and product engineering have vital links to social and human factors within a holistic approach to modeling. Major infrastructure projects continually reinforce a more complete view than that which is often taken by process and product engineers. In this chapter the vision of modeling within the process or product life cycle is expanded to see just what has been achieved and where the challenges lie for the future. An introductory chapter (Section 3, Chapter 7) on the supply chain (SC) network has already presented the elementary principles and systematic methods of supply chain modeling and optimization. In Chapter 3 of Section 4, the need for and integrated management of the SC is further emphasized and novel challenging solutions are presented. As seen, supply chain management (SCM)comprises the entire range of activities related to the exchange of information and materials between costumers and suppliers involved in the execution of product and/or service orders in an extremely dynamic environment. A successful management of the supply chain management requires direct visibility of the global results of a planning decision in order to include this global perspective. This requires significant integration across multiple dimensions of the planning problem for nonconventional manufacturing networks and multi-site facilities over their entire supply chain. Objectives such as resources management, minimum environmental impact, financial issues, robust operation and high responsiveness to continuous needs must be simultaneously considered along with a number of operating and design constraints. All integrated applications needed to design and operate a plant during its whole life cycle need to access reliable physical properties for all chemicals and materials occurring in the process. Chapter 4 presents an overview of the thermophysical properties needed for CAPE calculations and describes the major sources of such data currently available. Several databases for pure component physical properties are described. Phase equilibrium data collections are also reviewed. The quality of data inside the data calculation modules is essential: inaccurate data may lead to very expensive misjudgements whether it is to proceed with a new process or modifica-

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tion of it. Inadequate or unavailable data may cause a promising and profitable process can be delayed or in the worst case be rejected, only for the reason that it was not properly modelled in a simulation. The text provides also up-to-date references to information sources available on the Internet. The lack of software standards in computer-aided process and product engineering has been a subject of concern for years, as a source of unnecessary costs, delays and inconsistencies between data produced and consumed by different nonintegrated systems using different bases, different calculation principles, different units of measurements, running on different computers under different operating systems and written in different languages. Chapter 5 introduces software standards intended to remove these problems by providing the desired interoperability between software tools, platforms and databases. With appropriate machine-to-machine interface standards, using the best available tools together becomes a matter of plug-and-play,supposedly as easy as connecting USB devices or hi-fi systems. Moreover, not only do these standards enable the putting together of several software pieces available on your local PC, but they allow interoperating heterogeneous software modules available on your organisations’ intranet, or on the Internet. The chapter starts with a discussion on the concepts of openness and of open standards; then some of the most significant operational standards in computer-aided process and product engineering are examined. Following this, the authors look at some of the current software interoperability technologies that will power future systems, namely web services, service-orientedarchitectures and ontologies for the Semantic Web. The chapter concludes with a brief look at the organisational and economic consequences of the trend towards interoperability and standards in CAPE. Section 5 presents tutorial examples and case studies aiming to illustrate typical problems that can be solved using state of the art computer-aided tools. The goal is not only to show the benefits of using CAPE methods, but also to indicate what are some current limitations and point out areas where future research and developments should be directed. Chapter 1 analyses the increased use of computers in chemical engineering education. The authors present a set of computer-aided educational modules that have been specially developed with the aim to avoid the dangers of misuse of the software. The motivation is to help the students to not only understand the concepts but also to appreciate how the theory can be applied to solve chemical engineering. The computer-aided educational modules presented here involve property prediction (suitable for a course on thermodynamics or product design), extractive distillation-based separation (suitable for courses on separation processes, distillation, or process design) and model derivation and solution (suitable for courses on modeling, simulation and/or numerical methods). The students are encouraged to first assemble modules for corresponding calculation steps into their own software for simple problems and then use specialized software for larger more complex problems. At this stage, an integrated computer-aided system also becomes very useful and data transfer between the various calculation steps and the corresponding software options take place automatically, thereby sav-

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I Introduction

ing considerable time, which can be spent instead to understand the problem better and to analyze the results. Chapter 2 describes industrial case studies in plant operation and process monitoring. The application and benefits of data validation is illustrated by several examples, taken in a range of industrial environments: oil and gas, chemicals, power plants. Besides plant monitoring and fault detection, the on-line evaluation of key performance indicators is also illustrated. It is shown how the use of more detailed models (e.g., starting with component mass balance, adding energy balances, and later equilibrium constraints) contributes to improving the result quality. Chapter 3 describes the application of a production planning method for a multiproduct manufacturing plant, which optimizes profit under uncertainties in product demands. Flexibility refers exclusively to the planning problem here and it is not coupled with a plant re-design. The development and the application of the method have been highlighted by means of a case study taken from a food additives plant. This method is considered practical for plant management, because the required input data for the demand and process models and the profit function is easy to get by the users of the method. The generated output facilitates an easy interpretation of sensitivities of the optimized production planning in terms of common economic and product demand specification parameters. Overall, the theoretical and practical aspects of the computer-aided process engineering covered in this book should find wide use in libraries and research facilities, and a direct impact in the chemical industry, particularly in production automation, utility networks, supply chain and business management with embedded computer integrated process engineering. The editors would like to acknowledge the many authors that have made this book a reality. We would also like to thank everyone who has assisted in producing the material for this book, and in particular Waltraud Wust at Wiley-VCH for her help in editing the final copy. Luis Puigjaner Georges Heyen

References 1 Pekny, /. Venkatasubramanian V. Reklaitis G. V. L. Prospects for Computer-Aided Process

Operations in the Process Industries, in Puigjaner, L., Espuiia A. (eds.) Computer Oriented Process Engineering, Elsevier, Amsterdam, (1991) pp. 427-434

2 Westkiimper E. Business Challenges in

Industrial Production, in Proceedings of Mdti Supplier Operations: Strategies, Management and Techniques for Improving the Performance of Supply and Distribution Chains, European Community Conference, Stuttgart, Germany 1992

Section I Computer-aided Modeling and Simulation

Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

I

Section 1 presents a review on actual trends and shows new advances in mathematical modeling and digital simulation techniques that permit practitioners to solve the complex scenarios that describe real engineering systems. The material in this section is organized i n seven chapters covering basic methods and techniques needed to develop and solve models: these extend ffom large-scale algebraic systems found in steady-state process systems (Chapter 1 ) to partial differential equations (PDEs) or partial diferential algebraic equations (PDAEs) encountered i n distributed dynamic models (Chapter 2). In Chapter 1 a description of direct substitution gradient and Newton‘s basic methods isfollowed by the presentation of more elaborated alternative methods of the quasi-Newtonfamily and continuation methods. Distributed dynamic models are dealt with i n Chapter 2, which involve the solution of PDEs or PDAEs incorporating convection, dijksion, reaction and/or thermodynamic property terms. Since there only exist analytical solutions in few cases due to nonlinearity and complexity, computational methods (or numerical analyses) are i n general required to solve such distributed dynamic models. This chapter ends with a n approach for combining computational fluid dynamic (CFD) technology with process simulation, which is illustrated and discussed through motivating case studies. Chapter 3 presents molecular modeling concepts involved i n the multiscale modeling approach for process study i n a broader jarnework that promotes computer-aided integrated product and process design. Molecular modeling is presented as a n emerging discipline for the study of energetic interaction phenomena. A molecular simulation performs numerical experiments that obtain accurate physicochemical data provided sampling, and energyforcejeld issues are addressed carejhlly. Still, computer-demanding molecular modeling tools will likely not be used “online” or be incorporated in process simulators. But, rather like computerJuid dynamics tools, they should be used i n parallel with existing eficient simulation tools i n order to provide information at the molecular scale on energetic interaction phenomena and increase the knowledge of processes that must manufacture ever more demanding end products. Modeling jameworks for complex processing (specijcally separation) systems with emphasis not only on the models themselves but also on specialized techniques for the e&cient solution of these models are considered in Chapter 4. Specijcally, modelingjameworks on pressure-swing adsorption and membrane-based processes for gas separations, crystallisation, and grinding processes are presented due to their increased industrial interestThe increased complexity and size of process models requires appropriate methods and tools for their tuning, discrimination, and verijcation. An extension to model validation and tuning, model checking and initialization is made in Chapter 5,followed by a coverage of rnultiscale modeling through a discussion of the origins of such phenomena i n process

13

and product engineering, as well as discussing the approaches to the modeling of systems that seek to capture multiscale phenomena (Chapter 6). Finally, Chapter 7 of Section 1 addresses one of the current challenges facing the modeling community:the description of regulatory networks in micro-organismsas examples constituting entire autonomous chemical plants.

Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

1 Large-Scale Algebraic Systems Cuido Buzzi-Ferraris and Davide M a m a

1.1 Introduction

In this section, we address the solution of a system of N nonlinear equations:

f(x) = 0

(1)

in N unknowns, x, with particular attention given to large systems. It is worth noting that the equations of system (1) must not necessarily be algebraic but may originate, for example, from the solution of a differential system with some initial conditions, or by the evaluation of the upper limit of an integral equation. The solution of nonlinear equations is therefore significant not only as a problem per se, but it is also connected to the solution of differential-algebraicequation (DAE) and ordinary differential equation (ODE) stiff problems. In the following, we will describe some iterative methods for the solution of system (1).With the term “iteration”we mean that given a previous point xi the following one is determined by the equation: Xi+l

= Xi

+ aipi

(2)

The numerical methods for the solution of Non Linear Systems, NLSs, are characterized by the selection of direction pi and by the amplitude of the movement (xi along pi. Some methods require the evaluation of the Jacobian matrix defined as:

In the following, fi represents f(xi)and Ji represents J(xi). As far as large systems are concerned, it is instinctive to try to reduce the dimensions of the problem. Using this idea, some previous numerical techniques, such as tearing and partitioning, were developed to automatically rearrange the system in order to minimize the Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

16

I

1 Large-Scale Algebraic Systems

number of equations to be solved simultaneously. Unfortunately,the following problems should not be underestimated 0

0

It is not certain that the solution of a small NLS requires less time then a larger one. In an NLS, the role of unknowns within an equation is not symmetric. In other words, a function can be easily solved with respect to a variable but it can be difficult to find the solution when another unknown is involved. In spite of the original NLS being well-conditioned, the reduced NLS obtained from the original one can be ill-conditioned.

The first problem arises from the fact that it is not possible to evaluate the nonlinearity of a system of equations. In other words, contrary to the linear case, it is not possible to determine a priori the time required to solve an NLS as a hndion of its dimension. For example, system (4)was shown to be easier to solve than the smaller system (5) by Powell (1970).

The second problem is once again bound to the nonlinearity of the system. An example is given by the conversion c in an adiabatic reactor as a function of the reaction temperature T. Often, it is possible to write the following energy balance: c =g m (6) Evidently, it is trivial to determine the conversion when T is assigned. Conversely, it may not be easy to evaluate the temperature that produces a specific conversion. The third problem is due to the fact that a rearrangement of the NLS can introduce an ill-conditioningthat was not originally present. Let us suppose we have to solve the following system: x1+ 1000x4 = 1001

+ x2 = 1001 + x3 = 1001 + + x3 + = 4

1000x1 1000x2 x1 xz

(7)

x4

whose solution is: x1 = x2 = x3 = x, System (7) can be rearranged into: XI

x2 x3

= 1001 - 1 0 0 0 ~ ~ = 1001 - 1000x1 = 1001 - 1000x2

f4 = x1+ x2 + + x4 X)

-4

=0

=

1.

1.2 Convergence Tests

The new problem (8), although being characterized by only one equation with unknown x4 has an extremely ill-conditionedform. As a matter of fact, if x4 is evaluated numerically as: x4 = 0.99999, then the following values are obtained: x1 = 1.010014, x2 = -9.013580, x3 = 10,014.58 andf(x,) = 10,003.58. It should be emphasized that system (8)is linear (therefore a simpler problem) and that it is reduced to a very simple equation in one unknown: x4. Quite often, it is better to avoid any manipulation of the system if the goal is to reduce its dimensions. Actually, it is advisable to leave the NLS in its original form since it comes from modeling a physical phenomenon. Doing so, there are more guarantees that the numerical system is well-posed because it describes a real problem. What should be done is something that is apparently similar to rearranging the system but it is conceptually quite different. It is advisable to try exploiting the structure of the system without manipulating it. A very simple example is represented by the solution of a steady-state distillation column. The liquid-vaporequilibria and the material balances of the unit should not be solved stage by stage, in a top-bottom sequence, while iterating towards convergence through the overall material balance of the column so to have the input/output flowrates consistent. By doing so, the physical structure of the problem would shift to obey to a sequential mathematical algorithm that would solve several apparently simplified subproblems. Such an approach would not respect the physical structure of the equilibrium stage in the sense that it would be equivalent to solving a flash problem starting from the known composition of one output stream to determine the compositions of the input and second output streams. On the contrary, the intrinsic structure of a distillation column brings tridiagonal organization to the correlated mathematical problem. Such a tridiagonal structure should be exploited to efficiently solve the numerical problem.

1.2 Convergence Tests

Working at the implementation of a numerical algorithm for the solution of NLSs, a quite important matter must be addressed. How do we determine if the new estimate, xi+l.is better or worse than the previous one, xi? The typical approach is to accept the new value, xi+,,if: (9)

Ilf(xi+l)ll2 < Ilf(xi)lll

or, equivalently, if there is a decrease in the merit function: 1

N

1 2

@(x) = - C 4 2 ( x ) = -fTf

j=1

18

I

I Large-Scale Algebraic Systems

The criterion represented by Eqs. (9) and (10) should be avoided within a generalpurpose program whenever the functions fare unbalanced and have significantlydifferent orders of magnitude. In those cases, the equations with lower orders of magnitude do not contribute to the merit function. As an example, we can report the evaluation of a flash, or the modeling of a distillation column. The stoichiometric equations (order of magnitude: 1) stay together with the enthalpy balance equations (order of magnitude: l.EG-l.E9) and significant differences in terms of orders of magnitude are present in the resulting NLS. An improvement of the previous criterion is given by weighting each equation with a suitable weight, wj.Consequently, the merit function becomes: . N

By introducing the diagonal matrix, W, which has elements equal to the weights, 9, the matrix notation follows: 1 1 @,(x) = -(Wf)T(Wf) = -fTW2f 2 2

(12)

More generally, the weights can vary with the iterations. Consequently, the weight matrix becomes Wi. A reasonable criterion for the definition of the weights is to make all equations have the same order of magnitude. To do so, it is sufficient to use weights equal to the inverse of the order of magnitude of the corresponding equations. This criterion can be implemented in the following ways: 0 0 0

The user directly writes the equations in an adimensionalized form. The user assigns the weights to be used by the numerical solver. The numerical solver evaluates the weights of Eq. (11).

Another approach is to consider whether vector, x, is sufficiently near the solution of the problem. Let us suppose we have the following linear system:

Ax=b

(13)

The distance of a point xi from one of the planes, j: ajlxl

+ aj2x2 + . . . +

ajNxN

= bj

(14)

is determined by calculating the point at which the orthogonal line passing through xi intersects the plane itself. The square of the distance between that point and xi is: dj

=

[ajl(xl)i

+ aj2(x2)i + . . . + ajN(xN)i - hi] 2 N

m=l

(15)

7.2 Convergence Tests 119

By adopting the following weights: 1

2

w.= J

N

m=l

every term of summation (11) evaluated at point x, represents the square of the distance between such a point and the planes of system (13). As far as NLSs are concerned, matrix A becomes an approximation of the Jacobian, J. Since the Jacobian matrix changes with the iterations, the weights should also be modified. This strategy may be adopted to automatically evaluate the weights in Eq. (11). Unfortunately, the aforementioned strategy does not benefit from the following property (Buzzi-Ferrarisand Tronconi, 1993): Given a merit function, F(x), applied to a linear system, it is assumed that if F(x,+,) < F(x,) then point x , + ~is closer to the solution than point x,. The criteria described so far do not benefit from this property except for the linear system (13) consisting of orthogonal planes. with reference to system (13),let us suppose we know the exact solution, x,, that makes the residuals null:

b-Axs=O

(17)

Given a point xi, other than x,, we have the residual:

b-Axi = f i

(18)

by subtracting Eq. (17) from Eq. (18) we obtain: A ( x ~- Xj) = f j

(19)

Formally, the Euclidean norm of the distance xi - x, is:

and the geometric interpretation of Eq. (20) is that the quantity IIA-'f,llz measures the distance of xi from the solution x,. With regards to NLSs, Eq. (20) is a measure of the distance of point xi from the solution of the linearized system, where A = Ji represents the Jacobian matrix evaluated using xi. Finally, the distance of a new point xi+, from the solution of the same system is:

Whenever a nonlinear system is concerned, the new point

must be accepted if

given that, in the case of linear systems, Eq. (22) means that xi+l is closer to the solution than xi is. It is worth highlighting that the Jacobian of Eq. (22) is kept constant while f, and f,+, are the residuals at points xi and xi+'.

20

I

I Large-Scale Algebraic Systems t

If Newton’s method is adopted to solve the NLS (see subsequent paragraphs) then the Jacobian matrix J i has already been factored to solve the linear system produced by the method itself. The evaluation of the merit function using the two points, xi and xi+l, is therefore straightforward and manageable. Often, besides normalizing the functions, it is also advisable to normalize the variables. A practical way to implement the normalization is to scale the variables by multiplying them for a coefficient so that all the variables have the same order of magnitude. By indicating with D a suitable diagonal matrix of multiplying coeficients, the proposed transformation is: z = DX

(23)

Consequently, the merit function with the new variables becomes: &D(z)

1 2

= -f(D-*z)

T

W2f(D-*z)

1.3 Substitution Methods

Before applying the substitution method to the solution of an NLS it is necessary to transform the equations into the following formulation:

h(x) = q(x)

(25)

where system h(x) should be easily solvable if the value of q(x) is known. The method consists of applying the iterative formula: h(xi+l) = q(xi)

(26)

where xi+l is obtained from xi. The easiest iterative formula is: x j = &(XI, x2, . . ., + l ,

xj+l,

. .., X N )

(27)

where each variable is obtained explicitly from the corresponding function. The procedure shown in (27) has the same shortcomings as the monodimensional case. Moreover, it is quite difficult to find a proper formulation that converges to the solution.

1.4 Gradient Method (Steepest Descent)

The gradient of a function is a vector. The function changes more rapidly in the direction of the gradient. With reference to the merit function (lo),the gradient in xi is given by:

Consequently, vector

P ( x ~= ) pi = -gi = - J T f i

(29)

describes the direction where the merit function (10) decreases more rapidly. Obviously, the direction of the gradient changes whenever a different merit function is adopted. When the merit function (11) is involved, the search direction becomes:

If the variables are also weighted and the merit function (24) is adopted, then the search direction becomes:

The evaluation of the space increment, ai,is performed by a monodimensional search. The procedures that adopt the gradient (steepest descent) method, as the search direction, have major limits if used alone. Actually, such methods are efficient only at the initial steps of the solving procedure. The gradient method may be efficiently coupled with Newton’s method since it is quite complementary to it. Newton’s method is rather efficient in the final steps of the search while the gradient method is efficient in the initial ones.

1.5 Newton’s Method

If the fi of the NLS can be expanded in terms of a Taylor series:

f(xi

+ di) = f i + Jidi + O( lldi 112)

(32)

and point xi is rather close to the solution, it is possible to stop the expansion to the first order terms. In this case, the correction vector, Di, to be summed with point xi comes from the solution of the system:

The following iterative procedure represents the elementary formulation of Newton’s method:

where di comes from the solution of the linear system (33):

J1. d1 .-- - f . 1

(35)

22

I

7 Large-Scale Algebraic Systems

Consequently, Newton's method has the following search direction: pi = di

(36)

and a,= 1 Whenever Newton's method converges, its convergence rate is quadratic. It is possible to identify one difference between the solution of nonlinear systems and multidimensional optimization. Usually, the Jacobian matrix of system (35) is not symmetric. Thus, it is not possible to either solve the linear system with the Cholesky algorithm or to halve the memory allocation. The most efficient methods adopted for the Jacobian factorization require twice as much time as the Cholesky algorithm. The correction, di, obtained from system (35) is independent from either a change of scale in the variables or the merit function. In fact, by introducing the scale change: y=cx+c

(37)

the new Jacobian, with respect to the variables y, becomes: Jy = JS1

(38)

and the Newton's method estimate for the x variables is: X;+I

= X; - Ji'fi

As a result, the Newton's method estimate is invariant with respect to a linear transformation using the variables x as well as the merit function (22). Vector di represents a direction where all the merit functions decrease. Actually:

g'd; = (JTfi>' (-JL'fi) T

= -fTJjJi'f; = -fTfi

gTd, w 1 -- (JTWtfi) (-Jr'fi)

= -fTWfJiJi'fi =

g%,di = (JTWtfiDF2)T (-JL1fi) =

i0

-fTWtfi < 0

-Di2fTWffi < 0

Such a property is valid if the following two conditions are satisfied: 1. the Jacobian matrix is not singular, i.e., the inverse matrix must exist; 2. the Jacobian in Eq. (41)must be a good approximation of the true Jacobian matrix and not a generic matrix Bi, otherwise:

1.5 Newton's Method

J,B;' # I (45) and the previous Eqs. (42-44) may not be true. It is also possible to outline another difference between the solution of nonlinear systems and multidimensional optimization. As far as multidimensional optimization problems are concerned, matrix Bi may also be a bad approximation of the Hessian (provided it is positive and definite) and at the same time be able to guarantee a reduction of the merit function. Conversely, matrix Bi involved in the solution of NLSs should be a good estimate of the Jacobian. Besides the previously mentioned advantages, Newton's method also presents some disadvantages that suggest not using it with the trivial iterative formulation of Eqs. (34) and (35). Three different categories for the classification of the problems related to Newton's method can be outlined: 1. Problems related to the Jacobian matrix 0 The method undergoes a critical point if the Jacobian is either singular or illconditioned. 2. Problems related to the convergence of the method 0 The method may not converge to the solution; 0 The new prediction may be worse than the previous one with respect to all merit functions. 3 . Problems related to the Jacobian evaluation and the linear system solution 0 Every new iteration requires the evaluation of the Jacobian matrix. If the Jacobian is evaluated numerically, this means that the nonlinear system (1)is called N times; 0 Each new iteration requires the solution of the linear system (35).

The algorithms derived by the original Newton's method may be divided into two classes depending on how the Jacobian matrix is evaluated. The first class comprises Newton's modified methods where the Jacobian is evaluated analyhcally or numerically approximated at point xi. The second class comprises the quasi-Newtonmethods that update the Jacobian by means of the information gathered during the iterative process. For both classes, as soon as the Jacobian matrix has been either evaluated or updated, it is recommended to immediately execute a Newton's iteration in order to exploit the efficiency of such a method. Consequently, both of the aforementioned classes first verify the point:

xi+l = xi

+ di = xi - Ji'fi

Such a point is accepted if it satisfies at least one of the following tests:

(46)

I

23

24

I

I Large-Scale Algebraic Systems

ffflW2fi+l < fTWZfi(1 - y )

(48)

The y parameter guarantees a satisfactory improvement of the merit functions.

1.6 Modified Newton's Methods

In these methods the Jacobian matrix is evaluated analytically or is approximated numerically. Since the Jacobian is recalculated at each iteration it is also necessary to solve the linear system (35). Several expedients and precautions are necessary to reduce the drawbacks of Newton's method.

1.6.1 Singular or Ill-conditioned Jacobian Matrix

The solution of system (35) is performed through the factorization of the Jacobian matrix. Whichever factorization is adopted, it is mandatory to evaluate the condition number of the system and to properly operate if such a number is too high. Actually, it is possible to introduce the auxiliary function:

The minimum of function (49) is:

which is equivalent to the prediction of Newton's method (35) applied to the solution of the nonlinear system. If the Jacobian is well-conditioned then the correction, Di, is achieved by solving system (35) instead of system (SO). Conversely, the use of function (49) becomes interesting when the Jacobian matrix is quite ill-conditioned or even singular. As a matter of fact, the system matrix, JTJi, is the Hessian of function (49) and it is symmetric. By using function (49) instead of the merit function (lo),there is the advantage of knowing the Hessian without having to evaluate the second derivatives. At the same time, it is possible to apply to matrix JTJi, all the expedients exploited when an ill-conditionedminimum problem is involved. Two algorithms implement the aforementioned idea: 0

The Levenberg-Marquardtmethod modifies system (50) in the following way:

1.G Modified Newton’s Methods

The ,u parameter may be chosen so to transform matrix (jTiji+ p ~ in) a wellconditioned matrix. Besides being an artifice to reduce the ill-conditioning of the Jacobian,the LevenbergMarquardt method is also an algorithm that couples Newton’s method to the gradient one. Actually, if the Jacobian is not singular, the solution of system ( S l ) , with p = 0, is equivalent to Newton’s estimate. Conversely, when high values of parameter p are involved, the search direction tends to the gradient of the merit function (10). The Gill-Murray criterion represents the second alternative. The idea is to make the diagonal coefficients of matrix J:Ji positive. If the Jacobian J is QR factored and matrix R is worked out in order to avoid any zeros in the main diagonal, then matrix JTJ = RTR is symmetric positive definite. Buzzi-Ferrarisand Tronconi (1986)showed a new methodology for the modification of the Jacobian matrix. If the Jacobian is ill-conditionedor singular then some equations in system (35) are linearly dependent. Consequently, it is possible to eliminate those linearly dependent rows. Since the resulting system becomes underdimensioned, it is appropriate to adopt the LQ factorization that produces the solution with minimum Euclidean norm for vector di. Thus, it is possible to avoid an excessively large correction on such a vector. The numerical solution satisfies not only the subsystem but also the equations that were removed, since, if compatible, they are almost a linear combination of the others. This criterion is often efficient and is preferable to the previous one since it is not influenced by the merit function. At the same time, it does not produce a false solution. By the term false we mean a solution of the minimum problem that is not the solution of the NLS. 1.6.2 The Convergence Problem

As mentioned, the Newton’s estimate is not satisfactory whenever point xi+ldoes not meet conditions (47) and (48).In such a case, it is possible to adopt the following strategies: 0

0 0

A monodimensional search is performed in the same direction as the Newton’s one. The region where the functions are linearized is reduced. An alternative algorithm to the Newton’s method is adopted.

Before addressing these points, the following feature should be emphasized: an NLS may not have a solution. Moreover, if a solution exists, we are not sure that it will be possible to determine it. A numerical program should warn the user about its incapability of solving the problem. 1.6.2.1 Monodimensional Search

Usually, since the monodimensional optimization is both not time-consuming and quite efficient, it is adopted in all-purpose solvers. Normally, the monodimensional

26

I

I Large-Scale Algebraic Systems

search algorithm is not pushed to the extreme. Actually, the optimization is intended to identify a new point where Newton’s method might easily converge. The merit function that is usually adopted is the weighted one (12).At the outset, the following data are known: 0 0

the value of Qw at point xi; the gradient g, at point xi and that in the direction di, g:di; the value of Qw at point xi + di.

Since there are three data in the direction di, the merit function Qw may be approximated by the parabola:

Y ( t ) = Y(0)+ tY’(0) + t2 [ Y ( U - y(0)

- Y’(O)]

(52)

Tne minimum of the parabola is:

In the following,we will assume to have a good estimate of the Jacobian matrix. Consequently, the following equation may be adopted:

and the minimum of the parabola becomes: U =

fTWffi f;+,Wff;+, fTW;fi

+

(55)

It is recommended to check that u is not too small by imposing: u > 0.1

(56)

Since point xi+l does not satisfy Eq. (47),an upper limit for u is automatically set. If the minimization is not successful then a new artifice should be exploited, otherwise the program must stop with a warning. 1.6.2.2 Reduction o f the Search Zone

The Levenberg-Marquardt method may be considered from three distinct perspectives: 0 0 0

as an artifice to avoid the ill-conditioned problem of the Jacobian matrix; as an algorithm that couples Newton’s method to the gradient one; as a method exploiting either a reduced step or a confidence region.

The third point is the most interesting when a reduction of the search zone is concerned. In this case, it is required to identify the correction di that minimizes the auxiliary function (49)with the constraint:

1.7 Quasi-Newton Methods

where 6 has a specified value. A valid alternative to the Levenberg-Marquardt method is represented by the dog leg method, also known as Powell’s hybrid method (1970). Once again, such a method couples Newton’s method to the gradient one. The original version of Powell’s method was close to the concept of either confidence region or reduced step. Powell proposed a strategy for the modification of parameter 6 subject to both the successes and failures of the procedure. 1.6.2.3 Alternative Methods

Whenever Newton’s method fails, it is necessary to switch to an alternative method. For this reason, the most commonly used method is the gradient of a merit function. There are several alternatives. It is possible to perform a monodimensional search along the gradient direction. Even better, the two methods may be coupled as it happens with the Levenberg-Marquardt algorithm or the dog leg method. Another choice is to perform a bidirectional optimization on the plane defined by both search directions. Unfortunately, there are no heuristic methods for the solution of NLSs. Only for very specific problems can a substitution method be expressly tailored and coupled to Newton’s method. As an example, in the field of chemical engineering, the boiling point (BP) method may be implemented and applied to distillation columns. In the following,we will introduce the continuation methods. Such methods transform the functions of the NLS and solve an equivalent and dynamically easier problem.

1.7 Quasi-Newton Methods

Let Bi be an approximation of the Jacobian matrix at point xi.As mentioned before, matrix Bi must be a good approximation of the Jacobian. Consequently, also in the case of quasi-Newton methods, it is necessary to evaluate either analytically or numerically the Jacobian matrix. If during the search of the solution the rate of convergence should decrease, reevaluating the Jacobian is recommended. Therefore, it is not possible to implement a quasi-Newton method without a modified Newton’s method. During the solution procedure the values of functions f, and f,+lare known in the points xiand xi+l. Such points must not necessarily correspond to previous Newton’s method estimates. Given:

AX;= ~ i + l xi

(58)

if the distance between the two points is not significant, it is possible to link the function values f, and fi+l through a Taylor expansion:

127

28

1

I Large-Scale Algebraic Systems

fi+l = fi

+ BAxi

(59)

where the Jacobian, B, is evaluated in a suitable point between xi and xi+l. Specifically, it is possible to impose that the Jacobian satisfies the following condition:

f i+l = f i

+ Bi+l AX;

(60)

Equation (GO) does not allow univocal evaluation of all components of the Jacobian when the number of equations N > 1. In this case, N - 1 more conditions are necessary. In 1965 Broyden proposed to choose the conditions to be added to Eq. (GO) in order to keep invariant the product between the Jacobian, evaluated in xi and in xi+l, and an orthogonal vector to Axi. Generally, for any given vector, qi, with: qTAxi = 0

(61)

it must result that:

This condition is reasonable ifwe consider Eq. (59). Actually, it is possible to modify the Jacobian in the direction Axi so as to satisfy condition (GO). On the contrary, in a direction orthogonal to the previous one, there is no additional information and the behavior of the Jacobian,with respect to a Taylor expansion in that direction, should be invariant. By coupling conditions (62) and (GO), it is possible to univocally identify the Jacobian in xi+l: Bi+l = Bi

+ (fi+l

-

fi -

B~AX~)AXT

AX:AX~

1.8 Large and Sparse Systems

When the number of equations and variables is quite large, often each equation depends on a reduced set of variables. As far as the Newton’s and quasi-Newton methods are concerned, it is necessary to exploit the sparsity of the Jacobian matrix so as to reduce the memory allocation while saving CPU time. In particular, the following expedients are essential: The solution of system (35) should be made by the method that best exploits the Jacobian sparsity and structure. If the Jacobian has no specific structure that can be directly exploited, it is worthwhile rearranging both the variables and equations so as to reduce the CPU effort and memory allocation required by the factorization of the Jacobian matrix. The null Jacobian components should not be evaluated. This happens automatically if the Jacobian is evaluated analytically. Conversely, whenever the Jacobian matrix is approximated numerically, the following computations:

1.8 arge and Sparse Systems

Jik =

J(Xi

+ hkek) -j(xi) hk

(64)

should be avoided if it is a pAoA known that: x ( x i + h&) =f;(xi) It is possible to exploit some formulas to update the Jacobian, which are able to preserve its sparsity. At the same time, if some elements are constant, they should not be updated by those formulas. Schubert (1970) proposed a modification of the Broyden formula (1965), while Buzzi-Ferraris and Mazzotti (1984) proposed a modification of the Barnes formula (1965). These formulas take into account the coefficientsthat are known and do not modify them. The update is performed only on the coefficients that are unknown. 0 If the Jacobian is evaluated numerically, it is not convenient to increment a variable one at a time and to perform a call to the nonlinear system. This point must be emphasized. If Eq. (64) is adopted to evaluate a Jacobian matrix that is supposed to be full, then vector ek is the null array except for position k, where the element is equal to 1. In this case, system (1) is called N times to evaluate the derivatives of the functions with respect to the N variables. Let us now consider the following sparse Jacobian matrix, where the symbol x represents a nonzero element (see Fig. 3.1). It can be observed that when the system is called to evaluate the derivatives with respect to variable xl, the only functions to be modified arefi and&. If at the same time variable x2 were modified, it could be possible to evaluate the derivatives with respect to this variable since it only influences functionsh andf7. Going on with the reasoning, it is possible to show that only three calls to the system of Fig. 3.1 are sufficient to evaluate the whole Jacobian matrix. In fact, with the first call it is possible to increment variables x1x2x3x4x6x9.With the second call we increment variables x5x8. Finally, with the third call we increment variables ~7x10.When the system is sparse, the total number of calls necessary for the evaluation of the Jacobian matrix can be drastically reduced. It is not easy to identify the sequence of variable groupings that minimizes the number of calls to the nonlinear system. Curtis, Powell and Reid (1972) proposed a heuristic algorithm that is often optimal and can be easily described. We start with the first variable and identify the functions that depend on it. We then check if the second variable does not interfere with the functions with which the first variable interacts. If this happens, we go on to the third variable. Any new variable introduced in the sequence also increases the number of functions involved. When no additional variables can be added to the list, this means that the first group has been identified and we can go on with the next group until all N variables of the system have been collected. It is evident that the matrix structure of the Jacobian must be known for this procedure to be applied. This means that the user must identify the Boolean of the Jacobian, i.e., the matrix that contains the dependencies of each function from the system variables (see Fig. 3.1).

29

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I Large-Scale Algebraic Systems Figure 3.1 The Boolean matrix describes the Jacobian structure and the function dependency from the variables of the nonlinear system.

1.9 Stop Criteria

When the problem is supposed to be solved or when there are insurmountable problems, there are some tests to bring the iterations to an e n d It is advisable to implement a limitation on the maximum number of iterations. If the weighted function (12) is lower than an assigned value, there is a good chance a solution has been reached. The procedure is stopped if the estimate of Newton’s method, di, has all components reasonably small. With multidimensional optimization it is not sufficient to check whether a norm of vector di is lower than an assigned value. On the contrary, it is advisable also to to check the following relative:

Even if a quasi-Newton method is used, a good approximation of the Jacobian is known. Consequently, this criterion is adequately reliable. Nonetheless, it should be emphasized that this test is correct only when the difference between two consecutive iterations, di = xicl - xi, comes from a Newton-like method and the Jacobian is not singular.

1.10 Bounds, Constraints, and Discontinuities

Some problems have solutions that are not acceptable since they belong to unfeasible regions. In these situations, it can be worthy assigning some bounds to the variables in order to avoid the solution from falling in those unfeasible regions. This issue requires the adoption of specifically tailored numerical algorithms that are able to

1 . 7 7 Continuation Methods

depart from the unfeasible attractor while moving towards the feasible region. Similar considerations apply when discontinuities are involved. In this case, the numerical algorithm should be able to work across the discontinuity while avoiding a crisis due to its presence. The discontinuity can be either in the function itself or in its derivatives (Sacham and Brauner 2002).

1.11 Continuation Methods

Let us suppose we have a nonlinear system of N equations whose solution is quite difficult. For some reason that will be explained in the following we suppose we have a vector of adjoint parameters z, of M elements, in the equations of the system. Therefore, the NLS can be rewritten as:

f(x, 2 ) = 0

(66)

The system must be solved with respect to the N unknowns, x, given a specific value, z = zF,of the parameter vector. Let us now suppose we know another system, q(x, z) = 0, in some way related to the previous one, whose solution, for a given value of parameters z, is quite easy. In this case it is possible to write a new system that is a linear combination of the previous two:

The parameter t in Eq. (67) is called the homotopy parameter. When the parameter t varies in the interval 0, ... 1, system h is solved for a value of x that satisfies both systems q and f. The parameters z can be a function of parameter t in any way, provided that for t = 1 we have z = zF. The most straightforward functional dependency between t and z is the linear one: z = ZO

+ (ZF

-

zo)t

(68)

where zo corresponds to the initial value of the parameters. Another functional dependency is the following one:

There are several alternatives for the auxiliary system q(x, z). The common characteristic is that for t = 0 the solution of system q ( q , zo) = 0 should be effortless. The following are some choices: 0

Fixed point homotopy: q(x, z)

=x -q

+ z - zo

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7 Large-Scale Algebraic Systems

h(x, Z, t ) = tf(x, Z) 0

+ (1 - t) [(x

ZO)

+ (1- t)J(%,

f ( q , zo) (71)

=

J(%. ZO) [(x - %) +

+

20) [(x - ~ g ) ( 2

Parametric continuation method: q(x, Z) h(x, Z, t ) = f(x, Z) = 0

-

(70)

=0

Homotopy with scale invariance: q(x, z) h(x, Z, t) = tf(x, Z)

0

- ZO)] = 0

Newton or global homotopy: q(x, z) = f(x, z) h(x, Z, t ) = f(x, Z) - (1 - t)f(Xg,

0

+

- ~ g ) (Z

- Z0)I = 0

(Z

- ZO)]

(72)

=0

(73)

The fourth criterion (73) deserves some explanation since one could think that the original problem has not been modified. In many practical cases, a problem may have a simple solution in correspondence to a value zo of the parameters, while there are numerical difficulties for zF. In this case the system is solved by setting t = 0 and z = zo. We then get a first solution ~0 that satisfies: h(xo, zo,O) = f h , 20) = 0

(74)

Successively, we change t from 0 to 1in order to modify continuously the parameters from zo to zF.By doing so, several intermediate problems are solved through a stepby-step procedure. It is worth highlighting two cases: 1. The parameters, z, correspond to some specifications that should be satisfied. Often, the problem can be easily solved if the specifications are mild, while it becomes hard when the requirements are tight. A typical example is represented by a distillation column. The continuation parameter can be the product purity. If the product purity is quite high, there can be some problems concerning the numerical solution. In this case, it is recommended to start with a lax specification. Once the solution has been evaluated, the problem is slightly modified by tightening the specification. A new solution is performed by adopting as a first guess the previous solution. By continuing the procedure, it is possible to reach the final product purity. 2. The problem can be solved easily by introducing some simplifications. The continuation method modifies continuously the simplified hypotheses, carrying the system towards the detailed model. In the case of separation units, one of the problems may be the evaluation of the liquid-vapor equilibrium constants, k. If the k vector strongly depends on the compositions then it can be difficult to identify the first guess values that make the Newton’s method converge. In this case, it is convenient to consider the system ideal. By solving the simplified problem under the hypothesis of ideal k values, a solution is easily obtained. Such a solution becomes the first guess for the continuation procedure that takes the system towards the hypothesis of nonideal liquid-vapor equilibria. The parameter vector z comprises the k values as follows:

7. 7 7 Continuation Methods

Initially, when t = 0, all parameters are equal to the ideal k values. The homotopy parameter, t , evolves from 0 to 1. By doing so the k values continuously change from the ideal to the real hypotheses. The same reasoning can be applied to the enthalpies of the mixture. The main advantage obtained by the continuation method is that the intermediate problems have a physical implication. Consequently, each intermediate problem has a solution where the variables take up reasonable values. Another approach to the solution of the continuation problem is to implement an ODE system that integrates the x variables and the z parameters from an initial condition (easy problem) to a final time (difficult problem). Seader and coauthors (Kuno and Seader 1988; Seader et al. 1990; Jalali and Seader 1999; Gritton et al. 2001) have worked extensively on this approach.

References 1 Barnes J. C. P. An Algorithm for Solving

2

3

4

5

6

7

8

9

Nonlinear Equations Based on the Secant Method. Comput. J. 8 (1965) p. 66 Broyden C. G. A Class of Methods for Solving Nonlinear Simultaneous Equations. Math. Comput. 21 (1965) p. 368 Broyden C. G. A New Method of Solving Nonlinear Simultaneous Equations. Comput. J. 12 (1969)p. 94 Buzzi-Ferraris G. Mazzotti M. Orthogonal Procedure for the Updating of Sparse Jacobian Matrices. Comput. Chem. Eng. 8 (1984) p. 389 Gritton K. S. Seader J. D. Lin W,]. Global Homotopy Continuation Procedures for Seeking all Roots of a Nonlinear Equation. Comput. Chem. Eng. 25 (2001) p. 1003 Buzzi-Ferraris G. Tronconi E. BUNLSI - A Fortran Program for Solution of Systems of Nonlinear Algebraic Equations. Comput. Chem. Eng. 10 (1986) p. 129 Buzzi-Ferraris G. Tronconi E. An Improved Convergence Criterion in the Solution of Nonlinear Algebraic Equations. Comput. Chem. Eng. 17 (1993) p. 419 Curtis A. R. Powell M./. D. Reid j . R On estimation of Sparse Jacobian Matrices. Report TP476 AERE Hanvell 1972 Jalali F. Seader J . D. Homotopy Continuation Method in Multi-phase Multi-reaction Equilibrium Systems. Comput. Chem. Eng. 23 (1999) p. 1319

10 Powell M . J. D. A Hybrid Method for Nonlin-

ear Equations. In: Numerical Methods for Nonlinear Algebraic Equations. Ed.: Rabinowitz G . Breach London, 1970 11 Schubert, L. K. Modification of a QuasiNewton Method for Nonlinear Equations with a Sparse Jacobian. Math. Comput. 24 (1970) p. 27

Further Reading 1

2

3

4

5

6

Allgower E. L. Georg K. Homotopy Method of Approximating Several Solutions to Nonlinear Systems of Equations. In: Numerical Solution of Highly Nonlinear Problems, Foster, Amsterdam, 1980 Davidenko D. On a New Method of Numerically Integrating a System of Nonlinear Equations, Doklady Akademii Nauk USSR 88 (1953) p. 601 Eberhan /. G.Solving Equations by Successive Substitution - The Problems of Divergence and Slow Convergence. J. Chem. Edu. 63 (1986) p. 576 Gupta Y. P. Bracketing Method for On-line Solution of Low-Dimensional Nonlinear Algebraic Equations. Ind. Eng. Chem. Res. 15 (1995) p. 239 Jalali F. Process Simulation Using Continuation Method in Complex Domain. Comput. Chem. Eng. 22 (1998) p. S943 /sun Y. W. Multiple-step Method for Solving Nonlinear Systems of Equations. Comput. Appl. Eng. Edu. 5 (1997) p. 121

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1 Large-Scale Algebraic Systems 7 Karr C. L. Weck B. Freeman L. M. Solutions

8

9

10

11

12 13

14

to Systems of Nonlinear Equations via a Genetic Algorithm. Eng. App. Artif. Intell. 11 (1998) p. 369 Kuno M. Seader]. D. Computing all Real Solutions to Systems of Nonlinear Equations with a Global Fixed-Point Homotopy. Ind. Eng. Chem. Res. 27 (1988)p. 1320 Neurnaier A. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge 1990 Paloschi]. R. Bounded Homotopies to Solve Systems of Algebraic Nonlinear Equations. Comput. Chem. Eng. 19 (1995) p. 1243 Paterson W. R. A New Method for Solving a Class of Nonlinear Equations. Chem. Eng. Sci. 41 (1986) p. 135 Rice R. J. Numerical Methods, Software and Analysis. Academic Press, London, 1993 Seader]. D. Kuno M. Lin W. J. Johnson S. A. Unsworth K. Wiskin]. W. Mapped Continuation Methods for Computing all Solutions to General Systems of Nonlinear Equations. Comput. Chem. Eng. 14 (1990)p. 71 Shacham M. Kehat E. Converging Interval Methods for the Iterative Solution of a Nonlinear Equation. Chem. Eng. Sci. 28 (1973) p. 2187

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Shacham M. Brauner N. Numerical Solution of Non-linear Algebraic Equations with Discontinuities. Comput. Chem. Eng. 26 (2002) p. 1449 Shacham M. Brauner N. Cutlip M. B. A Web-based Library for Testing Performance of Numerical Software for Solving Nonlinear Algebraic Equations. Comput. Chem. Eng. 26 (2002) p. 547 Schnepper C. A. Studthm M. A. Application of Parallel Interval Newton/Generalized Bisection Algorithm to Equation-Based Chemical Process Flowsheeting. Interval Comput. 4 (1993) p. 40 Sundar S. Bhagavan B. K. Prasad S. Newtonpreconditioned Krylov Subspace Solvers for System of Nonlinear Equations: a Numerical Experiment. Appl. Math. Lett. 14 (2001) p. 195 Wayburn T. L. Seaderj. D. Homotopy Continuation Methods for Computer-Aided Process Design. Comput. Chem. Eng. 11 (1987) P. 7 Wilhelm C.E. Swanq R. E. Robust Solution of Algebraic Process Modeling Equations. Comput. Chem. Eng 18 (1994) p. 511

Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

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2.1 Introduction

Chemical and biotechnical processes are often described by distributed dynamic models, that is, partial differential equations (PDEs) or partial differential algebraic equations (PDAEs) incorporating convection, diffusion, reaction and/or thermodynamic property terms. The PDAE models represent temporal as well as spatial variation of state variables. Since analytical solutions only exist in few cases, due to nonlinearity and complexity, computational methods (or numerical analyses) are generally required to solve such distributed dynamic models. In this chapter numerical methods for solving PDEs are reviewed in the following three sections, first treating semidiscretized (method of lines) and fully discretized methods before discussing adaptive and moving mesh methods. Several applications of distributed models appearing in preparative chromatography, futed-bed reactors, slurry bubble columns, crystallizers and microbial cultivation processes are treated in section 2.6 as a means to introduce various relevant aspects for the solution of PDE/PDAE models for chemical and biotechnical processes. Finally in section 2.7 an approach for combining computational fluid dynamic (CFD) technology with process simulation is illustrated and discussed. 2.2 Partial Differential Equations

Chemical and biotechnical processes often take place in spatially distributed systems and are therefore most appropriately described by distributed dynamic models, that is, partial differential equations (PDEs)incorporating convection, diffusion, reaction and thermodynamic property terms (Heydweiller et al. 1977; Kohler et al. 2001). For example, the material, energy, and momentum balances on moving fluid phases result in PDEs with respect to time and one or more space dimensions. Partial time derivatives occur as a direct consequence of the transient operation, while convective Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 200G WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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2 Distributed Dynamic Models and Computational Fluid Dynamics

and diffusive (or dispersive) effects normally lead to first and second order partial space derivatives, respectively. Material and energy balances on stationary phases (e.g., the solid adsorbent in a packed-bed adsorberlreactor) may not involve any convective or diffusive terms and are therefore free of partial space derivatives.The properties of such a stationary phase at any single point obey ordinary differential equations (ODES).Algebraic equations (AEs) are often used to define chemical equilibria, physical properties (e.g., enthalpy in terms of temperature, pressure and composition), or other intermediate quantities appearing in the differential equations. Therefore, physical models are generally expressed as PDEs coupled with AEs, i.e., socalled partial differential algebraic equations (PDAEs) submitted to initial conditions and boundary conditions. For the purpose of this review, a PDAE system with one spatial coordinate can be expressed as follows:

where u(t, x ) is the state variable as a firnction of time (toIt Itf) and space (xo I x 5 xf),F(u)is the convection flux, D is the diffusion (or dispersion) coefficient, r(u, 8) is the reaction rate equation depending on state variables (u) and parameters (8), and g(u) is a nonlinear algebraic equation. On the right-hand side of Eq. (la), the first, second and third terms take into account convection, diffusion and reaction, respectively. The partial differential equations govern a family of solutions. A particular member of the family of solutions is specified by the auxiliary conditions like initial and boundary conditions. For a PDE containing a first-order time derivative, one initial condition (IC) is required at an initial time level, t = toalong the space ( x ) : U(t0,

x ) = uo

(2)

For a PDE containing a second-order spatial derivative like Eq. (la), two boundary conditions are required at the physical boundaries of the solution domain. For example, the well-known Danckwert’s boundary condition (BC) can be imposed for Eq. (la):

=0

at x = xf for all t

where f(u) 11 in is the inlet flux predescribed by the operating condition. In the literature, Eq. (3b) is called the Neumann BC and Eq. (3a)is a mixture of the Dirichlet BC and the Neumann BC. Proper specification of auxiliary conditions is a necessary condition to obtain a well-posed problem (Hoffman 1993).

2.2 Partial Differential Equations

Physical mathematical models like the partial differential equations (PDEs)have a continuous form, while theose for solution purposes have to be discretized into a semidiscrete form (e.g.,using only spatial discretization, Ax) or a fully discrete form (e.g., combining temporal and spatial discretization, At and Ax) in order to represent the models in the temporal and spatial (or computational) domain. Among the large number of numerical methods developed for the solution of PDE or PDAE systems, the following is a well-established classification: 0

0

0

Method of lines (MOL),including finite difference methods, finite element methods and finite volume methods (Finlayson 1980; Schiesser 1991; Leveque 1998; Lim et al. 2001a, Mantzaris et al. 2001a and 2001b). Fully discretized methods (Hoffmann 1993; Chang 1995 and 2002; Lim et al. 2004). Adaptive mesh refinement (AMR) or adaptive grid methods (Berger and Oliger 1984; Berger and LeVeque 1998; Vande Wouwer et al. 1998). Moving grid methods (Miller and Miller 1981; Dorfi and Drury 1987; Huang and Russell 1997; Li and Petzold 1997; Lim et al. 2001b).

The semidiscretized method is called MOL, where PDEs (or PDAEs) are converted into a system of ODES (or DAEs) with respect to time by spatial discretization (see Section 2.3 for details). The main advantage is that well-established time integrators, e.g., Runge-Kutta or backward differentiation formula (BDF) methods, can be used for solving a large set of ODES or DAEs. A main drawback is, however, that it is difficult to control and estimate the impact of the spatial discretization error (Oh 1995). For fully discretized methods (Section 2.4), a system of nonlinear algebraic equations is obtained after temporal and spatial discretization. Adaptive and moving grid methods seem to be most promising since the idea is to use a numerical method in which nodes are automatically positioned in order to follow or anticipate steep moving fronts (Section 2.5). The node positioning may be achieved by using two basic strategies, namely AMR (i.e., local mesh refinement) and moving mesh methods (i.e., continuous spatial redistribution of a futed number of mesh points). These two types of methods are appropriate for solving PDEs in the presence of steep moving fronts or discontinuities. One of the key challenges facing process modeling today is the need to describe the interactions between fluid flow and phenomena models such as chemical reactions, mass transfer and phase equilibrium (Bezzo et al. 2003). Process simulations taking convection, diffusion and reaction into account and using computational fluid dynamics (CFD)for fluid hydrodynamics are important tools for the design and optimization of chemical and biochemical processes. The two technologies are largely complementary, each being able to capture and analyze some of the important process characteristics (Bezzo et al. 2000). Their combined application can therefore lead to significant modeling and simulation benefits, as will be discussed in Section 2.7. Before proceeding with this review, several preliminary concepts are summarized to facilitate presentation of the numerical methods.

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2.2.1 ODE (or DAE) Integration

In the MOL framework, the numerical solution of PDEs (or PDAEs) is obtained by the time integration of the ODEs (or DAEs) resulting from spatial discretization. The general form of ODEs is expressed

(4)

M(t)u = h(t, U )

where u is the time derivative. When the matrix M(t) is singular, Eq. (4)represents a DAE system rather than an ODE system. Solving the DAE system is more complicated than solving the ODE system (see Section 2.2) because the DAE system only has a solution if the initial conditions ~0 are consistent in the sense that the equation M(to) uo = h(b,ug) has a solution, &, for the initial slope. Computations in a DAE integrator does not require M(t) to be nonsingular (Ascher and Petzold 1998). If the time dependent ODE has a condition number that is large, then the problem

a

is stiff. In other words, system (4)is stiff if the Jacobian J = - au (in the neighbor-

I

h

IL >>1. For the stiff ODE/DAE hood of the solution) has eigenvalues A,, where 3 lilminl

systems, implicit BDF time integrators, such as DASSL (Petzold 1983), LSODI (Hindmarsch 1980), and DISCO (Sargousse et al. 1999) and o d e l b in Matlab (The Mathworks Inc., MA, USA) are used for accurate evaluation of time derivatives.

2.2.2 Accuracy and Computational Performance

How close the numerical solution is to the true solution (or analytical solution, if that exists) at the finite temporal and spatial stepsizes (At and A x ) is assessed by evaluating the accuracy of the discretization. As At and A x converge to zero and as the approximation order of derivatives increases, the approximation error generally diminishes. However, one must also account for computational efficiency in terms of the computational time that increases as the accuracy rises. In this context, there is a tradeoff between accuracy and computational efficiency.Thus, to simultaneously minimize the approximation error and the computational time, it can be considered as a multiobjective problem (Lim et al. 2001a). The set of AEs or ODEs obtained after discretization of PDEs differs from the original PDE by the presence of the truncation error terms, which implicitly contribute to the numerical difision or dissipation. The truncation error related to accuracy is always present in the finite approximation of a PDE. An appropriate numerical method should be selected for a given PDE system in order to meet a tolerable numerical error within a reasonable computational time.

2.2 Partial Diferential Equations

2.2.3 Automatic Differentiation

The numerical methods employed in the solution of many scientific computing problems require the computation of derivatives of some functions. Probably the best known are the gradient methods for optimization (e.g., successive quadratic programming, Powell 1971; see also Section 2.4), Newton’s method for the solution of nonlinear algebraic equations (see Section 2.1), and the numerical solution of ODEs and PDEs. Both the accuracy and computational requirements of the derivative computation are usually of critical importance for the robustness and speed of the numerical solution (Bischof et al. 1992). Taking a system of ODEs converted from a partial differential equation with MOL (see Section 2.1) as an example, the system is given as: du = h(t, AX, U )

(5)

dt

where the state variable u = [u1 ... u N and the nonlinear function h = [h, ... hNare represented (or approximated) on N discrete spatial mesh points (or finite elements). ODE solution methods, such as implicit Runge-Kutta and BDF methods, require a

(3

( N x N) Jacobian - , which is either provided by the user or approximated by a

difference quotient also called divided differences. In fully discretized methods (e.g., the conservation element and solution element (CE/SE) method, see Section 2.2) for the numerical solution of a PDAE, a nonlinear system is obtained as a function of time, spatial stepsizes, and state variables. 0 = h(At, AX, U)

(6) For a futed time and spatial stepsizes, Eq. (6) is solved by a Newton-type iteration dh requiring the Jacobian -. Therefore, the computation of derivatives (or JacodU

bian) is a crucial ingredient in the numerical solution of PDEs or PDAEs. Hand-coding is increasingly difficult and error prone, especially as the problem complexity increases. Numerical approximation by divided differences has the advantage that the function is only needed as a black box. For example, a central divide difference is expressed as:

The main drawback of divided differences is that their accuracy is difficult to assess. In addition, they are computationally expensive. The basic idea of automatic differentiation (AD) is to avoid not only numerical approximations, which are expensive and contain rounding errors, but also handcoded differentiation, which is error prone. Automatic differentiation techniques rely on the fact that every function, no matter how complicated, is evaluated on a com-

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puter as a sequence of elementary operations such as additions, multiplications and elementary functions (Bischof and Hovland 1991). By applying the chain rule

over and over again to the composition of those elementary operations, one can compute derivative information of h(u)exactly and in a completely mechanical fashion. Several AD packages such as Automatic Differentiation in FORTRAN (ADIFOR) (Bischof et al. 1998) are available from the AutoDiff organization Web site (http:/ /www.autodiff.org/)). 2.2.4 Fixed, Adaptive and Moving Grids

The numerical study of evolutionary PDEs with steep moving fronts has demonstrated the need for numerical solution procedures with time and space adaptation. Over recent years, a great deal of interest has developed in adaptive mesh methods (Vande Wouwer et al. 1998).The objective of such approaches is to obtain solutions as accurately as could be obtained if a fine mesh was used over the entire physical domain, but at significantly lower computing cost. One would normally like to concentrate a large proportion of the nodes in regions where the solution exhibits rapid variation with respect to space. In the solution of many chemical engineering problems, steep moving profiles also appear. Common examples are (1) concentration breakthrough curves in fixed-bed absorbers (Kaczmarski et al. 1997), (2) particle (or crystal) size distribution governed by a population balance equation (Kumar and Ramkrishna 1997),and (3) heat conduction problems with a phase change (Mackenzie and Robertson 2000). The futed grid method uses the constant spatial mesh size (Ax) during time integration, whereas the moving mesh method continuously moves a fued number of nodes to the regions of rapid solution variation over time. In the adaptive grid method (or AMR),meshes are locally added or removed at certain time levels according to solution steepness. In Section 2.3 adaptive and moving mesh methods are reviewed and compared. 2.3 Method of Lines

Time-dependent PDEs can be solved by means of the following two-stage procedures. First, the spatial variables are discretized on a selected spatial mesh so as to convert the original PDEs (or PDAEs) into a system of ODES (or DAEs) with time as an independent variable. Secondly, the discretization in time of the ODE/DAE system then yields the required fully discretized scheme, normally using an ODE/DAE solver. This two-stage approach is often referred to as the method of lines (MOL) in the literature.

2.3 Method of Lines

The spatial discretization means that the physical spatial domain (x E Rd, in d dimensions) is discretized, replacing the analytical domain by its discrete equivalent domain (computational domain, 5 E Rd) satisfylng the original PDE in a finite number of discrete points distributed over the physical domain. The discretization of PDEs on spatial domains normally leads to a Jacobian matrix whose elements lie within a narrow band (band Jacobian matrix). But, the bounded structure will be destroyed by the equations resulting from the boundary conditions, recycle streams or other nonlinear features. Hence, a sparse matrix would often be seen. In the ODE/DAE solver, the user will define the appropriate type of the Jacobian matrix to be evaluated by numerical difference, user-provided code or automatic differentiation, as discussed above. The discretization techniques are important since not satisfying local conservation equations will give meaningless results. The numerical scheme has to closely mimic the behavior of the original PDEs and guarantee local conservation of flow properties. To achieve this, it is necessary to not only use conservative formulation of the governing equations but also a conservative numerical scheme. The discretization of the spatial derivatives in Eq. (1) can be accomplished using three main categories: the finite difference method (FDM),the finite volume method (FVM) and the finite element method (FEM). The grid system may be a fixed grid, an adaptive grid or a moving grid. 2.3.1 Finite Difference Methods

The finite difference approximation of Eq. (la) can be expressed in a simple way on N mesh points as follows:

where the first-order spatial derivative is approximated by a first-order upwinding

af 2 0 (or positive convective flow) and the secondscheme under the condition I 3U order derivative by a central difference scheme. For i= 1 and N, boundary conditions such as Eq. (3) are applied. The spatial discretization of the parabolic PDE (1) may cause stiffness due to second-order spatial derivatives, while that of the convection-dominated PDE (i.e., large convection velocity relative to diffusion coefficient) may cause instability, which is associated with oscillatory behavior of their solution due to first-order spatial derivatives (Finlayson 1980).The instability is encountered in using central schemes and higher-order upwinding schemes. To improve the accuracy of the FDMs for the PDEs, numerous attempts have been focused on approximating the first-order spatial derivatives, x i n Eq. (la). Some guidance is provided in the selection of ax

upwind methods in the FDM solution (Saucez et al. 2001).

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In the traditional finite difference discretization (i.e., the fixed-stencil approach to be introduced later), the stencil (Si) used to approximate spatial derivatives is fixed in both size (number of grid points) and position of the stencil points over the discretization procedure. Fixed-stencil (FS) approximations may not be adequate near discontinuities or steep fronts where they may give rise to oscdlations. These problems have motivated the idea of an adaptive stencil (AS) (Shu and Osher 1989) and a weighted stencil (WS) (Jiangand Shu 1996). Namely, the left stencil shift (r) changes with the location xi (see Fig. 2.1), but retaining the total number of points in the stencil. We consider the cell-centered grid rather than vertex-centered grid, as shown in cell centers (xi), and stencils (Si) in one spatial dimension are Fig. 2.1. Cells (Ci), defined by Ci

= [xi-1/2,

+ xi+1/2),

xi = 0 . 5 ( x i - 1 / 2 Si[Ci-r,

(10)

xi+1/21,

. ., Ci, C i + l , . . . , Ci+sl =

Ci-r+l,.

Si[xi-r-1/2,

(11)

~i-r+1/2,

. . ., x i - 1 / 2 , 3 ~ i + 1 / 2 ,

. . .,~ i + s - 1 / 2 ,

(12)

xj+,+l/Z]

where rand s denote the left and right stencil shifts, respectively. The approximation order (k)is defined as: k-l=r+s Consequently, FS approximations can be classified according to the left stencil shift (r) at the given @-order accuracy (see Table 2.1). For the AS methods (e.g., essentially nonoscillatory (ENO) schemes, Shu and Osher 1989), the left stencil shift (r) changes with locations (xi) in order to avoid including a discontinuous (or steep front) cell (Ci)if possible. Just one stencil is selected out of some candidate stencils changed by r when doing the reconstruction, retaining the same order of accuracy.

i

Figure 2.1

;

I I

I

I'

I

fl-

i r=3 , s=3 -v '1 r

-

I

1

Right stencil shift (s)

Left stencil shift (r)

I

,-,'ci. ci+,.c

Si=[C,-,,Ck2,c

Stencil (5,) and cell (C,) structures in one dimensional problems

2.3 Method of Lines

As a result, there are no oscillations and peaks are sharpened. The weighted stencil (Jiang and Shu 199G), however, uses all candidate stencils, each being assigned a nonlinear weight that depends on the local smoothness of the numerical solution. For convective conservation laws, the one-dimensional hyperbolic PDE is expressed by, Ut

= -A

(14)

where the subscripts t and x indicate temporal and spatial partial derivatives (du/dt and df/dx), respectively. If a function h(x) satisfies at a discrete point xi,

then, its derivative with respect to x (i.e.,

fx) can be expressed as follows:

Therefore, if a numerical flux f;+1/2 approximates h ( ~ , + to ~ /a ~kth-order ) accuracy, the convection term can be discretized into k"-order accurate conservative forms:

where fi+llz and are numerical upflux and downflux, respectively, and the uniform mesh size (Ax)is used. Note that in the spatial direction, Eq. (17)can be considered as the finite volume discretization (see Section 2.3.3). The two numerical fluxes are exactly symmetrical with the one mesh distance. So, we only define the numerical upflux, in this text. 2.3.1.1 Fixed-Stencil Approach

In the FS approach, the stencil ( Si) is fixed both in number and position. The numerical flux is approximated in a conservative manner from the flux point values with constants to meet k*-order accuracy:

The constants ci are shown in Table 2.1 only for r 2 0 (Shu 1997). Note that the constants ci are obtained from the derivative of the Newtonian interpolation polynomial (see Eq. (25) for details). For instance, FS-upwind-1stands for a method of the futed stencil with the first-order accuracy (k = 1) in the upwind direction. When the convection velocity is 2 2 0 in Eq. (14), its numerical upfluxldownflux are given by dU

Eq. (18) and Table 2.1:

I

43

44

I

2 Distributed Dynamic Models and Computational Fluid Dynamics Table 2.1 Accuracy order

The constant ci up to fifth-order accuracy ( r 2 0)

Left stencil

(4 (3

j=i

=O

j=z

j=3

0

2

0 1

1I2 -112

112 312

0

113 -116 113

516 516 -716

-116 113 1116

2 3

114 -1112 1/12 -114

13/12 7/12 -5112 13/12

-5112 7/12 13/12 -23112

1/12 -1112 114 25/12

0 1 2 3 4

115 -1120 1/30 -1120 115

77/60 9/20 -13160 17/60 -21120

-43160 47/60 47/60 -43160 137160

17/60 -13160 9/20 77/60 -163160

4

5

1 2 0 1

fx= (J +AI x)

Reference name in this section

FS-upwind-1

1

3

j=4

I1

FS-central-2 FS-back-Z(TPB) FS-upwind-3

FS-central-4

-1112 1/30 -1120 115 137160

FS-upwind-5

af

for - > 0 au -

which was also introduced in Eq. (9). When the convection velocity is a f <0, Eq. dU (20a) is modified symmetrically in the opposite direction:

The first-order upwind scheme (FS-upwind-1)in Eq. (20)gives a very stable solution while poor accuracy because of low accuracy (k = 1).When k = 2 and r = 0, the FScentral-2 is obtained -J-1) fx= (J+l 2A x

which is called the second-order central scheme. The FS-back-2 is equivalent to the three-point backward (TPB) method (Wu et al. 1990). In approximating the firstorder spatial derivative, the central difference formulas (e.g., FS-central-2 and FScentral-4) tend to induce phase errors that appear in the form of numerical oscillations, as mentioned earlier. Higher-order upwinding schemes (e.g., FS-upwind-3

2.3 Method ofLines

I

45

and FS-upwind-3in Table 2.2) cannot remove the numerical oscillatory behavior in steep regions (Lim et al. 2001a). 2.3.1.2 Adaptive Stencil Approach

Finite difference E N 0 schemes were developed by Harten et al. (1987). They employed adaptive stencils in order to obtain information of solution gradients from smooth regions near discontinuities. This provides a sharp, E N 0 shock transition coupled with a formal uniformly high-order accuracy in smooth regions. Shu and Osher (1988, 1989) have proposed an efficient implementation of E N 0 schemes on the basis of fluxes rather than cell averages. The numerical are evaluated using high-order interpolating polynomials constructed from adaptive stencils in the upwind direction. The primitive of h(x)from Eq. (15), N ( x ) ,can be defined by:

Once H ( x ) is approximated by a k*-order Newtonian interpolation polynomial P(x), using central divided differences (DD) at the k + 1 points, k

j-1

,

We can obtain the numerical flu J+1/2

=

through the derivative of the above equation.

dP(xi+1/2)

dx

The O* degree divided differences (DDdi('))are defined by: (0)

DDi

=J

(26)

[~i-1/2.x i + l / ~ I

and in general the kth degree divided differences, for k . DD(k)tx,-r-1/2,

~i-r+1/2,.

DD'"''[~i-~+lp

. ., xi-r+j-1/2,

~i-r+j+1/21

2

1, are defined by:

=

. . . x;-,+j+1/2] - DD(k-1)[~i-r-1/2 . . . ~ i - ~ + j - ~ p ] Xi-r+j+1/2

- xi-r-1/2

(27)

46

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2 Distributed Dynamic Models and Computational Fluid Dynamics

For first-, second-, and third-degree divided differences, DD1)i+l/l,DD!2)and DDJ)i+lj2 are defined, respectively, as flux point values (6):

In the case of a fmed left stencil shift (r),Eq. (25)becomes equal to Eq. (18).Note that the constants ci (see Table 2.1) are obtained from Eq. (25), when the accuracy order (k) and the stencil shift (r) are given. In the AS approach, the stencil shift (r) is adaptively chosen in Eq. (25). Since a smaller I D D ~ ~ implies that the function is smoother in that stencil, a smaller one (i.e., r) is chosen through the comparison of two relevant divided differences (e.g., IDDil and IDDi-ll). The E N 0 schemes are nonlinear even for linear problems and are especially suitable for problems containing both shocks and complicated smooth flow structures (Shu and Osher 1989). The E N 0 schemes also have some drawbacks. One problem is the freely adaptive stencil, which could change by a rounding error perturbation near zeroes of the solution and its derivatives. Also, this free adaptation of stencils is not necessary in regions where the solution is smooth. Another problem is that the E N 0 schemes are not cost-effective because the E N 0 stencil selection procedure involves many logical statements (i.e., if/then statements). 2.3.1.3 Weighted Stencil Approach

The weighted stencil (WS)method (i.e., WENO scheme) is an approach used to overcome the aforementioned drawbacks while keeping the robustness and high-order accuracy of E N 0 schemes. The idea of WENO is to use a convex combination of all candidate stencils instead of approximating the numerical flux by using only one of the candidate stencils (Jiang and Shu 1996). For the third order WENO scheme in the upwind sense, two candidate stencils are used to define the numerical upflux That is, based on the FS approximation at k = 2 (see Table 2.1), two numerical fluxes (i.e., qo and ql) from the two stencils (Si and Si+lfor r = 0) are incorporated with the weighting:

where, wo =

a0

ffo + f f l ~

and w1 =

ff1

ffo + f f l ~

.

2.3 Method of Lines

qr is obtained as in the FS approach of Eq. (18) for k = 2 in Table 2.1:

C 1

40

=

Qjfi+j= COO&

j=O

+ ~ l f i + =l (J; + ~ + 1 ) / 2

1

41 =

C

Cljj+j-1

I

= c10J-l

+ SIJ

= (-J-I

+ 3J)/2

j=O

The question now is how to define the weighting parameters (aoand al), such that the E N 0 property is achieved. The weighting parameters are calculated by divided from Eq. (28): differences (DDi)i+112 213

(334

1/3

where E is a small positive real number that is introduced to avoid the denominator becoming zero (often, E = lo-' lo-"'). It is suggested in Jiang and Shu (1996) that dJ+l2 . is the power p = 2 is adequate to obtain E N 0 approximations. If a flow speed 1

-

dX

negative, the numerical flux is defined in the reverse order as fi, fi-l). Thus, the WENO scheme is a type of upwinding scheme. The WENO scheme has the following properties: (1)it involves no logical statements, which do appear, however, in the basic E N 0 schemes, (2) the WENO scheme based on the (k - 1)* order EN0 scheme is a k* order approximation in smooth regions, (3) it achieves the E N 0 property by emulating EN0 schemes at discontinuis a smooth function ities, (4)it is smooth in the sense that the numerical flux fi+1/2 and (5) the WS method combines the FS method with the AS method. Hence, the WENO scheme (Jiang and Shu 1996) improves on the EN0 scheme in robustness, smoothness of fluxes, convergence properties and computational efficiency (Shi et al. 2000). 2.3.1.4 Comparison of FS, AS and WS Approaches

Table 2.2 displays the formulation of 12 spatial discretization methods. The notation

X-Y-k is used, where X stands for stencil type (FS, AS or WS), Y indicates the high-

lighted characteristics (upwind, central or backward), and k is the approximation (or accuracy) order. Table 2.3 shows how the stencil structure (i.e., mesh points used for the numerical upflux) changes the position of a shock. In the FS and WS approaches, the position and number of the mesh points do not vary with the shock position. However, a stencil xi, xi+l] 0fj+lj2for the WS scheme is composed of two substencils Si-1/2[x,-l,

47

If-&)/2 h (3f;-+f;., +f;r)/2 h (2f;+, + 3f;- Gf; +f;-2)/6 h

(-f;+2+ 8f1 - 8 x 1 +X2)/12 h

FS-central-2 FS-back-2 FS-upwnd-3 FS-central-4

AX DD&:

=

&+I

Name

Numerical flux

Adaptive stencil (AS)

L ( x), refer to Eq. (28). 2 - x,

+ 15J-2 -

first order divided differences, e.g., DD!&

(- 3f;+z + 30J+l + 20J 2f;-3)/60 h

v;-f;-l)/h*

FS-upwnd-l

FS-upwind-5

Spatial discretization (fJ

Name

Fixed stencil (FS)

Classification of the flexible stencil methods (FS, AS and WS) in the upwinding sense, n the flow speed is positive

le 2.2

Numerical flux &+,,')

-

Name

-

Weighted stencil (WS)

2.3 Method of Lines Table 2.3 a

Stencil structures o f third order FS/AS/WS methods for numerical upflux in a positive flow velocity FS (FS-upwind-3)

shock Right

t

n

t" XI1

*

XI1

+

x,,

x,

X,+,

l

AS (AS-upwind-3)

WS (WS-upwind-3)

XI*?

~

XI1

~

XI1

x,

I

XI

x,+,

X,+?

XI1

xi2

xi,

x, x*,

X,+i

x] and Si+l12[x,xi+l] weighted with respect to the magnitude of neighboring divide differences (DD). In the third-order AS scheme in Table 2.3, the position of mesh points for the numerical upflux shifts to avoid a cell involving the shock. That is, the stencil adapts to solution variations. To compare numerical performance for the three approaches, a linear equation of the conservation law is tested with an initial condition (uo) of various wave forms:

+ u,

= 0, -1.0 < x < 1.0,

(34)

u(x, 0 ) = uo(x)

(35)

where the initial condition

u'=

I

(Jmax(1 - 1OO(x - 0.495)2,0)

I' 10

49

j--#fl~j-j--f-&

The arrows indicate the position of shocks.

Ut

I

fJmax(1 - 1OO(x - 0.505)2, 0 ) +4Jmax(l - 1OO(x - 0.5)2,0 )

0.4 5 x 5 0.6;

otherwise

50

I

2 Distributed Dynamic Models and Computational Fluid Dynamics

contains a smooth but narrow combination of Gaussians (e-"(x-m2),a square wave, a sharp triangle wave and a half-ellipse (Jiang and Shu 1996). Since this PDE only has a linear convective term, its analytic solution shows the same shape as the initial condition, i.e., u(x, t) = uo(x- t). Using a BDF ODE integrator (DISCO, Sargousse et al. 1999), the ODE system is solved on a PC. The band Jacobian matrix is numerically evaluated. To check accuracy at a given time level, the L1-erroris measured: L1-error =

(U(x)analytical- U(x)numerical

sx=l x=-1

I dx

(36)

Table 2.4 shows the benchmarking results achieved from the 12 discretization methods on uniform 200-mesh points ((Ax = 21200). The L,-error is measured at t = 0.4 s and the computational time required is for time integration of 0.4 s. Instability is indicated by spurious oscillatory behavior in the numerical solutions. Figure 2.2 depicts numerical performance within L1-error vs. CPU time spaces from the data of Table 2.4. In general, as the approximation order increases, the error decreases. However, the fourth-order central discretization method (FS-central-4) produces much error over the FS-upwind-3 because of strong oscillations near the shock. Minimizing both the L1-errorand the computational time simultaneously, the six methods (FS-upwind-1,AS-upwind-314, and WS-upwind-3/4/5) are selected as the effective methods with consideration of the stability of the numerical solution (Lim et al. 2001a). It is found that the AS-upwind-2takes an abnormally long computation time due to excessive iterations required for convergence. In Fig. 2.3, the numerical solutions of FS-upwind-1(shortest computational time) and WS-upwindd (smallest L1-error)are compared to the FS-upwind-5.The numerical results of FS-upwind-1are stable but not accurate due to the truncation error. The Table 2.4 Accuracy, computational performance, and stability evaluation of the FDM in a linear convection equation ( N = 200, L,-error at t = 0.4 s and CPU time during 0.4 s integration time) Stencil Type

Name

Accuracy ( b enor)

Computational performance (CPU time, s)

FS

FS-upwind-1 F S-central-2 FS-back-2 FS-upwind-3 FS-central-4 FS-upwind-5

0.2696 0.1878 0.1345 0.0538 0.1219 0.0379

0.8

AS-upwind-2 AS-upwind-3 AS-upwind-4

0.0961 0.0548

9.1 4.6 4.8

0 0 0

WS-upwind-3 WS-upwind4 WS-upwind-5

0.0841

3.5 4.9 5.7

0

AS

ws *

0.0440

0.0452 0.0421

Stability evaluation: 0 (stable), X (not stable)

Stability*

1.1 1.1 1.1 1.3 1.4

0 0

-4

10 -

-(l)FS

8-

(2) AS -9-(3)WS

-

h

J

-A-

6-

.a 2 48

* 2

07

-Analytic solution 0 FS-upwind-1 -4 FS-upwind-5 0 WS-upwind5

Figure 2.2 Error and computational time comparison of FS, AS and WS methods for the linear convection law (each curve follows the point from right to left). (1) FS: FS-upwind-1 + FS-central-2 + FS-back-2+ FS-upwind-3 + FS-central-4 + FS-upwind-5, (2) AS: AS-upwind-2 + AS-upwind-3 + AS-upwind-4, (3) WS: WS-upwind-3 + WS-upwind-4 + WSupwind5

n -0.2 Axial direction (x)

Figure 2.3 Numerical solutions of a linear convection equation according t o discretization methods on 200 fixed-grid points at t = 0.4 s and -1

.o 5 x 5 1.o

FS-upwind-5 method yields a stable solution in smooth regions, but produces some oscillations near discontinuities. One of the WS approaches, WS-upwind-5,is stable and accurate over all regions but computationally somewhat prohibitive (see Table 2.4).

2.3.2 Finite Element Methods

The finite element method (FEM) divides the physical domain into many smaller subdomains (elements) and applies weighted residual methods within each element. Each physical variable over the entire domain is expressed as a sum of finite elements. Additional restrictions are introduced to ensure various degrees of continuity

52

I

2 Distributed Dynamic Models and Computational Fluid Dynamics

of the solution at the element boundaries. In principle, any weighted residual methods can be combined with the finite element concept to yield a corresponding finite element method. In practice, the most commonly used method is the orthogonal collocation on finite elements (Finlayson 1980). 2.3.2.1 Orthogonal Collocation Method on Finite Elements

The method of orthogonal collocations on finite elements was presented in Villadsen and Michelsen (1978).An orthogonal collocation method approximates the solution by weighted combinations of orthogonal polynomials of degree M, and demands that the describing equations be satisfied exactly at a finite set of points called collocation points, which are the zeroes of an orthogonal polynomial. Table 2.5 lists the normalized collocationpoints for the orthogonal Legendre polynomials of degree of 2,3 and 4 (Finlayson 1980). Table 2.5 Normalized collocation points for orthogonal Legendre polynomials of degree 2, 3 and 4 Degree of polynomial Collocation points

0, 0.5, 1 0, 0.21132, 0.78868, 1 0, 0.1127, 0.5, 0.8873, 1

2 3 4

In many areas, such as reaction engineering, the orthogonal collocation method has proved to be a powerful method leading to accurate results. However, when the solution has steep gradients, it is more beneficial to use it in conjunction with a finite element approach. As shown in Fig. 2.4, the physical domain is divided into a number of elements and an orthogonal collocation method is applied in each element. This gives rise to the orthogonal collocation method on finite elements. The position of thefh point in element i is denoted by xp The approximated solution G ( x ) in the element i can be given by:

;(xi)

M

zz

U(~y)Ly(g), i = 1.. . N

(37)

j=O

-

--

--

--

2.3 Method of Lines

92 - sk is the Lagrange interpolation polynomial of degree M, where ~ ~ ( 3 GE0 2 ) J

k#j

xj

-

xk

N is the number of elements, and is the normalized position within the element i: - xi0

X"

osi="

Ax

(1 -

where Ax is the equidistant element length From Eq. (37), the first-order derivative of the approximated solution +I) k in element i becomes:

l M du(xik)x U(xij)A$, dx A x j=o

i = 1 . ..N ,k = 0 . . . M

at position

(39)

where AYk is a constant (M + 1) X (M + 1)matrix defined by dLy(92k)

AM

~

Jk

d32

, j , k = O ...M

From the definition of the Lagrange polynomial and the normalized collocation points in Table 2.5, the constants AMfor M = 2, 3 and 4 are evaluated:

[

A'=

-3.0 fl

-1.0

-7 8.19615 A =[ -2.19615 1

1 ;4]

(414

-2.73205 1.73205 1.73205 -0.73205

0.73205 -1.73205 -1.73205 2.73205

2.19615 -8.19615 7

-13 -5.32379 14.78831 3.87298 -2.66667 2.06559 1.87836 -1.29099 -1 0.67621

1.5 -3.22749 0 3.22749 -1.5

-0.67621 1.29099 -2.06559 -3.87298 5.32379

3

1 -1.87836 2.66667 -14.78831 13 I 1

(4V

The second-order derivative of the approximated solution a ( x ) at position k in element i can be obtained using a similar procedure: M

d2k(X;k) 1 x-cU(~jj)B$, dx2 Ax2 j=o

i=l

... N , k = O ... M

(42)

where BYk is a constant (M + 1) x (M + 1) matrix defined by BM Jk

d2LJM(?k) , j , k = O ... M d922 =

(43)

I

53

54

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2 Distributed Dynamic Models and Computational Fluid Dynamics

Using Eq. (37), the formula for the integral of ii(x) over any one of the subintervals Xi,k+lI can be derived as: Xi,k+l

M

G ( x ) dx x AX

C ii(~ij)C’,

i = 1 . .. N , k = 0.. . M - 1

j=O

l i k

where Cykis a constant (M + 1) x M matrix defined by:

Using the orthogonal collocation method on finite elements, Eq. (la) can consequently be approximated by:

It is interesting to note that the FDM and FEM presented here can be derived in a very similar fashion, namely by defining and manipulating interpolating polynomials over a finite set of points (see Eqs. (24) and (37)), despite their apparent differences. In both cases, spatial derivative approximations at a point xi (or integral approximations over an interval xi+llz])involve the values of the function at a set of neighboring points. The main difference between the two methods is in the composition of the set. For the FDM, this normally involves a fixed number of points (i.e., stencil shift in Section 2.3.1) to the left and right of the current point xi. For orthogonal collocation of the FEM, it involves all points within the element to which xi belongs (Oh 1995). 2.3.2.2 Continuity at Element Boundaries

An important facet of all finite element methods is the treatment of the boundaries between elements. In general, the solution values are assumed to be continuous at the element boundaries, and this normally corresponds to physical reality. One could also make the first spatial derivative continuous across the interface, thus resulting in continuous solution approximations throughout the domain. However, in some cases (e.g.,for inhomogeneous domains),it may be more appropriate to enforce continuity of some other quantity at the element boundaries, for instance, dispersive mass flux or conductive heat flu (Oh 1995). The continuity of the first derivative at each boundary can be written as:

and can be described in the computational domain as follows: M

M

j=O

j=O

C i i ( ~ q ) A g= C U(xi+lj)A$,

i = 1 . .. N - 1

2.3 Method of Lines

- --

-

--

In general, the discretization of a PDE and its associated boundary conditions using the orthogonal collocation method on finite elements results in three different dasses of relation being applied at three different types of points (see Fig. 2.5): 0

0

0

the appropriate boundary conditions are applied at the physical domain boundary such as Eq. (3) + 2 AEs (or 2 ODEs). the discretized PDE is enforced at the collocation points within each element + N(M - 1) ODES element boundary continuity is enforced at the boundaries between elements -+ (N- 1) ODES

Therefore, (NM + 1) DAEs (or ODEs) are obtained for one PDE. 2.3.3 Finite Volume Methods

For conservation laws, it is often preferable to use a finite volume method rather than a finite difference method in order to ensure that the numerical methods conserve the appropriate quantities of physical PDEs (Leveque 1998). Consider the numerical solution of time dependent one-dimensional conservation laws,

for (x,t ) ES2 = (xL,xR) x (0,r), where r(u) accounts for all considered source and sink terms. We consider that the domain S2 is partitioned into strips such that

where Ntimeis the number of time steps. Each strip is made up of two spatial grids in the case of nonuniform spatial grid, while in the fxed grid strips of control volume are rectangular as shown in Fig. 2.6.

I

55

56

I

2 Distributed Dynamic Models and Computational Fluid Dynamics Figure 2.6 Time and space control volume filled with dots (Q:) and its path line (any) used on a uniform fixed grid

X

The midpoints of spatial grids (xi+li2) are defined at t" simply as

The finite volume approximation of Eq. (49) over the control volume Q: is derived from the original integral expression of the conservation law

1 (e+ ""> at

ax

dx dt =

J, r( u) dx dt

Application of the divergence theorem based on Green's theorem' to Eq. (52) yields a line integral along the boundary, 8Qp.Performing the line integral on the left hand side of Eq. (52) yields:

is obtained. For the right hand side, the source term is simply approximated: ~ ( udxdt ) = T(u)AxiAt,

(54)

where Axi = xi+lj2- ~ i - ~ / ~ Atn a n d= t"+l- P. A n approximation of Eq. (53) is obtained using a numerical quadrature. A number of possibilities are available that give rise to either explicit or implicit methods. For example, using the midpoint formula to integrate along the bottom and top edges of Q y we get the approximations: 1) 1 Green's theorem: Let R be a closed region bounded by C in the xy-plane. Let P(x,y) and Q(x,y) be functions

defined and continuous first partial derivatives.Then

2.3 Method of Lines

For the right and left hand side edges, we use the following family of approximation:

p

Finally, the numerical results of the line integral yield: aay

[f(u) dt - udx] = axi (u:+~ -

.r>

+A;;;*

+ 0.5 (&2

-fi’lt1/2

-A:;;2)

(59)

For the explicit form of f(u), approximatingf”+1i+l/2 and fl+’i-1/2 to f ? + l p andf?-1,2, respectively, we can obtain a simple numerical form of the conservation law as follows,

which is a fully discrete formula for Eq. (49)where the convection term is discretized by the second-order central scheme (i.e., FS-central-2 in Section 2.3.1). Therefore, it seems that the explicit FVM for one-dimensional conservation law has almost the same formulation as the conservative FDM of Eq. (15). However, there are differences between the FVM and the FDM in accordance with the definition of the numerical f l w t e ~ f ? +In ~ ~Eq. ~ . (60),f?+l12is in fact an approximation to the average flux at x = along the finite volume Q::

and for the conservative FDM at t

= t,,

like Eq. (15):

Note that a complete FVM (namely the CE/SE method) in space and time domains is introduced in Section 2.4.3.

I

57

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2 Distributed Dynamic Models and Computational Fluid Dynamics

2.3.3.1 Spatial Finite Volume Method

Rather than attempting to discretize simultaneously in space and time, our attention is paid to discretization of spatial derivatives in the MOL using an adaptive time integrator, e.g., a BDF ODE solver. A naturally conservative spatial discretization procedure is provided by finite volume methods, where the discrete value viewed as an in Fig. 2.1, approximation to the average value of f(xi, P)over a cell Ci[xi-l,z, rather than as an approximation to a point-wise value of f(xi,t") (Leveque 1998).

where

The advantage of the semidiscretized approach is to achieve high accuracy in space. The cell average is simply the integration of f(x, t) over the cell divided by its area, so conservation can be maintained by updating this value based on fluxes through the cell edges. Although the derivation of such methods may be quite different from that of the conservative FDM,the resulting formulas are identical to Eq. (17). The flux function$+l,2 delivering high-order accuracy in space can be obtained by using higher-order interpolation polynomials like E N 0 schemes and WEN0 schemes (see Section 2.3.1).

2.4 Fully Discretized Method

The generic PDE with convection, diffusion and reaction terms can also be solved by temporal and spatial discretization of the original PDE. The time discretization procedure can be explicit or implicit. Several Mly discrete schemes are introduced in finite difference and finite volume approximations.

2.4.1 Explicit Time Discretization

In this section we consider a PDE with the flow velocity (a) and diffusivity (D) like in Eq. (la): ut = -au,

- Dux, - r(u)

(65)

The above equation is discretized by the forward-time methods. For example, the Leapfrog scheme can be expressed on equidistant At and A x as:

.;+" = UP ,-

where v

2.4 Fully Discretized Method

V

- (UP ,+I -

+P

u:-1)

-

2u:

+

+ Atr (ur)

(66)

a At = - is

called the Courant-Friedrichs-Lewy (CFL) number or convection Ax number and p = DAt is the diffusion number (Hoffman 1993). The partial time Ad derivative (u,) is approximated by a first-order forward difference and the partial space derivatives (u. and uxx)are approximated by a second-order central difference. Since the central scheme of the first-order spatial derivative is unconditionally unstable as mentioned in Section 2.3.1, an upwind scheme is given by the equation below when a 2 0:

The method is shown to be convergent but only conditionally stable. It introduces significant amounts of implicit numerical dissipation into the solution in the presence of steep fronts. The Lax-Wendroff scheme (1960)is a very popular explicit finite difference method for hyperbolic PDEs (i.e., D = 0 in Eq. (65)).To suppress numerical instability caused by the central discretization, an artificial diffusion term is introduced by a secondorder Taylor expansion in time:

ur'" x ur

1 + ( u t ) l A t + -(utt):At2 = u: 2

- a(ux)lAt

1 2 (u,,)lAt + -a 2

2

(68)

where the time derivatives ut and uttare determined directly from ut = -au,. Applying central discretization to Eq. (68),Lax-Wendroff scheme is given for Eq. (65):

+

(69) + u,?_,)+ Atr ( u r ) From a stability analysis, the method is stable only if1 ' 1 5 1. However, it is not V

un+" = UP - -

(UP 1+1 -

u:,)

V2

(uG1 - 24'

often used to solve convection-diffusion PDEs (Hoffman 1993). The Dufort-Frankel method (1953) proposed a modification of the Leapfrog scheme Eq. (GG), which yields a conditionally stable explicit method. In this modification, u? is replaced by the approximation u l =

+

At

U:fl

-

$1

2

(r(uf+l)

+ +-I))

Solving Eq. (70) for u:+' yields:

+

(1 2y)u;+l

=

-v

( U S " - U,P_")

+ (1

-

At +2 (r(uf+') + r@-1))

2y)uy-l

+2 p

-

(71)

I

59

60

I

2 Distributed Dynamic Models and Computational Fluid Dynamics

The scheme cannot be used for the first time step because of the term u7-l. The 5 1. However, large values of p result in inaccurate solumethod is stable only if tions and a starting method is required to obtain the solution of the first time step. The MacCormark method (1969)is based on the second-orderforward time Taylor series as Eq. (68).

(YI

u:+'

u:

1 + (ut):At + -(utt);At2 2

2:

u:

1

+ -2

((ut):

+ (u,):")

At

(72)

where (h); is obtained from the approximation (u)C= (-au? + D(u.)"&. The method is composed of a predictor and a corrector step. In the predictor step, ur+' is approximated by the first-order forward difference: ;;+I

= u: - v (u? 1+1 - ur)

+ p (u,?+'- 24' + u,?-,) + Atr(ur)

(73)

For the corrector step, u?' is given: l

=1

2

(.:+ q + 1 )

- "(;;+I

ui-')

- *n+l

+ ?p(":

- 26";

+ a;::) + Atr (uF)(74)

The two-sep method that shows second-orderaccuracy in both time and space is very popular for solving Eq. (65) and is conditionally stable. A numerical solution can be convergent only if its numerical domain of dependence contains the true domain of dependence of the PDE, at least in the limit as At and Ax go to zero. The necessary condition is called the CFL condition. All fully discrete explicit schemes require fulfillment of the CFL condition:

For stiff PDEs, implicit time discretization is usually preferred. The method of lines (see Section 2.3) using implicit time integrators is originally motivated to solve stiff PDEs, as mentioned above. In the next section, implicit methods fully discretized in both time and space are presented.

2.4.2 Implicit Time Discretization

The implicit Euler central difference method is: U!+1

p(

= UP - -

.?+I 1+1 - .?+I 1-1

) + p(uz:

Rearranging Eq. (76a)yields:

-(i v +

p ) u:

+ (1+ 2p)u:" +

-

2u7"

)::+7~

:':4.1

= ur

+ Atr (u:)

(76a)

+ Atr (ur)

(76b)

2.4 Fully Discretized Method

Eq. (76) cannot be solved explicitly for un+'i,because the two unknown neighboring ~ i ~ "++ ' ~i + also ~ appear in the equation. Due to the implicit feature, this values ~ ~ + and scheme, which has first-order accuracy in time and second-order accuracy in space, is unconditionally stable and convergent (Hoffman 1993). This implicit Euler method yields reasonable transient solutions for modest values of Y and p. The Crank-Nicolson central difference scheme is constructed by a second-order approximation in both time and space: .?+I

=

V un, - [(u::;'

-

- 2u;+l

+

At 2 [r(u;+l)

+

qy)+

(U>l

-

+ u;-+;) + (u;+l

.;-,)I

- 2u;

+ u;-l)]

(77)

I($)]

The Crank-Nicolson method is also unconditionally stable and convergent. The implicit nature of these methods yields a set of nonlinear algebraic equations, which must be solved simultaneously. Therefore, the iterative calculation requires substantial computational time, especially for multidimensional problems. Recently, an explicit fully discrete method called the CE/SE method has been developed as a finite volume approach to solve fluid dynamics problems. The CE/SE method enforces flux conservation in space and time, both locally and globally. The method is explicit and, therefore, computationallyefficient. Moreover, it is conceptually simple, easy to implement and readily extendable to higher dimensions. Despite its second-orderaccuracy in space, this method possesses low dispersion errors (Ayasoufi and Keith 2003).

2.4.3 Conservation Element/Solution Element Method

The CE/SE method has many nontraditional features, including a unified treatment of space and time, the introduction of conservation element (CE) and solution element (SE) and a novel shock capturing strategy without special techniques. Spacetime CE/SE methods have been used to obtain highly accurate numerical solutions for l D , 2D and 3D conservation laws involving shocks, boundary layers or contacting discontinuities (Chang 1995; Chang et al. 1999). The CFL number insensitive Scheme I1 (Chang 2002) has recently been proposed for the Euler equation (i.e., convection PDEs for mass, momentum and energy conservation). Stiff source term (e.g., a fast reaction) treatment for convection-reactionPDEs (Yu and Chang 1997) is also presented for the space-time CE/SE method. The extension to a PDAE system (Lim et al. 2004) derived from the original CE/SE method (Chang 1995) and Scheme I1 (Chang 2002) is presented in the following. Consider a PDE model like Eq. (65): Ut

=

-6 - p ( u )

(78)

I

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2 Distributed Dynamic Models and Computational Fluid Dynamics

where the flux (F, implying convection and difision is defined as

f = au - D U X

(79)

Thus, Eq. (78) is identical to Eq. (G5). By the divergence theorem the equation is equal to flux conservation as follows:

(80)

V.h=p

. By using Gauss's divergence theorem

(or Green's

theorem) in a space-time E,, it can be shown that Eq. (80) is the differential form of the integral conservation law:

where S (V) is the boundary of an arbitrary space-time region V in E2, and ds = d (a . n with d a and n,respectively, being the area and the outward normal vector of a surface element on S(V). Note that, because h . ds is the space-time flux of h leaving the region Vthrough the surface element ds, Eq. (81) simply states that the total space-time flux of h leaving Vthrough S (V) is equal to the integral of p over V. Also, since in E,, dais the length of a differential line segment on the simple closed curve S (V), the surface integral on the left-hand side of Eq. (81) can be converted into a line integral. In fact, Eq. (81)is equivalent to (Chang 1995):

i;?;

(-udz+fdt)

=

s,

pdv,

where the notation C.C. indicates that the line integral should be carried out in the counterclockwise direction. In Fig. 2.7, the mesh points (e.g., points A, C and E) are marked by circles. They are staggered in space-time. Any mesh point 0,n ) is associated with a solution element S E 0 , n) and two conservation elements CE-0, n) and CE+(I',n). By definition, SEO, n) is the interior of the shaded space-time region depicted in Fig. 2.7a. It includes a horizontal line segment, a vertical line segment, and their immediate neighborhood (Chang 1995).Also, by definition, (1) CE-(j, n) and CE+(j, n), respectively, are the rectangles ABCD and ADEF depicted in Fig. 2.7a and b; and (2) CE(j, n) is the union of CE_(j,n) and CE+(j, n),i.e., the rectangle BCEF. Let the coordinate of any mesh point (j,n) be (3,t") with xj = j A x and t" = nAt. Then, for any ( x , t ) E SE(j, n),u(x, t ) , f ( x , t ) and h(x, t ) , respectively, are approximated by a first-order Taylor expansion: .(xj, t " )

= uj" + (U.)j"(X

- Xj)

+ (ut)j"(t- t " )

(83)

2.4 Fully Discretized Method 1-112

1-1

I

1+1l2

l+1

Br

-

C

CE+W

Figure 2.7 Solution element (SE) and conservation element (CE) atPh position and nfhtime level (Chang 1995). (a) Space-time staggered grid near SE(j, n). (b) CE-(j, n) and CE+(j, n)

so that, q x j , t")

= (J'(Xj,

t"),

qzj, t")) .

(85)

Here u;, (ux);, (ut)T,J, (f")?and v;); are constants in SEG, n). In the CE/SE framework, (ut);,J, &); and, v;)) are considered as functions of (u); and (ux);.These functions will be defined as follows. According to Eq. (79),one has:

fJ " = QU; - D(U,);

(86)

Also, by neglecting the contribution from the second-order derivative, 6); may be obtained using the chain rule:

In order that (ut)? can be determined in terms of (ux);,it is assumed that for any (x, SEG, n),

t) E

v .q*j,

t") = 0

63

f-lF E

CEUA

I

(88)

Thus, within SE(j, n),the contribution of the source term (p) that appears in Eq. (80) is not modeled in Eq. (88).Note that (1)because it is the interior of a region that covers a horizontal line segment, a vertical segment and their immediate neighborhood, as shown in Fig. 2.7, SEO, n) is a space-time region with an infinitesimally small volume; and (2) as will be shown, the contribution of source terms will be modeled in a numerical analogue of Eq. (82). As a result, Eq. (88)implies:

Distributed Dynamic Models and Computational Fluid Dynamics

(J)Jn

=

(”In (y a. 25

- (f.);

. (f.); (&);

-a 2 (ux)jn

at

Note that, by using Eqs. (86), (87), (89) and (9O),J, V;)j”, (ut)j”and v;): can be determined explicitly in terms of uj” and (.&. 2.4.3.1 Iterative CE/SE Method

The approximated conservation flux,

Fn =

#

C.C.

S(CE(j,n))

in Eq. (82), is defined within CElj, n):

(-iidx+fdt)

With the aid of Eq. (83) and (84), the line integral in Eq. (91) results in:

The approximated source term flux (Pj”) is obtained within V(CE(j, n)) as:

?,‘f

~fv(p)jndV

The volume integral in Eq. (94) leads to: Ax

y=pjnl

d x l

At/2

dt=-

AxAt 2 Pj”

The numerical analogue of Eq. (82) becomes:

With the aid of Eqs. (92) and (95), Eq. (96) implies that:

Equation (97) is a nonlinear algebraic equation in terms of uy, which originates from a nonlinear source term (py), Since this system of equations should be solved iteratively (e.g., using a Newton’s iteration method), it is called the iterative CE/SE method, where Jacobian matricesf, and pu are required in Eq. (90) and (97). Here, A x and At are user-supplied parameters. How their values should be chosen is problem-dependent.A small spatial step size (Ax)should be chosen for a problem associated with steep moving fronts. Also, a small CFL number

2.4 fully Discretized Method

ferred for a problem that is stiff with respect to time. Note that the stability of a CE/ SE scheme requires that the CFL number IYI < 1 (Chang 1995), as mentioned for explicit schemes in Section 2.4.1. Without using special techniques that involve ad hoc parameters, the numerical dissipation associated with a CE/SE simulation with a fixed total marching time generally increases as the CFL number decreases. As a result, for a small CFL number (say J Y ~< O.l), a CE/SE scheme may become overly dissipative. To overcome this shortcoming, a new CFL number insensitive scheme, i.e., the so-called Scheme 11, was introduced in Chang (2002).The new scheme differs from other CE/SE schemes only in how (ux)Yis evaluated. Refer to Chang (2002) or Lim et al. (2004) for the detailed formulation. 2.4.3.2 Noniterative CE/SE Method

The noniterative CE/SE method is simply obtained from a first-order Taylor approximation of the source term (Molls and Molls 1998; Lim and Jarrgensen2004).

jP J =p"p"(X-xj)+p;(tJ xj

J

t")

(98)

Using the above equation, Eq. (95) is replaced by

where the time and space derivatives of source terms (p) are reformulated through the chain rule: p[

ZE

ap a u au at

- - =puU[

(100a)

(100b) With the aid of Eq. (99), Eq. (97) evaluated on CE(j, n) is replaced by:

At where w;;l)lzz= - (4p;;;%z+ AxpZ;'{h+ Atp$#$). (u,),!'is also evaluated by Scheme I1 8 proposed by Chang (2002). Thus, two unknowns (u5 u5) are obtained from four known values (u;!y: at the previous time level (tn-'I2).u; in Eq. (101) is obtained without nonlinear iteration procedure. This scheme is a noniterative CE/SE scheme, where Jacobian matrices fu and pu evaluated at the previous time level ( t = t"-'I2) are required in Eq. (101).

U;;Y,~)

I

65

66

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2 Distributed Dynamic Models and Computational Fluid Dynamics

2.4.3.3 Boundary Conditions

Boundary conditions (atj = 1 and Nmesh)for state variables (u) and its spatial derivatives (u,.) are needed only at each integer-time level (n = 0, 1,2, 3, ...) because of the staggering mesh structure and the intrinsically space-time triangle computational elements (see Fig. 2.7). At each half-time level (n = 1/2, 1 + 1/2, 2 + 1/2, ...), the Values of u and u, for all mesh points (j = 1 + 1/2, 2 + 1/2, ..., NmeSh-1/2)are calculated on the basis of the values at the previous integer-time level without requiring boundary values. When the Danckwert boundary condition Eq. (3) is applied, conservative boundary conditions (BCs) at x = ~0 and x = xfcan be constructed within CE+(I, n) and CE-(Nmesh, n), respectively. Performing a line integral along CE+(l, n) and using Eq. (3a), the boundary condition at x = xo (i.e., j = 1) for the iterative CE/SE method is obtained: At 2 J

u? - - p ? + q ? J

J

= u n-112 . - n-112 J + ~ P 'j+1/2

(102a) (102b)

Ax At --fyj At2 andfk is the inlet flu predefined by the 4 A$' 4Ax operation condition. Eq. (102)leads to a nonlinear equation with respect to two variables, urand uz'when j = 1. The boundary condition at x = xf (i.e.,j= Nmesh) for the iterative CE/SE method is obtained in the same way but by performing a line integral along CE-(N-h, n):

where q;

=

-u

(103a) u,", = 0 J

(103b)

At Here, since q; can reduce to q;= - - with the aid of Eq. (90) and (103b), ur is A J computed from Eq. (103a)through a nonlinear iteration. For the noniterative CE/SE simulation, Eq. (102a) and (103a)are replaced, respectively, by:

When other boundary conditions are imposed, appropriate formulations can be derived in a conservative manner within the conservation elements (CE, (j, n)).

2.4 fully Discretized Method

2.4.3.4 Comparison of CE/SE Method with Other Methods

The iterative CE/SE method, at each time level, is associated with a block diagonal Jacobian matrix. Let the number of PDEs and spatial mesh points be N p D E and Nmesh, respectively. The maximum number of nonzero Jacobian elements for the CE/SE method, JgZLsE, is: = (NPDE x NPDE) x Nmesh

(106)

In the case of linear source terms or noniterative CE/SE simulations, the Jacobian matrix is further reduced to a diagonal form: CE/SE -

Jmi,

- (NPDE x 1) x Nrnesh

(107)

When an implicit ODE integrator is used in the MOL framework for Eq. (78),a band matrix is obtained. Let the length of the upper and lower band matrix be M U and M L dependent on the spatial discretization and nonlinearity of the PDE considered. The maximum number of nonzero band-Jacobian elements for the MOL is known as (Lim et al. 2004):

JEzL= NPDE

'

Nrnesh(ML

+ M U + 1) - -21M L ( M L + 1)

-

1

-MU(hfU 2

+ 1)

(108)

For example, in the simple case where the convection term is discretized by a firstorder backward scheme and the diffusion term by a central scheme like Eq. (9),ML = M U = N p D E . The smallest number of nonzero Jacobian elements in this case,Jip, can be approximated at each time step: (2NPDE X NPDE)X Nmesh

(109)

As a result, the following relation can be derived CE/SE

Jmi,

<

]:YE IJ:p IJgL

JizL,

Eq. (110) means that the number of nonzero Jacobian elements for the MOL, is not less thanJ$EfE. The computational time is normally proportional to the number of nonzero Jacobian elements u) multiplied by the number of time steps (N,im), i.e., J x Ntime. Therefore, it is expected that the computational time of the CE/SE method is shorter than the MOL for the same number of time steps. Especially for nonstiff systems (e.g., chromatographic adsorption problems), the CE/SE method will save computational time because a small number of time steps can be used (Lim et al. 2004). In Section 2.6, the MOL and the CE/SE methods are compared for several PDE problems in terms of accuracy and computational efficiency. In Chang et al. (2000), the CE/SE method is compared with the Leapfrog, LaxWendroff, DuFort-Frankel and MacCormarck schemes (see Section 2.4.1).Here, the CE/SE method shows promising performance compared to these fully discrete methods.

68

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2 Distributed Dynamic Models and Computational Fluid Dynamics

While the implicit ODE integrator has a self-adaptive feature, i.e., variable order and time stepsize (At), the present CE/SE method has a fixed value of At satisfylng the CFL condition. Thus, for stiff problems a main disadvantage of the CE/SE method could be the futed time step (At).

2.5 Advanced Numerical Methods

Adaptive mesh methods can improve the accuracy and efficiency of the numerical approximations to evolutionary PDE systems that involve large gradients or discontinuities. As the solution changes in an evolutionary PDE, the mesh must also change to adaptively refine regions where the solution is developing sharp gradients, and to remove points from regions where the solution is becoming smoother (Li 1998). Over the past years, significant interest has been devoted to adaptive mesh methods. Various sophisticated techniques have been proposed. For example, adaptive mesh refinement (AMR) removes/adds the nodes at discrete time levels and moving grid methods function by moving the nodes continuously over time (VandeWouwer et al. 1998). Adaptive mesh methods have important applications for a variety of physical and engineering problems (e.g., solid/fluid dynamics, combustion, heat transfer, etc.) that require extremely fine meshes in a small part of the physical domain. Successful implementation of the adaptive strategy can increase the accuracy of the numerical approximation and also decrease the computational cost. This section addresses two different strategies: AMR and the moving mesh method.

2.5.1 Adaptive Mesh Refinement

The AMR approach (Berger and Oliger 1984; Berger and LeVeque 1998) has been shown to be one of the most effective adaptive strategies for PDEs and refines in space and/or time. The AMR process is composed of three steps: error estimation, mesh refinement and solution interpolation. An AMR package called the conservation laws package (CLAWPACK) from the University of Washington is available from http://www.amath.washington.edu/claw/.

-

2.5.1.1 Error Estimation

One way to estimate errors is to use a weighted combination of first and second solution differences. The error (Ei) at x = xi is estimated to be: Ei =

NPDE

C k=l

W1 Iuk,i+l - 4 , i l

+ w2 Iuk,i+~

-

24.i

+ Uk,i-ll

1

i = 2 . . . (Nm&

-

1) (111)

2.5 Advanced Numerical Methods

where NpDEdenotes the number of PDEs and the weighting factors w, and w 2are user-defined. In the adaptive algorithm, the mesh is refined in portions of the physical domain where the inequality Ei 2

F

is satisfied and where E is a user-specified error tolerance.

(112)

2.5.1.2 Mesh Refinement

AMR adds new refinement grids where the error is estimated to be large. The refine-

ment grids are usually aligned with the underlying base grid. The refinement grids are arranged in a hierarchy, with the base grids belonging to level one, the next grids being added to level 2 and so on. Grids on level m are refined by a refinement ratio r, (usually 2 or 4)from the grids on level (m - 1).The grids are normally properly nested so that a grid on level m is completely contained in the grids on level (m - 1). A hierarchical block grid structure for two-dimensional AMR with r = 2 is shown in

overall structure

t

I

I

I

I

I

I

I

r

>,

e

f

.I

Figure 2.8

A hierarchical block grid structure of AMR

I

I

70

I

2 Distributed Dynamic Models and Computational Fluid Dynamics

Fig. 2.8 (Li 1998). Each refinement level consists of several blocks. A block is a logically rectangular grid. After each refinement, the refined cells are clustered into several blocks. Buffer zones and ghost boundaries may be added to each block. All the blocks are managed by a hierarchical data structure. The data structure in the AMR algorithm is complex due to the existence of several levels (m).If u(xi,t", m) were used to store the data, it would be a waste of memory, because at higher refined levels, only a small part of the grid is used. In order to efficiently manage all the discrete points at the same level, we need to cluster them into several disconnected segments, called patches, and treat the patch as the basic data unit (Li 1998). The patches are building blocks of the hierarchical grid structure. 2.5.1.3 Solution Interpolation

The regridding includes computing the physical locations for each fine grid and copying or injecting the solution from the old grid to the new grid. The physical mesh positions are easy to compute by linear interpolation. The solution needs more attention. Although the solution can be obtained from the coarse grid by injection or interpolation, a more accurate solution is obtained from the old grid at the same level, which partially overlays the new grid. One of the secrets behind the success of the AMR algorithm is that flow discontinuities always fall within the overlay regions between the new and old grid. Thus the adaptation process cannot introduce further errors in these problem regions by solution interpolation. The method used to interpolate the solution from the coarse grid needs to be chosen with care. Conservative interpolation is useful in regions near a discontinuity. The coarse grid solution is assumed to be piecewise linear. The slopes for each grid are found by applying a MinMod limiter function to the forward and backward slopes between cell centers (Li 1998). So, for a coarse cell i, ui+lp - ui-lp = MinMod(ui+l - ui, ui - ui-1) where MinMod(a, b) =

("

sign(a) . min(la1, Ibl),

ifab < 0 elsewhere

2.5.1.4 Boundary Conditions

There are two types of boundaries in an AMR system: external boundaries and internal boundaries. External boundaries are given by the problem definition and internal boundaries are generated by refinement. Each patch in one-dimension has two ends: the left and the right. The boundary values are often collected only at the backward time t"-' just before the integration from t"-' to t". This causes a problem when the time integration is performed by an implicit or higher-order MOL, because the boundary values at t" are usually required to compute the intermediate time derivatives of the boundary cells in an MOL approach. This problem can be solved by collecting the values for the internal boundaries from the parent coarse grid at the forward time t" before integrating the current time level.

2.5 Advanced Numerical Methods

2.5.2 Moving Mesh Methods

Although AMR or local mesh refinement is quite reliable and robust, Furzeland et al. (1990)stated that it is cumbersome in some cases to apply it because of the interpolation procedure, nonfxed number of grid points, restart of integration at certain time steps, etc. The moving grid methods (Miller and Miller 1981; Dorfi and Drury 1987; Huang and Russell 1997), where the number of mesh points is kept unchanged, could be very powerful due to the continuous grid adaptation with the evolution of the solution. In the MOL framework, Furzeland et al. (1990) consider the moving finite difference (MFD) approach (Do15 and Drury 1987; Huang and Russell 1997) as promising with respect to reliability, efficiency and robustness. The moving finite element (MFE) approach (Miller and Miller 1981; Kaczmarski et al. 1997; Liu and Jacobsen 2004) that enables one to handle more complicated physical domains may be considered difficult to use because of tuning parameters and to be computationally inefficient. Moving mesh methods have traditionally used a finite difference method (normally with a simple three point central difference) to discretize both the physical PDE and moving mesh PDE (MMPDE). The MFD approach using the central discretization proposed by Huang and Russell (1997) and Dorfi and Drury (1987) is still unstable in some cases because of the central discretization of first-order derivatives. The E N 0 and WEN0 methods (Shu and Osher 1989; Jiang and Shu 1996) are uniformly high-order accurate right up to the discontinuity. Moreover these methods may well be applied to the moving grid method due to the reliable numerical results of first-order derivatives. Li and Petzold (1997) presented a combination of the moving grid method of Dorfi and Drury (1987) with the E N 0 schemes (Shu and Osher 1989) in order to improve stability and accuracy in the discretization procedure. We are interested in the numerical solution ofwell-posed systems of PDEs, e.g., in Eq. (la). If meshes are moving continuously with time, i.e, xi= xi@),by the chain rule, the solution of Eq. (la) satisfies the following equation (Dorfi and Drury 1987):

where U and x denote the time derivatives of u and x , respectively,when nonuniform physical meshes (xi)are transformed into uniform computational meshes (5;).Mesh movement is governed by m(u, x, 3;). The PDE and the mesh equation are intrinsically coupled and are generally solved simultaneously. 2.5.2.1 Equidistribution Principle

The grid equation, m(u, x, X), is induced from the equidistribution principle (EP), which means that the grids are spaced in order to make each arc length of discrete solutions equally distributed at each grid step. Therefore, the nodes are concentrated

I

71

72

I

2 Distributed Dynamic Models and Computational Fluid Dynamics Figure 2.9

of the solw

Arc length (ds =

tion a t a time level (t)

0

1

X

in steep regions. The one-dimensional EP can be expressed in its integral form with the computational coordinate (0 I 5 5 1)and the monitor function, M(x, t), as a metric of each arc length (see Fig. 2.9):

where M (x, t) is called the monitor function. For example, as shown in Fig. 2.9, the arc length monitor function is given: ds M ( x ,t ) = - = dx

d

m 1

Let the total arc length of a numerical solution at a time t be 0 ( t ( = J-M ( x , t ) dlj. 0 Therefore, Eq. (115) is replaced by:

where lji = i/Nmcshis the uniform computational coordinate mentioned above. O(t) is fmed at a given time regardless of 5 and is unknown. As it is difficult to treat this unknown term, O ( t ) , in the numerical procedure, it is eliminated by differentiating Eq. (117)with respect to once and twice. A quasistatic EP is so obtained

From equation (118)one can obtain various mesh equations involving node speeds (X), the so-called moving mesh PDE (MMPDE),which are employed to move a mesh having a fxed number of nodes in such a way that the nodes remain concentrated in regions of rapid variation of the solution (Huang and Russell 1997). For most discretization methods of Eq. (114),abrupt variations in the mesh will cause deterioration in the convergence rate and an increase in the error (Huang and Russell 1997). Moreover, most discrete approximations of spatial differential operators (e.g., u, in Eq. (114)) have much larger CFL condition numbers

2.5 Advanced Numerical Methods

(e.g., =

($+ k

73

) e in Eq. (114))on an abruptly varying mesh than they do on a

gradually varying one. The ill-conditioned approximations may result in stiffness in the time integration in the framework of MOL. Robust mesh equations spatially smoothed are proposed by Dorfi and Drury (1987) and Huang and Russell (1997). As an illustration, the well-known Burgers’ equation with a smooth initial condition is considered: U t = -uux U(X,

+ 10-~~,, o 5

0) = sin(2nx)

I 1

(119)

+ sin(rrx)/2

(120)

The boundary condition is given as u(0, t ) = 0.0 and u(1, t ) = 0.0. A uniform grid structure is used as the initial grid position. The solution is a wave that develops a very steep gradient and subsequently moves towards x = 1. Because of the zero boundary values, the wave amplitude diminishes with increasing time. This is quite a challenging problem for both fmed and moving mesh methods. Proper placement of the fine mesh is critical, and a moving grid method tends to generate spurious oscillation as soon as the mesh becomes slightly too coarse in the layer region, just like nonmoving mesh methods with a central difference (Lim et al. 2001b). Figure 2.10 shows numerical results of the Burgers’ equation solved by the MOL with the third-order W E N 0 scheme on 40 moving grid points (Lim et al. 2001b). The mesh points are well concentrated on the moving front and adapt to physical fluid flow. In Fig. 2.11, grid evolution with time is shown for this case. The mesh points move continuously according to variation of the solution. The moving grid method attains a resolution corresponding to 5000 equidistant grid points near the shock, and to 7.5 equidistant grid points near the smooth regions. 1.5 1 .o

a

2

0.5

Y

a 0.0 -0.5 -1.0 1

I

Figure 2.10 Numerical solutions of Burgers’ equation on 40 moving grid points

74

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2 Distributed Dynamic Models and Computational Fluid Dynamics

1.5

1.25

1

0.75 E .w

0.5

0.25

0

0.2

0.6

0.4

0.8

1

mesh, x Figure2.11 Grid evolution with time on 40 moving grid points

2.5.3

Comparison between AMR and Moving Mesh Method

In the numerical analysis of PDAEs, discretization methods on fixed mesh points are generally more robust and easy-to-use than those on adaptive mesh points. In the cases involving steep moving fronts, the adaptive mesh methods are efficient with respect to accuracy and computational time. AMR and moving mesh methods are two of the most successful adaptive mesh methods. However, some care is needed to successfully use them. AMR has been developed for explicit temporal integration, while the moving mesh method works efficiently for implicit temporal integration (e.g., MOL in Section 2.3) because of stiffness of the grid. Moving-grid methods use a fixed number of spatial grid points, without need for interpolation. Moving mesh methods implemented via implicit time integration take advantage of the fully automatic adaptation of temporal and spatial stepsize (At, and Ax,). However, simultaneous solution procedures of physical and mesh equations typically suffer from the large computation time due to highly nonlinear coupling between the two equations, often requiring an excessive Newton iteration at each time step. This problem is further exacerbated by the dense clustering of mesh points near discontinuities, which degrades the convergence of the iteration (Stockie et al. 2001). Moreover, the extension of moving mesh methods from one dimension to higher dimensions in not straightforward (Li 1998).The two-dimensional moving mesh equation is much more complicated, because it includes many factors such as temporal smoothness, orthogonality and skewness of the mesh (Huang and Russell

2.6 Applications

1999). Simplicity and efficiency for the extension from 1 D to 2D/3D motivate development of the local refinement method such as AMR (Li 1998). Data structure and algorithms in AMR for a one-dimensional grid can be extended to higher dimensions without difficulty. Adaptive mesh methods also introduce overhead. For the moving mesh, such overhead includes evaluation of the monitor function, regularization of the mesh function, computation of the mesh velocity and solving additional mesh equation for the node positions. Compared with the moving mesh method, the overhead for the AMR method is much less. The evaluation of the monitor function is much cheaper and there is no need for regularization of the mesh function. Most of the overhead comes from the refinement and management of the hierarchical data structure (Li 1998). Recently, the combination of the two adaptive mesh strategies was presented (Hyman et al. 2003) and the two methods are compared and reviewed.

2.6 Applications

This section illustrates applications of the introduced numerical methods for the solution of PDE or PDAE systems in several dynamic chemical/biochemical processes. In Table 2.6, the five examples to be presented are characterized according to the type of equations and physical dominant phenomena. Each of the five problems is described by a time-dependent process model within one-dimensional space. First, chromatography columns modeled by a PDAE system are presented in Section 2.6.1. Here, numerical performances of several MOL methods and the CE/SE method are compared for both linear and nonequilibrium adsorption. In the fmedTable 2.6

Classification o f application examples

Section

Type o f equations

Physical meanings related

Characteristics

2.6.1

Chromatography

PDAE with source term

Convection, diffusion, and adsorption

Steep moving fronts

2.6.2

Fixed-bed reactor

PDE with source term and recycle

Convection, diffusion, and reaction

Mass and heat recycle and oscillation profiles

2.6.3

Sluny bubble column reactor

PDE with source term

Convection, difFusion, and reaction

Chemical reaction related to three-phase hydrodynamics

2.6.4

Population balance equation

Integro-PDE with source terms

Growth, nucleation, agglomeration, and breakage

Dynamic behaviors of the particle size with discontinuous fronts

2.6.5

Cell population dynamics

Integro-PDE with source terms

Growth and cell division

Oscillatory behaviors of cell populations

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2 Distributed Dynamic Models and Computational Nuid Dynamics

bed reactor model (see Section 2.6.2), oscillatory behaviors of state variables caused by mass/energy recycling are examined and several numerical methods are compared. In Section 2.6.3, a slurry bubble column reactor for Fischer-Tropschsynthesis is considered, where three-phase hydrodynamics are modeled by empirical equations given by De Swart and Krishna. (2002). The dynamics of gaslliquid concentrations and temperature are predicted at the beginning of operation. A population balance equation modeling crystal growth, nucleation, agglomeration and breakage is solved in Section 2.6.4, where discontinuous moving fronts appear due to initial seed crystals. Finally, Section 2.6.5 considers cell population dynamics in microbial cultures described by cell population balance equation (PBE) coupled to metabolic reactions relevant to extracellular environment (Zhu et al. 2000).

2.6.1 Chromatography

Packed-bed chromatographic adsorption between the stationary and mobile phases leads, for each component, to a partial differential algebraic equation (PDAE) system involving one partial differential equation (PDE), one ordinary differential equation (ODE) and one nonlinear algebraic equation (AE) (Lim et al. 2004): (121a)

dn _ - k(n* - n)

(121b)

0 = g(C, n*)

(121c)

dt

where vLis the interstitial velocity, D,, is the axial dispersion coefficient, a is the volume ratio between the two phases, and k refers to the mass transfer coefficient. The liquid and solid concentrations for each component are referred to as C and n, respectively. n* is the equilibrium concentration (or adsorption isotherm). Since the Peclet number (ratio of convection to diffusion, Pe

=

YLL

Da,

where L, is the column

length) is often large in chromatographic processes (Poulain and Finlayson 1993), Eq. (121) is classified as a convection-dominatedparabolic PDAE system. The padted-bed chromatographic problem in Eq. (121), is solved for one component with the volume ratio a = 1.5, the fluid velocity v L = 0.1 m/s, the axial dispersion coefficient D, = 1.0 x lo-’ m’/s, and the adsorption rate coefficient k = 0.0129 s-’. A linear adsorption isotherm is used for the algebraic Eq. ( 1 2 1 ~ ) . n* = 0.85C

(122)

The column length is in the interval 0 5 z 5 1.5 and the integration time is 0 5 t 5 10 s. As the initial condition, C(0, z )= 0, n(0,z ) = 0 and n*(O, z ) = 0 for all z except z = 0 and z = 1.5.

2.G A p p h t i o n s

Suppose that the Danckwert's boundary condition for Eq. (121a) is imposed as below: aC At z = 0 and Vt, U L ( C- Cin) = D,, . (123a)

ac --

Atz=l.SandVt,

az

(123b)

az - 0

where Ci, is a known feed concentration just before entering to the column. Here, an inlet square concentration pulse is considered as follows: Ci, = 2.2, for 0 5 t 5 2.0s

( 124a)

Ci, = 0.0, for 2.0s 5 t 5 10.0s

(124b)

The numerical solutions are obtained on 201 equidistant spatial mesh points. The CFL number for the iterative CE/SE method (Lim et al. 2004; see also Section 2.4.3) is given at Y = 0.4. The reference solution is obtained on 401 equidistant mesh points through the iterative CE/SE method. The error is estimated using Eq. (36). Table 2.7 reports numerical performance on accuracy, computational efficiency and stability for the chromatographic adsorption problem with axial dispersion on 201 mesh points. The second-order central and fifth-order upwinding schemes give spurious oscillatory solutions near steep regions. Thus, the two methods seem to be inadequate for convection-dominatedproblems as mentioned in Lim et al. (2001a). The first-order upwinding scheme (called first-order upwind, or FS-upwind-1)is not accurate because of its low order of accuracy. The two WEN0 schemes (third-order and fifth-order) enhance accuracy and stability but at the cost of longer computation time. The CE/SE method gives, in this case study, the most accurate solution with very short calculation times in a stable manner. In Fig. 2.12 numerical solutions of the fluid concentration (C) are depicted near z = 0.9 at t = 10 s for the adsorption problem. The reference solution is a smeared square profile at z = 0.8 and 1. The CE/SE method shows the best solution without Table 2.7 Accuracy, temporal performance and stability evaluation for a chromatographic adsorption PDAE with axial dispersion and square input concentration on 201 mesh points Accuracy (LI error)**

MOL

FS-upwind-1 FS-central-2 FS-upwind-5 W S-upwind-3 WS-upwind-5

Iterative CE/SE

(CFL= 0.4)

**

Unstable numerical solution. L1 error at t = 10 s.

*t*

CPU time during 10-s integration time.

-L

CPU time (s)***

0.2075 0.0979" 0.0060" 0.0449 0.0168

1.6 1.9 2.9 11.3

0.0087

1.3

7.7

I

77

78

I

2 Distributed Dynamic Models and Computational Fluid Dynamics Reference Solution ..... .. FS-upwind-1 FScentral-2 W S-upwind-5

I

+

2.5 1

A

2.0 1.5

v

CFISE 0 . 4

C

.=0 1.0

e c 8 0.5

c.

0.0 0 -0.5

0

Axial

M

Figure 2.12 Fluid concentration (C) profiles for different numerical schemes around z = 0.9 at t = 10 s for the single component chromatographic adsorption problem with axial dispersion (Dax= 1.0 x lo-') and square input concentration on 201 mesh points

spurious oscillation of the six schemes tested. As expected, first-order upwind (or FSupwind-1) is not accurate and second-order central (or FS-central-2)is highly oscillatory. The MOL with fifih-order WENO (or WS-upwind-5)and the CE/SE with CFL = 0.4 exhibit similar resolution in steep regions. Figure 2.13 shows the propagation of steep waves with time. Note that the fifthorder WENO scheme (circles)and the CE/SE method (solid line) have a nondissipative

0

c'

.-

0

.I-

2

.I-

C 0,

2

8

o.:i1 0

0

0.5

1

0 r^ 0 ..a-

2

.a-

C

Q

0 C

8

1.5

axial direction, z

axial direction, z

I

2

0

c-

2

-

0

1.5

.-0 2 1 C

C

CI

0 .c.

w

4-

2 0)

8

E l C

a,

2

0.5

0

1.5

0

0.5

1

axial direction, z

1.5

8

0.5

0

0.5

0

1

axial direction, z

Figure 2.13 Fluid concentration (C) propagation with time for a chro. matographic adsorption problem with axial dispersion on 201 mesh points (dashed line: first-order upwind; circles: fifth-order WENO; solid lines: CE/SE with CFL = 0.4)

2.6 Applications

I

79

feature owing to conservative discretization, since the waves do not widen with time. In contrast, the peak of the first-order upwinding scheme (dashed line) broadens continually as time increases.

2.6.2 Fixed-Bed Reactor

Recycling is often used in industrial processes to reduce the costs of raw materials and energy. The nonlinear effects of introducing recycling on a futed-bed reactor are considered in plant-wide process control and bifurcation analysis (Recke and Jmgensen 1997). This nonlinearity has the most pronounced effect around bifurcation points, i.e., points where the system solutions change stability and/or number of possible solutions. The fEed-bed reactor we consider here is a packed-bed tubular reactor with a single irreversible exothermic reaction of hydrogen’s catalytic oxidation to form water (Hansen and Jnrrgensen 1976).

The reactor is mass- and heat-integrated, which means that unconverted reactants are recycled and the reactor effluent is used to preheat the reactor feed in an external heat-exchanger, as shown in Fig. 2.14. The reaction is assumed to be first order in oxygen concentration with Arrheniustype temperature dependence. The model describing the reactor with both mass and energy recycling is given by:

I

Fresh feed

Figure 2.14 Schematic drawing of the fixed-bed reactor with mass/ energy recycles

80

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2 Distributed Dynamic Models and Computational Fluid Dynamics

where the ratio of mass to thermal residence time x = 1/600, the dimensionless flow rate Y ) 1.0, the axial dispersion mass Peclet number Pe, = 270, the Damkohler number Da = 0.376, the Arrhenius number y = 9.0, the axial dispersion heat Peclet number PeH = 118, the Biot number Bi = 0.5, the dimensionless surrounding temperature 0, = 0.79, and the dimensionless heat reaction Be = 0.49 are used for simulation. The variables t, 6,y and O are the dimensionless time, axial direction, oxygen concentration and temperature, respectively. The mass and energy recycling are assumed to follow first-order dynamics:

where ,z and re denote mass and energy recycle time lag constants with the units [t-'1, respectively. The above two ordinary differential equations have analFc solutions as follows: Yrec

= ~ C =-I (Yrec,o

erec =

e,=l

- YO=I,O)

e

-rmt

- (&c,o - ec=l,o)ecTet

(130) (131)

where yreC,(,and y5=l,o are the initial conditions for yrc0 and ybl and Orec,Oand OE=~,Oare those for Ore, and E = ~Tqe . ~ e q w q h tipe s hay q o v m a v t a z, = 30 and teare used in this Simulation. The Danckwert boundary conditions at the inlet point (6= 0) are expressed as:

where the dimensionless feed oxygen concentration ( y f e d ) and temperature ( Ofeed) are given as y f d = 1.0 and Ofe, = 0.8 and the mass and energy recycle ratios are assumed to be cr, and ae. The boundary conditions at the outlet point (ij= 1)are given as:

The bed is initially set to no reactant (i.e., yo = 0 for all g at t = 0). The initial bed temperature is Oo = 0.79 for all 5. The model is solved by the MOL and the noniterative CE/SE method for solution comparison. In the framework of the MOL, the convection term is discretized on uniform 201-mesh points by the first-order upwinding scheme (FS-upwind-1)and

2. G Applications

the third-order WEN0 scheme (WS-upwind-3),and the diffusion term by the central scheme (see Section 2.3.1). The boundary conditions Eqs. (132)-(134) are converted into nonlinear algebraic equations (AEs) by spatial discretization. The band Jacobian structure is broken by mass and energy recycles. The resulting system is thus a set of DAEs with a sparse Jacobian matrix. Using the noniterative CE/SE method, the two coupled PDEs are fully discretized on uniform 201-mesh points at CFL = 0.6 The resulting system has 402 linear algebraic equations at each time level for uyand u respectively.

zj

-y

+y -y

x

*

0

02

04

(FS-upmd-1) (WS-up~d-3) (WSEmethod) theta (FS-upwmd-I) theta (WS-upwmd-3) theta (WSEmethod)

06

Reactor length

(u

04

06

08

I

08

I

2

? iu-

15

n =

A-

Y?!

@ a o e 1

0

c g 2kE

gg

05

0

0

02

Reactor length ( x ) Figure 2.15 Comparison o f numerical solutions for dimensionless oxygen concentration ( y ) and temperature (0) variations with respect to the reactor length (x) at (a) t = 3 and (b) t = 3 5

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2 Distributed Dynamic Models and Computational Fluid Dynamics

Figure 2.15 shows the spatial distribution of dimensionless oxygen concentrations (y) and temperatures (8) at two time levels, t = 3 and t = 3.5, for each of the three numerical methods. Even though smooth fronts move with time, the solution profiles depend highly on the numerical methods used due to mass/energy recycling. A different numerical method provides a different oscillatory frequency, phase degree, and/or amplitude. In Fig. 2.16, it is shown that a steady-statesolution is reached differently depending on the numerical method used.

2

-yy (FS-upwind-1) (WS-upwmd-3) U

-9-y

1

2

3

( W S E method)

5

4

6

time (t)

2

(b) 1 5 4 4 0

y (F'S-upwind-1) UY (WS-upwind-3)

d

y (WSEmethod) theta (FS-upwind-1) x theta (WS-upwind-3) o theta (WSEmethod)

___

0

15

16

17

18

19

time (t) Figure 2.16

Comparison o f numerical solutions for dimensionless oxygen concentration ( y ) and temperature (0) variations with respect to time, (a) 1 < t < 6 and (b) 1 5 < t < 20, at the reactor outlet point (5= 1)

20

2. G Applications

Liu and Jacobsen (2004)stated that some discretization methods such as finite differences and finite elements can result in spurious bifurcation and erroneous prediction of stability. To minimize discretization error, they proposed a moving mesh method (see Section 2.5.2), i.e., an orthogonal collocation method on moving finite elements, for solving a futed-bed reactor model with energy recycling. For the futed bed reactor system, there is clearly a need for checking the approximation error of spatial and temporal derivatives as the fronts move. This would mean for the CE/SE method that an adaptive mesh method is applied. In addition, it is questionable whether the exit boundary conditions Eq. (134) are physically reasonable, especially when steep fronts are moving out of the reactor.

2.6.3 Slurry Bubble Column Reactor

Slurry bed reactors are applied increasingly in the chemical industry. The specific example selected here focuses on Fischer-Tropsch (FT) synthesis. FT synthesis technology, such as fluidized bed, multitubular futed-bed, and three-phase slurry bed, forms the heart of many natural gas conversion processes that have been developed by various companies in recent years (e.g., SASOL, Shell, Exxon, etc.). The FT reaction converts the synthesis gas (H,+CO) into a mixture of mainly long straight chain paraffins. This example concerns the Fe-based (or Co-based)catalyk slurry bed reactor, as shown in Fig. 2.17. Unconverted gas

t

I+

Slurry

Model

-

I

-

Synthesis gas Figure 2.17 Hydrodynamic model of slurry bubble column reactor (SBCR) in the heterogeneous flow regime (Van der Laan et al. 1999)

I

83

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2 Distributed Dynamic Models and Computational Fluid Dynamics

A highly exothermic reaction takes place on the Fe-based catalytic surface at high temperature (about 250°C): 1 + (1 + E)H2 + -CnH, + H2O + 165kJ/mol n CO + H 2 O ++ C 0 2 + H 2 + 41 kJ/mol

CO

(135)

where n is the average length of the hydrocarbon chain and m is the number of hydrogen atoms per carbon. Since hydrogen is considered the limiting component, balance equations can be set up for hydrogen only (De Swart et al. 2002). The complex hydrodynamics of gas bubbles are simplified by gas holdups (&&big) of large bubbles (dg,big= 20-80 mm) and those ( E ~ , ~ , , , ~of U ) small bubbles (dg,smll= 1-6 mm), as shown in Fig. 2.18. The dimensionless mass and energy balances for hydrogen are described for the three phases: 0

gas phase

ay big & g , b i g F = -

0

liquid phase

0

solid phase

(1

1f

acont

+ acont . Ybig) 2

(usg

- udf) aYbig ugo

at

The dimensionless energy balance for the slurry phase is:

The dimensionless variables are denoted ybig = CH2,g,big/ CH2g07ysmall= C H ~ , ~ , ~ ~ xU / C H ~ , ~ ~ ~ = mCH,,L/CH,,g@8 = T/T,, = h/H and z = tugO/H,where the initial hydrogen concentration CHz,go = 0.38412 kmol/m3,the distribution coefficient of hydrogen between gas and liquid phases m = 5.095, the heat-exchanger wall temperature T, = 501 K, the slurry reactor height H = 30 m, and the inlet gas velocity ugo= 0.14 m/s are preliminarily given for simulation.

2. G Applications

For the gas phase mass balance, the gas holdup of large bubbles ( ~ ~ , can b ~ be ~ ) estimated by the following relation with the gas superficial velocity (usg= 0.14 m/s), the small bubble superficial rising velocity (u& the gas density (pg 7 kg/m3 at P 40 atm and T- 500 4, and the reference gas density (& 1.3 kg/m3 at P = 1 atm and

-

-

-

The gas holdup of the small bubbles is given where the transition from the homogeneous to the churn turbulent flow regime occurs (van der Lann et al. 1999):

where the small bubble holdup in solids-freeliquid is .&= 0.27 and the solid holdup is given as q,= 0.25. The gas contraction factor is assumed to be a,,,, -0.5 in Eq. (136). The gas phase Peclet numbers for large and small bubbles are assigned to be Peg,big 100 and Peg,small= u@G/EL 80, respectively. The Stanton numbers of the large bubbles (St,b, = kL,H2,bigabigH/m/ug0 4.51) and the small bubbles (St,,,lI = kL,Hl,smallClsmallH/m/upo 24.7) are calculated as an empirical correlation proposed by Calderbank and Moo-Young (1961).The superficial velocity of small bubbles, udf, is defined as:

-

-

-

-

-

Pa . s, surface tension u= 0.019 Pa . m, liquid density 7.0 kg/m3 and gravity g = 9.81 m/s2.

= 680

kg/tn3, gas density

=

For the liquid phase mass balance, the liquid hold up is determined by: &L =

- Ep -

(&g,big

+ Eg.small(1 - &g.big))

(144)

The liquid phase Peclet number (PEL = u@H/Er) is assumed to have the same value as the small bubble Peclet number. The liquid Stanton number for large bubbles and small bubbles are defined as Stg,big = ~,~~,bipa~,i~H/u@ (- 22.97) Stg,rmall = kL,H2,srnaiias. mallH/~gO (- 125.87), respectively. The superficial slurry velocity is equal to us,= 0.01 m/s in the simulation and the average catalyst concentration fraction is = 0.25. The Damkohler number as the dimensionless pre-exponential kinetic factor is defined as:

rs

Da = AmELH/ugo

(145)

where the preexponential kinetic factor (or collision frequency factor) is A = 5.202 X 10” s-’. The Arrhenius number is given from the kinetic data (De Swart et al. 2002):

I

85

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2 Distributed Dynamic Models and Computational Fluid Dynamics

1.175 x lo5 J/mol = 28.209 8.314 J/mol/K. 501 K

E,

y=-=

RT,

For the liquid phase energy balance, the heat transfer Peclet number ugo H %ff a, (Pe, = -x 7Q)the heat transfer Stanton number ( S ~ H= EL

~

Ps cps ugo

x 7.0),

and the dimensionless heat reaction (Be = -AHRCH2’go x 0.0488) are from mpsCpsTc De Swart et al. (2002). The model described by the above set (i.e., Eqs. (136)-(140)) of partial differential equations (PDEs), which include convection, diffusion and reaction, is solved with initial conditions and Danckwert’sboundary conditions (see Eq. (3)). De Swart et al. (2002) solved the above model numerically using the MOL with finite difference method (FDM) and a BDF ODE integrator. We here use the noniterative CE/SE method (see Section 2.4.3) to solve the model. Figure 2.18 shows dynamic contours 0.4,

d C d

= -

J

0.3

’$ 6

8% 2 2 0.2

$n a

-P

C

a 0

5

35 a g

0.1

0 0

0.2

0.4

0.6

column height (5)

0.8

0.25 0.2

0.15

3

0.1

R(

0.05 0

0

1

0.2

0.4

0.6

column height (5)

0.8

250,

0

8

sf

Y

P 0

0.2

0.6 column height 0.4

(5)

0.8

1

1

I

245 240 235 230 225‘ 0

0.2

0.4

0.6

column height (5)

Figure 2.18 Unsteady-state concentration contour o f (a) large bubble gas concentration; (b) small bubble gas concentration; (c) liquid concentration; and (d) slurry temperature with respect to the reactor height, within 5 min

0.8

I

1

2.G Applications

of large bubbles, small bubbles and liquid concentrations of hydrogen, and bed temperatures along the reactor height. These profiles show how to reach steady state within 5 min. From the model-based dynamic simulation we can predict conversion ratio and temperature changes with feed composition, heat-exchanger temperature, feed flow rate, and catalyst types. The optimal operating conditions can be obtained for a given objective function (e.g., cost-benefit function) using the method of nonlinear programming (NLP, see also Section 2.4). The three-phase bubble column shows complex hydrodynamics of reactant gas bubbles at elevated pressures (e.g., 10-40 atm). Several recent publications have established the potential of computational fluid dynamics (CFD) for describing the hydrodynamics of bubble columns (Krishna and van Baten 2001). Using a commercial CFD code (CFX, AEA Tech., UK) to solve mass/momentum conservation equations in the three phases, fiishna and van Baten (2001)predict the gas holdup and the liquid velocity within a cylindrical two-dimensional reactor at different column dimensions, pressures, and superficial gas velocity. The empirical correlations of the gas holdups in Eqs. (141)and (142)can be verified or predicted for different column dimensions by the CFD simulation results (Krishna et al. 2000). In Section 2.7, we will present in detail a combination of process simulation and CFD.

2.6.4 Population Balance Equation

The population balance equation (PBE) has been demonstrated to describe the particle size distribution (PSD) in various chemical/biological engineering problems such as crystallization, polymerization, emulsion, and microbial cultures. Indeed, modeling with the PBE provides a good description for parameter identification, which may be used for determination of operating conditions, and for process design and control. In crystallization processes, the PBE, which governs the crystal size distribution (CSD), is solved together with mass/energy balances and crystallization kinetics such as nucleation, crystal growth, breakage, and agglomeration. The system, which often leads to hyperbolic-like integro-partial differential equations (IPDEs), is complex due to a lot of feedback relationships between the equations (Wey 1985). To determine the CSD, all equations (e.g., PBE, mass, and energy balances) must be solved simultaneously. An inaccurate solution of a PBE will affect particle nucleation and subsequently particle growth and results in an incorrect CSD. Therefore, a numerical procedure to obtain an accurate solution of PBEs is necessary (Lim et al. 2002). The crystal size distribution (CSD) is usually expressed as the crystal number (N, no.) or number density (n,no./m, or no./m3)with respect to the crystal size (L,m) or volume (v,m3).A simple relationship between the crystal number (N) and the crystal number density (n) is given as follows using the finite volume approach:

I

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2 Distributed Dynamic Models and Computational Fluid Dynamics

Ni =

Li

Li+l

ndL

n;(Li+l - Li)

or Ni =

Jci

"if1

ndu x ni(ui+l - ui)

(147)

Both bases (i.e., Nand n) can give a good description of the CSD. However, the CSD based on the number (N) is often preferred, for conservation of the mass and the number of crystals, in the cases involving agglomeration and breakage kinetics (Kumar and Ramkrishna 1997). A number-based PBE as a governing equation ofthe CSD is usually described in terms of the birth of nuclei, their growth, agglomeration, and breakage:

+ (-ddtNi )breakage

dt where

Nrnesh

yoAL.

(148)

La

2. N.J'

i=l

j=i+l Lj '

yo AL .

La Li

La

. N;

Nmesh

La

j=i+l

Lj

+ 2y0A L .

yoAL. 2 . Ni - yoLq . Ni, Li

. N j - y0Lq . Ni,

i = 2 . . . (Nmesh- 1) = Nrnesh

(152)

2.G Applications

2.6.4.1 Method of Characteristics

It is well known that for the scalar linear conservation law (e.g., PBE considered here) there usually exists a unique characteristic curve along which information propagates. If the solution moves along the path line of propagation, the convection term a(GN) in the PBE disappears. Hence, numerical error and instability caused by 8L approximation of the convection term is removed. Kumar and Ramkrishna (1997) derived a modified MOC formulation for the PBE:

!Lo-($) ,

dt

nucleation

+(%)

(153a)

agglomeration

dL; dt

(153b)

- = G;

where Eq. (153b)is the mesh movement equation. The MOC formulation is numerically solved by using the MOL. To overcome the nucleation problem, a new mesh of the nuclei size (Ll) is added at given time levels. The system size can be kept constant by deleting the last mesh at the same time levels. Since the number of crystal nuclei can vary with the number of mesh points added or deleted, a proper number of added mesh points should be selected according to stiffness of nucleation. Suppose that a stiff nucleation takes place only at a minimum crystal size ( L , = as a function of time: n(t, L1) = 100

+ lo6 exp (lOP4(t

- 0.215)2)

(154)

5 L 5 2.0, the nuclei grow and the crystals aggregate as Within the size range well as break for 0.0 5 t 5 0.5. A square initial condition as seeds is also given:

n(0, L ) = 100, for 0.4 5 L 5 O h n(0, L ) = 0.01, elsewhere

(155)

The kinetic parameters are given: G = 1 (linear growth rate), = 1.5 X (constant agglomeration kernel) and y = 1.0 x L2 (breakage kernel). See Lim et al. (2002) for details. The discretized PBE based on the crystal number (N,)or the crystal density (ni) is solved by using the implicit BDF O D E integrator in the framework of the MOL. 2.6.4.2 Nucleation and Growth

When the PBE with the nucleation and growth terms is considered on the basis of the crystal density (n),its analFc solution is derived from the MOC: n(t, L ) = 100

+ lo6 exp (-

2

104((Gt - L ) - 0.215) ), for 0.0 5 L p Gt,

n(t, L ) = 100, for 0.4 5 ( L - Gt) 5 O h ,

(156a)

(15Gb)

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2 Distributed Dynamic Models and Computational Fluid Dynamics

-Analytic solution WS-upwind-3 0 WS-~pwind-SA 0

1.E-02h 0.0 0.5

1 .o

MOC-5%

Crystal size ( L )

1.5

2.0

Figure 2.19 CSDs for the stiff nucleation case without agglomeration and breakage

n(t, L ) = 0.01, elsewhere

(156c)

In the solution, a discontinuous front (due to square seed) and a narrow wave (originating from nucleation) move along the propagation path line, L = LI+Gt. The numerical tests are carried out on the 200 fEed grids for both the numerical MOC and the WENO schemes. In Fig. 2.19, the numerical results of the WS-upwind-3/5A (see Lim et al. (2002) for details) and MOC-50p (i.e., numerical MOC with an additional 50 mesh points) are compared to the analybc solution Eq. (155) at the end time (t = 0.5). While moving fronts are smeared near discontinuities using the WENO schemes on the weighted stencil, the numerical MOC-50p shows a quite good resolution even at the discontinuous fronts. 2.6.4.3 Nucleation, Growth, Agglomeration, and Breakage

When the agglomeration and breakage kinetics are added to the previous PBE, the analytic solution can not be derived. The numerical solution of Eqs. (148) or (153) is obtained on 101 points of the uniform grid, using MOC-2Op or WS-upwind-5. Employing the MOC-2Op (insertingldeleting 20 mesh points), the following mesh equations are used dL1 -=O,

dt

d Li -=1 dt

f o r i = 2 . . . 101

(157)

In Fig. 2.20, CSD changes obtained from MOC-20p are depicted according to the kinetics used. The solid line is the analytic solution for the pure growth problem without agglomeration and breakage. Since numerical diffusion error is small, high resolution is observed at the corners of steep fronts. Due to the agglomeration term, the CSD spreads out and the population of large crystal sizes increases (see Fig.

2.6 Applications

(b) Grow th+Agglome ratio

c

.5 rn

5

-0

102

loo 10'

0.5

0 1o6

1

1

1.5

0

0.5

0

0.5

1

1.5

2

1.5

2

crystal size, L

(c) Growth+Breakagyl

I

0

2

crystal size, L

0.5

1

crystal size, L

1.5

1 2

1

crystal size, L

Figure 2.20 CSD changes obtained by the MOC-20p on 102 meshes according to the growth, agglomeration, and breakage terms (solid line: analytic solution for pure growth)

2.20b). In contrast, the breakage term increases the population of small crystal sizes (see Fig. 2.20~).The CSD of the PBE with four kinetics is dispersed more broadly, as shown in Fig. 2.20d. Using the fifth-order WENO scheme (WS-upwind-5),Fig. 2.21 also shows effects of the growth, agglomeration, and breakage terms on the CSD. Considerable numerical dissipation is found in steep regions (or discontinuities) in Fig. 2.21, and also shown in Fig. 2.19. However, comparing Fig. 2.20d with Fig. 2.21d, the two solutions are similar due to the effects of agglomeration and breakage on the CSD. Though the modified MOC gives more accurate numerical results than the WENO scheme, there are some limitations to using it such as the need for careful determination of addingldeleting time levels and a unique mesh velocity equation (or growth rate, see Eq. (153b)).Using the spatial discretization methods (e.g., MOL with WENO schemes) to circumvent these limitations, attention must be paid to discretization of the growth term (convection term), which can cause much numerical error and instability in the presence of steep fronts or discontinuities.

92

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2 Distributed Dynamic Models and Computational Fluid Dynamics lo6

(a) Pure growth

,

h

I

'

(b) Growth+Agglomeratio

1o4

1o4 C

2 v)

C

W

U

1o2 1oo 10"

I

0

0.5

I

1

1.5

2

crystal size, L

I 0

0.5

0

0.5

I 1

1.5

2

1.5

2

crystal size, L

1o6

1 o6

(c) Growth+Ereakage

1o4 C

C

.25

.2;

u)

u)

C

C

W

W

U

U

1o2

1oo

10" I

0

0.5

I

1

1.5

crystal size, L

2

1

crystal size, L

Figure2.21 CSD changes obtained by the WS-upwind-5 scheme on 101 meshes according to the growth, agglomeration, and breakage terms (solid line: analytic solution for pure growth)

2.6.5

Cell Population Dynamics

Cell cultures are composed of discrete microorganisms whose population dynamics play an important role in bioreactor design and control. The cell cultures are known to exhibit autonomous oscillations that affect bioreactor stability and productivity. To increase the productivity and stability, it is therefore desirable to derive a dynamic model that describes the oscillatory behavior and to develop a control strategy that allows modification of such intrinsic reactor dynamics (Henson 2003). As a model example for cell culture dynamics, consider a segregatedlunstructured modeling based on the cell population balance equation (PBE) coupled to metabolic reactions that are relevant for the extracellular environment (Zhu et al. 2000; Mhaskar et al. 2002). The segregatedlunstructured model provides a realistic description of the cell cycle events that lead to sustained oscillation in cell cultures under the assumption that oscillations arise as a result of interactions between the cell population and the extracellular environment.

2.G Applications

The cell population dynamics including cell growth and cell division is described

X w , newborn-cell birth term (I 2 p r w d m ) , mother-cell division death term (-rW)and dilution loss by a partial differential equation including a convection term

(a,)

(-DW):

a w ( m , t ) - - a(v,(s’). ~ ( mt ),) at am +

I”

2p(m,m’)r(m’, S’) W(m’, t ) dm‘ - [ D + r ( m ) ] W(m, t )

(158)

where W(m, t) is the cell number concentration as a function of mass (m) and time ( t ) ,vS(S’is the overall single cell growth rate at the substrate concentration ( S ’ ) , p(m, m’) is the newborn-cell mass distribution function with newborn-cell mass m and mother-cell mass m’, r ( m ’ , S’), is the division intensity function, and D is the dilution rate. Detail models and their parameters are given below on the basis of Mhaskar et al. (2002) for a S. cerevisiae (or yeast) culture. For convenience, all of the g] and the cell number concentration ( W (m, t ) has masses have the units [ x the units [ X 10-l~no./g]. The division intensity function is introduced to account for the probability nature of cell division:

where m“, is the cell transition mass, mo = 1 is the minimum cell mass for division, and m“’d is the division mass. E = 5 and y = 200 are the constant parameters that determine the transition rate and the maximum intensity value, respectively. Sustained oscillations are generated through the introduction of a synchronization mechanism in which the transition and division masses are functions of the nutrient concentration. The following saturation functions are used (1GOa)

(1GOb)

where the substrate concentration is S’ = G’ + E , and the constants are given as Sl = 0.1 g/l, S ~ = ~ , O ~ , ~=, ,4.55, , ~ O mdo = 10.75, Kt = 0.01 . 1-l and & = 3.83 * 1-’.

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2 Distributed Dynamic Models and Computational Fluid Dynamics

The newborn cell probability function p (m,m’)has the form: p ( m , rn’) =

(y

. e-~(m-m;)2

lo.

+

. e-~(m--m’+m;)2 , m’> rn and rn’ > rn: elsewhere

+ mg

(161)

Here the constants are set to a = and p = 40. This function yields two Gaussian peaks in the cell number distribution, one centered at mgrtcorresponding to mother cells and one centered at m*,- m’corresponding to daughter cells. Oscillatory yeast dynamics are observed in glucose-limited growth environments. Under such conditions, both glucose and the excreted product ethanol can serve as substrates for cell growth. The following reaction sequence accounts for the relevant metabolic pathways, glucose fermentation, glucose oxidation, and ethanol oxidation:

where G’ and E represent intracellular glucose and ethanol concentrations, respectively, and 0 is the dissolved oxygen concentration. hgf= 30( x lo-’’ g/h),hgo = 3.25 ( x 10-13 g/h), and he,, = 7(x g/h) are maximum consumption rates, Km&= 40 g/l, KMgo= 2 g/l, Kmgd = 0.001 g/l, K,, = 1.3 g/l, and Kmed = 0.001 g/l are saturation constants, Kinhibit = 0.4 g/l is a constant that characterizes the inhibitory effect of glucose on ethanol oxidation. The overall single cell growth rate vg(g(s’)is the sum of the growth rates due to the three metabolic reactions. vg(S’) = Kgf(G’)

+ Kgo(G’, 0)+ Keo(G’, E’, 0)

(164)

For intracellular glucose and ethanol concentrations (GI, E ) and liquid oxygen concentrations (0)in Eq. (163),the mass balance equations of these substrates are d G’ dt

- = kg(G - G’)

d E’ dt

- = ke(E - E’)

do dt

-=

( 192 Kgo(G’) 180 Ygo

koa(O*- 0) - -~

96 Q,(E’)) +--46 Ye,

Ntotal

2.6 Applications

where G and E are extracellular concentrations, kg and k, = 20 h-1 are glucose and ethanol uptake rates, respectively, k,a = 1500 h-’ is the oxygen mass transfer rate, and Ygo= 0.65 g/g and Ye,= 0.5 g/g are the yield coefficients in the glucose oxidation and of microorganethanol oxidation reactions, respectively. The total cell number ( isms is defined by

The saturation oxygen concentration, 04, is obtained from the oxygen solubility, RT which is assumed to be governed by Henry’s law: 0” = H , -Ooul with the HenMW.02

ry’s rate constant (H, = 0.0404 g/l/atm), the gas constant ( R= 0.082057 1 . atm/mol/ K ) , the absolute temperature ( T = 298 K ) , the molecular weight of O2 (M,,o, = 32) and the oxygen concentration in the gas exhaust stream (Ooul). The gas phase oxygen balance is: V

* dt

= F(Oi, - Oout)- koa(O* - 0) . y

where Vg = 0.9 l and V, = 0.11 are the gas phase and liquid phase volumes, respectively, F = 90 l/h is the volumetric air-feed flow rate, and Oi, = 0.275 g/l (= 0.21 atm) is the oxygen concentration in the air-feed stream. For extracellular glucose and ethanol concentrations (G and E), the substrate mass balance equations are:

where D = 0.18 h-’ is the dilution rate, Gf= 30 g/l and Er= 0 g/l are the feed glucose and ethanol concentrations, respectively, and Yd= 0.15 g/g is the yield coefficient in the glucose fermentation reaction. The total cell number for microorganisms related to ethanol excretion is denoted in Eq. (171) as:

Experimental data suggests that key products, such as ethanol, are excreted primarily by budding cells. This behavior is modeled byf(m):

where ye = 1.25, E,

=

15 and me = 1.54 are constant parameters.

I

95

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2 Distributed Dynamic Models and Computational Fluid Dynamics

The liquid phase carbon dioxide balance (C) is

where k,a

=

1500 h-' is the C 0 2 mass transfer rate and C? is the saturation C02 con

centration modeled by c;' = H,RT C,, with the Henry's rate constant (H, = Mw,co* 1.48g/l/atm at pH = 5.0), the molecular weight of C02(Mw,co,= 44) and the COz concentration in the gas exhaust stream (C,,,,). The gas phase C02 balance is:

V

dt

= F( Ci, - Gout) - kca( C" - C) . Vl

(175)

where Ci, = 0.00054 g/l (= 0.0003 atm) is the carbon dioxide concentration in the airfeed stream. In summary, this cell PBE model is described by a PDAE system containing one PDE for the cell population (W(m, t ) )and eight ODES for eight substrate variations (G, E, G', E , 0, C, O,,,, and COu,).The single cell growth rate (vg(G',E , 0))is computed in Eq. (164). The initial condition of W(m, t) is set to W(m, 0) = 0.5 e-S.(m-G)', 1 5 m 5 11.The boundary condition of W(m, t ) is also given as W(11, t) = 0 for 0 5 t 5 6 hr. For the eight-substrate concentrations, their initial values are G' = G = 0.8, E = 0.01, E = 0.0001, 0 = 0.008, C = C,, = 0.003, and O,,, = 0.275. For the solution of cell PBE models, Zhu et al. (2000) and Mhaskar et al. (2002) used the orthogonal collocation FEM (Finlayson 1980). Motz et al. (2002) reported that the CE/SE method gives better performance in terms of accuracy and computational time than a flux limited finite volume method. Mantzaris et al. (2001a,b)compared several numerical methods such as finite difference methods (FDM) with explicit/implicit time integration and finite element methods (FEM) with explicit/ implicit time integration, where they suggested (i) the time-explicit scheme (e.g., Runge-Kutta method) is better for computational-efficiency than the implicit timeintegration scheme (e.g., BDF-types) and (ii) the finite difference method is preferred for multidimensional PBEs to the finite element method due to computational eficiency. Fortunately, this model can be solved by the modified MOC (see Eq. (153)),since there is a unique growth rate Eq. (164). The integration terms appearing in Eqs. (158), (168),and (172) are simply evaluated by the Trapezoidal rule:

where Nmsh is the number of mesh points. When the numerical MOC is used in the framework of the MOL, it is not easy to provide its Jacobian full matrix with hand-coding due to strong nonlinearity and it will be prohibitive to numerically evaluate the Jacobian because of the large system size (i.e., the number of equations is (2 X N,& + 8), and the Jacobian matrix is 2 X

2.6 Applications

N,& + S)2).The automatic differentiation technique is appreciated in this case for accuracy and computation efficiency. Figure 2.22 shows the dynamics of the cell population number density (W(rn,t ) ) is solved by the numerical MOC on 82 mesh points (N,,&,) where 135 mesh points are added at rn = 1 and also deleted at rn = 11. Due to cell division, the cell number density of small sizes tends to increase with time and the oscillatory behavior of the cell number density are regularized after about t = 6 hr. The oscillatory behaviors of the cell number (Ntotal) in Eq. (168) and the cell mass

(= c

Nrnesh-l

Ni

(mi

+ mi+l)

are shown in Figs. 2.23a and b. The cell number varia-

i=l

tion affects the extracellular glucose/ethanol (G and and oxygen/carbon dioxide (O,,, and C,,,) concentrations in the gas exhaust stream (see Figs. 2.23~-f,respectively). As ethanol is excreted, primarily by budding cells (see Eq. (173)),it is shown that the extracellular ethanol concentration ( E ) slowly reaches a regular oscillatory state in Fig. 2.23d. Figures 2.23g,h depict the dynamics of the consumed oxygen concentration ratio 100 x 100 x

(9)

and evolved carbon dioxide concentration ratio

, respectively.

800

-

600

m

2 D

z

7

z

400

v

200

0

12

5

6

13

mass, m (x10

Figure 2.22 Distribution of cell population concentration W(rn.t) over mass and time

g)

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2 Distributed Dynamic Models and Computational Fluid Dynamics

-

4.2

1

8

0

0.27361

,

2

4

6

8

W t ' hr

1

0 I

1

I

00001

2

1

0.0003

L - - . . . L r

0

2

4

I

6 time ( t l hr

8

10

12 I

'

I

0

0 0029

.

0 0028

8

00026

0 0025

s 6 3 N-

2 82 0

h

cu'

62 61 60

0

2

4

6 time (t),hr

8

10

12

Figure 2.23 Oscillatory behaviors in time of (a) cell number; (b) cell mass; (c) extracellular glucose (C); (d) extracellular ethanol (4; (e) oxygen in exhaust gas stream (Oout); (f) carbon dioxide in exhaust gas stream (&); (g) evolved oxygen ratio; and (h) evolved carbon dioxide ratio

All of the parameters used for this simulation need to be adjusted to experimental data, as shown in Mhaskar et al. (2002).

2.7 Process Model and Computational Fluid Dynamics

A multiscale model is a composite mathematical model formed by combining partial models that describe phenomena at different characteristic length and time scales. For example, modeling of a packed-bed catalytic reactor involves microscale chemical kinetics at the active sites on the catalyst, mesoscale transport processes through the pores of the catalyst pellets, and macroscale flow and heat exchange at the reactor vessel level. Computational tools such as molecular dynamics (MD), computational fluid dynamics (CFD), and process simulation have been used to help fill particular

22.7 Process Model and Computational Nuid Dynamics

length- and timescale gaps. In general, despite their obvious connection, phenomena at different characteristic scales have usually been studied in isolation (see Section 2.5). One of the key challenges facing process modeling today is the need to describe complex interactions between hydrodynamics and the other physical/chemical phenomena. This is particularly important in the case of complex systems (e.g.,polymerization, crystallization, and agitated bioreactors) in which the constitutive phenomena interacts with mixing and fluid flow behavior. Process simulation tools, which play an increasingly central role within most process engineering activities are able to represent (i)multicomponent, multiphase, and reactive systems, (ii) individual unit operations, multiple interconnected units, or entire plants, and (iii)thermodynamic properties. However, most of the models used by process simulation tools either ignore spatial variations of properties within each unit operation (invoking the well-mixed tank assumption) or are limited to simple idealized geometries. Moreover, the treatment of fluid mechanics is usually quite rudimentary (Bezzo et al. 2003). CFD techniques solve fundamental mass, momentum, and energy conservation equations (e.g., the Navier-Stokes equation) in complex three-dimensional geometries. From CFD simulation, some valuable information (e.g., mass flow rate, heat transfer coefficient, velocity, etc.) for process simulation can be obtained. However, CFD’s ability is still limited in application to complex reactive systems and multiphase processes with multicomponent phase equilibria. Furthermore, performing realistic dynamic simulation often requires excessive computational time. In view of the above, CFD and process simulation technologies are highly complementary (Bezzo et al. 2000). Combination of process simulation and CFD can therefore lead to significant advantages in accurate modeling of processes.

2.7.1 Computational Fluid Dynamics

The CFD technique has focused on the solution of PDEs representing conservation equations describing fluid flow over domains of often complex geometry. There are several commercial CFD packages such as Fluent (Fluent, Inc.), CFX (AEA Tech., Hanvell, UK) and FemLab (COMSOL). The CFD packages usually comprise three distinct elements, namely preprocessing (geometry specification, model selection, parameter specification, and grid generation), numerical solution procedure and post-processing (visualization and data treatment). In the solution procedure, mass/momentum/energy conservation equations are solved within the specified geometry. Generic conservation equations may be described by PDEs with advection, viscosity/diffusivity and source terms:

-a4 + - axa (~ 4 - ra ~ -- s 4 = o at a )l

(177)

I

99

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2 Distributed Dynamic Models and Computational Fluid Dynamics

where @ is a conserved quantity such as mass, energy or momentum, rQ the viscosity or difisivity, sg the source or sink and x the set of spatial dimension variables. The models are called the compressible (or incompressible) Navier-Stokesequation. When the viscosity/diffusivity terms can be neglected due to relatively small influence on the result, the inviscid Euler equation is obtained. Let p, u,p, and e be the mass density, velocity, pressure, and energy per unit volume, respectively. The inviscid Euler equation of a perfect gas can be expressed as:

where u = (p, p,e)' andf= (p, p + eu2,ue +

We may write e = @eint,,r

1 +-

2 p2, where eintemris the internal energy per unit mass. Therefore, this equation is the three-dimensional hyperbolic PDE. Most CFD packages do not use the MOL approach (see Section 2.3) of reducing PDEs into ODEs in time. Instead, they choose to discretize both temporal and the spatial dimensions (i.e., fully discrete methods, see Section 2.4), thereby reducing the PDEs into a set of nonlinear algebraic equations (Oh 1995). The process simulation models mentioned in Section 2.6 represent specific and simplified conservation equations related to the macroscopic process level. Here, complex fluid dynamics is lumped by parameters or simple empirical equations (e.g., the three phase bubble column model in Section 2.6.3). Microscopic chemical reactions are simplified by kinetic equations as a function of temperature, pressure, and concentrations. If we use a full CFD simulation including complex reactions, thermodynamics, population dynamics and hydrodynamics, it would be practically infeasible because of high computing load and lack of existing tools. To effectively take into account interactions between hydrodynamics and the other physicallchemical phenomena, a hybrid approach, namely, multizonal/CFD simulation (Bauer and Eigenberger 1999 and 2001; Bezzo et al. 2003),is proposed (see Fig. 2.24). Several zones, assumed to be well-mixed compartments, are described by process models (e.g., AEs, ODEs, PDEs or PBEs) with the exception of the fluid-flow ones, and CFD simulation provides each zone with the mass flow rate at interzonal interfaces and additional fluid dynamical properties such as mass transfer coefficient and turbulent energy dissipation rate (Bezzo et al. 2003).

2.7.2 Combination o f CDF and Process Simulation

Figure 2.24 shows a structure of the general multizonal/CFD model. The spatial domain of interest is divided into several zones (z1-z5).Each single zone (2)is considered to be well mixed and homogeneous. Two zones can interact with each other via an interface that connects a port (p) of one zone with a port of the other. The flow of material and/or energy across each interface is assumed to be bidirectional. The transient behavior of a zone is described by a set of algebraic equations (AEs),ordi-

22.7 Process Model and Computational Fluid Dynamics

Multizonal model (all phenomena except fluid dynamics)

physical properties

I

1

I

1

I

I

l

l

I

r

l

l

l

l

l

l

l

CFD model (total mass and momentum conservation only) Figure 2.24 al. 2003)

l

l

1

Structure of the general multizonal/CFD model (Bezzo et

nary differential equations (ODES),partial differential equations (PDEs) or population balance equations (PBEs).The multizonal model uses detailed dynamic modeling of all relevant physical phenomena, with the exception of fluid-flow, over a physical domain divided into a relatively small number of zones. Mixing parameters and interzonal mass flow rates are determined by solving a detailed CFD model over the same physical domain. The CFD model focuses solely on fluid-flow prediction, trylng to do this as accurately as possible by dividing the space into a relatively large number of cells and solving the total mass and momentum conservation equations. Thus, the CFD model does not attempt to characterize intensive properties such as composition, temperature or particle size distribution. The transient behavior is ignored, based on the assumption that fluid-flow phenomena operates on a much shorter time scale than all other phenomena. The solution of the CFD model will require knowledge of the distribution of physical properties (e.g., viscosity, density, compressibility factor, etc.) throughout the physical domain of interest. These properties are usually a function of the systemintensive properties and are computed within the multizonal model. The hybrid model is formed by the coupling of the multizonal model with the CFD mode, both representing the same spatial domain. The mapping between the zone and cell is achieved by means of appropriate disaggregation and aggregation procedure (Bezzo et al. 2003). The multizonal concept and CFD simulation is applied to bubble column reactors (Bauer and Eigenberger 1999, 2001) and to a bioreactor processing and mixing a highly viscous fluid (Bezzo et al. 2003).

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2 Distributed Dynamic Models and Computational Fluid Dynamics

2.8 Discussion and Conclusion

In many chemical and biotechnical processes, partial derivatives result from a consequence of dynamic behaviors of mass, energy and momentum in space. The derivative or algebraic terms describe fluid flow, physical phenomena and/or constitutive relations in the different phases. The description of convective and diffusive (dispersive) fluxes introduces first and second order spatial derivatives. Mass exchanges between the fluid and stationary phases (e.g., reactions and adsorptions) are described by time-dependent differential equations. Equilibrium relations and physical properties are described with algebraic equations (AEs).Thus process models are in general represented as partial differential equations (PDEs)coupled with algebraic equations (AEs),i.e., PDAEs with pertinent initial and boundary conditions. In this chapter, a large class of one-dimensional PDAE models is presented in Section 2.6 to introduce the most frequently applied numerical methods for their solution (Sections 2.3-2.5). Finally, the complementary relations between process simulation and computational fluid dynamics most often employed to solve models with fully developed flow field are demonstrated (Section 2.7). A very important element for a correct discretization is to ensure that the discretized formulation (or numerical approximation) indeed converges to the continuous formulation (or physical model), as the discretization (or spatial and/or temporal stepsize) is refined. However, this issue contains several subtleties and is not dealt with in detail in this chapter. To improve accuracy and efficiency of numerical solutions, it is desirable to select appropriate numerical methods according to the physical models considered. The method of lines (MOL),which includes time integration and spatial discretization, is adequate to solve stiff problems such as diffusion-dominated models and fast reaction problems (Section 2.3). In the presence of steep moving fronts, the MOL incorporating adaptive and moving mesh methods to capture large spatial variations provides efficient solutions (Section 2.5). When there is a unique solution propagation path line, the method of characteristics (MOC) can be formulated. The MOC formulation can be solved in the framework of the MOL (Section 2.6.4). Since the solution moves along the propagation path line, numerical error and instability caused by approximation of the convection term is avoided. In solving convection-dominatedmodels, the conservation element and solution element (CE/SE) method is appreciated for accuracy and efficiency due to a finite volume approach and explicit time integration (Section 2.4.3). The CE/SE method possesses low numerical dissipation error but a fine time stepsize is needed for stiff models. Most CFD packages use fully discrete methods (Section 2.4) rather than the MOL (Section 2.3) for solving conservation laws for fluid flow within an often complex multidimensional geometry. Combination of process simulation and CFD is useful to describe complex interactions between fluid hydrodynamics and other physical/ chemical phenomena (Section 2.7).

2.8 Discussion and Conclusion

Although many efficient methods have emerged for solving PDEs/PDAEs, several challenges remain. The numerical method of PDEs with stiff nonlinear source terms is one of the currently active research areas (Ahmad and Berzins 2001; Hyman et al. 2003), where mesh refinement by an efficient control of space-time errors is needed. Simulation methods for the processes considerably influenced by fluid hydrodynamics should be improved by combining CFD technologies. Hybrid dynamic systems that exhibit coupled continuous and discrete behaviors have also attracted much attention (Ma0 and Petzold 2002). When a reactant within a phase of a spatially distributed reactor disappears and appears, special considerations are required for the hybrid system to be properly handled by the numerical methods.

References 1 Ahmad 1. Berzins M . MOL solvers for hyper-

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bolic PDEs with source terms, Mathematics and Computers in Simulation 56 (2001)p. 115-125 Ascher U. M . Petzold L. R. Computer methods for ODES and DAEs, SIAM, Philadelphia 1998 Ayasouj A. Keith T. G. Application of the conservation element and solution element method in numerical modeling of heat conduction with melting and/or freezing 13(4) (2003) p. 448-471. Bauer M. Eigenberger G . A concept for multiscale modeling of bubble columns and loop reactors Chem. Eng. Sci. 54 (1999) p. 5109-5117 Bauer M. Eigenberger G. Multi-scale modeling of hydrodynamics, mass transfer and reaction in bubble column reactors Chem. Eng. Sci. 56 (2001) p. 1067-1074 Berger M. /. Oliger J. Adaptive mesh refinement for hyperbolic partial differential equations J . Comp. Phy. 53 (1984) p. 484-512 Berger M . J. Leveque R. J. Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM J . Numer. Anal. 35(6) (1998) p. 2298-2316 Bezzo F. Macchietto S. Pandelides C. C. A general framework for the integration of computational fluid dynamics and process simulation, Computers and Chemical Engineering 24 (2000) p. 653-658 Bezzo F. Macchietto S. Pandelides C. C. General hybrid multizonal/CFD approach for bioreactor modeling AlChE J. 49(8) (2003)p. 2133-2148 BischofC. Houland P. Using ADIFOR to compute dense and sparse Jacobians, Tech-

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nical memorandum ANL/MCS-TM-159, Mathematics and computer science division, Argonne National Laboratory, USA 1991 Bischof C. Carle A. Corliss G. Griavank A. Hovland P. ADIFOR: Generating derivative codes from Fortran programs, Scientific Programming l(1)(1992) p. 11-29 Bischof C. Carle A. Khademi P. Mauer A. Hovland P. ADIFOR 2.0 user’s guide, Technical report ANL/MCS-TM-192, Mathematics and computer science division, Argonne National Laboratory, USA 1998 Calderbank P. H. Mooyoung M. B. The continuous phase heat and mass-transfer properties of dispersions Chem. Eng. Sci. 16 (1961) p. 39-54 Chang S. C. The method of space-time conservation element and solution element-A new approach for solving the Navier-Stokes and Euler equations J. Comput. Phys. 119 (1995) p. 295-324 Chang S. C. Courant number insensitive CE/ SE schemes, 38* AIAA joint propulsion conference, AIAA-2002-3890,Indianapolis, USA 2002 Chang S. C. Wang X. Y. To W. M . Application of the space-time conservation element and solution element method to onedimensional convection-diffusion problems J. Comput. Phys. 165 (2000) p. 189-215 Chang S. C. Wang X. Y . Chow C. Y.The space-time conservation element and solution element method: A new high-resolution and genuinely multidimensional paradigm for solving conservation laws J. Comput. Phys. 156 (1999)p. 89-136 De Swart S. W.A. Krishna R. Simulation of the transient and steady-state behavior of a bubble column slurry reactor for Fischer-

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Tropsch synthesis Chem. Eng. Processing 41 (2002)p. 35-47 Do@ E. A. Dmry L. O’C. Simple adaptive grids for 1-D initial value problems J. Comput. Phys. 69 (1987) p. 175-195 Dufort E. C. Frankel S. P. Stability conditions in the numerical treatment of parabolic differential equations, Mathematical Tables and Other Aids to Computation 7 (1953) p. 135-152 Finlayson B. A. Nonlinear analysis in chemical engineering, McGraw-Hill, NewYork 1980 Furzeland R. M . VemerJ. G. Zegeling P. A. A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines J. Comput. Phys. 89 (1990) p. 349-388 Hansen K. ]orgensen S. B. Dynamic modeling of a gas phase catalytic futed-bed reactor 1-111, Chemical Engineering Science 31 (1976)p. 473-479,579-598 Harten A. Engquist B. Osher S. Chakravarthy S. Uniformly high order essentially nonoscillatory schemes I11 J. Comp. Phy. 71 (1987) p. 231-303 Henson M. A. Dynamic modeling of microbial cell populations, Current Opinion in Biotechnology 14 (2003)p. 460-467 Heydweiller]. C. Sincovec R. F. Fan L. T. Dynamic simulation of chemical processes described by distributed and lumped parameter models Comp. Chem. Eng. 1 (1977)p. 125-131 Hill P. /. Ng K. M. New discretization procedure for the breakage equation AIChE J. 41(5) (1995)p. 1204-1216 Hindmarsch A. C. LSODE and LSODI: two new initial value ordinary differential equation solvers, ACM SIGNUM newsletter, 15 (1980) p. 19-21 H o f i a n ] . D. Numerical methods for engineers and scientists, McGraw-Hill Inc., Mechanical engineering series Part 111. Partial differential equations (1993) p. 371-774 Huang W. Russell R. D. Analysis of moving mesh partial differential equations with spatial smoothing SIAM J. Num. Anal. 34 (1997) 1106-1 p. p. 126 Huang W. Russell R. D. Moving mesh strategy based on a gradient flow equation for two-dimensional problems SIAM Sci. Comput. 20(3) (1999)p. 998-1015

32 Hyman]. M. Li S. Petzold L. R. An adaptive

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moving mesh method with static rezoning for partial differential equations Computers and Mathematics with Applications 46 (2003) p. 1511-1524 jiang G. Shu C. W. Efficient implementation of weighted E N 0 schemes J . Comp. Phy. 126 (1996) p. 202-228 Kaczmarski K. Mazzotti M . Storti G. Morbidelli M. Modeling fuced-bed adsorption columns through orthogonal collocations on moving finite elements Comp. Chem. Eng. 21 (1997) p. 641-660 Kohler R. Gerstlauer A. Zeitz M. Symbolic preprocessing for simulation of PDE models of chemical processes Mathematics and Computers in Simulation 56 (2001) p. 157-170 Krishna R. van Baten]. M . Eulerian simulations of bubble columns operating at elevated pressures in the churn turbulent flow regime Chemical Engineering Science 5G (2001) p. 6249-6258 Krishna R. van BatenJ. M. Urseanu M. I. Three-phase Eulerian simulations of bubble column reactors operating in the chumturbulent regime: a scale up strategy Chemical Engineering Science 55 (2000) p. 3275-3286 Kumar S. Ramkrishna D. On the solution by discretization balance equations by discretization-111. Nucleation, growth and aggregation of particles Chem. Eng. Sci. 52(24) (1997) p. 4659-4679 Lax P. D. Wendrof B. Systmes of conservation laws Comm. Pure and Appl. Math. 13 (1960) p. 217-237 Leveque R. 1.Finite difference methods for PDEs Lecture note Department of Mathematics University of Washington 1998 Li S. Adaptive mesh methods and software for time dependent PDEs PhD thesis University of Minnesota 1998 Li S. Petzold L. Moving mesh methods with upwinding schemes for time-dependent PDEs J. Comput. Phys. 131 (1997) p. 368-377 Lim Y. I. Jsrgensen S. B. A fast and accurate numerical method for solving simulated moving bed (SMB)chromatographic separation problems Chem. Eng. Sci. 59(10) (2004) p. 1931-1947 Lim Y. I . Chang C. S. ]srgensen S. B. A novel partial differential algebraic equation

2.8 Discussion and Conclusion

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(PDAE) solver: iterative conservation element/solution element (CE/SE) method Comput. Chem. Eng. .28(8) (2004) p. 1309- 1324 Lim Y. I. Le Lann]. M. ]oulia X.Accuracy, temporal performance and stability comparisons of discretization methods for the solution of Partial Differential Equations (PDEs) in the presence of steep moving fronts Comp. Chem. Eng. 25 (2001a)p. 1483-1492 Lim Y. I. Le Lann J . M. Joulia X. Moving mesh generation for tracking a shock or steep moving front Comp. Chem. Eng. 25 (2001b)p. 653-663 Lim Y. I. Le Lann J. M. Meyer X. M . Joulia X , Lee G.B. Yoon E. S. On the solution of Population Balance Equations (PBE) with accurate front tracking methods in practical crystallization processes Chem. Eng. Sci. 57 (2002) p. 177-194 Liu Y. Jacobsen E. W. On the use of reduced order models in bifurcation analysis of distributed parameter systems Computers and Chemical Engineering 28 (2004)p. 161-169 MacCortnark R. W. The effect of viscosity in hypervelocity impact cratering American Institute of Aeronautics and Astronautics paper (1969) p. 69-354 Mackenziej. A. Robertson M. L. The numerical solution of one-dimensional phase change problems using an adaptive moving mesh method J. Compt. Phys. 161(2) (2000) p. 537-557 Mantzaris N. V. Daoutidis P. Srienc F. Numerical solution of multi-variable cell population balance models: I. Finite difference methods Computers and Chemical Engineering 25 (2001a)p. 1411-1440 Mantzaris N. V. Daoutidis P. Srienc F. Numerical solution of multi-variable cell population balance models: 111. Finite element methods Computers and Chemical Engineering 25 (2001b)p. 1463-1481 Ma0 G. Petzold L. R. Efficient integration over discontinuities for differential-algebraic systems Computers and Mathematics with Applications 43 (2002)p. 65-79 Mhaskar P. Hjorts0 M. A. Henson M. A. Cell population modeling and parameter estimation for continuous cultures of S. cerevisiae Biotechnol. Prog. 18 (2002) p. 1010-1026 Miller K. Miller R. N. Moving finite elements I SIAM J. Numer. Anal. 18 (1981) p. 1019-1032 Molls T. Molls F. Space-time conservation method applied to Saint Venant equa-

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tions J . Hydraulic Eng. 124(5) (1998) p. 501-508 Motz S. Mitrovic A. Gilles E.-D. Comparison of numerical methods for the simulation of dispersed phase systems Chem. Eng. Sci. 57 (2002) p. 4329-4344 Oh M. Modeling and simulation of combined lumped and distributed processes, PhD thesis, University of London 1995 Petzold L. R. A description of DASSL A differential/algebraic system solver in scientific computing , eds. R. S . Stepleman et al., North-Holland, Amsterdam (1983)p. 65-68 Poulain C. A. Finlayson B. A. A Comparison of Numerical Methods Applied to Nonlinear Adsorption Columns” Int. J. Num. Methods Fluids 17(10) (1993) p. 839-859 Powell M. J. D. On the convergence of the variable metric algorithm J. Inst. Math. Appl. 7 (1971)p. 21-36 Recke B. Jargensen S. B. Nonlinear dynamics and control of a recycle fmed bed reactor Proceedings of the 1997 American Control Conference vol4 (1997) p. 2218-2222 Sargousse A. Le Lann I. M. Joulia X. Jourdu L. DISCO:un nouvel environnement de simulation orient objet; Proceeding of MOSIM 1999 61-66, (1999) p. Nancy, France Saucez P. Schiesser W. E. van de Wouwer A. Upwinding in the method of lines Mathematics and Computers in Simulation 56 (2001) p. 171-185 Schiesser W. E. The numerical method of lines-integration of partial differential equations Academic press New York 1991 ShiJ. H u C. Shu C. W. A technique of treating negative weights in WEN0 schemes Brown University Scientific Computing Report Series BrownSC2000 (2000) p. 15 USA Shu C. W. Osher S. Efficient implementation of essentially non-oscillatory shockcapturing schemes J. Comp. Phy. 77 (1988) p. 439-471 Shu C. W.Osher S. Efficient implementation of essentially non-oscillatory shockcapturing schemes 11 J. Comp. Phy. 83 (1989) p. 32-78 Shu C. W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws ICASE Report No. (1997) p. 97-65 Stockiej. M. Mackenziej. A. Russell R. D. A moving mesh method for one-dimensional hyperbolic conservation laws SIAM J. Sci. Comput. 22(5) (2001) p. 1791-1813

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Van der Laan G. P. Beenackers A. Krishna R. Multicomponent reaction engineering model for Fe-catalyzed Fischer-Tropsch synthesis in commercial scale slurry bubble column reactors Chem. Eng. Sci. 54 (1999) p. 5013-5019 72 Vande Wouwer A. S a m z P. Schiesser W. E. Some user-oriented comparisons of adaptive grid methods for partial differential equations in one space dimension App. Num. Math. 26 (1998) p. 49-62 73 Villadsen J. Michelsen M. L. Solution of differential equation models by polynomial approximation. Prentice-Hall Englewood Cliffs New Jersey 1978 74 W q J .S. Analysis of batch crystallization processes Chem. Eng. Commun. 35 (1985) p. 231-252 71

Wu]. C. Fan L. T. Erickson L. E. Three-point backward finite-difference method for solving a system of mixed hyperbolic - parabolic partial differential equations Comp. Chem. Eng. 14 (1990) p. 679-685 76 Yu S. T. Chang S. C. Treatment of stiff source terms in conservation laws by the method of space-time CE/SE AIAA 97-0435 35" Aerospace Sciences Meeting, Reno, USA 1997 77 Zhu G.-Y. Zamamiri A. Henson M. A. Hjortse M. A. Model predictive control of continuous yeast bioreactors using cell population balance models Chem. Eng. Sci. 55 (2000) p. 6155-6167

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Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8 I107

3 Molecular Modeling for Physical Property Prediction Vincent Cerbaud and XavierJoulia

3.1 Introduction

Multiscale modeling is becoming the standard approach for process study in a broader framework that promotes computer-aided integrated product and process design. In addition to the usual purity requirements, end products must meet new constraints in terms of environmental impact, safety of goods and people, and specific properties. Engineering achievements can be startling from the user perspective like aqueous solvent paint that is still washable after drying! This can only be done by improving process knowledge and performance at all scales, right down to the atomic scale. Current experimental and modeling approaches assess with difficultly such submicronic scales. In experiments, there is the question of how to conceive experimental devices small enough and how to introduce them in molecular systems without irreversibly affecting the phenomena that they look at. In modeling and simulation, the questions are: which hypotheses are still relevant? How does one handle boundary effects? Numerical difficulties may arise along with the necessity of defining new parameters. They will be adjustable ones as no experiments can obtain them. This latter statement is particularly true for energetic interaction parameters like binary interaction parameters in current liquid-vapor equilibrium macroscopic thermodynamic models based on the activity coefficient approach or on the equation of state approach. The study of any process, phenomena attributed to energetic interactions has always been left for another time, but that time has come. Indeed, molecular modeling is a field of study that is interested in the behavior of atomic and molecular systems subject to energetic interactions. It is then a natural complement of experimental and modeling approaches to expand multiscale approaches towards smaller scales. Besides, process flows primarily concern molecules from raw materials to end products. Therefore, at any process development step, the challenge of knowing the physical properties and thermodynamical state of molecules is critical. However, the future of this challenge is dim when one thinks Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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about the millions of chemical compounds referenced in the chemical abstract series. Neither experimental approaches nor current themodynamic models can handle the combination of properties needed. In some cases experiments are not even practical because of material decomposition or safety issues. Universal group contribution methods are a pipe dream and existing ones are efficient but are restricted to specific areas like petrochemical and small molecular systems. As providers of accurate physicochemical data, molecular modeling methods offer an alternative to an intensive and expensive experimental campaign once molecular models are available, which is becoming increasingly the case (Case, Chaka, Friend et al. 2004). But the first goal is nothing compared to the main interest of molecular methods, that is, probing matter at the molecular level (Chen and Mathias 2002; De Pablo and Escobedo 2002; Sandler 2003). Indeed, molecular modeling can be seen as a “third way to explore real matter” (Allen and Tildesley 1987). Like a theoretical approach, it is based on a model system of the real one. But unlike theory, no hypothesis and no transcription of key phenomena into equations or correlations is performed. Rather, molecular modeling performs numerical experiments to simulate directly the behavior of the model system. The concept of numerical experiment is strong. First, the model system is made of a boundered molecular system and of an interaction model analogous to an experimental sensor that enables one to compute the internal energy of the model system. Second, think of the pseudoconstant thermometer temperature and of the Brownian motion of atoms in a liquid that generates a fluctuating temperature. More generally, any macroscopic property value measured by an experimental probe is a timeaverage over many instantaneous fluctuating values. Statistical thermodynamics postulates that this time-average equals an ensemble average over a statistically significant number of model system configurations. Molecular modeling generates them numerically using methods like molecular dynamics or Monte Carlo methods. Any property of interest is then derived using thermodynamical laws from instantaneous property value averages and correlation factors. Thirdly, numerical standard deviation associated to the ensemble average is the equivalent of experimental accuracy. This chapter presents molecular modeling concepts so as to demystify them and stress their interests for chemical engineers. Multiscale approach including molecular modeling are not illustrated due to restricted space. Rather, routine examples on the use of several molecular techniques suitable for acquiring accurate vapor-liquid equilibrium data when no data is available are provided. 3.2 What is Molecular Modeling?

Molecular modeling includes computer theoretical chemistry and molecular simulation. Computer theoretical chemistry calculations are carried out at 0 K and solve Schrodinger’s equation to obtain nuclear and electronic properties such as conformation, orbital, charge density, and electrostatic potential surface in fundamental or excited states. Computation time is huge, being proportional at best to N2.Selectrons,

3.2 What is Molecular Modeling?

which restricts its use to small systems. The precision of the results is significant because the only assumptions are linked to approximations carried out to solve Schrodinger’s equation. In particular, there are no adjustable parameters. Besides, it provides crucial information on the electronic distribution that enables one to evaluate electrostatic interactions in molecular simulation. Molecular simulation is a numerical technique used to acquire the physicochemical properties of macroscopic systems from the description, on an atomic scale, of the elementary interactions and from the application of statistical thermodynamics principles. It concerns the calculation of a model system internal energy at a positive temperature. Computation time is proportional to Nmolecules, which makes it a technique adapted to the study of real systems: phase properties, interfaces, reaction, transport phenomena, etc. Molecular simulation carries out dynamic modeling of the system subjected to realistic temperature and pressure conditions thanks to an adequate sampling of the system configurations. A configuration is a set of particle coordinates and connections. Inaccuracy may arise from the energetic models that contain fitted but physically meaningful parameters or from system configuration sampling techniques that must comply with statistical thermodynamic principles. Molecular simulation offers the most potential for process engineering. Wherever energetic-interaction-relatedphenomena have a prevalent place, molecular simulation deserves to be considered for use in studying and loolung further into the knowledge of the phenomena in the heart of the processes. In particular, it is suitable for the study of phase equilibrium, interfacial properties (specific adsorption on catalyst), transport coefficients, chemical reactivity, activity coefficients, etc.

3.2.1 Scientific Challenges of Molecular Modeling in Process Engineering

The use of molecular simulation in process engineering lies mainly in the difficulty of establishing the link between the macroscopic properties and their energetic description or that of significant parameters at mesoscopic or molecular scale. The micro-macro relation can be simple: in distillation, the knowledge of phase equilibrium data enables one to run an extensive study and design of the process. In tablet processing, the relation is more complex: the tablet properties (compactness, friability, dissolution) are related to the pellet’s cohesion and to the substrate’s solubility. Obviously, the energetic interaction is a key phenomenon and is taken into account through solubility parameters, which can be broken down into primary energy contributions (van der Waals repulsion-attraction, Coulombic interaction, etc.), precisely the applicability of molecular simulation. But particle size and solvent effects on the aggregate size and homogeneity are equally important, notwithstanding operational process parameters, and are still difficult to address at a molecular scale. So, identifying the limiting phenomena is a priority before any molecular simulation. The size of the model systems is not an unsolvable problem as periodic boundary conditions can be applied to replicate the original system box and mimic a homogeneous macroscopic phase. Rather, the scientific challenges concern issues often

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encountered in experiments, the sensor challenge, the sampling challenge, and the multiscale challenge. 3.2.1.1 The Sensor Challenge

For data-oriented simulations, accurate force fields/sensors are needed to evaluate precisely energetic interactions. The study of highly polar systems, reliable and relevant extrapolation of carefully set force field parameters, and the absence of temperature dependency of these parameters are key improvements of molecular simulation models over existing macroscopic models. The model system is usually a parallelepiped box filled with particles whose energetic interactions are described by a force field enabling one to compute the system internal energy. In order to mimic a homogeneous phase, the box is usually replicated in 3-D by applying periodic boundary conditions. The typical size ranges from 20 to 1000 8, and may vary during simulations. Edge effects are to be envisaged and can be attenuated by increasing the box size. The development of a force field requires a strong collaboration with theoretical chemists and physicists. Indeed, different lunds of force fields can arise: some based on quantum chemistry concepts and some based on molecular mechanics (Sander 2003). Quantum-based models are used in static modeling and naturally in computer theoretical chemistry calculations. Solving the Schrodinger equation, they provide the nuclear and electronic properties system and consequently the true energy of the system (e.g., the energy of ionization) that is physically measurable. Molecular mechanics models are used in molecular simulation to calculate intensive properties (T, P) and extensive properties, among which the internal energy of the system, which is not directly measurable by experiment, but enables one to calculate other thermodynamic properties by using thermodynamic laws. Properties like vaporization enthalpy connected to differences in internal energy are computed and can be compared to experiments. Quantum models (QM)are practical on a few tens of atoms at best and are being used more and more in combination with molecular mechanics models for some part of the system where accurate electronic distribution is needed, e.g., a reactive zone or to provide a description of the electronic distribution. Molecular mechanics (MM) models are the most used and are based on a springs and beads mechanistic description of the intermolecular interactions and of the intramolecular bonds. They allow calculations on several hundreds of particles, which enable one to model real systems in a satisfactory way. They contain physical parameters evaluated from quantum calculation but also empirical parameters, which must be regressed from experimental data. However, this empiricism is attenuated by some physical significance attributed to the parameters. Moreover, M M force fields show amazing properties. Valid over a large pressure and temperature range, they can be used to compute many properties and all molecules can be described from a small set of parameters if careful parameterization is conducted, which constitutes the first challenge.

3.2 What IS Molecular Modelmg?

ir I

Figure 3.1

Process engineering and molecular modeling

3.2.1.2 The Sampling Challenge

The second challenge requires a strong involvement of process engineers. Novel and smart methods must be developed to sample specific states of the model system, which are of great interest for process engineering, for instance, transition states that set the reaction energetic barrier, azeotropes that affect strongly the distillation process feasibility and design, dew points, etc. Usually, existing molecular simulation methods sample nonspecific states like a vapor-liquid equilibrium point. Unlike measurement time, its experimental equivalent, numerical sampling can be advantageously biased to sample the specific state of interest but it requires expertise to comply with statistical thermodynamics principles, which permit a bridging of the microscopic and macroscopic scales. Furthermore, for existing methods based on molecular dynamic or Monte Carlo methods, sampling efficiency should be improved, in particular for complex molecules like macromolecules, even if the alternate solution of running more simulations is still the leading choice as computer power increases. With this second challenge, process engineering finds a new use for molecular modeling: it cannot be solely data-oriented, but also discovery-oriented and assumes its status of numerical experiment. 3.2.1.3 Molecular Modeling in a Multiscale Approach

The integration of molecular modeling in applicable models for the study of macroscopic systems and their properties is of the utmost importance for process engineering. Indeed, often considered as decisive, phenomena related to energetic interactions have often been left aside during a process study because of a lack of suitable tools or incorporated into parameters. Thermodynamic models used in phase equilibrium calculations are a good example: binary interaction parameters must be found empirically despite their solid physical meaning. The first illustrative example addresses the issue of calculating binary parameters by molecular modeling methods.

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Process engineering models are knowledge-based models. In most domains, process study requires a multiscale approach. As a technique of experimentation, molecular modeling makes it possible to visualize on a molecular scale physicochemical phenomena. It can thus be used to develop or revisit theories, models or parameters of models and therefore improve our knowledge of processes and increase the capacity of predictions and extrapolation of existing models.

3.3 Statistical Thermodynamic Background

Suggested Readings 1

2

McQuarry D. A. Statistical thermodynamics. Harper and Collins. New York, 1976. ISBN 0060443669 Allen M. P. Tildesley D. J. Computer Simdation of Liquids. Oxford University Press, Oxford, UK. 1987. ISBN 0198556454

3

Frenkel D, Smit B Understanding Molecular Simulation. From Algorithms to Applications. Academic Press, San Diego, 1996. ISBN 0122673700

3.3.1 A Microscopic Description of Macroscopic Properties

Traditional thermodynamics and statistical thermodynamics address the same problems but differ in their approach: thermodynamics provides general relations without any consideration of the intrinsic constitution of the matter, while statistical thermodynamics supposes the existence of atoms, molecules, and particles to calculate and interpret thermodynamic properties at the molecular level. The objective of statistical thermodynamics is to describe the behavior of a macroscopic system in terms of microscopic properties of a system of molecular entities. The main idea is to evaluate an average property value and its standard deviation from a statistically significant number of configurations, much like a real experiment. Indeed, the temperature reading on a thermometer appears falsely constant. At the molecular level, a positive temperature is the result of atomic vibrations and collisions occurring on a time scale (e.g., IO-' s.) much lower than the sampling s.) (Fig. 3.2). Using statistical thermodyperiod of the experimental sensor (e.g., namic concepts, molecular simulation will do the same and perform a numerical experiment. Each instantaneous configuration (atomic positions and moments) of the system exists according to a probability distribution. The most probable will have the largest contribution to the computed average value. For the experimental system the macroscopic property X value is a time-average over a set of configurations r (t) sampled during the measurement time tmeas:

3.3 Statistical Thermodynamic Background

TO,

Measurement time (10-3s) >> fluctuations ( Figure 3.2

Measurement of a “mean” temperature and its relation with instantaneous temperature

s)

But knowing all configurations T(t)is impractical because the number of particles (6.023 x for a mole) and thus the number of positions and moments are incommensurable. Statistical thermodynamics was developed to solve this problem statistically. The first postulate of statistical thermodynamics is that “the value Ofany macroscopic property is equal to its average value over a sample of the model system conjgurations,” as shown here:

where ttotal is the number of sampled configurations. The notation ( )ensemble refers to a statistical ensemble. By definition it consists in a significant number of subensembles having the same macroscopic properties. The thermodynamic state of a macroscopic system is perfectly specified by a few parameters, for example the number of moles N,the pressure P, and the temperature T. From them, one can derive a great number of properties (density, chemical potential, heat capacity, difhsion coefficient, viscosity coefficient, etc.) through equations of state and other thermodynamic relations. Reproducing conditions occurring in experiments, the “canonical” NVT, and the “isobar-isothermal”NPTensembles are quite useful. The notations NVT and NPT mean, respectively, that the number of moles N + volume V + the temperature T and the number of moles N + the pressure P + the temperature T, are kept constant for each system configuration during simulations run in those ensembles. One considers that the postulate of statistical thermodynamics applies during simulations in a statistical ensemble on systems with a few thousands of particles replicated by periodic boundary conditions and that averages are made on a few million configurations. The sampling size and quality are often the Achilles’ heel of molecular simulations. 3.3.2 Probability Density

Equation 2 states that configurations have the same weight, on average, and the same probability of existence. This is the second postulate of statistical thermodynamics: “Allthe accessible and distinct quantum statesfrom a closed system offixed energy

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(‘microcanonic’NVE) are equiprobable.” Equation 2 is therefore rewritten: X(r(t))h,,

= (X)ensemble =

(x(r(r)) Pensernble(r(r))) ’

Second postulate

(3)

r(r)

where pensemble(I’)is the probability density, which is the probability of finding a configuration with positions and moments r(t).In the NVT ensemble, any configuration probability density is connected to its energy E and to the total partition function, namely the sum over all configurations, by the Boltzmann formula:

am,

Two points are noteworthy: 1. The knowledge of the partition function allows the calculation of all thermodynamic properties. But this can never be done fully but rather imperfectly through the generation of a statistically representative number of configurations. 2. A model is required to evaluate any configuration energy in order to calculate the partition function. This is done through a force field.

3.3.3 Average, Fluctuations, and Correlation Functions

Equation 2 is the usual mean formula to calculate an average value (molar fractions, etc.). Other properties (heat capacity, etc.) are calculated from the variance expressing the fluctuations around the mean:

( x ) Variance ~

(5)

Correlation coefficients give access to properties describing the dynamic state of the system. The nonnormalized form of the correlation coefficient over T configurations is:

C x(tO).x(to+~)Correlation coefficient

Ittotal

correlxx(r) = ( x ( T ) . x (=~~))

ro=l

(6)

The integration of the nonnormalized correlation coefficients enables one to directly calculate macroscopic transfer coefficients (diffusion,viscosity, or the thermal diffusivity coefficient). Their Fourier transform can be compared with experimental spectra.

3.4 Numericd Sarnp/ing Techniques

3.3.4 Statistical Error

Molecular simulation is a numerical experiment. Consequently, the results are prone to systematic and statistical errors. The systematic errors must be evaluated, and then eliminated. They are caused by size effects, bad random number generation, and insufficient equilibration period (see below). Statistical errors are inversely proportional to sampling and are thus zero for infinite sampling. On the assumption that the Gauss law applies, the statistical error is the variance (Eq. (5)). However, sampling a large but finite number of configurations induces a correlation between the Ttotd configurations that persist during a certain number of successive Configurations. A statistical factor of inefficiency s is introduced to evaluate the number of correlated successive configurations. The ttotal configurations are cut into nb blocks of ‘Gb configurations upon which the average ( 4 b and its variance ( S * ( a b ) are computed. By selecting several increasing values of o b , the statistical inefficiencys and the statistical error 02((X)total) is evaluated:

3.4 Numerical Sampling Techniques

The generation of a statisticallyrepresentative sample of the model system configurations is mainly done by two techniques: molecular dynamics and the Monte Carlo method. They obey the principles summarized in Fig. 3.3. Both methods differ in their applications. The Monte Carlo method is adapted for the study of static phenomena (equilibrium and static interface) while molecular dynamics is suitable for the study of dynamic phenomena (shear induced flow). A phase equilibrium easily computed using Monte Carlo methods would be difficult to reach in molecular dynamics because of the time needed and of boundary effects near the interface.

3.4.1 Molecular Dynamics

Molecular dynamics generates a trajectory by integrating the classical equation of motion over time steps 6t starting from an initial configuration whose particle positions and velocity are known (Fig. 3.3). The n* configuration can be traced down the initial one by reverse integration. In the equation of motion (Fig. 3.3), the forces Fi acting on the particle of mass mi are equal to the derivative of the Vi(r)potential describing the interactions of the particle i with its surroundings.

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Trajectory

C b-= m ddV, t

+

1 1 ?&

I

Figure 3.3

i

Molecular Dynamics

Monte Carlo

Random configurations

I _.

.:,

...

I ;j‘:.’

1

Basic concepts of the Monte Carlo method and molecular dynamics

The integration of the differential set of equations is carried out mainly by Verletlike algorithms rather than by Gear-like algorithms, which are widespread in process engineering. The Verlet algorithm calculates the new particle positions r(t) using a 3rdorder Taylor expansion and replacing the second derivative by the forces thanks to the equation of motion, one gets a formula with no velocity term: r(t

F ( t ) tit2 + tit) = 2r(t) - r(t - tit) + . - + o(tit4) m 2!

(9)

Velocities are computed afterwards: dr(t) dt

-

u ( t ) = --

r(t

+ tit) - r(t - tit) 26t

+ o(st2)

(10)

This algorithm shows several interesting characteristics: (1)It is symmetrical with regards to 6t, which makes the trajectory reversible over the time. (2) It preserves the total energy of the system over long periods of integration, a key point to get long trajectories and deduce with accuracy some correlation functions. In particular, it is more precise than the Gear-like algorithms for large 6 t (the reverse is true for small 6 t ) making it suitable to simulate long trajectories, which is our goal. (3) It requires less data storage than Gear-like algorithms. Transport coefficients (self-diffusion,thermal diffisivity, and viscosity) are computed from autocorrelation coefficients, the “Green-Kubo”formulas, for instance, the coefficient of self diffusion Diis related to the relative particle velocities:

Similarly, viscosity is obtained from the shear stress tensor autocorrelation coefficient related to the pressure exerted on the particle and the thermal diffusivity is obtained from the energy flow autocorrelation coefficient.

3.4 Numericd .Samp/ingTechniques

A challenge with molecular dynamics run in a statistical ensemble where the temperature is set constant, is keeping it constant when moving and interacting particles inevitably heat the system. A solution is to place the system in a large thermostated bath periodically set in contact with the model system through techniques like the Andersen or Nose-Hoover methods.

3.4.2 Monte Carlo Method

The Monte Carlo method generates system configurations randomly. The n* configuration is related to its preceding one but it is impossible to go back to the initial configuration. First of all, randomness is particularly critical and has given its name to the method in reference to the Monte Carlo casino. The advice is to always use a published robust random number generator and never try to build one or use the falsely random precompiled “Ran”function on a computer. Systematic deviation and repetitive sequences can be checked by running simple tests. The second key issue is sampling (Fig. 3.4). Uniform sampling allows a good estimate of the partition function needed to compute all macroscopic properties, but at the expense of sampling high energy and thus improbable configurations. Preferential sampling (or metropolis sampling) samples the configurations with the largest contribution in the calculation of the partition function and of averages. The disadvantage of metropolis sampling is that the partition function (equivalent to the surface under the curve) is no longer correctly evaluated. Thus, the question arises of finding how to generate the configurations with a correct probability distribution without having to calculate the function of partition that occurs in the definition of the probability density (Eq. (4)). The solution is to obey the microscopic law of reversibility. Given an old (0)and a new (n) configuration, their probability densities p are proportional to exp(-E(,,)/kBT)and exp(-E,,,/kBII) in the NVTensemble (Eq. (4)).Defining the transition probability M(o+n) of going from (0)to (n), the microscopic reversibility states that at equilibrium the number of transitions from (0)to (n) and from (n) to (0)corrected by the probability densities must be equal.

In addition, an acceptance criterion is introduced acc(o + n) along with a ( o-+ n) an (0)to (n),which is supposed to be symmetrical (a(o+ n) = a ( n + 0)):

a priori probability of trying to go from

M(o + n) = a(o + n) . acc(o -+ n)

(13)

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Then by exploiting the symmetry of a , Eq. (12) becomes:

(-(%

acc(o -+ n) - pg& acc(n -+ 0) = eXP

&

kbT

)

(14)

In this equation, the partition function that is so difficult to calculate no longer appears. The metropolis idea is to choose acc(o+n) asymmetrically. As indicated in Fig. 3.5:

If the new configuration energy is lower than the old one, the transition is always accepted If the new configuration energy is higher than the old one, one picks a random number between 0 and 1: o if = 5 exp(-(E(,) - E(,,))/keT), the transition is accepted, o if 5 = > exp(-(E(,, - E(o,)/kBTj,the transition is rejected.

c

Applied to Eq. (14), this acceptance criterion enables one to define the acceptance probability of a random displacement in general and in the NVT ensemble:

Last, a symmetrical a(o + n) is chosen to allow a sampling effective in terms of acceptance and efficient in terms of the configuration space. Usually one defines a maximum value associated with the transition, like a maximum displacement d,that is fxed in order to satisfy a rate of 50 % of accepted transitions. If several movements are possible (e.g., translation, rotation, and volume), the type of movement will be chosen randomly from a predetermined statistical distribution. Again, one insists on the randomness of the choices of particle and of the type of movement in order to respect the microscopic reversibility. In ensembles other than NVT, probability densities are corrected with respect of the microscopic reversibility law.

probability

1

r\

probability

ILAttt

Uniform sampling Figure 3.4 Uniform and preferential sampling

ttt

Preferential sampling

configurations

r.

0 Figure 3.5

E,,,-E,,

AE

Metropolis preferential sampling. Criterion of acceptance

3.4.3 Phase Equilibrium Calculations using Cibbs Ensemble Monte Carlo

The Gibbs ensemble was developed by Panagiotopoulos in 1987 to simulate vaporliquid equilibria. Simulations are carried out in a NVT ensemble on two microscopic boxes located within two homogeneous phases far from any interface. Each box is simulated with periodic boundary conditions. Constant total volume V and total N particles are divided between the two phases V,, N1 and V,, N2. The temperature is set constant in the simulations and random movements are performed to satisfy the phase equilibrium conditions as described in Fig. 3.6: 0

0

0

Displacements (translation, rotation) within each phase to ensure minimal internal energy; A change of volume proportional between the phases: AVl = -AV2 so that the total volume is constant. This should satisfy the pressure equality. Transfer of the particle from one box to the other to equalize the chemical potentials.

The acceptance probabilities of the various movements in the case of a single component system are given below. For the translation in each area:

For the change of volume, V, is being increased by A V and V2 is being decreased by just as much:

with A V chosen by generating a uniform random number 5 between 0 and 1 ; 6 V, being the change of maximum volume adjusted to obtain a fixed percentage (e.g., 50 %) of acceptance of the move: AV =

< . SVma, . min(V1, V2)

(18)

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For the transfer of a particle of area 2 to area 1:

One of the main difficulties of the Gibbs ensemble Monte Carlo method resides in the transfer of particles to satisfy the chemical potentials equality because of the difficulty to insert polyatomic molecules in the dense phase. An alternative is to seek open spaces where insertion is eased into the particle. This affects randomness and introduces a statistical bias like the configurational bias method, which consists of inserting segment by segment a molecule in a phase. The probability of acceptance of the transfer of the particle represented by Eq. (19)is then modified by introducing the energy differences A Ei into weighting factors Wi that represent the total energy of interaction with the surroundings of the inserted molecule. For an L segment molecule inserted in m possible directions:

More generally, the introduction of a bias consists of defining an a pliori probability a(o+n), which is no longer symmetrical. Equations 14 and 15 become:

Real bulk phases

Initial

Gibbs Ensemble Monte Carlo

state

Mavement type

Internal l.-placements

-.

Equilibrium conditions Figure 3.6 ensemble

AV, = -A&

............ .-

E, minimal

Principles of phase equilibrium simulations in the Gibbs

Partide lransfer .........

P, = P,

&,I = K,2

3.5 Interaction Energy I 1 2 1

To conclude this section, both molecular dynamics and Monte Carlo methods require the calculation of the interaction energy; for molecular dynamics to derive forces exerted on the system particles and for the Monte Carlo method to calculate the acceptance criterion. The following section reviews the main features of the force fields enabling to calculate intra and intermolecular interaction energies. 3.5 Interaction Energy Suggested Readings 1

Leach A. R. Molecular Modelling, Principles and Applications. Longmann, Harlow, UK, 1996

2

Karplus M. Porter R. N. Atoms and Molecules. Benjamin inc., New York, 1970

3.5.1 Quantum Chemistry Models

Quantum chemistry models are never used alone in molecular simulation because of the still prohibitive computation time. However, they must be considered as they can provide less-sophisticated molecular mechanics models with partial electronic charges and various dipoles useful for computing Coulombic and dipolar interactions. They can also provide accurate spring constant values describing the bonding intramolecular interactions associated with the various oscillatory modes within the molecules (stretching, bending, and torsion). In quantum chemistry, only atomic nuclei surrounded by revolving electrons are considered. Calculations provide the nuclear and electronic properties system and the true total energy of the system. Total energy is related to the general time dependent wave function Y (r, t) by means of the generalized Schrodinger equation: where H is the Hamiltonian, a mathematical operator with kinetic and potential energetic contributions. Apart from an analytical solution for the sole hydrogen atom, Schrodinger equation solutions are always approximate to some degree because a compromise must be made between computation time and accuracy. Three levels of approximation are considered, namely ab initio methods, mean field methods, like density functionnal theory (DFT),and semiempirical methods. Among ab initio methods, configuration interaction (CI) methods are the most accurate, but the slowest. Calculated energy values have a precision comparable with

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experimental ones (0.001 ev). CI solutions are obtained by minimizing a linear combination of the wave functions associated with the system fundamental state and all excited states. The self consistent field molecular orbital concept considers atomic orbitals that represent wave functions of electrons moving within a potential generated by the nucleus and by an average effective potential generated by the other electrons. The best such wave functions are Hartree-Fockwave functions and solve the Schrodinger equation for a given electronic configuration (e.g., the fundamental state) without any empirical parameter. They can be used for CI calculations. Atomic orbital wave functions are approximated using Gaussian functions, which leads to peculiar denominations like STO-3G Hartree-Fock calculations (use of a basis set of three Gaussian functions). The larger the basis set, the longer and the more accurate the calculation. The semiempirical methods are the most approximated quantum methods: Hiickel calculations can be done on a sheet of paper; finer semiempirical models enable one to obtain with good precision the ionization energy, optimal conformations, and electronic surface potential. However, they present the disadvantage of calculating the wave functions approximately by replacing various integrals by fitted empirical parameters. Between the two levels of approximation, one finds the mean field methods of the popular density functionnal theory. The idea is rather than to seek to solve the exact Hartree-Fock problem in an approximate way, one could seek to solve an approximate problem in an exact way. That consists of modifylng the Hamiltonian operator and replacing the term of exchange of correlation accounting for multiatomic orbital interactions by the electronic density pi. The results are obtained with satisfactory accuracy and much faster than Hartree-Fockcalculations, enabling one to even study periodic systems of interesting size. Nevertheless, all quantum mechanical calculations are performed for a static configuration of the system under 0 K conditions. But as a provider of key properties like electronic distribution, they should be systematically used in any molecular simulation aiming to be quantitative.

3.5.2 Molecular Mechanics Models

Molecular simulation uses molecular mechanics models to calculate the internal energy of the system. It considers that the molecules can be represented by centers of forces like beads and the bonds can be represented by springs (Fig. 3.7). As Fig. 3.7 shows, the total internal energy is the sum of intramolecular or bonding interactions and of intermolecular or nonbonding interactions. The set of molecular mechanics parameters is called a force field. Intramolecular energy takes into account vibration phenomena between bonded centers of forces. As the beads and spring model suggests, they are described by harmonic functions and handle stretching, bending, torsion as well as improper rotation

3.5

/flterOdOfl

Energy

if needed. Average parameters lo, +o and harmonic constants ki and are usually fitted to accurate vibration energy calculations made with quantum mechanical methods. Intermolecular energy takes into account the two-body interactions between the centers of forces. Three-body interactions are rarely included. Short range interactions can be described by a van der Waals potential modeled by a 12-6 Lennard-Jones function. The 1/rI2term represents the repulsive contribution, which becomes significant below 3 A. The 1/4term represents the attractive contribution related to the dispersive effect of induced dipoles. More rigorous forms may include 1/$ or l/rl’ terms or other functional forms (Buckingham potential, “exponential-6”potential, etc.). Electrostatic interactions are a major contribution to intermolecular energy as they are long-range interactions felt up to a distance of 25 A for multicharged ions. Permanent dipole and multipole are rarely included, but Coulombic interactions related to partial atomic charges qiand q, are a must. All electronic parameters (dipoles, partial charges) should be fitted to electronic surface potentials computed by quantum mechanics to improve quantitative predictions of molecular simulations. Hydrogen bonding interactions are either modeled explicitely by a 12-10LennardJones function or assumed to be implicitely taken into account in the van der Waals interaction. The functional form of the molecular mechanics energy shows that it is not a true energy that can be measured experimentally. Rather, for a single molecule, it is zero at its most stable conformation, whereas true zero energy corresponds to the protons, neutrons, and electrons infinitely split apart. Molecular mechanics predictions of conformations are in excellent agreement with experimental ones. Nevertheless, the practical use of molecular mechanics is great because for a system of several molecules, it computes the thermodynamic internal energy from which many interesting properties can be derived. Force fields can be of all-atoms (AA) type, as shown in Fig. 3.7, in which there is a center of force on each atom. Their names are Dreiding, Universal Force Field, Compass, OPLS, etc. But other types exist where atoms are grouped (e.g., -CH3) under a single center of force in order to reduce the computing time of short range interactions. This leads to united atoms (UA) force fields. In all cases, long-range electrostatic interaction is split in as many centers as possible, usually on all atoms and sometimes on virtual centers. A similar idea is at the origin of polarizable force field like the anisotropic united atoms (AUA), which intends to take into account the electronic cloud shift when two particles approach: the charged center is displaced along the resultant of the nearby bonds. Since all intramolecular parameters and electrostatic parameters are systematically derived from quantum mechanical calculations, molecular simulations using molecular mechanics force fields have greatly improved their accuracies. However, even if for a particle i, Lennard-Jones parameters ui and E~ are, respectively, associated with the collision diameter (the distance for which energy is null) and with the potential well. They must still be fitted to some extent, as will be shown later using experimental data (enthalpies, formation energies, densities, etc.).

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I-stretching

4,s;

Figure 3.7

2-bending

3-torsion

--

%Van der Waals 5-Coulomb

Typical molecular mechanics force field.

For multi component systems, the diameter ay and the energy parameter E Y are obtained from pure substances using traditional mixing rules like those of LorentzBerthelot:

These very simple rules have rarely been questioned, more proof of the strong physical basis of molecular simulation. Furthermore, they highlight that the study of a system with M different centers of forces only requires the knowledge of 2M parameters, whereas a traditional approach with a thermodynamic model with binary interaction parameters would require M(M+1)/2 such parameters. Even if the main functional forms of the potentials (stretching, bending, torsion, van der Waals and Coulomb) are present in all quoted force fields, the choice must be made by knowing the type of experimental data used to regress the Lennard-Jones parameters and the way electrostatic interaction are described. Mixing LennardJones parameters from several force fields without any confrontation to experimental data is acceptable only if qualitative results are sought. 3.6 Running the Simulations

How should one represent the behavior of macroscopic systems when the model system typically contains only a few thousand particles?The problem is solved by adopting periodic boundary conditions that duplicate in all directions identical images of the model system. In molecular dynamics, care should be taken that any particle moving through one wall of the main image will reenter at the opposite wall with the same velocity. Notice that the interaction energy of a particle must include interaction with all included replicated particles. However, for long-range interactions this would require too much computer effort and limiting techniques are implemented: rough ones, such as a “cutoff distance beyond which the interaction is supposed to be null, or more accurate ones like the Ewald summation. In the case of a cutoff, it is necessary to include long range corrections.

3.7 Applications

The initial particles in the box are usually set along a periodic network to avoid overlaps that would result in an infinite energy. Then, a statistical ensemble and a sampling technique are chosen. Force field parameters are associated with all force centers and for molecular dynamics simulations, a statistical distribution of initial velocities is set. Finally, the simulation is launched. It consists of a phase of equilibration and a phase of production. The purpose of the phase of equilibration is to bring the system from an initial configuration to a configuration representative of the system: random distribution of the molecules and the velocities within a system with imposed thermodynamic conditions (that of the chosen statistical ensemble). In molecular dynamics under fured temperature T,the system in gradually heated to the T set value. The phase of production starts when key properties like potential energy, pressure, and density fluctuate over mean values. Each configuration then generated is kept to calculate the macroscopic properties from averages of fluctuation to coefficients of correlation. As statistical error decreases when the number of configuration increases, at least 10' configurations should be generated. 3.7 Applications

Vapor-liquid equilibrium calculations are a major field of investigation because of the importance of processes like distillation. Too often, data are missing. We present two approaches that use molecular modeling to obtain such data. The first example aims at computing binary interaction parameters occurring in the UNIQUAC activity coefficient model. The second example directly computes the equilibrium compositions using a Gibbs ensemble Monte Carlo method.

3.7.1 Example I: Validation of the UNIQUAC Theory 3.7.1.1 Overview o f UNIQUAC Model

The practical calculation of vapor-liquid equilibrium (Eq. (26)) involves an activity coefficient (yi to describe the nonideality of the liquid phase due to energetic interactions.

By applying the thermodynamic relation of Gibbs-Duhem, one connects the individual coefficients of activity yi with the excess gibbs energy GE:

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The UNIQUAC model proposes an expression for the G Ewith two contributions: a combinatorial part that describes the dominant entropic contribution and a residual part that mainly occurs due to the intermolecular forces responsible for the mixing enthalpy. The combinatorial part is related to the composition xi and to the molecule shape and size. It requires only pure component data. The residual part depends, in addition, on the interaction forces embedded into two binary interaction parameters Ay and Aji. GE

RT

combinatorial

residual

Parameters ri, qi,and qi' are molecular constants for each pure component i, related respectively to its size, its external geometrical surface, and its interaction surface. q' can be different from q, in particular for polar molecules. The model system upon which the UNIQUAC theory was developped considers interacting molecules. Then, the two binary interaction parameters Ay and A,i can be expressed in terms of interaction energies Uvbetween dissimilar molecules i and j, and Uii between similar molecules i:

where NAis the Avogadro number. The Wilson activity coefficient model also proposes two binary interaction parameters. It is a simplification of the UNIQUAC model in which parameters r, q, and q' are all set to unity. Interaction surfaces are not taken into account and molar volumes are eliminated in the equation. The relationship between the Wilson and UNIQUAC parameters is as follows:

where V; and y a r e the molar volumes of components i and j. The traditional approach consists of regressing Av and A,i from experimental data, with all drawbacks associated with this approach: data specific parameters, poor extrapolation capacity, temperature and pressure dependency, and the need of experimental data. A few years ago, an attempt to directly calculate the binary interaction parameters was made and is reported below.

3.7 Applications

3.7.1.2 Calculation Using Molecular Mechanics UNIQUAC Binary Interaction Parameters

In 1994, Jonsdottir, Rasmussen and Fredenslund (1994) computed interaction energies between isolated couples of molecules. They used molecular mechanics models not in a molecular simulation perspective, but rather like a quantum mechanical approach. For a given orientation of the two molecules, an energy minimization was run to reach a stable conformation. Many orientations are selected and the mean interaction energy Ui,and U, is evaluated by weighting each value using its Boltzmann factor exp(- Uti/kBT). This corresponds to a rough sampling, obviously not statistically representative as only a few hundred couples are investigated. This questions the validity of the first statistical thermodynamic postulate that equals the ensemble average and macroscopic time-average value. Rightfully, the authors claim to perform a molecular static approach in between quantum mechanics and molecular simulation approaches. The consistent force field parameters are optimized for the alkanes and ketones that are the molecules of interest but no value in particular and no partial atomic charge values are provided. Alkane conformers are taken into account, however, which is an advantage of molecular modeling approaches over classical parameter fitting. The Uiiand U,interaction energies are computed as the difference between the molecules couple energy and the energy of each isolated molecule:

Simulation results are used with the UNIQUAC equation to predict vapor-liquid equilibrium data (Eq. (26)),which are compared with experimental ones: 0

0

For the alkane/alkane systems (n-butaneln-pentane; n-hexane/cyclohexane (Fig. 3.8) and n-pentane/n-hexane), the relative error ranges from 1.1 to 4% for the pressure and the absolute error tandis ranges from 0.011 to 0.042 for the molar fractions. For the alkanelketones (n-pentane/acetone; acetone/cyclohexane (Fig. 3.8); cyclohexane/cyclohexanone), the relative error ranges from 4.3 to 17.6 % for the pressure and the absolute error ranges from 0.016 to 0.042 for the molar fractions.

In conclusion, the error increases along with the molecule polarity. One may question the force field ability to handle electrostatic interaction in addition to likely insufficient sampling of the system configurations. Finally, the authors tested the Wilson model and found errors four times greater for the n-pentane/acetone system. They concluded that the UNIQUAC equation has a better physical basis than that of the Wilson equation. This result was foreseeable since the Wilson model is a simplified form of the UNIQUAC model.

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n-hexane I cyclohexane

12%!0

'

0:2

'

0:4

'

0:s

'

40.0

0:s

'

1jO

acetone I cyclohexane

XI

XI

1 .o I

- 30.0 sn m

20.0

10.0' 0.0

'

'

0.2

'

'

0.4

'

'

0.6

'

'

0.8

'

'

1.0

0.0 0.0

'

s

0.2

'

'

'

"

0.4

XI

"

0.6

0.8

1.0

XI

Bubble curve at 298.15 K for the n-hexane/cyclohexane (top) and acetone/cyclohexane (bottom) systems. Solid line: simulation. Stars: Figure 3.8

experimental data (reprinted from Jonsdottir, Rasmussen and Fredenslund (1994) with permission from Elsevier)

3.7.1.3

of UNIQUAC Binary Interaction Parameters Compared to Jonsdottir, Rasmussen and Fredenslunds (1994)work, Sum and Sandler (1999) used quantum mechanics models to improve the representation of electrostatic interactions. Often in such calculations, no rigorous sampling is performed, the emphasis being made on minimizing the total system energy. For each binary system, eight molecules are considered (four of each). A stable system conformation is found by minimization using semiempirical methods. Then the energy is minimized using ab initio methods (Hartree-Fock method with an extended basis set 6-3119<*G(3d, 2p)). Couples of molecules are then isolated and their interaction energy is computed according to Eq. (31). The average interaction energy is computed at best on 10 pairs of molecules. Then binary interaction parameters are derived for the UNIQUAC and Wilson equation, which enables one to com-

Ab lnitio Calculation

3.7 Appkations

pute vapor-liquid equilibrium data using Eq. (26).The systems studied are highly polar: water-methanol;water-ethanol;water-formicacid; water-acetic acid, and wateracetone. Simulation data are then compared with experimental data and with prediction using the activity coefficient group contribution method UNIFAC. The results obtained for the Wilson model are never quantitative and are even qualitatively wrong as it does not manage to reproduce the azeotropic behavior of the water-ethanol mixture (Fig. 3.9). On the other hand, despite the poor sampling, simulations with the UNIQUAC model give good quantitative results, comparable with experimental data and UNIFAC predicted data. Two points are significant: no experimental data were used at any stage and no temperature or pressure conditions were set, which is an advantage over regressed binary parameters. Indeed, the same set of water-acetone parameters is used to generate accurate data over a large temperature and pressure range (Fig. 3.9). In conclusion, when no experimental data are available, vapor-liquid equilibrium data can be predicted using UNIQUAC binary interaction parameters directly computed with molecular modeling methods. If the sampling issue is not yet settled, quantum mechanics methods, which accurately describe electronic distribution, have demonstrated their use, while force field approaches did not for polar systems. The next example shows that accurate predictions can be made with carefully set force field approaches using efficient sampling of phase equibrium systems thanks to the Gibbs ensemble Monte Carlo method. 5.00

.--

4.00

T = 523.15 K

-z 3.00 ep!

' $

I

2.00

0.08 0.06 0.04

0.021 0.0 0.2

0.4

0.6

XI, Y1

0.8

I

1.0 -PPRSV-WS

0.0 0.2

water I ethanol Figure 3.9 Vapor-liquid equilbriurn at 298.15 K for the water-ethanol

and water-acetone systems (reprinted with permission from Sum and Sandler (1999)

0.4 0.6 XI, Y1

0.8

water I acetone

1.0

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3.7.2

Example 2: Direct Prediction of Nitrile Vapor-Liquid Equilibrium

As highlighted before, the development of molecular simulation as a systematic provider of accurate physicochemical data is impeded by the availability of accurate force fields. In the 1980s, force fields were derived to reproduce physicochemical data of monophase systems or devoted to macromolecules of biological interest (amino acids, proteins, etc.). But the simulation of multiphase systems was neglected until the Gibbs ensemble Monte Carlo method and the active development of new AA-, AUA-, and UA-type force fields like, OPLS by Jorgensen, Madura and Swenson (1984),Trappe by Martin and Siepmann (1998), NERD by Nath, Escobedo and de Pablo (1998),Exp6 by Emngton and Panagiotopoulos (1999),and AUA by Toxvaerd (1990, 1997) added to this world effort. As stated before, force field development consists of deriving short range van der Waals interaction parameters like the (T and E Lennard-Jones parameters. But the challenge is to obtain generic values that can be used for many molecules, much like in a group contribution approach, and for many properties with various sampling techniques, such as phase equilibrium data (using the Gibbs ensemble Monte Carlo method, transport coefficients, and molecular dynamics) and absorption isotherms (Monte Carlo). By comparison, no existing macroscopic model can compute such a wide variety of properties using so few parameters. In the AUA4 model, generic parameters have been derived for linear, branched, and cyclic alkanes, aromatics, hydroxyl, carboxyl, and thiol groups (Delhommelle, Granucci, Brenner et al. 1999; Ungerer, Beauvais, Delhommelle et al. 2000; Delhommelle, Tschinvitz, Ungerer et al. 2000 Bourrasseau, Ungerer, Boutin et al. 2002; Bourrasseau, Ungerer and Boutin 2002). For the nitrile group -C=N, we proceeded as follows (Hadj-Kali, Gerbaud, Joulia et al. 2003): 1. Quantum mechanics calculations using DFT for the acetonitrile molecule, for which many experimental data are available, to find a stable conformation, determine harmonic constants for the intramolecular contribution of the force field potential, and determine discrete partial atomic charges from quantum electrostatic surface potentials. 2. Setting up the acetonitrile (CH,CN) force field for which CH3 Lennard-Jones parameters are taken from the generic databank of the AUA4 force field. The same general expression shown in Fig. 3.7 is used. 3. Running Gibbs ensemble Monte Carlo simulations to identify missing ( E ~ oN) , and ( E C . ac) Lennard-Jonesparameters of the nitrile group. The acetonitrile molecule is fully flexible and long-range electrostatic interactions were evaluated with a cutoff and tail corrections. Reference experimental data (Francesconi, Franck and Lentz 1975; Chakhmuradov and Guseinov 1984; Kratzke and Muller 1985; Warowny 1994) are the saturated vapor pressure ln(Pat)at 433.15 and 453.15 K, the vaporization enthalpy AH,,,, and the liquid density elis at 273.15, 298.15, 433.15, and 453.15 K. The optimization method is a simple gradient method and the objective function used is a square mean root function with uncertainty values set equal to 0.1 for ln(Psat),1 kj mol-' for AHvap,and 10 kg m-3 for elis.

3.7 Applications

4. Once aN,ON) and EC, ac) values reproduce acetonitrile data accurately, their genericity is evaluated by predicting, with no further parameter adjustment, vaporliquid equilibrium data of other linear nitriles (propionitrile, butyronitrile). In these molecules aN, u N ) and ac, ac) are taken as being equal to the values obtained for acetonitrile, whereas CH2 and CH3 Lennard-Jones parameters are extracted from the AUA4 databank. The harmonic constant, partial charges, and stable conformations are obtained from quantum DFT calculations.

DFT electrostaticsurface potentials are fitted to partial atomic charges using the simple Mulliken population analysis, which equally splits the electronic distribution according to the van der Waals radius, or using the MEP method, which mimics the electrostatic potential surface with a least square method. As shown below, the MEP analysis gives the best results, but does not pass the genericity test. Each atom bears a partial charge. All quantum calculated conformations and dipolar moments agree with experimental data (Goldstein, Buyong, Lii et al. 1996) and harmonic constants agree with literature reference values (Goldstein, Buyong, Lii et al. 1996; Ungerer, Beauvais, Delhommelle et al. 2000). Parameter regression requires that each optimization cycle (two are used) performs 16 Gibbs ensemble Monte Carlo simulations to compute the gradients varying [(EN

6 E N , ON); (EC, OC)]? [(EN,

6ON); (EC, OC)], [(EN, ON); (ECt OC

+ 6cC, ac)]for each of the four temperatures considered.

6aC)], [(EN! ON);

(EC

Each simulation takes 20 h on a Linux Pentium IV, 1.9 GHz with 512 Mb RDRAM. Equilibration period requires lo6 configurations and the production period ranges from 2.3 X lo6 to 4.5 X lo6 configurations. Results of the optimization of the Lennard-Jonesparameters are shown in Table 3.1. As indicated below (Fig. 3.10), the set of MEP parameters gives the best results for the acetonitrile with a mean standard deviation over all reference values of 1.9 % and a very good estimate of the critical point. With an underestimation of the vapor densities, an overestimation of liquid densities at elevated temperature and a poor estimation of the critical point, the Mulliken set gives an error of 3.1 %. But MEP generic character is poor, whereas the Mulliken one is excellent for propionitrile and butyronitrile vapor pressure predictions (Fig. 3.11). A possible explanation for the poor MEP predictions lies in the charge values computed for the propionitrile and n-butyronitrile (Hadj-Kali2004). The least square fitting of the quantum calculated electrostatic potential surface has in that case led to unphysical values with positive nitrogen atomic charge, in contradiction with the well-known electronegativitycharacter of this atom. Also, the MEP nitrile uc parameter value, which is too elevated compared to other uc values associated with other carbonated chemical groups of the AUA4 force field (Table 3.2). Correctly, the oc value with the Mulliken distribution follows a decreasing trend as the carbonated chemical group size decreases. So, two criteria for a generic set of Lennard-Jonesparameters are able to model the van der Waals interaction are: (1)physically meaningful values of the Lennard-Jones u and E parameters and (2) physically meaningful set of atomic charges representing the electrostatic potential surface of the molecule. The importance of representation of the electrostatic potential surface has also been acknowledged in COSMO

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approaches for the computation of physical properties, which recently won the first industrial fluid properties simulation challenge (Sander 2003; Case, Chaka, Friend et al. 2004). Critical points in Fig. 3.10 are obtained using the king method (Frenkel and Smit 1996). For direct simulations near the critical point, it is difficult to achieve convergence. As shown in Fig. 3.12 for the density versus configuration plot for a simple Lennard-Jonesfluid, the fluctuations increase and boxes interchange as the reduced temperature nears 1. Experimental observations of a similar phenomenon are well known and demonstrate that the molecular simulation is indeed a numerical experiment that can not only compute accurate physicochemical data, but also behave as an efficient sensor of system behavior on a molecular scale.

3.8 Conclusions

Molecular modeling is an emerging discipline for the study of energetic interaction phenomena. A molecular simulation performs numerical experiments that enable one to obtain accurate physicochemical data provided sampling and energy force field issues are addressed carefully. Still computationally demanding, molecular modeling tools will likely not be used “online” or be incorporated in process simulators. However, rather like computer fluid dynamics tools, they should be used in parallel with existing efficient simulation tools in order to provide information on the molecular scale about energetic interaction phenomena and increase the knowledge of processes that must manufacture ever more demanding end products.

+

300

4)

A simulations (Mulliken charges)

b

3.8 Conc/usions A

I

133

n-butyronitrile - Chakhmuradov 84

A n-butyronitrile (Mulliken charges) t

propionitrile - Chakhmuradov 84

0 propionitrile (Mulliken charges)

t

A

I

A

1 100

1I380

II570

lnemperature (K].'

Figure 3.1 1 Predicted saturated vapor pressure for propionitrile and butyronitrile. Mulliken charges

Tr = 0.8

Tr = 0.98

Tr =0.94 '

I

600

600c

400 200 OO

2 . M

Figure 3.12 Density versus configuration number in the vicinity of the critical point for a Lennard-Jonesfluid Table 3.1

Optimal ( E ~ uN) , and (EC,(JC) parameter values for the nitrile group. Mulliken charges

(K)

(A)

E C / ~(K)

€NlkO

Charges MEP optimization

50.677

65.470

3.5043

3.3077

Charges Mulliken optimization

95.52

162.41

3.2183

3.5638

~~

UC

uN

~

Table 3.2

Comparison of o Cparameters for various chemical groups in the A U M force field ~~

AUQ chemical group

(Jc

(4

-CH,

=CH,

=CH

-C MEP charges

-C Mulliken charges

3.6072

3.4612

3.3625

3.5043

3.2183

!je+o6

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References 1 Allen M. P. Tildesky D. J. Computer Sirnula-

2

3

4

5 6

7

8

9

10

11

tion of Liquids. Oxford University Press, Oxford, UK, 2000. Bourrasseau E. Ungerer P. Boutin A. Fuchs A. Monte Carlo Simulation of Branched Alkanes and Long Chain n-Alkanes with Ansotropic United Atoms Intermolecular Potential. Molec. Sim. 28(4) (2002) p. 317 Bouvasseau E. Ungerer P. Boutin A. Prediction of Equilibrium Properties of Cyclic Alkanes by Monte Carlo Simulations-New Anisotropic United Atoms Intermolecular Potential-NewTransfer Bias Method. J . Phys. Chem. B 106 (2002) p. 5483 Case F. Chaka A. Friend D. G. Frurip D. Golab J. Johnson R. Moore J. Mountain R. D. Olson J. Schiller M. Storer J. The First Industrial Fluid Properties Simulation Challenge. Fluid Phase Equilibria 217(1) (2004)0,1 (also see http://www.cstl.nist.gov/FluidSimulationchallenge/) Chakhmuradov C. G. Guseinou S. 0. Iz. Vys. Uc. Zav. 1984 65-69 (in Russian) Chen C.C. Mathias P. M. Applied Thermodynamics for Process Modelling. AIChE J. 48(2) (2002) p. 194 Delhommelle J. Tschirwitz C. Ungerer P. Granucci G. Millie P. Pattou D. Fuchs A. H. Derivation of an Optimized Potential Model for Phase Equilibria (OPPE) for Sulfides and Thiols. J. Phys. Chem. B 104 (2000) p. 4745 DelhommelleJ. Granucci G. Brenner V. Millie P. Boutin A. Fuchs A. H. A New Method for Deriving Atomic Charges and Dipoles for Alkanes: Investigation of Transferability and Geometry Dependence. Mol. Phys. 97(10) (1999)p. 1117 De Pabb J. J. Escobedo F. A. Perspective: Molecular Simulations in Chemical Engineering: Present and Future, AIChE Journal 48(12) (2002) p. 2716 Errington /. R. Panagiotopoulos A. 2. A New Potential Model for the n-Alkanes Homologous Series J. Phys. Chem. B 103 (1999) p. 6314 Francesconi A. 2. Franck E. U. Lentz H. Die PVT-daten des Acetonitrils bis 450c und 2500 bar. Ber. Bunsen-Ges. Phys. Chem. 79 (1975) P. 897 (in German)

12 Frenkel D. Smit B. Understanding Molecular

13

14

15

16

17

ia

19

20

21

Simulation. From Algorithms to Applications. Academic Press San Diego, CA 2002. Goldstein E. Buyong M. A. Lii J. H. Allinger N. L. Molecular Mechanics Calcdations (MM3)on Nitriles and Alkynes. J. Phys. Org. Chem. 9 (1996) p. 191 Hadj-Kali M. K. Gerbaud V. Joulia X. Lagache M. Boutin A. Ungerer P. Mijoule C. Dufaure C. Prediction of Liquid-Vapor Equilibrium by Molecular Simulation in the Gibbs Ensemble: Application to Nitriles. Comput. Aided Chem. Eng. 14 653-658 proceedings of 6& 1 Cheap ed.: Pierucci S. (Pisa, Italy, 06/08-11/03) 2003 Haaj-Kali M. K. Application de la simulation moleculaire pour le calcul des equilibres liquide vapeur des nitriles et pour la prediction des azeotropes. These de Doctorat en Genie des Procedes Institut National Polytechnique de Toulouse Toulouse, France (in French) Jonsdottir S . 0. Rasmussen K. Fredenslund A,. UNIQUAC Parameters Determined by Molecular Mechanics. Fluid Phase Equilibria 100 (1994)p. 121-138 Jorgensen W.L. Madura J. D. Swenson C. J. Optimized Intermolecular Potential Functions for Liquid Hydrocarbons. J. Am. Chem. SOC.106 (1984) p. 6638 Kratzke H. Muller S.. Thermodynamic Properties of Acetonitrile 2. (P, @, T) of Saturated and Compressed Liquid Acetonitrile. J. Chem. Thermodyn. 17 (1985)p. 151 Martin M. G. Siepmann J. I. Transferable Potentials for Phase Equilibria. 1. United Atom Description of n-Alkanes. J. Phys. Chem. B 102 (1998)p. 2569 Nath S. K. Escobedo F. A. de Pablo J. J. On the Simulation of Vapor-Liquid Equilibrium for Alkanes. J. Chem. Phys. 108 (1998) p. 9905 Sandler S. I. Quantum Mechanics: A New Tool For Engineering Thermodynamics. Fluid Phase Equilibria 210(2) (2003) p. 147

3.8 Conclusions 26 Sum A. R Sandler S. I. A Novel Approach to

Phase Equilibria Predictions Using Ab-Initio Methods. Ind. Chem. Eng. Res. 38 (1999)p. 2849 27 Toxvaerd S. Equations of State of Alkanes 1. J.Chem. Phys. 93 (1990)p. 4290 28 Toxvaerd S.Equations of State of Alkanes 11. J. Chem. Phys. 107 (1997)p. 5197

29

Llngerer P. Beauvais C. Delhommelle 1.Boutin A. Rousseau B. Fuchs A. H. Optimisation of

the Anisotropic United Atoms Intennolecular Potentiel for n-Alkanes. J . Chem. Phys. 112(12)(2000)p. 5499 30 Warowny W . Volumetric and Phase Behavior of Acetonitrile at Temperatures from 363 K to 463 K. J.Chem. Eng. Data 39 (1994)p. 275

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4 Modeling Frameworks of Complex Separation Systems Michael C. Georgiadis, Eustathios 5. Kikkinides, and Margaritis Kostoglou

4.1

Introduction

Process modeling has always been an important component of process design, from the conceptual synthesis of the process flow sheet, to the detailed design of specialized processing equipment such as advanced reaction and separation devices, and the design of their control systems. Recent years have witnessed the traditional modeling approach being extended to the design of complex processes such as fuels cells, hybrid separation systems, distributed systems, etc. Inevitably the process modeling technology needed to fulfil the demands posed by such a diverse range of applications on different scales of complexity (Marquardt et al. 2000; Pantelides 2001). Years ago at the Foundations of Computer Aided Process Design Conference 1994, Pantelides and Britt (1995)presented a comprehensive review of some of the early developments in the area of multipurpose process modeling environments, i.e., software tools aiming at supporting multiple activities based on a common model. Recently, Pantelides and Urban (2004)presented a critical review of the progress achieved over the past decade and identified the key challenges for the next decade (Pantelides and Urban 2004). In recent years, complex processing systems such as periodic pressure-swing adsorption processes, zeolite membranes and hybrid separations processes have been finding increasing applications as energy efficient alternatives to other traditional separation techniques (such as cryogenic separation), and much progress has already achieved in improving their performance with respect to both the process economics and the attainable purity of the products (see, for instance, Ruthven et al. 1994). The performance of these processes is critically affected by a number of design and operating parameters (design of the processes, duration of the various processing steps, operating levels at each step, etc.). Therefore, their accurate modeling in a compact and robust way is a necessity so as to minimize the capital and operating costs of the process while ensuring that minimum purity and throughput specifications are met. Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright @ 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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This chapter presents a review of modeling frameworks for complex processing systems with an emphasis not only on the models themselves but also on specialized solution techniques related to these models. More specifically,due to their increased industrial interest, a general modeling framework for adsorption-diffusion-basedgas separation processes is presented in Section 4.2 with a focus on pressure-swing adsorption and membrane-based processes for gas separations. The subsequent sections present a critical review of models and specialized solution techniques for crystallization and grinding processes. Finally, concluding remarks are drawn in the Section 4.4.

4.2

A Modeling Framework for Adsorption-Diffusion-based Gas Separation Processes 4.2.1 General

Gas separation is important in many industries ranging from the development of natural gas and oil resources to petrochemicals and foodstuffs. Moreover, separation and recovery from gaseous industrial effluents are issues of considerable environmental significance to a world-widelevel and constitute a major problem demanding efficient solutions. It is generally accepted that the greatest energy consumption generally derives from the separation sections of the processes, which may also account for in excess of 50% of the total capital costs. The principal gas separation technologies include absorption, fractional distillation and adsorption-diffusion-based processes. The market leaders are absorption and distillation, both of which are capital and energy intensive. Adsorption-diffusion-based processes, compared with the other two processes, possess several advantages: 0 0 0 0 0

low energy requirements, small, easily operated, low cost units, compactness and light weight, non-labor-intensive, modular design allowing easy expansion or operation at partial capacity.

The selection of separation techniques depends primarily on the process scale. Distillation and, to a lesser extent, absorption exhibit large economies of scale. Conversely, adsorption-diffusion-basedseparation techniques are modular with relatively fixed capital/throughput ratios for a given separation and hence are favored for smaller scale operations (Yang 1987; Ruthven et al. 1994). The basic requirement in an adsorption separation process is the existence of an appropriate material (adsorbent) that preferentially adsorbs one component from a gas mixture. The selectivity of each adsorbent depends on a difference in adsorption equilibrium or kinetics (diffusion through the pore space of the adsorbent). All adsorption separation processes involve two major steps: (1) adsorption, during which the preferentially adsorbed species are captured from a feed mixture by the

4.2 A Modeling Frameworkfor Adsorption-D~usion-basedGas Separation Processes

adsorbent, and (2) desorption during which the adsorbed species are removed from the adsorbent in order to regenerate the material. It is evident that the emuent during the adsorption step corresponds to the light (weakly adsorbed) product of the separation process (often called the raffhate), while the effluent during desorption corresponds to the heavy (strongly adsorbed) product of the process. The need for process commercialization has lead to the use of cyclic or periodic adsorption separation processes where fxed beds packed with adsorbent operate at a certain sequence and are periodically regenerated by total or partial pressure decrease (pressure-swing adsorption (PSA), vacuum-swing adsorption (VSA)),or less often by temperature increase (temperature-swing adsorption (TSA)).Periodic adsorption processes are thus dynamic in nature and operate in a periodic mode having fued adsorption and desorption cycle times. The periodic excitation is achieved by regular periodic variation of the boundary conditions of certain properties of the gas mixture (temperature, pressure, concentration, velocity) at the two ends of each bed and the connectivity between two or more beds that operate in a certain sequence depending on the complexity of the process. After a certain number of cycles each bed approaches a so-called “cyclic steady state” in which the conditions at the beginning and at end of each cycle are identical to each other. Over the last two decades PSA-VSA processes have gained increasing commercial acceptance over TSA, which is preferred only if the preferentially adsorbed species is too strongly adsorbed imposing high vacuum demands for adequate adsorbent regeneration (Ruthven et al. 1994). An alternative technology employed in adsorption-desorption processes for gas separations is membrane technology. Membranes are thin films ranging from a few micrometers down to the order of several nanometers, which are made of organic (polymers) or inorganic materials and can be nonporous (dense) or porous. Inorganic membranes have considerable advantages in many gas separation processes, which are required to operate under demanding conditions, such as high temperatures and in corrosive environments. Recently there has been considerable interest in the potential of microporous zeolite membranes because of their regular and controlled pore size and geometry. Commercialization of membrane technology has prompted the growth of coherent, crack-free membrane films on top of planar or cylindrical macroporous supports (pore size of the order of 0.1-10 pm) that provide mechanical strength and do not significantly affect the separation performance of the membrane (Burggraaf 1996; Baker et al. 1997; Strathman 2001). Consequently a membrane unit or module is made using standard geometric arrangements (hollow fiber, spiral wound, etc.). The simple design of a membrane enables straightforward expansion of capacity compared to periodic adsorption processes. The concept of a membrane process is straightforward: the separation is achieved through preferential permeation of a species from a gas mixture through the membrane. The key parameters that determine membrane performance are the selectivity towards the gas to be separated and permeate flux or permeability. The former is related to product purity and recovery while the latter is related to throughput or productivity and determines the membrane area required.

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4.2.2 Process Description 4.2.2.1 The PSA Processes

A typical PSA-VSA process consists of a high pressure, adsorption step during which the gas is fed through the bed co-currently and separation is achieved followed by the recovery of the light product (raffinate),and a low pressure, desorption step, where the bed is regenerated (usually in counter-current fashion) with the possible simdtaneous recovery of the heavy product (extract). These two basic steps are interconnected through the necessary depressurization (blow down) and pressurization steps that are employed either co-currently or counter-currently in the respective beds, depending on the specific needs of each particular application. These four basic steps constitute a single PSA or VSA cycle, which is repeated until cyclic steady conditions are achieved. Note that the desorption step is achieved by purging the bed with a fraction of the light product at low pressure (PSA) or by evacuation of the bed using pumps (VSA). The former method is favored in terms of energy-savings since the use of a pump is avoided, but on the other hand it produces a light product with significantly reduced recovery. The basic four-step cycle described above requires the use of only two beds and is shown schematically in Fig. 4.1.In practice more beds

Feed

Blowdown

Purge

Pressurization

Bed1

Purge

II -1

Pressurization

Feed

Blowdown

Bed 2

Figure 4.1 A typical four-step two-bed (Skarstrom) PSA cycle

4.2 A Modeling Framework for Adsorption-Dgusion-based Gas Separation Processes

are normally employed in typical industrial applications, based once again on the specifications of each application and the economics of the process. The performance of a PSA process is assessed on the basis of several important output quantities. These are, the (light) product purity collected during the adsorption step, the (light) product recovery defined as the amount of light product collected in the adsorption step minus the amount of product used to purge the bed in the desorp tion step normalized by the amount of the light product in the feed. If this amount is normalized on the basis of the amount of adsorbent used in the bed then one defines the (light) product productivity per unit time. Although in most cases PSA processes involve the recovery of the light product in a gas mixture there have been a few theoretical and experimental studies in the literature that deal with the additional recovery of the heavy product from the exhaust, during the blow down and purge steps (Ritter and Yang 1989; Kikkinides and Yang 1991, 1993; Kikkinides et al. 1993, 1995). The performance of the PSA process is critically affected by a number of design and operating parameters. The first category includes the size of the bed(s) in the process and the physical characteristics (e.g., particle size) of the adsorbent. On the other hand, important operating parameters include the duration of the various steps and the overall cycle and the pressure and/or temperature levels in each step. The process designer is therefore confronted with an optimization problem typically aiming to minimize the capital and/or operating costs of the process while ensuring that minimum purity and throughput specifications are met. In view of the large number of degrees of freedom, a mathematical programming approach to the optimization of PSA appears to be highly desirable, but this has to address the intrinsic complexity of the processes being studied and in particular the complications arising from their periodic nature. To this end, the optimization of periodic PSA systems has received some attention by the process systems engineering community. Smith and Westerberg (1991)presented a mixed-integer nonlinear program (MINLP)to determine the optimal design of PSA separation systems (operating configuration, size and operating conditions) using simple models and simple time-integrated balances to describe the initial and final concentrations and temperature profiles for each stage. The work of Nilchan and Pantelides (1998)is a key contribution to the optimization of periodic adsorption processes. They presented a rigorous mathematical programming-based approach to the optimization of general periodic adsorption processes. Detailed dynamic models taking account of the spatial variations of properties within the adsorption bed(s) are used. A new numerical method was proposed for the solution of the optimization problem and the calculation of the cyclic steady state, employing simultaneous discretization of both spatial and temporal variations to handle the complex boundary conditions. The approach is capable of handling interactions between multiple beds. Bechaud et al. (2001) investigated stability during cyclic gas flow with dispersion and adsorption in a porous column, as encountered during PSA. KO et al. (2003)presented a mathematical model and optimization procedure of a PSA process using zeolite 13X as an adsorbent. Serbezov and Sotirchos (2003) investigated a semianalyhcal solution of the local equilibrium PSA model for multicomponent mixtures. The solution involves simple algebraic and ordinary differential equations and can provide the basis for quick evaluation of different design alter-

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natives and optimization studies. Recently, Cmz et al. (2004) presented a strategy for the evaluation, design, and optimization of cyclic adsorption processes. Jiang et al. (2003) developed a direct determination approach using a Newton-based method to achieve fast and robust convergence to cyclic steady state of PSA processes. An eficient, flexible and reliable optimization strategy that incorporates realistic detailed process models and rigorous solution procedures was investigated. 4.2.2.2 Membrane Processes for Gas Separations

Contrary to the PSA-VSA process where there is a certain degree of complexity in synchronizing the cyclic operation of two or more fured beds, the case of a membrane separation process is much simpler. In the latter case the membrane unit has a simple geometrical arrangement operating at steady state conditions. In this arrangement there are two main compartments: the retentate at high pressure where the feed is introduced, and the permeate at low pressure where the product is collected. The two compartments are separated by the membrane layer, which controls the separation performance of the process and the production rate of the product. Since the driving force for permeation is the difference between the partial pressures of the product in the two compartments, it is obvious that there will be a back-pressure effect that deteriorates the separation performance of the membrane. In order to reduce the back-diffusion effect one needs to reduce the partial pressure of the product in the permeate side as much as possible. This is once again achieved by either evacuating the permeate section with the use of a pump or by sweeping the product away with the use of a sweep or purge inert gas that lowers the partial pressure of the permeate product. In most commercial applications the sweeping is done countercurrently to the feeding achieving the maximum possible separation performance for the membrane unit. A typical single-stagemembrane unit for separation is shown in Fig. 4.2. Again practice has prompted the use of more complex configurations using recycle streams or membrane cascades depending on the specifications and the economics of each application.

membrane Sweep

Retentate Permeate

dFb (a) Co-currentoperation

Feed

Retentate

Retentate Sweep

(b) Counter-currentoperation

Figure 4.2 Single stage (a) co-current and (b) counter-current membrane processes

4.2 A Modeling Framework for Adsorption-D~usion-basedGas Separation Processes

The optimization of membrane-based gas separation systems has received limited attention. This can be mainly attributed to the complexity of the underlying mathematical models. Tessendorf et al. (1999) presented various aspects of modeling, simulation, design and optimization of theses processes. A membrane module model was developed capable of handling multicomponent mixtures and considering effects of pressure drop and energy balance. The module has implemented and tested in an external process simulators. Kookos (2002) proposed a superstructure representation of the membrane-based gas separation network along with a targeting approach to the synthesis of membrane networks. Using simple models, the membrane material is optimized together with the structure and the parameters of the network. Vareltzis et al. (2003) presented a mathematical programming approach to optimize complex structures of zeolite membranes using detailed models. Various tradeoffs between different optimization objectives were systematically revealed. The impact of detailed modeling on the optimization results was investigated through a comparison with corresponding results obtained using simple models.

4.2.3 A General Model of Adsorption-Diffusion-based Gas Separation Processes

In modeling the separation performance of each process we will assume for the sake of generality that the same material is used to develop the microporous adsorbent particles and the membrane layer. Furthermore, we will assume that essentially the same material is used to make the macroporous binder of the adsorbent particles and the macroporous support on top of which the thin membrane layer is formed. This assumption will enable us to uncouple any effects that depend on the physicochemical characteristics of the materials used in the two separation processes from the effect of the inherent process characteristics of each process. Of course this assumption is often difficult to hold in practice since there are materials that are easier to make in the form of adsorbent particles than coherent membrane layers and vice versa. Mass Balance in the Fixed Bed and/or the Membrane Compartments

For the sake of simplicity we will consider 1D transport along the axial direction neglecting any radial variations. This assumption is seen to be valid for the majority of theoretical and experimental cases found in the open literature. Thus, the mass balance for component i in the interstitial fluid is given by the following equation:

For i = 1, N where N is the total number of species. Alternatively it is equivalent to write the above equation for the first N - 1 species and include an overall material balance. The total concentration of the gas mixture C is related to the temperature T and total pressure P from an equation through an equation of state, which in most cases is represented by the ideal gas law:

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4 Modeling Frameworks ofComplex Separation Systems N

c = c c j = -P j=1

RT

The flux term kJi(Ci - CRPi)in the above equations accounts for interparticle (film) diffusion in the gas phase of component i,transported from the interstitial fluid to the surface of the adsorbent particles or the membrane layer. For a membrane process, Eqs. (1)and (2) hold for the case of gas transport through a Ci the retentate and permeate units by simply neglecting the accumulation term, at ’ due to the steady state operation of the process, and by putting &b = 1 since in this case the compartments are completely empty. Also in this case the values of the mass ~ k,-i will be different for the same reason. transfer coefficients D L ,and Equilibrium Adsorption Behavior at the Particle or Membrane Surface

In many cases the adsorption equilibrium behavior of a multicomponent gas mixture at the surface of the adsorbent is adequately represented by the Langmuir isotherm:

j=1

The advantage of the Langmuir equation is that it is relatively simple and can be easily inverted and solved for the gas phase concentrations, while the parameters qyt and bi can be evaluated from the respective simple component equilibrium isotherm data. The main drawback of the above equation is that is predicts a constant separation factor an assumption that is often violated especially for the case of nonideal gas mixtures (Krishna 2001; Karger and Ruthven 1992). In the latter case more involved modes based on the ideal adsorption solution theory (IAST) should be employed. Mass Balance in the Microporous Particle or Membrane Surface

Where Ni is the flux of component i transported through the pore space of the particle. It is straightforward to show that:

Where denotes the volume-averaged adsorbed phase concentration. For a membrane process, Eqs. (4) and (5) hold after neglecting the respective accumulation terms, due to steady state operation.

4.2 A Modeling Framework for Adsorption-Dguusion-basedGas Separation Processes

Evaluation of the Flux Terms

Evaluation of the flux terms Ni in Eqs. (4) and (5) requires the identification and description of the major transport mechanisms that take place in the pore space of the adsorbent particles. Transport in the pores can take place through various mechanisms, depending on the strength of the interaction of the gas molecules of one species with the molecules of the other species and with the pore walls, and by the relative magnitude of three different length scales characterizing, the size of the molecules, the distance between the pore walls and the fluid density in the pores, respectively. In many cases the adsorbent particle consists of two interpenetrating networks of pores, one representing the pore structure of the particle crystallites (e.g., zeolites, silica gel, etc.) and consists of micropores (0.1-1.0 nm) according to the IUPAC classification and another one that represents the structure of the binder or support and consists ofmeso- (1-50 nm) and macropores (50-5000 nm) in the IUPAC classification (Gregg and Sing 1982). It is evident that different mass transfer mechanisms prevail at each pore network, with the micropores providing the necessary features for gas separation by selective diffusion through the micropores and/or adsorption at pore surface. The effect of the binder or support, on the other hand, is an additional resistance to the mass transfer of the species with no selective features in either adsorption or diffusion through the macropores of the material. Thus it is desirable to minimize as much as possible the effect of transport through the support in order to achieve a better separation performance. Fortunately, in many cases the resistance to mass transport through the micropores of the crystallites in much stronger compared to that through the support and thus the latter can be either completely ignored or approximated through a linear driving force expression assuming fast diffusion or permeation kinetics in the pore space of the support. In this case the additional diffusion resistance is incorporated into the film diffusion coefficient kJi. The generalized Maxwell-Stefan (GMS) equations provide an adequate basis for the accurate description of multicomponent mass transfer in porous media with minimum unary data (fiishna and Wesselingh 1997; Kapteijn et al. 2000; Krishna 2001; Karger and Ruthven 1992).The basis of the Maxwell-Stefan theory is that the driving force for movement, acting on a species, is balanced by the friction experienced by that species and each friction contribution is considered to be proportional to the difference in the corresponding diffusion velocities. The application of this theory on microporous or surface diffusion yields: 8;

--vjLi

R.T

=

i c e j ~-eiiNj +-

j=l

qTtqTtD.. tr

N~

qT'Di

The driving force for diffusion is the chemical potential gradient ( V p i ) .Parameters DVand Di are the Maxwell-Stefan surface diffusivities and represent inverse friction factors between molecules and the solid surface, respectively. In of a binary mixture with adsorption equilibrium behavior represented by the Langmuir isotherm and after some algebraic manipulations the surface flux Ni is given by the following expression:

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Note that if we assume DQ+ 00 the above equation becomes: [(I - ej)]vei + eivej N. - -N. 1 10 (1 - ei - ej)

The above equations correspond to the GMS (DQ+ m) model, which basically assumes negligible difhsional adsorbate-adsorbate interactions and has been frequently employed to describe diffusion of binary mixtures in zeolites (Ruthven et al. 1994). Finally, in the limit of dilute systems (4;<< l),Eq. (8)becomes: Ni = --NioVtli

(9)

which is the classic Fick's law applied in the microporous adsorbent. For the case of fast diffusion kinetics in the pore space of the microporous particle it has been shown that the adsorbed phase concentration has a parabolic profile in space. Combining this assumption with Eqs. (3), (5),and (7) we come up with the linear driving force approximation (LDF) often employed to describe diffusion kinetics in the adsorbed phase:

Heat Effects

The exothermic nature of the adsorption process can result, under certain conditions and system sizes, in significant temperature variations resulting in the heating and freezing of the bed during adsorption and desorption, respectively. Considering 1 D adiabatic heat transport along the axis and assuming negligible variation between the temperature in the solid and fluid phase the following heat balance equation holds in the bed:

Where .p8 and cpa are the total density and heat capacity of the mixture in the gas phase and .pSand c ~ ,are ~ , the density and heat capacity of the adsorbent. Note that several alternative heat balance models with different levels of complexity can be developed, depending on the available degree of information for the above quantities. For the case of a membrane unit the heat effects are in most cases negligible and the process can be safely considered as an isothermal one. In the rest of the manuscript and without loss of generality we will consider isothermal operation in order to

4.2 A Modeling Frameworkfor Adsorption-Diflirsion-based Gas Separation Processes

have a better comparison between the two types of processes. Inclusion of the heat balance is straightforward and only adds more unknown parameters without significant changes in the solution approach during the simulation and optimization procedures. Pressure Drop Effects

In many applications the use of long beds and/or small adsorbent particles can induce a pressure drop of appreciable magnitude that results in certain changes in the propagation of the concentration and temperature waves through the bed. The pressure drop along the axial direction of the adsorption bed is usually determined by the Ergun equation:

Where p is the viscosity of the gas mixture and dp is the diameter of the adsorbent particles. For the case of a membrane unit the simpler Hagen-Poiseuille equation for parallel laminar flow is used to calculate the pressure drop through the axial direction of each section (Pan 1983, Giglia et al. 1991).Nevertheless, the pressure drop effects are many cases negligible and can be safely ignored. Boundary Conditions

The appropriate boundary conditions for the solution of the problem are described by the following set of equations: 1. Adsorption step or retentate compartment

a ci = o -

atz= L

u(0) = Uf

atz=O

P ( 0 ) = PH

atz=O

az

(13a)- (13d)

2. Desorption step or permeate compartment

a ci = o -

atz=O

u(L)= up

atz= L

P ( L ) = PL

atz= L

az

(14a)- (14d)

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3. Blow down step

-a =cio

atz=Oandz= L

u(L) = 0

atz= L

az

(15a)- (15d)

atz=O 4. Pressurization step

atz=O

-a =cio

atz= L

u(L) = 0

atz= L

az

(16a)-(16d) atz=O

Note that the pressure histories at the feed end of bed during the pressurization and blow down steps, respectively, are known functions of time.

4.3 Modeling of PSA Processes in gPROMS

The advanced distributed process modeling capabilities of gPROMS (trademark of Process Systems Enterprise Ltd.) permit a detailed description of the complex phenomena taking place inside adsorption columns. A major advantage of gPROMS is its ability to describe detailed operating procedures and to handle discontinuities arising from major changes in the structure of the underlying models. In the context of PSA processes this is particular important since the boundary conditions depends on the operating stage. Furthermore, the entire operation involves successive transitions between the various operating stages thus introducing extra discontinuities in the model. For the simple case of one PSA column but with no loss of generally boundary conditions can be efficiently implemented in gPROMS as seen in Fig. 4.3. The implementation of the PSA operating schedule is also a complex simulation task given the transition between the various processing steps. Figure 4.4illustrates the process scheduling task for a specified number of cycles (one PSA column).

4.4 Efliccient Modeling ofCrystallization Processes

;ELECTOR OperationMode AS (Pressurization, Depressurisation)

Figure 4.3 Boundary conditions o f a single PSA column in gPROMS

IOUNDARY :At the feed end :ASE OperationMode OF #Pressurization Step WHEN Pressurization FOR i:=l TO NoComp DO Heed * Yfeed(i) = C(i,O) * R * Tfeed; END #FOR P(0) = Heed; #Depressurisation Step WHEN Depressurisation FOR i:=l TO NoComp DO PARTIAL(C(i,O), Axial) = 0; END #FOR P(0) = h a s t e ; :ND # Case At the product end Pressurization I Depressurisation step 'ARTlAL(C, BedLength), Axial) = 0;

J(BedLength) = (Qvol*Patm) / (BedArea*P(BedLength));

4.4 Efficient Modeling of Crystallization Processes 4.4.1 General

Crystallization from solution is one of the oldest and economically most important industrial separation processes. It is applied both as a large-scale continuous process for the production of inorganic (e.g., ammonium sulphate) and organic (e.g., adipic acid) material and as small-scale batch processes for the production of high purity pharmaceuticals or fine chemicals (e.g., aspartame). In order to optimize and control the crystallization process, reliable mathematical models are necessary. Detailed modeling of the crystallization process requires knowledge of phenomena on a microscopic as well as on a macroscopic scale. On the microscopic scale the basic phenomena are the primary (heterogeneous or homogeneous) nucleation, secondary

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SCHEDULE SEQUENCE Cycle := 1 ; WHILE Cycle <= NoCycles DO SEQUENCE CONTINUE FOR CycleTirne/2 SWITCH Column.OpexationMode := Co1umn.Depressurisation; END # Switch CONTINUE FOR CycleTime/2 SWITCH Column.OperationMode := ColumnPressurisation ; END # Switch RESET # Oxygen product purity Column.Purity := OLD(Column.Mgroduct(2)) I SIGMA(OLD(Co1umn.M-product)); # Oxygen product recovery Column.Recovery := OLD(Column.M-product(2)) / OLD(Co1umn.M-fed(2)) ; END # Reset Cycle := Cycle + 1 ; REINITIAL Co1umn.M-fed, Co1umn.M-product, Co1umn.M-waste WITH Co1umn.M-fed = 1E-6 ; Co1umn.M-product = 1E-6 ; Co1urnn.M-waste = 1E-6 ; END # Reinitial END # Sequence END #While END # Sequence END #Task OperateColum Figure 4.4

Operating schedule o f a single PSA column in gPROMS

nucleation, crystal growth, coagulation between crystals, and crystal fragmentation. A great variety of models with different degrees of complexity have been presented in the literature for the above processes. On the macroscale, the macromixing in the crystallizer is very important. Coagulation and fragmentation phenomena depend on local energy dissipation, which can be varied by orders of magnitude in a stirred tank.

4.4 Efficient Modeling ofcrystallization Processes

The modeling of crystallization processes poses special problems not encountered in more conventional process operations. The state of such systems is usually characterized by particle size distribution functions instead of, or in addition to, standard point properties such as concentrations. Moreover, the steady state and dynamic behavior of these systems is described by population balance equations rather than simple mass balances. Finally, the physical properties of solids encountered in crystallization processes are generally much less well characterized than those of fluids. Traditionally, most process modeling and simulation tools have been aimed primarily at the mainstream chemical and petrochemical industry. Commercial steady state simulation packages have now reached a high degree of sophistication, encompassing extensive libraries of unit operation models, as well as large compilations of physical property data and calculation techniques. However, given the differences outlined above, it is hardly surprising that the area of crystallization and grinding processes has not been served well by tools now used routinely by process engineers in other areas. It has been long realized that even with relatively sophisticated general process modeling tools, the modeling and simulation of particulate processes still presents serious difficulties. One key problem is the mathematical complexity of the models: population balances invariably lead to partial differential equations, and these are often coupled with other equations describing the evolution of properties in the fluid surrounding the particles through integral terms. This results in systems of integral-partial differential equations, which may be very difficult to solve. In fact most current equation-oriented modeling frameworks cannot even describe directly such distributed parameter systems (Pantelides and Oh 1996).To this end the rest of this part of the chapter will focus on presenting state-of-the-arttechniques for reducing the modeling complexity of crystallization and grinding processes, without any loss of accuracy and generality, to a level where standard modeling tools can be used for simulation and optimization purposes.

4.4.2 A Comprehensive Modeling Framework of Crystallization Processes

A generally accepted concept for the modeling of dispersed phase systems is the population balance approach introduced for the particular problem of crystallization by Randolph and Larson (1971). Each crystal in the system is described by a vector of properties x (internal coordinates) and its position in the crystallizer r (external coordinates). A very detailed model of the process would require as much internal coordinates as possible (which can be supported by experimental findings) and must be spatially distributed. The most extensively used crystallization model until today is based on a spatially homogeneous population balance (zero external variables) with one internal variable (crystal volume or linear size). Recently, several efforts have been made towards an increase of the number of internal or external variables to describe more accurately the process. Here, this relatively simple model will serve as the basis for a comprehensive discussion of the difficulties, existing solutions techniques and possible extensions of current crystallization process models.

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Apart for the common variables used for modeling of any chemical reactor (temperature, concentrations, etc.) an additional variable used for the crystallizer is the which is described by the differential crystal volume crystal size distribution (CSD), distribution functionf(x, t) where x is the crystal volume andJx, t)dx is the number concentration of crystals with volumes between x and x + dx. The evolution of the CSD (f(x, t)) is determined by the following population balance equation:

where the vector c contains composition and temperature of the liquid phase and is used to denote dependence of the undergoing phenomena on this. The function G ( x , c) is the volumetric growth rate of a crystal with volume x. The function K(x, y; c) is the so-called coagulation kernel defined such that the expression K ( x , y; c)f(x, t ) f ( y , t)dxdy is the rate of coagulation events per unit fluid of volume between a crystal with volume in [ x , x + dx] and a crystal with volume in [y, y + dy]. This is in general a symmetric function with respect to x and y. The function b(x) is the fragmentation frequency for a crystal of volume x while B(c) is the nucleation rate (i.e., the rate of generation of nucleus which are crystals with size a(c)).The function p ( x . y) is called the fragmentation kernel and is such that p ( x , y)dx is the probability for having a fragment of volume in [x, x + dx] as a result of fragmentation of a crystal with volume y. Finally z is the residence time in the system and%(%)is the inlet particle size distribution function. The initial condition for the solution of the Eq. (17) isf(x, 0) =fo(x). The above equation is rather comprehensive in the sense that includes processes with different features and different computational requirements (continuous versus batch system, precipitation versus crystallization). For the sake of clarity of our discussion it would be useful to discriminate between the process of crystallization (Mullin 1993) and precipitation (Sohnel and Garside 1992). Although the physical phenomena are the same (precipitationis a type of crystallization) the features exhibiting by the two processes (e.g., crystal sizes, supersaturation, etc.) are quite different requiring a different modeling approach. From the practical point of view precipitation is the crystallization of sparingly soluble substances (mainly salts). The mass of the active species in precipitation is small and thus the final volume fraction of the solid phase is also small and the crystal radius doe not usually exceed 1 pm. On the other hand in crystallization the mass fraction of the solid phase can be large and the size of crystals is of the order of millimeters. We can now focus our attention on the modeling of the different phenomena described in generic Eq. (17). Generally, the nucleation rate is the sum of the primary nucleation rate (given by the theory of homogeneous nucleation) and the secondary nucleation rate (production of small crystals from the fragmentation of the large ones) (Dirksen and Ring 1991).In case of precipitation there is no crystal fragmentation and secondary nucle-

4.4 EfJiccient Modeling ofcrystallization Processes

ation so B(c), a(c) can be directly computed from the homogeneous nucleation theory (a slight modification is needed for heterogeneous nucleation which is usually the case). In the case of crystallization, the nucleus size is extremely small in comparison with the mean crystal size in the system so it can be assumed equal to zero and the nucleation term in Eq. (17) can be replaced by the following boundary condition on particle size distribution (PSD).

From the mathematical point of view this is an important simplification which weakens the problem of multiple crystal size scales existing in Eq. (17).Although the secondary nucleation can be rigorously simulated using an appropriate fragmentation kernel (of attrition type), it is more convenient to be included in B(c) as a term proportional to total solid mass concentration (Mahoney and Ramkrishna 2003). The coagulation rate is the product of the collision frequency and the collision efficiency. For the case of precipitation the collision between the crystals is due to their Brownian motion and the carrier fluid flow field. The collision efficiency is the result of the microscopic interaction between the crystals (given by the DLVO theory) and does not depend on concentration c. Rigorous models based on first principles can be derived for the above phenomena (Elimelech et al. 1995). On the other hand, in case of crystallization the coagulation phenomenon is included in a purely phenomenological manner to achieve a fit of the model to the experimental data. The collision rate is usually assumed constant and the coagulation efficiency is associated with the creation of solid bridges between the collided particles so it depends on growth rate and thus on c. The growth rate for the case of precipitation is computed rigorously taking into account the bulk diffusion and surface reaction steps for each substance participating to the crystal growth (e.g., Kostoglou and Karabelas 1998).The growth rate used for the case of crystallization is of empirical nature and several expressions can be found in the literature (Abegg et al. 1968). In many cases a surface reaction dominated growth rate is assumed in combination with a diffusive (in crystal size coordinate) term to account for the stochastic nature of the crystal growth phenomenon (Tavare 1985). Regarding the fragmentation kernel and rate several empirical functions have been used for crystallization modeling, whereas the phenomenon does not exist in precipitation processes. Several attempts to increase the number of internal coordinates of-the model h r a better description of the crystals have been made. To mention but a few are the use of the intrinsic crystal growth rate as the second internal coordinate (Janse and de Jong 1976)and the case of different growth rates for different faces of the crystal (Ma et al. 2003). The heterogeneity in the crystallizer may be very important and must be modeled in some way. In case of crystallization (usually a continuous process) the compartmental modeling is the appropriate compromise between accuracy and computational efficiency. The crystallizer is approximated with a few well-mixed regions interconnected with material streams. The flow rate of the streams can be found by CFD calculations. For the case of crystallization there may be two-way coupling of

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the CFD since the extent of crystallization influences the flow properties of the fluid. In case of precipitation (usually a batch process) a one-way coupling is always enough (due to small solid mass fraction).The nature of the process is such that the compartmental model is not appropriate and a fine grid, similar to that used by the CFD module is needed. So the direct implementation of the Eq. (18) in a CFD framework is necessary (Seckler et al. 1995).Furthermore, in the presence of strong nucleation the extremely strong nonlinearity of the nucleation term makes the usual averaging procedures for turbulent flows inapplicable,calling for the use of the complete probability density function (PDF) approach (Marchisioet al. 2002). It is worth noting the existence of a user-friendly software package (PARSIVAL particle size evaluation) designed for solving general integral-differential equations with one internal and zero external variables (Wulkov et al. 2001). The main application of the package is the simulation of crystallization processes. The algorithm behind the package is fully adaptive in both particle size and time coordinates. The size discretization is based on the Galerkin h-p method and the time discretization is of Rothe type. The package is not capable to handle control aspects.

4.4.3 Efficient Solution Approaches

In the general case Eq. (17)does not have an analytical solution and therefore it must be solved numerically. Its numerical solution is by no means a trivial task since the problem combines the following features: 0

0

0

0

an extraordinarily wide range for the independent variable x (particle volume) since the particle radius can be from the order of nanometers (nucleus) to order of millimeters; highly localized in the x variable domain nucleus size distribution imposing difficulties to the use of polynomials for approximation of PSD; convolution type integral and the associated nonlinearity imposed by the coagulation term; The hyperbolic form of the growth term and its ability to move discontinuities in the x domain makes its discretization difficult.

In the literature there are well known techniques to face each of the above problems but their simultaneous consideration is still a very challenging task. The conventional finite difference discretization it is not even capable to conserve integral properties of the system (e.g.. total particulate mass), which are of paramount importance for crystallization application so special techniques must be developed. In general the available methods for solving Eq. (17) can be divided into the following six categories: (1) analytical solutions, (2) finite element methods, (3) higher order methods, (4) Monte Carlo methods, (5) sectional (zero order) methods, and (6) methods of moments. A brief overview of each of these methods is presented below.

4.4 Eflient Modeling ofcrystallization Processes

AnalFcal solutions of Eq. (17) have been derived for certain simple forms of the growth rate (constant and linear) and coagulation kernel (constant and sum) certain combinations of the phenomena described by Eq. (17) and batch or steady state condltions. A special reference will be made in the work of Ramabhadran et al. (1976) deriving an analytical solution for combined nucleation growth and coagulation in batch conditions, and the work of Saleeby and Lee (1995) for the case of nucleation, growth and stochastic crystal growth dispersion in case of Continuous stired Reactors (CSTR).Although the value of analytical solutions for the simulation of realistic crystallization processes is limited, they have been extensively used as tools for the assessment of numerical techniques for the solution of Eq. (17). The finite element approach (for x discretization) to the solution of Eq. (17) is not a usual choice but some particular versions of the technique have been used over the last years. In particular Gelbard and Seinfeld (1978) used collocation on finite elements using third order polynomials with continuous first derivatives along the element boundaries, as basis functions. Tsang and Huang (1990) used Petrov-Galerkin finite elements to account for the hyperbolic character of the growth term. More recently, Nicmanis and Hounslow (1998) used a finite element Galerkin approach with Lagrangian third order polynomials for the steady state case. Rigopoulos and Jones (2003) developed a collocation finite element technique using linear basis functions. Finally, Mahoney and Ramkrishna (2002) used the Galerkin finite element techniques with linear basis functions to solve the linear size-based edition of the Eq. (17). Special care is taken to capture and follow discontinuities appearing in the PSD. In all the above approaches a geometric grid based on particle volume is used, except in the final one where the grid is linear and based on particle diameter. The higher order methods imply the global approximation of the PSD with a polynomial multiplied by a proper function. Lacatos et al. (1984)used a collocation procedure employing Laguerre polynomials. Recently, Hamilton et al. (2003) developed a collocation method based on Hermite polynomials defined on a grid moving in order to fulfill some integral conditions. Also collocation with wavelets as basis functions has been used. The higher order methods offer very high accuracy (on the cost of large computational effort and complex code implementation) but they have the drawback of requiring special treatment of singularities (e.g., a monodisperse initial distribution). The Monte Carlo method has a long history as a tool for the simulation of the particulate processes. Van Peborgh Gooch and Hounslow (1996),developed a stochastic approach for the particular process of crystallization with an arbitrary number of internal variables. Falope et al. (2001) also used another variant of the Monte Carlo method for crystallization with two internal variables. The significance of the Monte Carlo method increases sharply as the number of internal variables increases making the solution of the deterministic problem from difficult to impossible. On the other hand there is not a simple and efficient way to use the Monte Carlo method for the spatially distributed case. The sectional methods and the methods of moments are capable to transform Eq. (17) in a conventional ODE-DAE system that can be readily solved by existing integrators, and can be easily incorporated into existing modeling tools for flow sheet

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simulation including crystallization processes. Due to their practical importance both methods will be discussed in further details. 4.4.3.1 Sectional Methods

According to the sectional methods (equivalent to finite volumes) the particle volume coordinate is partitioned using a number of points vi (i = 0,1,2 ... L). Particles with volume between vi-l and v L belongs to the ith class and their number concentration

(

is denoted as Ni i.e.,

If(.,

,,

t)&)

. Until the 1990s the most widely used schemes for

the discretization of Eq. (17) were those of Gelbard et al. (1980) and Gelbard and Seinfeld (1980).The corresponding codes, although developed for aerosol processes, were extensively applied to crystallization processes. As regards to the coagulation terms, the particle number-based approach of the discretization scheme does not conserve the total particle mass and the grid must be geometric with ratio larger than 2 for an efficient implementation. To overcome this deficiency Hounslow et al. (1988) developed a discretization method for crystallization applications conserving both particle number and mass having the disadvantage that the only choice for the grid is geometric with ratio 2. The method of Hounslow et al. (1988)was extended for a geometric grid with ratio the qth root of 2 ( q is an integer) permitting grid densification (Litster et al. 1995).The most general discretization scheme is that of Kumar and Ramkrishna (1996),which conserves also particle number and mass but admits a completely arbitrary grid. The crystal growth term in Eq. (17) makes it of hyperbolic form. The inability of fxed grid (Eulerian) discretization to handle properly this type of problem is well known. The direct finite volume discretization does not conserve particle mass. Kostoglou and Karabelas (1995)developed first and second order schemes that conserves particle number and mass simultaneously. Their first order scheme (of upwind type) is unconditionally stable but suffers from numerical diffusion. The second order scheme shows much less diffusion but numerical dispersion appears (source for instability).An efficient treatment of the crystal growth terms requires a grid moving along the characteristicsof the hyperbolic Eq. (17) (Lagrangian approach).The implementation of the moving grid approach is easy in the absence of coagulation and fragmentation (Gelbard 1990)but for the numerical solution of the complete Eq. (17) the moving grid must be compatible with the discretization of the coagulation or fragmentation terms. This compatibility can be achieved only using the Kumar and Ramkrishna (1996)discretization scheme. Their method made for the first time possible the use of a sectional approach with moving grid for the solution of Eq. (17) (Kumar and Ramkrishna 1997).The generation of new particles by nucleation makes necessary the addition of more and more sections during a particular simulation. This is not a desirable feature for any kind of numerical algorithm. The advantage of having a futed number of ODES instead of a variable (possibly uncontrolled one) is very important and it is strongly believed that a fxed grid (Eulerian)approach is preferable than the Lagrangian one. The increase of computing efficiency made possible the use of a large number of sections (fewhundredths versus few decades as seen ten

4.4 Eflcient Modeling ofCvstallization Processes

years ago) leading to a great reduction of the numerical difision error of the Eulerian methods. The proposed discretized form of Eq. (17) is (the fragmentation terms are not shown since they can be found in the corresponding Section 4.4):

where the x, are such that Y , - ~= + x,)/2, 6 , is the Kronecker delta in L-dimensional space, the integer m is such that Y , - ~ a (c) i Y, and Ai =

2 (h+1 -

w) (h - Ut-1)

1, vi

G ( x , c ) dx

4.4.3.2 Methods of Moments

A proper discretization of Eq. (17)with the sectional method leads to a model with at least 50 degrees of freedom (number of ODES). This renders the sectional method computationally intractable for the case of spatially distributed problems. The handling of complex spatially distributed problems imposes the need for low degrees of freedom approaches to the solution of Eq. (17).This made the method of moments a necessity to efficiently solve the population balance equation. These methods have a longer history than the sectional ones and they are based on the transformation of the Eq. (17) in a system of equations for some moments of the unknown distribution. The system is getting closed by defining a closure relation, relating the moments appearing in the right hand side of the system to those of the left hand side. The method of moments achieves an enormous reduction of the computational effort (typically 3-8 degrees of freedom) sacrificing the information content and the accuracy of the solution. Only some moments of the PSD can be computed with accuracy less than that of the sectional method but they are considered adequate for practical applications. There is a great variety of method of moments. The older variants were used only for the case of very simple rate and kernel functions. Other methods assume a particular shape for the PSD (e.g., log-normal, gamma, Weibull, see Williams and Loyalka 1991) and the results are reliable only if the actual solution resembles the assumed form (e.g., problems with bimodal PSD cannot be attacked by these methods).

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Recently, a quite general method of moments (generalizedmethod of moments) was developed by Marchisio et al. (2003) and applied in crystallization systems. This method can be used without a restriction to the sophistication of the models for the occurring phenomena (i.e., rates and kernels) and to the actual shape of the PSD. The evolution equations for the moments of index ai of the PSD

+ (1 - s(ai))

c PI 2

x,?i-' ~ ( 3 c)wj ,

+ ~ ( c ) a a(c) i

j=1

The evolution of xj, wj (j= 1, 2, ... P/2) is given implicitly from the following nonlinear algebraic system:

j=1

The system of Eqs. (20) and (21) can be easily solved with a traditional ODE-DAE integrator. To derive the full crystallization model Eqs. (20) and (21) must be coupled with other equations describing the behavior of the crystallization process (component mass balances, energy balances, physical property models, etc.). For example, assuming a three-phase continuous mixed suspension mixed product removal crystallizer, by appropriate heating or cooling, a product is generated in the form of crystals. A vapor phase is also formed because of the evaporation of part of the liquid which comes into the crystallizer. The contents of the crystallizer are removed by means of the top outlet stream (vapor) and the bottom outlet stream (slurry with product crystals and solution). The material balance of each component is mathematically expressed as follows:

The energy balance has the following general form:

Supplementary relations for the mass balance equation involve calculation of the total amount of the solute in the crystallizer, the combined concentrations of the other components, the volume of the crystallizer, the mass fractions in the liquid and vapor phase, the total amount in the solid phase, vapor-liquid equilibrium relations, etc. In regards to the energy balance supplementary relations serve to calculate the specific internal energies for each phase, the fraction of the suspension volume and the rate of heat removal of the crystallizer. Due to space limitations details of these relations are not presented here.

4.4 Efficient Modeling ofCrystallization Processes

4.4.4 Modeling and Optimization of Crystallizers

The selection of the optimal method for solving the population balance equation strongly depends on the particular features of the mathematical problem under question. Some general guidelines for the selection of the appropriate method are given in Table 4.1. It is important to emphasize that the efficient solution of the population balance equation can provide the basis for model-based design, control and optimization studies of large-scale crystallizers. The presence of very little work in this area can be attributed not only to the lack of rigorous models but also, and perhaps more importantly, to the lack of techniques for efficient solving the population balance equation without loss of accuracy. It is clear that the population balance equation constitute the basic component of the overall crystallization modeling framework. Component mass balances, energy balance and auxiliary algebraic equations describing the physical properties complement the model. The design and optimization of crystallizes is a very challenging problem. There is a lack of systematic procedures for developing optimal operating policies and design options for complex crystallization systems. Kramer et al. (1999) presented a formal approach of design guidelines considering the influence of crystallizer geometry, scale, operating conditions and process actuators on the process behaviour and product quality. fiamer et al. (2000) developed a compartmental modeling framework to describe the crystallization process of evaporative and cooling suspension crystallizers. The framework has been implemented in the SPEEDUP environment and is capable of predicting a large supersaturation profile in a large-scale crystallizer. Ma et al. (2002) presented a rigorous compartmental crystallization model to achieve optimal control. Ge et al. (2000)have illustrated the application of mathematical optimization to the problem of batch crystallization. A targeting approach to the optimization of multistage crystallization networks has been investigated by Sheikh and Jones (1998). Bermingham et al. (2003) presented a formal optimization approach for the design of solution crystallization processes using rigorous models. A largescale industrial case study was used to illustrate the applicability and usefulness of the overall optimization methodology. Recently, Choong and Smith (2004a,b) proposed an optimization framework based on a stochastic optimization algorithm for

Table 4.1 Optimal method of solution for the crystallization model versus number of external and internal coordinates o f the model ~~~

~

External coordinates number Internal coordinates number

0

1

More

1

Sectional

Sectional, Moments

Moments

2

Monte Carlo, Moments

Moments

Moments

More

Monte Carlo

Not available

Not available

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optimizing batch cooling and batch, semi-batch and nonisothennal evaporative crystallization operations. The results demonstrate significant improvements over conventional approaches and heuristic rules. The deviation of the well-mixedbehavior in a crystallizer is primarily caused by the hydrodynamic conditions, which lead to temperature, supersaturation and participle concentration profiles in the crystallizer. An approach which can overcome the shortage of the well-mixed models is the employment of a multizonal representation which divides the equipment volume into a network of interconnected zones where the idealized mixing patter is assumed for each zone. Urban and Liberis (1999) used a hybrid multizonal/CFD modeling approach for the modeling of an industrial crystallizer. Each zone incorporates a detailed description of the crystallization phenomena in terms of population balance equations. Both homogenous and heterogeneous crystal nucleation are taken into account, the latter being a strong function of the turbulence energy dissipation. A CFD model of the process is used to determine the directionality and rate of flow between adjacent zones, and the mean energy dissipation rate within each zone. Zauner and Jones (2002) adopted a compartment mixing model to predict the mixing on crystal precipitation. The population balance is solved simultaneously with the mass balance using data obtained by CFD calculations. Recently, Bezzo et al. (2004) presented a formal multi-scale framework based on a hybrid multizonal/CFD model. The framework is applicable to systems where the fluid dynamics operate on a much faster time-scale than other phenomena, and can be described in terms of steady state CFD computations involving a (pseudo)homogenous fluid, the physical properties of which are relatively weak functions of intensive properties. A crystallization process was used to illustrate the overall modeling approach.

4.5 Modeling of Grinding Processes 4.5.1 A General Modeling Framework of Grinding Processes

Fine grinding of solid materials is of prime importance in many industrial applications. In addition to mineral processing, it is also widely used in the manufacture of paints, ceramics, pharmaceuticals etc. The grinding process can be performed under wet or dry conditions using a large variety of equipment. In general, the grinding of the feed material is made by mobile pieces (e.g., spheres, cylinders) or by large fued in space elements (e.g., rollers) made from hard material. Irrespective of the details of the particular grinding process, the scope is always the reduction of the particle size (the range can be from millimeters to micrometers) and the simulation approach is the same. Traditionally, the simulation of processes associated with the grinding of solids is based on the solution of a particular and rather simplified form of the population balance equation known as fragmentation equation. The main feature of grinding modeling is the application of a population balance, and the selec-

4.5 Modeling ofCrinding Processes

tion of appropriate independent variables. Hence, the usual approach taken in the literature is based on solution through the use of population balance equations to produce models which can simulate grinding. Today, a significant challenge which faces those who model grinding is the lack of a modeling tool which allows the easy implementation of the many different forms and evolutionary changes in the grinding models. This is coupled with the need for robust solution techniques for the various integral-partial-differential equations found in utilizing the population balance approach. Recent simulation examples in the literature include grinding of coal in ring-roller mills (Sato et al. 1996), the wet stirred ball attrition of alumina and synthetic diamond (Shinohara et al. 1999), the attrition of alumina hydrate in a tumbling mill (Frances and Laguerie, 1998),the dry stirred ball attrition of quartz (Ma et al. 1998) and the wet stirred bead attrition of carbon (Varinot et al. 1999).According to the particular formalism, the evolution of the particle size distribution (PSD) as it is described by the differential particle volume distribution functionflx, t ) can be calculated from the solution of the following linear integral-differentia1equation:

The first part of the right hand side of the above equation stands for the generation of particles of size x by fragmentation of larger than x particles and the second term for the loss of size x partides due to their fragmentation.The function b(x)is the fragmentation frequency for a particle of volume x (the term rate is also used instead of frequency). The function p ( x , y ) is called fragmentation kernel and is such that p ( x , y)dx is the probability for having a fragment of volume in [ x , x + dx] as a result of fragmentation of a particle with volume y. The grinding equipment usually operates under batch conditions so the corresponding form of the fragmentation equation (which can also be directly used for plug flow continuous operation) is examined here. The above formalism is purely phenomenological so there is not space for considerable improvements. The extension of the formalism using external coordinates is meaningless since the equation cannot be considered to describe a local phenomenon in the physical space. An axial external coordinate has been used in a purely phenomenological manner by Mihalyko et al. (1998) to account for the partial mixing (axial dispersion) during plug flow grinding process. Also attempts to add more internal variables for the characterization of the particles have not been made. In generally Eq. (22) is considered adequate to describe the grinding process. The challenge for the development of efficient techniques for its solution is coming not from the need to incorporate it (describing the “local”phenomenon) in CFD codes but for the need to incorporate efficient grinding equipment submodels to the large flow sheet simulators. The fragmentation functions b(x), p ( x , y) should satisfy the following requirements in order to give physically meaningful results: 1. lim b(x) = 0 to avoid the generation of smaller and smaller particles without limit. X

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2.

j xp(x, y)dx

y this is the mass conservation condition and stipulates that the

=

0

total volume of particles resulting from the breakup of a particle of volume y must be equal to y. 3. ~ ( y =)

jp ( x , y)dy, where ~ ( y is) the number of fragments generated during the 0

breakup of a particle with volume y. In all cases, it should be ~ ( y 2) 2. 4.

k

0

xp(x, y)dx 2

Pk

(y - x ) p ( x , y)dy for k

< y/2. This condition is usually overlooked

having as consequence the use of fragmentation kernels without physical meaning for fitting experimental data. The fragmentation frequency used in the grinding literature (Varinot et al. 1997, 1999) has the following form

This is a composite law resulting from the matching of the two asymptotes (power laws) b(x)= ( x >> xR)and b(x) = xb ( x << xR)at the region of x = x,. Usually b-a is a small number implying an almost size independent fragmentation rate for large particles. As the particle size decreases the relatively large exponent b dominates prevailing further fragmentation of the smaller particles. The most general fragmentation kernel employed for grinding simulation (Eyre et a]. 1998) is

The values of the parameters C1, C, al, a, should be carefully chosen in order to satisfy the above-mentioned requirements 2 through 4. This kernel exhibits the very important property of homogeneity. A breakage kernel is called homogeneous if the shape of the fragment size distribution does not depend explicitly on the parent particle size y but only on the ratio x/y. An homogeneous kernel can be written as p ( x , y) =-1

Y

p

($).

4.5.2 Solution Approaches 4.5.2.1 Sectional Methods

The solution methods for the Eq. (22) with frequency in Eq. (23) and kernel in Eq. (25) can be organized in five categories as follows: a) Analytical (Ziff and McGrady, 1986) and large time asymptotic solutions (Ziff, 1991) exist only for the particular case a = C1 = 0 in Eqs. (24) and (25).

4.5 Modeling of Grinding Processes

b) Stochastic (Monte Carlo) methods (Mishra 2000). In the case of grinding simulation the addition of extra internal variables is not an option so stochastic methods are of little importance. c) Higher order (polynomial approximation) methods. In particular the Galerkin weighted residual formulation using as basis functions B-splines (Everson et al. 1997) and wavelets (Liu and Tade 2004) have been used. d) Sectional (zero order) methods. e) Moment methods. The last two categories include methods which allow direct and unconditionally stable transformation of Eq. (22) to a system of ODES that can be solved and further processed by existing integration codes. For this reason these methods will be discussed in detail. According to the sectional method the particle volume coordinate is partitioned using the points vi (i = 0, 1, 2 ... L). The particles with volume between Y ; - ~ and Y, belongs to the ith class and their number concentration denoted as N,

(x.

,:

).

i.e., J f(x, t)dx ’

=

Yi-1 ~

The characteristic particle size for the class i is taken to be

+ Y,. The direct sectional (finite volume) equivalent of (22) is ( i = 1, 2, ... L):

2

bi = b(x;)

ny =

lvi Vi-1

p(x, x j ) dx

(27)

This system of ODES can be solved analytically (Reid, 1965) and has been extensively used in the grinding literature as the fundamental equation and not a simplification of the continuous form of Eq. (22). Experimental values of Ni can be directly supplied from sieve analysis and rag can be found by fitting the model in Eq. (25)to the experimental N;sequentially. The problem arises from the fact that the discretized form of Eq. (26)does not conserve integral properties of the PSD and in particular total particle mass. This problem was considered by Hill and Ng (1995)who used two sets of unknown constants multiplied by both terms of the right hand side of Eq. (25).These constants were computed by the requirement of internal consistency with respect to the total particle mass and total particle number. The term “internal consistency” of a characteristic scheme with respect to a particular moment of the PSD refers to the ability of the discretized system to reproduce the discretized form of the evolution equation for the particular moment. Although this is a highly desired property (i.e., for the total particle mass, this is equivalent to the total mass conservation),it cannot guarantee the exact computation of the moment. The procedure developed by Hill and Ng (1995) may require complex analytical derivations that depend on the particular form of the fragmentation kernel and on the particular grid used for the characteristic. The above authors made these deriva-

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4 Modeling Frameworks ofComplex Separation Systems

tions for three forms of the fragmentation kernel, two forms of the grid (equidistant and geometric) and exclusively for power law fragmentation rate. The major drawback of their procedure is that it cannot be directly generalized for arbitrary parameters and implemented in a computer code. Vanni (1999) improved the situation by replacing the requirement of internal consistency with respect to the total particle number, with a better handling of the second term of right hand side of Eq. (25) (death term). This new version (slightly less accurate than its predecessor) can be fully automated, i.e., computed numerically regardless of the fragmentation rate and kernel. A different approach for the development of a quite general sectional method with arbitrary fragmentation functions, arbitrary grid, exhibiting internally consistency with respect to two arbitrary moments has been investigated by Kumar and Ramkrishna (1996). In this case the internally consistency is achieved by the proper sharing of the fragments resulting from a fragmentation event to the respective sections. The coefficients ng for the particular case of internal consistency with respect to total particle mass are:

Z(a, b, c ) =

j bbb p- ( ~y ,

c) dy

a

where 6 is the Dirac delta function. Several improvements have been proposed for the above sectional approach. As an example, Attarakih et al. (2003) developed a method where the pivot (characteristic) size for each class is free to move between the boundaries of the class and in addition the grid is moving as a whole to capture better the features of the PSD. The improved methods of this type can be implemented only through custom codes and cannot cast the problem to the form of system of ODES directly solvable by commercial integrators. 4.5.2.2 Methods of Moments

The idea led to the development of the method of moments is that in some cases the amount of information on the PSD given by a sectional method can be sacrificed in favor of the reduction in the computational requirements. For example, in a large plant simulator the grinding submodel has to be solved as efficiently as possible even in the expense of having as the only output the total particle number concentration and the mean particle size. From the technical point of view the method of moments is a generalization of the methods of weighted residual having a trial function more general than a linear superposition of basis function, i.e.,f(x, t) = F ( x , t; c) where the function F has a known form and the vector c = (cl, c2, ... cp)contains P unknown time dependent parameters which can be found in the following way. The Eq. (22) is multiplied by the P power law test functions xu' (i= 1, 2, ... P) and then is integrated for x between 0 and ~0 to give the system of equations (assuming an homogeneous fragmentation kernel):

dMa.

2=

dt

(jai - 1)

/

4.5 Modeling ofCrinding Processes 00

b ( x ) F ( x , t ; c)dx

(29)

0

xaiF(x,t ; C) dx = Mai m

where Ja,

=

I x"' p (x)dx. The most widely used forms of the distribution F are the 0

lognormal distribution f ( x , t) = e tionf(x, t) =

C1 r (c2 + 1)

(?)@e-''' c3

x

p

[-

(E)]

the gamma distribu-

with (al, a2, a3) = (0, 1, 2) and

r

is the

gamma hnction (Kostoglouand Karabelas 2002; Madras and McCoy 1998). For the case of a power law fragmentation rate b(x),the integrations in equations can be performed in closed form leading to a simple system of ODES with respect to the a;. If b(x) is not of power law form, the integral in equation (29) must be computed numerically. The Hermite and Laguerre quadratures are ideally suited for the case of lognormal and gamma distribution respectively. A systematic way to improve the log-normal method is the so called interpolation between the moments method (Kostoglouand Karabelas 2002). This method can be applied only for power law breakage rate using the following set of ai (al,a2, ... ap) = (0, 1 , 2 ... P-1). An explicit form of F is not assumed and each moment z appearing in the right hand side can be calculated from the integer moments of the PSD by the following interpolation rule:

For P = 3 the log-normal method is recovered while improved results can be found using P = 4 and P = 5. Larger values of P cannot improve the solution because the amount of information contained in the higher moments of the distribution is limited. According to the generalized method of moments the PSD is approximated by a set of Dirac delta functions with unknown strength and location; i.e., (P is an even number)

c PI2

F(x, t ) =

j=1

W j q x - xj)

Substituting Eqs. (29) and (30) leads to the following system of ODES-DAEs:

1

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4 Modeling Frameworks ofComplex Separation Systems

PI2

!%5 dt

= (Jai - 1)

wjx,? b(xj)

(34)

j=1

This method is quite general and can be used for any fragmentation rate and kernel. It is developed for the solution of the aerosol growth equation (quadrature method of moments; McGraw 1997) and aerosol coagulation equation (generalized approximation method Piskunov and Golubev 1999) independently. Kostoglou and Karabelas (2002)used it for the solution of the fragmentation equation (generalized method of moments). Typical values of P are 4 (Kostoglou and Karabelas 2004) and G (Marchisio et al. 2003), and the best choice for aiseems to be (al, az, ... ap) = (0, 1/3, 2/3, ... (P-1)/3). The system of Eqs. (33) and (34) can be solved directly using an ODE-DAE solver (Kostoglou and Karabelas 2002,2004) or using an ODE solver for (33) simultaneously with special procedures from the theory of Gaussian integration to find the weights wjand abscissas xj (Marchisio et al. 2003).

4.5 Concluding Remarks

Compared to the traditional tools and approaches for modeling and simulation of complex separation systems significant progress has been achieved the last decade. Today’s modeling tools provide advanced modeling languages and frameworks, either based on process engineering concepts or on mathematical perspectives that are suited to represent complex structural and phenomenological aspects of process systems engineering. However, a number of issues must be considered open still today. Significant challenges remain in all of the specific processing systems reviewed and these have been identified in the corresponding sections. A more general challenge is how to allow the incorporation of ideas originated from academic research into tools for industrial use. The emergence of open software architectures now provides reasonable straightfonvard routes for academic developments in some areas, such as physical properties and numerical solvers, to be directly used in process modeling tools. However, the situation is more problematic in areas of research that are related to the fundamentals of process modeling as reviewed in this chapter for specific processes. Arguably, the task of testing academic ideas and, ultimately, transferring them to commercial use has become more difficult in recent years due to the complexity of modem process modeling software and the degree of advanced software engineering that it entails (Pantelides and Urban 2004). Von Wedel et al. (2003) emphasized that the development of complex chemical process models can be improved towards a formal theory to automatically generate, manipulate, and reason about these models. Such a theory will have a strong impact on the capabilities of hture modeling tools for complex chemical processing systems, but it constitutes an on-going open research issues. It will enable inexperienced researchers to effectively use model-based techniques for a wide range of

4.6 Concluding Remarks

applications such as parameter estimation, process control and optimization. The excellent book by Hangos and Cameron (2001) provides the basic principles towards this direction. As mentioned in the excellent review by Pantelides and Urban (2004)the increasing power of process modeling technology brings new perspectives to the development and deployment of model-based solutions throughout the process lifecycle, from the initial process development to the detailed design of individual items of processing equipment and entire plants, and their control systems. To a large extent, this has been a natural evolution of earlier trends in this area and it is particularly true for the processes reviewed in this chapter. For example, a very interesting development in recent years has been the increasing permeability of the boundary between “off-line”and “on-line”applications for crystallization processes. This permeability has two distinct but related positive aspects. First, the process models themselves are re-used for both design and operational tasks, although in many cases some simpler models may be required in view of the special efficiency and robustness requirements posed by real-time and other applications, as illustrated in the review of crystallization and grinding processes. Secondly, standard process modeling software tools such as gPROMS are employed for tasks on both sides of the boundary,

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51 Ma D. L. TaJi D. K. Braaz R. D. Comp.

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52 Madras G. McCoy B.J. AIChE J. 44 (1998)p.

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53 Mahoney A. W. Rarnkrishna D. Chem. Eng.

Sci. 57 (2002) p. 1107

54 Mahoney A. W. Doyle F.J. Ramkrishna D.

AIChE I. 48 (2002) p. 981

55 Marchisio D. L. Barresi A.A. Garbero M.

AIChE J. 48 (2002) p. 2039

T. Fox R. 0. Vigil R. D. Barresi A. A. AIChE J. 49 (2003) p. 1266 Marchisio D. L. Vigil R. D. Fox R. 0. J. Colloid Interface Sci. 258 (2003) p. 322 Marquardt W. Wedel L. von Bayer B. Perspectives on Lifecycle Process Modeling in Malone M. F., Trainham J. A., Carnahan B. (eds) Foundations of Computer-Aided Process Design, AIChE Symposium Series 323, vol. 96, pp. 192-214 2000 McGraw R. Aerosol Sci. Technol. 27 (1997) p. 255 Mihalyko C. Blickle T. Lacatos B. G. Powder Technol. 97 (1998)p. 51 Mishra B. K. Powder Technol. 110 (2000) p. 246 Nicmanis M . Hounslow M. J. AIChE J. 44 (1998) p. 2258 Nilchan S. Pantelides C. C. Adsorption 4 (1998)p. 113 Pan Y . AIChE J. 29 (1983) p. 545 Pantelides C. C. Oh M. Powder Technol. 87 (1996)p. 13 Pantelides C. C. Britt H . I. Multi-purpose Process Modeling Environments in Biegler L. T., Doherty M. F. (eds) Proceedings of Conference on Foundation of ComputerAided Process Design 1994, CACHE Publications, (1995)p. 128-141 Pantelides C. C. New Challenges and Opportunities for Process Modeling in Gani R., Jorgensen S . B. (eds) European Symposium on Computer-Aided Process Engineering 11, Elsevier, Amsterdam 2001 Pantelides C. C. Urban Z . E. Process Modeling Technology: A critical review of recent developments in Proc. Conf. On Foundation of Computer-Aided Process Design 2004, Princeton University, NJ 2004 Piskunov A V. N..Golubev I.]. Aerosol Sci. 30 (1999)p. S451 Ramabhadran T. E. Peterson T. W.Seinfeld]. H. AIChE J. 22 (1976)p. 840

56 Marchisio D. L. PikturnaJ.

57 58

59 60 61 62 63

64 65

66

67

68

69

70

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72 73 74 75

76 77

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80 81 82 83

84 85 86

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87 Tsang T. H. Huang L. K. Aerosol Sci. Tech-

nol. 12 (1990) p. 578

88 Urban 2. Liberis L. in Proceedings of

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90 91 92 93 94

95 96 97

98 99 100

Chemputers 1999 conference, Dusseldorf, Germany 1999 Van Peborgh Gooch]. R. Hounslow M. J. AIChE J. 42 (1996) p. 1864 Vanni M. AIChE J . 45 (1999) p. 916 Vareltzis P. Kikkinides E. S. Georgiadis M. C. Trans. IChemE Part A 81 (2003) p. 525 Varinot C. Berthiaux H. Dodds]. Powder Technol. 105 (1999) p. 228 Vannot C. Hiltgun S. Pons M.-N. Dodds]. Chem. Eng. Sci. 52 (1997) p. 3605 Von Wedel L. Marquardt W. Gani R. Modeling Frameworks in B. Braunschweig, R. Gani (eds) Software Architecture and Tools for Computer Aided Process Engineering, Elsevier Science, Amsterdam, pp. 89-126 2002 Williams M. M. R. Loyalka S. K. Aerosol Sci. Pergamon Press New York 1991 Wulkou M. Gerstlauer A. Nieken U.Chem. Eng. Sci. 56 (2001) p. 2575 Yang R. T. Gas Separation by Adsorption Processes, Butterworth, Imperial College Press and Word Scientific Publishers, Boston, MA 1987 Zauner R. Zones A. G.Chem. Eng. Sci. 57 (2002) p. 821 ZifR. M. McGrady E. D. Macromolecules 19 (1986)p. 2513 ZifR. M. J. Phys. A: Math. Gen. 24 (1991) p. 2821

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Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein I 1 7 1 Copyright 02006 WILEY-VCH Verlag GmbH 8

5 Model Tuning, Discrimination, and Verification Katalin M. Hangos and Rozblia Lakner

5.1 Introduction

Process mode-, are increasing in size an.. complexity in current computer-aided process engineering. Therefore the methods and tools for their tuning, discrimination and verification are of great importance. The widespread use of process models for design, simulation and optimization requires the proper documentation, reuse, and retrofit of already existing models that need the above techniques. This chapter deals with computer-aided approaches and methods of model tuning, discrimination and verification that are based on a formal structured description of process models. Basic assumptions. For the majority of process control and diagnostic applications, lumped dynamic process models are used. This model class, which is considered throughout this chapter, is obtained under the following basic modeling assumptions: 0

0 0

Only lumped models are considered (ordinary differential and algebraic equation models). only initial value problems are considered. All physical properties in each phase are assumed to be functions of the thermodynamic state variables (temperature T,pressure P,compositions Ck) only.

5.2 The Components and Structure of Process Models

The formal description of process models and their structure is the basis of any methods in computer-aided process systems engineering. If one considers process models as structured knowledge collection with underlying syntax and semantics then the formal methods of computer science can be applied for model discrimination and verification. The fundamentals of such an approach are briefly described in this section. Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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5.2.1 The Modeling Problem and the Modeling Goal

A process model is jointly determined by the process system it describes and by its modeling goal (Hangos and Cameron 2001). The specification of a process system includes the definition of the system boundaries and the way of interactions between the system and its environment together with the description of the internal stmcture (subsystems, mechanisms, etc.) of the system itself. The effect of the modeling goal is much less investigated despite its importance for constructing a process model. The modeling goal. Any process model is developed for a specific use or possibly

multiple uses. These uses influence the goals that the model must fulfill. For example, the application areas of process design, control, optimization or diagnosis usually lead to different model representations for the same physical system. Meeting the stated modeling goal provides a means of determining when the modeling cycle (see below) should terminate. A set of process models ishnctionally equivalent with respect to a modeling goal if every model of the set fulfils the inequalities in the modeling goal. The seven-step modeling procedure. Good modeling practice requires a systematic

way of developing the model equations 0 f a process system for a given purpose. Although this procedure is usually cyclic, in which one often returns back to an already completed step, the systematic procedure can be regarded as a sequence of modeling steps (Hangos and Cameron 2001) that include: 1. problem setup for process modeling; 2. selection of important mechanisms; 3. analysis of data; 4. construction of model equations;

5. model verification; 6. model solution; 7. model calibration and validation. Model tuning, discrimination and verification techniques are applied in the last four steps of this procedure.

5.2.2 The Model Equation Constructing Subprocedure and its Steps

The construction of model equations is the fourth step in the above procedure, which is a cyclic procedure in itself with the following steps: 1. system and subsystem boundary and balance volume definitions; 2. establish the balance equations;

5.2 The Components and Structure of Process Models

3. transfer and reaction rate specifications; 4. property relation specifications; 5 . balance volume relation specifications; 6. equipment and control constraint specifications; 7. selection of design variables. Incremental building of balance equations. The steps of the model building procedure should be carried out in a sequential-iterative manner. This means that the model equations are built up incrementally, repeating steps of the model equation constructing subprocedure in the following order of conserved extensive quantities:

0

0

Overall mass submodel. The terms and variables in the conservation balances for the overall mass in each balance volume appear in all other conservation balances. Therefore this subset of model equations is built up first. Component mass submodel. With the given conservation balances for the overall mass in each balance volume, it is easy to set up the conservation balances for component masses. This subset of model equations is added to the equations originated from the overall mass balances. Energy submodel. Finally the subset of model equations induced by the energy balances is added to the equations.

This way the kernel of the model equation constructing the subprocedure is repeated several times for every balance volume.

5.2.3 Model Equations, Initial and Boundary Conditions, and Model Parameters

The conservation balances of mass, component masses and energy are described by ordinary differential equations in a lumped process system model (Hangos and Cameron 2001). These are called conservation balance equations, and they are accompanied by suitable algebraic constitutive equations. Constitutive equations describe the underlying static relationships between model variables dictated by physics and chemistry. The process model is then a set of ordinary differential and algebraic equations (DAEs) where there are underlying semantic relationships between various variables, equations, and equation terms. In addition to the equations themselves, it is required to specify the initial conditions of the ordinary DAE system in order to solve the problem. Initial conditions set the values of the differential variables at the initial time (t=O). Note that in the case of distributed parameter systems, partial differential equations (PDEs) are used for describing the conservation balance equations of the model. In these models boundary conditions specifying the values of the differential variables for all time on each of the system boundaries and the specification of initial conditions for the whole spatial region of interest are also part of the process model.

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5.2.4 Hidden Components

Besides the model elements above, a systematically constructed process model contains elements that are usually not stated in an explicit way, but are important for model discrimination and verification. These are as follows: 0 0

0

Application domain determines the validity region of the model. Inequality constrains constrain the value of a parameter or variable often dictated by the underlying physics and chemistry (e.g., temperature should be positive). Modeling assumptions describe the decisions of the modeler in an explicit formal way.

The importance of the modeling assumptions is explained by the fact, that model building itself can be seen as a sequence of specifying, simplifying, or enlarging assumptions on the process system to be modeled (Hangos and Cameron 2001a). This way, a uniform assumption-driven approach can be developed where modeling assumptions are regarded as artifacts of the modeling steps and allow the rigorous formal description of the modeling process and its result.

5.3 Model Discrimination: Model Comparison and Model Transformations

Model discrimination is based upon systematically comparing different process models to find relationships between them. For this purpose we briefly review various model description forms and their transformations that form the basis of model discrimination.

5.3.1 Formal Representation of Process Models and their Transformations

Model elements. The differential-algebraicequation set that forms a lumped process model can be seen as a hierarchically structured knowledge collection constructed from the following main model elements: 0

0

0

The balance volumes are the basic elements in process modeling as they determine the regions in which the conserved quantities are contained. The conserved extensive quantities (differentialvariables) are the additive properties of a system and they are used for describing the conservation principles (such as mass, component masses and energy conservation) in the balance volumes. The balance equations reflect the conservation principles for each extensive conserved quantity.

5.3 Model Discrimination: Model Comparison and Model Transformations 0

0

The transport mechanisms such as convection, transfer, reaction, etc., correspond to an effect on the conserved extensive quantities so they appear as additive terms in the balance equations. Tne constitutive equations are algebraic relations that complete the model equations. They describe property relations, extensive-intensiverelationships, transfer and reaction rate relations, equipment and control relations, and balance volume relations. The algebraic variables are the nondifferential variables appearing in balance equations and constitutive equations in the form of thermodynamic state variables, transfer and reaction rate variables, equipment and control variables, constants, specification variables, etc.

Any process model can also be seen as a collection of mathematical elements,like variables and equations of the following type: 0 0

0

0

differentialequations that originate from the conservation balance equations; algebraic equations describing the constitutive equations, the transport mechanisms, etc., that are evoked by the conservation balance equations; diferential variables with their first time derivative present in the differential equations; algebraic variables including constants and specified (design) variables.

Finally, there can be other auxiliary elements,such as surfaces, for constructing a process model. Hierarchy of model elements. Driven by the role in the process model these model elements can be organized into the following natural hierarchy levels: 0 0

0 0

L1: balance volume level L2: balance equation level L3: transport mechanism level

L4: constitutive level

A simple process model of a jacketed tank reactor with all of the above-mentioned model elements is shown in Fig. 5.1. Modeling assumptions. A modeling assumption can be expressed in a natural language sentence and formally described by a triplet (Hangos and Cameron 2001a) given by:

variable-name relation keyword, where ‘variable-name’refers to a process model element described previously in this subsection, ,relation’ is an “=” (“equals”)or “is“ symbol in most cases, and ,keyword’ is a symbolical or numerical constant or another ‘variable-name’.Thus, a modeling assumption is understood as an assignment to the ‘variable-name’and usually translated into either additional mathematical relationships between model variables and parameters, or into constraints between already introduced variables and parame-

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CONSERVATION BALANCES Balance volume: tank - mass balance: M = const

du = vpcpTa- VpcpT+ Vr(

energy

)- Q

dt - component mass balances: dm, __ - vcAo- vcA-Vr dt

---vc,+Vr dm, Balance volume: cooler - mass balance: Mc= const - energy balance: dU,= dt

dt

v ~ P ~ c ~ ~ T ~+~Q - v ~ P ~ c ~ ~ T

CONSTITUTIVE EQUATIONS

r = kc, ~~

E

k = k,e RT mA =Vc,

u, = M'CJ M =Vp

M , = V,P,

ma = Vc, Figure 5.1

v, T, c , cB

ASSUMPTIONS - 2 lumped balance volumes (tank, cooler) - 3 components in tank (A, B, solver)

- 1 component in cooler - constant mass holdups - constant physico-chemical properties - A -> B first order exothermic reaction in tank

A simple process model example.

ters. The modeling assumptions can either be elementary assumptions consisting of just a single triplet or composite assumptions being the conjunction (logical and) of elementary assumptions. The model equations can be formally seen as a structured string obeying syntactical and semantic rules and the modeling assumptions can then be regarded asformal modeling transformations on these equations resulting in another set of model equations. The effect of an assumption on a given set of equations is computed following all of the implications ofthe assumption through the syntactical and semantic rules. Formally this is performed by substituting the assignment equations describing the assumption into all of the original model equations and then performing rearrangements using algebraic transformations. 5.3.2 Algebraic Transformations, Algebraically Equivalent Models

A set of functionally equivalent process models can be algebraically equivalent, when one can transform any member of the set to any other one using algebraic transformations. Algebraic transformations can be applied to model equations and to model variables (including both differential and algebraic ones). Examples of algebraic transformations on a set of process model equations are multiplying an equation by a constant number, adding two equations together, substituting one equation for another by expressing it as a variable and substituting that variable in every other equation. It is important to note that the variables do not change when applying algebraic transformations to the equations of a model, but the

5.3 Model Discrimination: Model Comparison and Model Transformations

CONSERVATION BALANCES Balance volume: tank - mass balance: M = const - energy balance: ._ dT v -=-(To dt V

- T ) + koe

RTCA(-AH)

p',

- component mass balances: dc, dt dcE

dt

-

v (cAo-cA)-k0e V

- 'c,+k,,e

Figure 5.2

V

Balance volume: cooler - mass balance: M-= const - energy balance: KA(T-T,)

dT,

V@,

dt

-

v,

V,

(TdJ-T )+

KA(T-T, I V c P$,'

__ E

R T ~ A

~-

'TcA

The substituted process model

model equations do. The formal description of algebraic transformations to a set of algebraic equations, together with a canonical set of primitive algebraic transformations and their effect on computational properties, can be found in Leitold and Hangos (1998).There it is shown that certain computational properties of DAE models, such as the differential index, does not change with algebraic transformations, but others, like the decomposition of the model may change quite drastically. Another type of transformation applicable to a model is when one applies a linear or nonlinear algebraic transformation to some of its variables and writes the model using these new transformed variables. This is analogous to coordinate transformation in geometry and is therefore called a coordinate transfornation. It is important to note, however, that the general locally invertible nonlinear transformations, which are useful and widely used in nonlinear system theory, are not well accepted in process systems engineering because they change the engineering meaning of the variables. The extensive-intensiveconstitutive algebraic equations, however, are widely used to transform a process model into its intensive variable form suitable for process control and diagnostic applications. Here we transform the set of differential variables in a balance volume originally equal to the set of conserved extensive quantities to be the set of overall masses (left unchanged) and the measurable intensive quantities (such as temperatures, compositions and pressures). Model classes. Algebraic transformations may change the mathematical form of a model, but an algebraicallytransformed model is the same from a process engineering point of view. Such algebraicallyequivalent models form a model class. Figure 5.2 shows an algebraically equivalent form of the simple process model of the jacketed tank reactor depicted in Fig. 5.1, where the constitutive equations have all been substituted into the differential ones.

5.3.3 Model Simplification and Model Building Transformations

Modeling assumptions can be regarded as representations of the engineering activity and decisions during the whole modeling process in constructing, simplifying and

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analyzing process models and they act as modeling transformations on the process models. Assumption-driven modeling works directly with modeling assumptions, thus enabling the definition and handling of process models as structured knowledge with defined syntax and semantics. Model building assumptions. The modeling assumptions applied in the model building phase determine the structure of the process model and the assumptions applied to an existing model modify the equations and may even change the structure of the model. The model building procedure is seen as a sequence of model building, specification assumptions, and their associated transformations, as well as algebraic transformations applied to a process model. This way of assumption-driven model building offers a systematic incremental way of constructing a process model in its canonical form. Model simplification assumptions. The model simplification assumptions, which can either be elementary (atomic) or composite, and are composed of a conjunction of elementary assumptions, can be formally described as model transformations. These transformations are projections in a mathematical sense and are often performed in two substeps :

1. adding the equality describing the assumption to the already existing set of model

equations and performing algebraic transformations (for example substitutions) to get a more simple form; 2. adjusting the set of differential, algebraic and design variables to satisfy the degree of freedom requirement. The effect of a simplification assumption on a given set of equations is computed following all of the implications of the assumption through syntactical and semantic rules. Formally this is performed by substituting the assignment equations describing the assumptions into all of the original model equations and then performing rearrangements using algebraic transformations. It is important to note that not every simplification transformation is applicable to a particular process model. Moreover, a transformation may influence only part of a process model and then this effect propagates through the relationships between the model elements. Forward reasoning can be applied to find all of the implications of a simplificationtransformation, and the effect of a composite transformation is computed by generating a sequence of simplified process models. It is important to note, however, that the resultant model may be different if the order of the assumptions is changed, because model simplificationtransformations may be related and noncommutative (Hangos and Cameron 2001a). In conclusion we can say that algebraic manipulations can be regarded as equivalence transformations, model simplification, and enrichment assumptions as general nonequivalence modeling transformations acting on process models that bring the process model out from its original model class.

5.4 Model Tuning

5.3.4 Model Discrimination and Model Comparison

Model discrimination aims to find an exact relationship between two given process models of the same process system. First, we have to determine if these models are developed for the same modeling goal, and if so, if they are algebraicallyequivalent. Unfortunately, there are no standard formal ways of performing the above two basic tasks, mainly because the lack of our knowledge in formal description and utilization of the modeling goal itself (see Lakner, Cameron and Hangos 2003). In the case of functionally equivalent models one should only perform model comparison to investigate if they are algebraically equivalent, and if not, give their relationship in terms of model simplification transformations that lead the more detailed model to a more simple one (see later Section 5.4.3). Canonical form. We have already seen in Section 5.3.1 that both functionally and

algebraically equivalent process models form a model class. It is then useful to define and use a canonicalform of a process model class, which is a member of the class having each model element in the form that has a clear engineering meaning (Hangos and Cameron 2001). The differential equations of a process model in canonical form are the conservation balance equations for the overall mass, total energy (or enthalpy) and all but one component masses in their extensive form containing terms for the convective, transfer and source transport. These are supplemented by constitutive algebraic equations of standard categories such as intensive-extensive relationships, reaction rate equations, thermodynamic state equations (such as the ideal gas law), etc. The simple process model shown in the left-hand side of Fig. 5.1 is in its canonical form. Comparison o f process models. In the case of algebraically different but functionally equivalent process models we aim at finding out if these models are from the same model class, or not. The general procedure of comparing these models is to first bring them into their canonical form and then compare them element-wise following the hierarchy of the model elements from top (balance volumes) to down (model variables and parameters). This approach requires that the models being compared are given with their entire model elements hierarchically arranged.

5.4 Model Tuning

Process models generated for a given modeling goal may often be over-simplifiedor over-complicated for another use, which is why there is usually a need to extend, to simplify, or to generate a new model in the worst case. In addition, process models usually contain unknown parameters to be estimated using measured data, when we need to calibrate the model to tune it to meet the modeling goal.

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In the model constructing step of the seven-step modeling procedure we may need to perform a model simplification phase for refinement of an already defined process model by additional simplifying modeling assumptions. These simplifylng assumptions can be described by triplets and are usually translated into additional mathematical relationships. The simplifying procedure itself consists of two main steps. These steps are the implication of the modeling assumptions on the model equations with the aid of syntactical and semantic rules, and the rearrangement of the resulting equations make use of formal algebraic transformations. Because the process model elements are related to each other by a well-defined syntax and semantic, a model element cannot be simplified independently of the others. In addition, the implications of a simplification assumption depend both on the assumption and the structure of the model. For example, when a modeling assumption is related to a balance volume, its implications can refer not only to the equations of the balance volume, but could modify other related balance equations in other balance volumes. The same way, a modeling assumption related to a term in a balance equation can imply modifications of other terms in other balance equations. Figure 5.3 shows how a modeling assumption on the mass convective term of a balance volume changes the energy and component mass convective terms when a simplification assumption is applied to the model of the jacketed tank reactor depicted in Fig. 5.1. The implications of an assumption on the model equations can be determined by forward reasoning where all of the implications of the assumption are computed by respecting syntactical and semantic rules. The set of resulting modeling equations at the end of the implication stage is rearranged to an easily-solvableform by using algebraic transformations (Lakner et al. 1999).

CONSERVATION BALANCES Balance volume: tank - mass balance: M = vp

- energy balance:

du= vpc,To dt

SIMPLIFICATION ASSUMPTION

- mass convective outlet in tank is negligible + Vr(

)-

Q

- component mass balances: dm

2= vcdo-Vr

dt dm,=Vr dt

Balance volume: cooler - mass balance: M, = const - energy balance: du, - vC~ , c , J ~ -v, o PF,T~ + Q

7

Figure 5.3

A simplified process model

CONSTITUTIVE EQUATIONS

Q=WT-T,J r = kc, E -_

k = k,e RT mA =Vc, mn

= VCB

U=Mc,T

u, =McccpTc M =Vp

M, = V,P,

5.4 Model Tuning

5.4.2 Model Extension

Model extension procedures are widely applied in process modeling in quite different contexts:

0

0

At the end of a modeling cycle, in the model validation step, it may turn out that the developed model fails to fulfill the modeling goal. Then one has to extend the model by including additional model elements that were originally neglected, such as balance volumes, balances, mechanisms, etc. The incremental assumption-driven model building (Williams et al. 2002) uses model extension procedures, too. The incremental building of balance equations in the model equation constructing procedure can also be seen as model extension (see Section 5.2.2).

There are two questions of critical importance in model extension procedures: the selection of default values and the methods of ensuring incremental consistence. Default values set the value of model elements belonging to a model element (the “children elements” in the model element hierarchy) that is just being created. Incremental consistency is ensured by allowing one to add only such new model elements to an already existing consistent model that are not in conflict or contradiction to any already existing model element. Details about these questions can be found in the literature on computer-aided process modeling (CAPM)tools (see, e.g., Jensen-Krogh 1998; Modkit 2000). Empirical model building. This is a special method of model extension using empirical data and grey box models (Hangos and Cameron 2001). It is a top-down approach of model extension where the model element(s)to be changed or extended is (are)determined in a heuristic black box way by using sensitivity analysis. The submodel of the new element is also constructed in a heuristic black box way from some general approximating model class with its parameters estimated using measured data (see also model calibration in Section 5.4.4).

5.4.3

Model Comparison by Assumption Retrieval

In order to avoid any inconsistency during model simplification and extension, it is extremely useful to register explicitly all modeling assumptions applied in the construction and modification of the process model. The documentation of the model contains these modeling assumptions (Hangos and Cameron 20014 in the ideal case, but this documentation can often be incomplete or even missing. In order to complete the model documentation with all modeling assumptions, an assumption retrieval procedure could be used. The retrieval of modeling assumptions from a pair of process models for model analysis and comparison is an important but unusual problem where not only efi-

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cient algorithms but the engineering understanding is lacking. The reason for this is that assumption retrieval can be regarded as the inverse task of model transformations where modeling assumptions are determined from two related (one detailed and one simplified) process models of the same process system. As model transformations are projections in mathematical sense, it is not possible in general to retrieve fully the original model from the transformed one and from the transformations. Because of this, the result of assumption retrieval from the original and the transformed models may not be, and in general will not be, unique. Because of the nonuniqueness of the assumption retrieval task, an intelligent exhaustive search algorithm (Lakner et al. 2002) is needed for its solution. 5.4.4 Model Calibration (Model Parameter Estimation)

Process models developed from first engineering principles almost always contain model elements, model parameters and/or other elements like reaction rate expressions, the value of which is unknown. While the modeling approach futes the structure of the model, these unknown elements make the model “grey,”that is, partially unknown. Measured data from the real process system to be modeled is used along with model parameter and/or structure estimation methods to fine-tune the model for meeting the modeling goal. This fine-tuning of process models is called model calibration and is a standard step in the seven-step modeling procedure (Hangos and Cameron 2001). There are several key points to take special care of when performing model calibration: 0

0

Selection of model parameters to be estimated. Besides the real unknown model parameters, one often has parameters or model elements with large uncertainty associated to their values. If the model is sensitive with respect to the value of these uncertain parameters, then it is advisable to consider them as unknown and estimate their values using measured data (NCmeth et al. 2003). Nonlinear parameter estimation. In most of the cases the parameters to be estimated enter the model in a nonlinear way and the model itself is dynamic. This makes the parameter estimation problem especially difficult and can only be solved by numerical optimization techniques (Hangos and Cameron 2001 ; Ailer et. a1 2002). Quality of the data and the estimated parameters. The statistical nature of model parameter estimation requires one to check carefully the following key ingredients and properties of the parameter estimation method: - quality of the measured data (steady-state,no outliers and gross errors, etc.) and the presence of sufficient excitation; - quality of the prediction error sequence (if this is a realization of a white noise stochastic process); - quality of the estimated parameters considering their nonbiasness, variance, and covariance matrix.

5.5 Model Verijcation

5.5 Model Verification

Having completed the model equation constructing step of the seven-step modeling procedure (see Section 5.2.2), one needs to perform model verification, that is, to check the model against engineering insight and expectations, before attempting its solution. Model verification includes checks of syntax and semantics, as well as the well-posedness of the model from mathematical sense, and analysis of computational and dynamic properties.

5.5.1 Formal Methods for Checking Syntax and Semantics

Before a mathematical model is used for solvability analysis or is solved, it is useful to check and ensure its consistency. There are several methods for consistency checking that are applicable both in computer-aided modeling tools and in process systems engineering practices: 0

0

0

Dimension analysis is a useful simple check for consistency of the model equations in terms of units of measure. This very useful but not widespread method is used in ASCEND (Evans et. a1 1979) and VeDa (Bogusch and Marquardt 1997) modeling languages, for example. Syntactical veriication methods (checking bracketing, vector operations, etc.) are especially important for computer-aided modeling tools in which the model equations can be directly defined by the users. An example for this is the ICAS/ModDev modeling system (Jensen-Krogh1998). Logical checking (hierarchical consistency, material characterization consistency, chemical reaction rate equation derivation, etc.) is used for examining the modeling assumptions’ consistency. This very important verification method, accomplished before generating the model equations, is used in the majority of computer-aided modeling tools.

It is important to note that the above-introduced model verification methods are applicable only for partial consistency checking and they cannot insure full model consistency.

5.5.2 Structural Analysis of Computational Properties of Process Models

The structural analysis of dynamic lumped process models form an important step in the seven-step modeling procedure and is used for the determination of the solvability and computational properties of the model. This analysis includes the determination of the degrees of freedom (DOF), the differential index and the structural components of the model.

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Analysis of DOF and differential index. In order to solve a mathematical model, a

sufficient number of variables have to be specified so that the number of unknown variables exactly equals the number of equations. The DOF, i.e., the difference between the number of unknown variables and the number of equations in the mathematical representation, is equal to the number of variables that must be specified for obtaining a solvable equation system. There are three possible values for DOF to obtain (Hangos and Cameron 2001): 0

0

DOF=O. This implies that the number of independent unknowns and independent equations is the same and a unique solution may exist. DOF>O. This implies that the number of independent variables is greater than the number of independent equations and the problem is underspecified. In this case some of the independent variables have to be specified by some external considerations in order for the DOF to be reduced to zero. DOF
It is important that the DOF analysis can be applied both on the entire equation system and on the subsets (mass-related equations, energy-related equations, etc.) of model equations separately. The diferential index of a DAE is defined as the minimum number of differentiations with respect to time that the algebraic system of equations has to undergo to convert the system into a set of ordinary differential equations (ODE) (Hangos and Cameron 2001). The index of a pure ODE system is zero by definition. When the Jacobian of the algebraic equation set of DAE is of full rank, then the index of DAE is one. In this case the initial values of the differential variables can be selected arbitrarily, and the DAE can easily be solved by conventional methods such as RungeKutta or Backward Differentiation methods. If, however, the index is higher than 1, special care should be taken in assigning the initial values of the variables, since some “hidden”constraints lie behind the problem specifications. Structural decompositions. Effective graph-theoreticalmethods have been proposed in the literature based on the analysis tools developed by Murota et al. (1987) for the determination of the most important solvability property of lumped dynamic process models (Leitold and Hangos 2001): the differential index and the structural components. The analysis is based on constructing the structural representation graph of the DAE model equations where the variables are represented as vertices and the equations as edges (dependencies) between vertices. Labels are associated with the vertices of the graph indicating the computational property of the associated variable. The reduced representation graph, together with the L-and M-components and their hierarchy, are determined by the analysis, which can effectively be used to select a suitable numerical solution method and to determine the computational path. In addition, one can artificially structure a DAE model by using algebraic transformations to be able to solve it more efficiently (Robertson and Cameron 1997).

5.5 Model Ver$cation

5.5.3 Analysis of Structural Dynamical Properties (Controllability, Observability, Stability)

One simple yet powerful method of model verification is to analyze the structural dynamical properties of the developed model and compare the result with engineering expectations. The first step of the analysis is to transform the lumped process model in its DAE form into a nonlinear state-spacemodel form, which is only possible for index 1 models. Structure graph. Then the so-called structure graph, a weighted (signed) directed

graph (SDG)of the model is constructed, which contains the state, input and output variables as vertices, and the model equations determine its directed edges in such a way that a directed edge points towards variable vi from vj when vj appears on the right-hand side of the equation that determines v,. The weight of the edges contains the sign of the effect (the sign of the partial derivative) the edge is associated with. Note that an SDG model corresponds to a class of process model with the same structure. We say that a structural dynamical property, such as structural controllability, observability, or stability, holds for a class of process models if almost every member of the class (with the exception of null-measure sets) possesses the property. Check o f structural properties. Given an SDG model, there are simple-to-check

combinatorial conditions for the underlying process model class to be structurally controllable or observable (Hangos and Cameron 2001; Hangos et al. 2001). For example, a process model is structurally controllable (observable)if its state structure matrix is of full structural rank and its SDG graph is input (output) connectable, that

Figure 5.4 The SDC of the simple process example in Fig. 5.2 (input: cAo,To,T& output: cg, T).

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is, there exists at least one directed path to every state (output)variable vertex from an input (state)variable vertex. The check of structural stability is more computationally demanding and requires finer qualitative information on the value of the model parameters. A simple, but not thorough enough, general method of checking structural stability of a process model class is the method of conservation matrices (Hangos and Cameron 2001). Here we use the state matrix of a locally linearized process model and check if it is a conservation matrix or not. A real square matrix is a conservation matrix if its diagonal elements are negative, all other elements are nonnegative and the diagonal elements are row (or column) dominants, i.e., the absolute value of the diagonal element is greater that the sum off the off-diagonal elements in every row (or column).

Acknowledgments

This research was partially supported by the Hungarian National Research Fund through contract numbers TO42710 and TO47198 and is gratefully acknowledged.

References

M. (2002) Parameter-estimation and model validation of a low-power gas turbine. Proceedings of the IASTED International Conference, Modelling, ident$cation and control, Innsbmck, Austria, pp. 604-609. 2 Bogusch, R., Marquardt, W. (1997) A formal representation of process model equations. Computers and Chemical Engineering 21, 1105-1115. 3 Evans, L. B., Boston, J . F., Britt, H. I., Gailler, P. W., Gupka, P. K., Joseph, B., Mahalec, V., Seider, W. D.. Yagi, H. (1979) ASPEN: An advanced system for process engineering. Computers and Chemical Engineering 3, 319-327. 4 Hangos, K. M.,Cameron, I. T. (2001) Process Modelling and Model Analysis. Academic Press, London, pp. 1-543. 5 Hangos, K. M.,Cameron, I. T. (2001a) A formal representation of assumptions in process modelling. Computers and Chemical Engineering 25, 237-255. 6 Hangos, K. M.,Lakner, R., Geizson, M. (2001) Intelligent Control Systems: An Introduction with Examples. Kluwer Academic Publishers, pp. 1-301. 7 Jensen-Krogh, J. A. (1998) Generation of Problem Specijc Simulation Modek; Withinan Integrated Computer-Aided System. PhD thesis, Danish Technical University. 1 Aibr, P., Szederkinnyi,G., Hangos, K.

8 Lakner, R., Cameron, I., Hangos, K. M. (1999) An assumption-driven case-specificmodel editor. Computers and Chemical Engineering 23,695-698. 9 Lakner, R., Hangos, K. M.,Cameron, I. T. (2002) Assumption retrieval from process models. Computer-Aided Chemical Engineering 9,195-200. 10 Lakner R., Hangos. K. M.,Cameron, I. (2003) Construction of minimal models for control purposes. Computer-Aided Chemical Engineering 14, 755-760. 11 Leitold, A,, Hangos, K. M. (1998) O n algebraic model transformations. In: CPM98.3rd IEEE European workshop on computer-intensive methods in control and data processing. Prague, Czech Republic, pp. 133-138. 12 Leitold, A., Hangos, K. M.(2001) Structural solvability analysis of dynamic process models. Computers and Chemical Engineering 25, 1633-1646. 13 ModKit (2000) Computer-Aided Process Modeling (ModKit), http:www.lfpt.rwth-

aachen.de/Research/Modeling/modkit.html.

14 Murota, K. (1987) Systems Analysis by Graphs

and Matroids. Springer, Berlin.

15 N h e t h , H., Pdkovics, L., Hangos, K. M.

(2003) System identification of an electropneumatic protection valve. Research Report SCL-o01/2003. Systems and Control Labora-

5.5 Model Ver$cation

tory. Computer and Automation Research Institute, Budapest, Hungary, http: daedalus. scl. sztaki. hu/PCRG/. 16 Robertson, G.A,, Cameron, I. T. (1997) Analysis of dynamic process models for structural insight and model reduction I: structural

identification measures. Computers and Chemical Engineering 21,455-473. 17 Williams, R. P. B., Keays, R., McGahey, S., Cameron, I. T., Hangos, R M. (2002) SCHEMA: Describing models using a model. ing object language. CHEMECA-2002, pp. 922.

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6 Multiscale Process Modeling /an T Cameron, Cordon D. Ingram, and Katalin M. Hangos

6.1 Introduction

This chapter covers multiscale modeling by discussing the origins of such phenomena in process and product engineering as well as discussing the approaches to the modeling of systems that seek to capture multiscale phenomena. The chapter discusses the development of the partial models that make up the multiscale model, particularly focusing on the characteristics of those models. The issue of partial model integration is also developed through the use of integrating frameworks. Those frameworks are analyzed to understand the important implications of model coupling and computational behavior. Throughout this chapter, reference is made to granulation processing, which helps illustrate the concepts and challenges in this important area.

6.2 Multiscale Nature of Process and Product Engineering 6.2.1 The Origin and Nature of Multiscale Engineering Systems

Multiscale systems, and hence their models, exist due to the phenomena they contain or seek to represent. This is due to the fact that thermodynamic behavior and rate processes undergird our main view of the scientific and engineering world. The underlying phenomena come into focus depending on the granularity of our perspective, which is influenced by the history of scientific investigation and, indeed, our own backgrounds. This perspective ultimately deals with length and time scales that can vary from atomic to global scales, or beyond. We can investigate time scales of nanoseconds to millennia, or length scales from nanometers to light-years,as seen in Villermaux (1996), who illustrated the typical scales dealt with in chemistry, Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright @ 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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physics, chemical engineering and astronomy. Hence, we are presented with a spectmm of scales depending on where we wish to view the system under study. Our modeling efforts are simply a mapping of our understanding of these phenomena into a convenient mathematical or physical representation. The amount of scale-related information we incorporate into our models determines the multiscale degree of that representation. In some cases we can work on a single scale of time and/or length, or incorporate two, three or more scales within our models. This latter case is the area of multiscale modeling, which we address here. As evidenced by the literature on the subject of multiscale engineering systems, there has been an explosion in interest since the mid 1990s (Li and Kwauk 2003). Papers on this topic at the start of the 1990s were very few. By 2000 a ten-fold increase in publications occurred and it continues to grow at a phenomenal rate. It is an area of intense research driven mostly by applications in science and engineering, especially materials science, mathematics and physics. Li and Kwauk (2003)and Glimm and Sharp (1997) provide examples from many disciplines. In fact, multiscale models are often multidisciplinary. Within chemical engineering, the multiscale approach facilitates the discovery and manufacture of complex products. These may have rnultiscale product specijcations, that is, desired properties specified at different scales. Biotechnology, nanotechnology and particulate technology - in fact, product engineering in general - are driving the interest in the multiscale approach (Charpentier 2002; Cussler and Wei 2003). In addition, despite the continuing increase in computing power, there are problems of practical interest that will remain intractable when tackled by direct, “brute force” methods. Multiscale techniques provide a way of making these problems feasible. There are a growing number of tools, methods and representations for engineering systems, yet little fundamental conceptual analysis leading to overall frameworks that help guide the modeling of multiscale systems.

6.2.2 Length, Time, Other Scales and their Representation

Because process engineering has its roots in physics and chemistry, the properties of models reflect the underlying time and length scales on which important phenomena occur. This can be from the quantum mechanical length scales of m with time scales of s to global scales of lo4 m and 10’ s and higher. At one extreme we are concerned about subatomic behavior, while the other extreme represents global processes that might have characteristic times of years or decades. Small scales are of significant interest in determining product properties whereas large-scale processes can be of interest to process engineers involved in areas such as climate change, environmental impact and supply chain management. Figure 6.1 shows a general scale map appropriate to process and product engineering. It is an adaptation of work by Grossmann and Westerberg (2000). It is noticeable that there is a general relationship between length and time scales that reflects the time constants over which phenomena occur at different lengths of behavior.

6.2 Multiscale Nature of Process and Product Engineering

mi:

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Figure 6.1 A general scale map

It is now widely appreciated that product quality is often determined at scales well below the scales applicable at the processing or macrolevel. Hence there is intense interest in micro- and nanoscale behavior in product design. For example, the development of granulation processes via drum or pan granulation is a multiscale operation, where final product quality is determined not only by the macroscale processing equipment level, but also at the microscale level of particle formation and interaction. A typical granulation circuit diagram is shown in Fig. 6.2, which highlights the principal operating equipment in the circuit. In this case, the circuit consists of the granulator where fine feed or recycle granules are contacted with a binder or reaction slurry. Growth occurs depending on a number of operational and property factors. Drying, product separation and treatment of recycle material then occurs. For this application, Fig. 6.3 shows a scale map from Ingram and Cameron (2004, which considers the key phenomena as represented by length and time scales within the processes. The scales represent individual particles through to agglomerates and then onto processing equipment and finally the complete circuit. Besides the length and time scales, a detail scale could also be considered, which seeks to develop models with varying degrees of fidelity in relation to the real world phenomena. This form of scale can consider such issues as: 0

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The granularity of the system view in terms of number and types of balance volumes and the degree of aggregation that takes place. The number of key mechanisms related to flows, reaction, heat, mass and momentum transfers within the model and their inclusion or exclusion. Of particular interest is the complexity and fidelity of the constitutive relations. These issues can have a significant impact on the validity region of the resultant process

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or product models, as different relations such as equations of state or property models are used. 0 Species identification and representation using, for example, “lumped” representations common in pseudo-component approaches to petroleum fractions. The detail scale complements the traditional time and length scales that are common in multiscale modeling.

6.3 Modelingin Multiscale Systems

6.2.3 Key Issues in Multiscale Process Modeling

We mention briefly some of the key issues in multiscale modeling before addressing some of those issues related to modeling and integration in Sections 6.3 and 6.4. The principal issues relate to:

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Scale identification and selection for a specific modeling goal: For any modeling task the issue of identifylng what scales are needed to be represented in the final model is an important consideration. As a first step, the literature contains a variety of scale maps and diagrams that show the hierarchical organization in specific application areas (for examples Alkire and Verhoff 1994; Maroudas 2000). There are several approaches that are more fully elaborated upon in Section 6.3. However, an understanding of the time scales of interest can often dictate the final scales of length and time that are needed in the model. Model representation: In what form does the model exist? This question is often answered by our understanding of the system under study and the phenomena we can identify. It is most often the case that grey-box models are used at several scales, because we often have some mechanistic understanding of the system phenomena. At the same time there are system parameters that are calculated via data fitting or, in some cases, are averaged values from another calculation at a lower scale. A spectrum of models exists from completely black box to mechanistic descriptions. Model integration: Model integration refers to linking the partial models that apply at a single scale into a composite, multiscale model. It still remains a challenging area and one where much is yet to be done to resolve this important issue. Section 6.4 discusses a number of these issues, with reference to several application areas. A number of integration frameworks exist, which possess distinct characteristics of information flow between scales and hence computational and other properties. Model solution: Solution of multiscale systems remains another major challenge, especially where distinct model forms are present within the composite or multiscale model. This is a huge area and beyond the scope of the present chapter. Some aspects are briefly discussed in Section 6.4.4.2.

6.3 Modeling in Multiscale Systems

The general approach for developing a process model for multiscale systems is an extension of that for conventional, nonmultiscale process systems (Hangos and Cameron 2001a). This approach, called the “seven-step modeling procedure”, included the following stages: 1. model goal set definition (modeling problem specification); 2. model conceptualization (identifylng controlling factors); 3. modeling data: needs and sources;

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4. 5. 6. 7.

model building and model analysis; model verification; model solution; model calibration and validation.

Thus, this section focuses on the special elements of the extended general approach that make it applicable for modeling multiscale systems as well. Similar to the conventional case, the modeling problem specification consists of the definition of the process system with its boundaries, subsystems, components, mechanisms together with the description of the modeling goal. In the multiscale case the process system description as well as the modeling goal may call for a multiscale model when there are order of magnitude differences in either length or time behavior between the system elements. A separate subsection below deals with the specialties of the modeling goal in modeling multiscale systems. An extended seven-step modeling procedure (Hangos and Cameron 2001a) can also be followed as a general approach in the multiscale case when some of the steps need special care and procedures that are described below.

6.3.1 Multiscale Modeling Strategies

Most often the need for developing a multiscale model arises in step 1 (Problem d e j nition) or step 2 (Identifi) controllingfactors) of the seven-step modeling procedure. Here, one usually identifies the necessary scales that become part of the problem definition. Thus, the first two steps should be repeated for each of the scales to develop individual modeling problem definitions and to identify the relevant controlling factors. This way a set of scale-driven related submodels is created. Interest in system behavior over a long period of time (steady-state properties) often excludes phenomena, and hence scales, operating on a very fast timeframe. This eliminates the fast components of the system. In other cases such as modeling startup and shutdown performance of processes the intermediate time scales are of main interest. Here, some exclusion of slow and very fast components can be made (Robertson and Cameron 1997a,b). In addition, modeling decisions should be made on how to organize the information flow between the partial models, that is, to determine the multiscale modeling framework (Ingram, Cameron and Hangos 2004) as discussed in Section 6.4. Having formulated the modeling problems for each of the partial models, identified the controlling factors and reviewed the data available,we can turn to constructing the model in step 4 of the seven-step modeling procedure. There are two fundamentally different approaches to doing this: the bottom-up and the top-down approaches. As their name suggests, bottom-up approaches start with constructing the partial model in the finest resolution scale and proceed towards the coarser scales. Alternatively,top-down approaches start with the coarsest scale partial model

6.3 Modeling in Multiscale Systems

and refine its elements using finer scale submodels if necessary. Two other approaches have also been suggested in the literature. A simultaneous approach, which has been used industrially in the context of new product design (Lerou and Ng 1996), involves developing models at each scale of interest at the same time, and then linking them together. Middle-out modeling is the method of choice in some multiscale biological applications (Noble 2002). It refers to building up a model by starting with the scale that is best understood and has the most data, and then working “outwards” (to finer and coarser scales) from that. In the following, the key elements of the extended modeling procedure, the development and role of the multiscale modeling goal set, and the specialties of the topdown and bottom-up strategies of model construction will be described in more detail. 6.3.1.1

The Role of the Modeling Goal

Any process model is developed for a specific use or possibly multiple uses. These uses influence the goals that the model must fulfill. It is, however, important to recognize that modeling goals normally change, are refined, deleted or added as the modeling cycle proceeds. This is clearly seen in the modeling process of multiscale systems, where the original modeling goal might indicate the use of a multiscale approach and then the modeling goal set of any partial model is established and refined incrementally as the modeling proceeds. Overall Modeling Goal

The modeling goal is typically a statement with three major components: 0 0 0

the need to develop a model in some relevant form; an application of the model for a given purpose; a reality that is being modeled.

These three aspects can in turn be decomposed into lower level goals that are applicable to the partial models. This can be seen in the overall modeling goals such as: “Developa model for evaluating control options for chemical vapor deposition (CVD) of...”. The overall modeling goal will determine the number and hierarchy of the scales and their integration framework. Multiscale models are needed if the process system has controlling factors or mechanisms with very different scales covering several orders of magnitude. The modeling goal or the requested modeling accuracy might require partial models of finer granularity. A goal might include inputs and outputs at different scales, e.g., in CVD, where reactor operating conditions (macro) might be achieved such that film microstructure has acceptable smoothness (micro) or simply feasibility. Alternatively,the modeling feasibility can dictate partial model inclusion. An example of such a case can be a dynamic modeling problem of an industrial granulator drum for fault detection and diagnosis purposes. Figure 6.4 shows some multiscale aspects of the granulation system and indicates some of the key informa-

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A rnultiscale view of linking granulation phenomena.

tion flows that exist. Because the malfunctions and faults in this equipment can be consequences of the granulation mechanisms, transport phenomena, fluid dynamics and operation procedures affecting the whole system, one needs to have several scales, a granulation particle scale and equipment scales integrated in a multiscale modeling framework. Modeling Goals for the Partial Models

The systematic decomposition of the overall modeling goal to the individual scales is difficult and has not yet been studied well in the literature. If we consider the modeling goal to be a multifaceted statement then some of the facets originate in the original overall modeling goal, i.e., those goal elements that are relevant to the partial model related to a particular scale are simply inherited from higher level goals. The other integrutingfucets in the modeling goal set of a partial model ensure its consistency and purposefulness from the viewpoint of the multiscale integrating framework applied to the modeling problem. This integrating part may contain variables in other partial models to be computed with a given accuracy, and may determine the data or other model ingredients to be used that are delivered by other partial models. Continuing the granulator drum example, the granulation particle submodel will inherit the goal facets related to the malfunctions caused by the granulation process

6.3 Modelingin Multiscale Systems

itself. That is, it should be dynamic and should describe the formation, growth, and breakage of the particles. The integrating facets in the modeling goal set ensure that the model to be developed should produce the source and kernel functions in the granule population balance in every time instance with a given accuracy that is needed to complete the conservation balance equations on the equipment scale. 6.3.1.2 Gradual Model Enrichment or Iterative Deepening

The philosophy of the top-down approach for multiscale model development is very simple: start from the overall system model on the coarsest resolution scale (largest time or space scale) and develop a new partial model on a finer resolution scale if any facet in the overall goal set requires it. This approach can be regarded as iterative model deepening, which is directed by the modeling goal and its sensitivity with respect to the elements of the process model being developed. The top-down approach has been viewed as the best method for process engineering because time and cost pressures favor quick application of the results, with a minimum of detailed modeling. Later on in the lifecycle of the process, model refinement can be applied as required. The iterative model deepening technique is applied in step 4 (Model building and model analysis) of the seven-stepmodeling procedure when the number of scales and the model integration framework have already been selected. We then start from the overall system model on the lowest resolution scale with the overall modeling goal and determine which facets of the modeling goal are not satisfied. By using sensitivity analysis, it is possible to determine which model elements influence the missing goal facets. The model is then enriched by a partial model on a finer resolution scale for the necessary model elements. Repeating the above deepening steps until the entire modeling goal set is satisfied constructs the final multiscale model. It is worth mentioning that the iterative model deepening procedure is similar to the approach applied for empirical model building (Hangos and Cameron 2001a). If we again consider the granulator drum example, we start constructing the overall multiscale model by developing the material and energy balances of the drum and find out that we need finer models for the convective and diffusive flows, together with the source terms describing the particle birth, growth, and breakage processes in the granule population balance. 6.3.1.3 Model Composition

The bottom-up approach of constructing multiscale process models is also applied in step 4 of the modeling procedure as an alternative method to the iterative modeldeepening procedure. The overall model construction starts by building the partial models on the highest resolution scale. These models are then integrated according to the selected multiscale integration framework to prepare the modeling problem statements for the submodels on the next, coarser resolution scale. Bottom-upmodel composition is a common way to build multiscale models, when the submodels originate from different sources and/or are based on different princi-

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ples. The advantage of this approach is that substantially different models may be integrated if a suitably selected framework is found. The drawback is that the resulting model is often not homogeneous in its approach, purpose, and accuracy or it may happen that a partial model cannot be suitably integrated. Model composition ensures that the upper level models have a sound fundamental basis and may consequently be more reliable than those developed by model enrichment (Section 6.3.1.2). However there is the risk of starting the modeling process at a level that is too fundamental, which may lead to accurate but inefficient modeling. 6.3.2 Partial Models: Approaches and Classification

In this section we focus on the partial models or submodels of a multiscale process model. For this purpose, we assume a well-posed modeling problem for each partial model, i.e., a system description and modeling goal specificationfor any partial model. This implies that the mechanisms and data available can be determined individually. The approaches of model building of the partial models, that is the development of the model equations, are essentially the same as in the classical case: we may apply mechanistic approaches based on first principles or a black- or grey-box model. Similarly, the classification of the resulting partial models goes along the same lines as in the general, nonmultiscale case. 6.3.2.1 Mechanistic Approaches for Process Models

The mechanistic approach of developing the model equations of a process model uses first principles to construct the ingredients of a partial model. The modelbuilding subprocedure (Hangos and Cameron 2001a) is followed in this case with the following substeps: 1. system and subsystem boundary and balance volume definitions; 2. define the characterizing variables (inputs, outputs, and system states); 3. establish the balance equations for conserved quantities: mass, energy, momentum and number, etc.; 4. transfer rate specifications; 5. property relation specifications; 6. balance volume relation specifications; 7. equipment and control constraint specifications; 8. modeling assumptions.

In the multiscale case, however, steps 1 and 2 need special care, because these are partially determined by the other partial model@)and by the integrating framework of the overall multiscale model. As the result of a mechanistic approach to partial model construction, we obtain a process model with standard ingredients (see Section 6.3.3.2). This makes it relatively easy to integrate the resulting partial model into a multiscale framework (see Section 6.4).

6.3 Modeling in Multiscale Systems Table 6.1 Classification of partial models. Type of Model

Criterion o f Classification

Mechanistic

Based on mechanisms/underlying phenomena

Empirical

Based on input-output data, trials or experiments

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Only defined for discrete values of time and/or space

I Containing continuous and discrete behavior

6.3.2.2 Black- and Grey-Box Modeling

Usually, either engineering knowledge or data are not available to construct a fully mechanistic so-called “white-box’’ model, but we obtain a partial model with unknown model parameters and/or structural elements. One can then use measured data from the real process to give an estimate of the unknown model parameters or to construct an empirical, so-called “black-box”model for the unknown model element. This way a fully determined model can be obtained in the model calibration and validation step of the modeling procedure. It is important to note, however, that in most of the cases it is rather difficult, if not impossible, to calibrate a grey-box partial model, because one only has measured data from the overall process system and not from its subsystems corresponding to partial models (see Section 6.4.4.3 for more details). In some cases a fully black-box model should be developed by using empirical model building (Hangos and Cameron 2001a) that is similar in its approach to gradual model enrichment. Typical of these models are Box-Jenkins and neural networks.

6.3.2.3 Model Classification

The classification of the resulting partial models is done similarly to the general case. As the characteristic of the different classes of models have a great impact on the solution techniques and on the application area we briefly recall the criterion of classification and the type of models in Table 6.1 (Hangos and Cameron 2001a).

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It is important to observe that a classification criterion generates a pair or triplet of model types and all classification criteria can be applied to a particular partial model. 6.3.2.4 Particular Modeling Techniques for Different Scales

We make a brief digression here to list some of the techniques that have evolved to describe specific scales. In approximate order of increasing scale, they include: 0

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computational quantum chemistry and molecular mechanics to deduce basic chemical properties on the electronic/atomic scale; molecular dynamics, Monte Carlo and hybrid methods that predict the ensemble behavior of many molecules; assorted techniques for front tracking, interface modeling, particle interactions and so on, grouped roughly as “mesoscale”models; computational fluid dynamics for detailed flow prediction; unit operation modeling and process flowsheet simulation, most familiar to chemical engineers, for vessel and plant scale studies; environmental simulation and business enterprise modeling on the “megascale”.

Each technique has many variations, both broad and subtle, and will likely require contribution from specialists in the field. This reinforces the cross-disciplinary nature of much multiscale modeling work.

6.3.3 Characteristics of Partial Models

This section deals with the characteristic properties and model elements of partial models in a multiscale process model with an emphasis on those model properties and ingredients that are important from the viewpoint of multiscale modeling. 6.3.3.1 Model Types

The classification of partial models and the resulting model types are already described in Table 6.1 in Section 6.3.2.3. There are some model types that often arise as partial models in multiscale systems and are therefore of special importance. Lumped and distributed parameter models. Most often both of these types of partial model can be found together in a multiscale model. The models in the finest scale are typically distributed parameter dynamic models, while the models on the coarsest scale are often lumped parameter models. The integration of such mixed-type partial models into a multiscale integration framework needs special care (see Section 6.4). Deterministic and stochastic models. There are some characteristic phenomena often found in multiscale systems, such as fluid dynamics, diffusion, heterogeneous

6.3 Modeling in Multiscale Systems

kinetics and the like that are often described by using stochastic models on a fine scale. The integration of such partial models into a framework of deterministic models is usually performed by using averages of different types (mean values for stochastic variables, time, and/or space averages). 6.3.3.2 Standard Ingredients of Partial Models

Partial models in a multiscale model have the same standard ingredients as usual process models. In this case, however, it is of crucial importance to have all the ingredients clearly specified in order to be able to integrate the partial models into a multiscale framework. The seven-step modeling procedure ensures that the resulting model possesses all ingredients in a consistent way. Therefore mechanistic partial models need no further effort to ensure this. Balance volumes. Homogeneous or quasihomogeneous parts of process systems over which conservation balances are constructed are called balance volumes. They are fundamental elements of a partial model in a multiscale system. The union of all balance volumes in a partial model spans the entire domain of the partial model, while the balance volumes in different partial models are related in a way determined by the multiscale integration framework. Model equations (differential and algebraic). The differential equations in a dynamic partial model usually originate from conservation balances and they are supplemented by constitutive algebraic equations that make the model complete from both the engineering and mathematical viewpoints (Hangos and Cameron 2001b). The algebraic equations are of mixed origin: they describe extensive-intensiverelationships, transfer and reaction rate equations, equations of state, physicochemical property equations, balance volume relations, and equipment and control relations. In addition, conservation balance equations are also algebraic equations in static partial models. Model variables and parameters. Variables in a dynamic partial model are time and possibly space-dependent quantities, which are either differential variables if their time-derivative appears in the model equations or algebraic variables otherwise. There are also model parameters present in a partial model; their value is regarded as constant. Some of the variables are given a value by using specification equations in order to make a model with zero degrees of freedom: specification equations are also part of the model. In dynamic partial models, some of the variables, the potential input variables, are also assigned a given “value,”a time-dependent function. Initial and boundary conditions. Similarly to any process model, partial models are sets of ordinary and/or partial differential (or integro-differential)and algebraic equations. In order to make the model well-posed from a mathematical sense, we need to give suitable initial and boundary conditions, which are also part of the model.

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Modeling assumptions. Although not always stated explicitly, modeling assumptions are key ingredients of any process model, because they document any decision the modeler had taken while developing the model. This way modeling assumptions are artifacts of process modeling and serve as key elements in model documentation, verification and consistency checking (Hangos and Cameron 2001a). 6.3.3.3 Particular Considerations for Integrating Multiscale Process Models

Some of the above standard ingredients of partial models have particular significance when integrating partial models into a multiscale framework. These are briefly listed in this subsection, while the way they are used is described in Section 6.4. Variables and parameters. Some of the characterizing variables of the partial models

in a multiscale process model, such as the conserved extensive quantities (mass, component masses, and energy), the thermodynamic state variables (temperature, pressure, and concentrations), and the rate variables, are usually related in a way determined by the integration framework. In addition, some of the model parameters in a partial model of coarser scale may be determined by the modeling output of partial models of finer scales. Such interscale relationships appear in a partial model in the form of additional specifications or equalities for the model parameters. A simple example of such an interscale relation can be the value of the porosity of an equipment scale model determined by another partial model on a particle scale. Initial and boundary conditions. Phenomena occurring at an interphase boundary often call for a partial model describing them in a finer scale. The result of such small scale modeling enters into the equipment-scale model as an expression for the boundary condition of that particular interphase boundary. Constitutive equations. Certain algebraic variables in partial models of coarser

scales, such as reaction or transfer rates, serve typically as connection points between partial models of different scales. Their determining “connecting”constitutive equations contain variables of different scale partial models, where the variables of finer scales are usually averaged to obtain the connecting algebraic variable on the coarser scale. A simple example of such a connection can be a reaction rate of a heterogeneous catalytic reaction, where the reaction rate equation serves as a connecting point between the equipment scale and the catalytic particle scale variables. As a result of the particle scale model, an algebraic relationship should be somehow determined that describes the average reaction rate in a point of the equipment as a function of the average concentrations and temperature at the same point in a time instance.

6.4 Multiscale Model Integration and Solution

6.4 Multiscale Model Integration and Solution

Model integration is the process of linking partial models that exist at different time and length scales into a coherent, composite multiscale model. Integration is the essence of multiscale modeling. If information did not flow between the scales of interest, the model would not be multiscale! Two models sharing information at the same scale does not constitute a multiscale system. In this section we will explore the challenges and status of model integration, compare alternative integration schemes, and address some implementation matters.

6.4.1 The Challenges and Status of Multiscale Integration

The multiscale modeler potentially faces several challenges in performing model integration because it may be necessary to link partial models that: 0

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Span a vast range oflength scales. For example, in electrodepositing of metal onto printed circuit boards, electrical function is influenced by lattice defects of O(lo-'') m, local hydrodynamic and mass transfer processes affecting the deposition rate occur over 0(104) m, and current distribution must be controlled over the entire job, which is 0(1) m in size (Alkire and Verhoff 1994). Operate on very dcrerent time scales. In semiconductor fabrication using chemical vapor deposition (CVD) for instance, individual diffusion and reaction events occur at atomic time scales of 0(10-13) s and thin film growth takes place over minutes 0(102)s, while the total processing time for a multilayer film may be hours or days 0(104)s (Jensen 1998). Have disparate natures. The models may have different dominant phenomena, be continuous or discrete, have different dimensionality, be deterministic or stochastic, exist in concept only or already be implemented in commercial software, and be drawn from different scientific disciplines.

These challenges raise several types of issues that need to be tackled for successful integration (Pantelides 2001): 0

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conceptual issues, for example, deciding what information should flow between the scales and how, or investigating the benefits of reformulating the partial models to allow better integration; mathematical issues, regarding how well-posed the problem is, for instance; numerical issues, such as the choice of numerical method for efficient and robust solution of the composite model; application issues, including the software engineering work needed to link diverse software across different platforms.

Despite the difficulties, there are many successful examples of multiscale model integration from a wide range of disciplines. It is partly the diversity of the applica-

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tion areas and the complexity of the implementation details that obscures the common features in these examples. There are the beginnings of a classification scheme for model integration methods, and there is some knowledge of their characteristics. However, we are not yet at the point of understanding, in a quantitative and general way, how the choice of integration method affects the resulting multiscale model nor how to select the best integration method for a particular modeling problem.

6.4.2 The Classification of Model Integration Methods

Multiscale models can be classified by the way the partial models at each scale are linked. The very act of classification is enlightening. It draws out the similarities and differences in the broad range of existing multiscale models. It also makes comparing alternatives easier. Several authors have proposed classification schemes, often just in passing. A widely accepted scheme has not emerged and there is a confusion of terms. We will summarize a few of these classification ideas then recommend a particular one in the next section. Maroudas (2000) divides multiscale models in materials science into two categories: serial and parallel. Serial multiscale models result from sequentially scaling up the partial models: the outputs of the finer scale models become the inputs of the coarser scale ones. By way of contrast, in parallel multiscale models, the partial models exist side by side and are solved together. Stefanovit and Pantelides (2000) identified three ways in which molecular dynamics (MD) for property estimation could be integrated with traditional process models. First, the MD model is run to generate “pseudo-experimentaldata,” which are then fitted to find macroscopic material parameters. The second possibility is sequential like the first, but instead the macroscopic parameters are found directly from the microscale simulations. In the third approach, the macroscopic model calls the microscopic one on demand to find a relationship between macroscopic variables. This technique eliminates the need for macroscopic parameters. Phillips (2001),whose interest is materials science, distinguishes two broad classes of multiscale models: those that work in information passage mode, and models with internal structure. Information passage models are of two subtypes. In the first, the fine-scalemodel is used to generate macroscopic parameters, which are then used by the coarse scale model. In the second, the microscale model is transformed into a macroscale model, that is, a new effective macroscale theory is derived from a microscale one. Models with internal structure, which in Phillips’ area are typically “mixed atomistic-continuum models,” are where the microscopic model is adjacent to or embedded within the macroscopic model and they are solved together. Guo and Li (2001) and Li and Kwauk (2003) are concerned with predicting the structure of dynamic, multiscale systems with competing mechanisms. They describe three modeling approaches of increasing complexity and illustrate them by referring to gas-solid fluidization. The first approach is descriptive: each scale of interest is modeled separately without attempting to link them together. The second is

6 4 Muhiscale Model lntegration and Solution

correlative, where fine scale information is scaled up for use in coarser scale models. Volume-averaging of parameters is a possible technique, for example. In the third approach, variational, the partial models are solved together subject to the minimization of some quantity that reflects the strengths of the competing mechanisms that influence the system structure. The last approach can handle regime transformations. These schemes represent different views of multiscale integration, which reflect their originators’ backgrounds and modeling objectives. They all distinguish between sequential and simultaneous application of the partial models, and most discriminate between variations on the sequential method. Pantelides (2001) presented a classification scheme for multiscale process modeling with four categories, serial, simultaneous, hierarchical and parallel, that encompass most of the ideas above. Ingram and Cameron (2002) refined and extended this scheme with an additional class in order to distinguish between different methods of simultaneously combining partial models.

6.4.3 A Framework Classification for Process Modeling

We introduce the termframework to describe the way partial models, which apply at different scales, are linked, or integrated, to form a composite multiscale model. Figure 6.5 shows the extended version of Pantelides’ classification scheme for multiscale frameworks that is proposed for process modeling. The main division is between decoupling and interactive frameworks. The serial and simultaneous frameworks decouple the solution of the partial models, so that one partial model is solved (in some sense) then the others are solved in turn. In contrast, the interactive frameworks-embedded, multidomain, and parallel-essentially involve simultaneous solution of the constituent partial models. Another view of the classification scheme, Multiscale models

‘Interactiveframeworks’ Transformation

‘Decoupling frameworks’ Figure 6.5 Classification scheme for multiscale models based o n the framework used to link the partial models.

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(b) Serial - Simolification (microscale)

(a) Simultaneous

(c) Serial - SimDlification (macroscale)

(d) Serial - Transformation

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(g) Embedded Key

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Information flow

(i) Parallel

Modelling step

Figure 6.6

Domain relationships and information flows in models with microscale (p) and macroscale (M)parts.

expressed in terms of the model domains and the information flows between them, appears in Fig. 6.6. The framework classification attempts to define the broad conceptual options for linking partial models. It is not intended to discriminate between the very specific techniques that are used in particular applications. The following sections explore the frameworks in more detail, by giving a short description, common application situations, some advantages and disadvantages, and the main challenges associated with them. The integration of two partial models at a time is discussed. The fine and coarse scale models are referred to as microscale and macroscale models, respectively.

6.4 Multiscale Model Integration and Solution

6.4.3.1 Simultaneous Integration Framework

Description

In simultaneous integration, the microscale model is used to describe the system in its entirety (Fig. G.6a). This approach corresponds, for example, to using discrete element modeling to predict the trajectory of every particle (microscalelevel) in a fluidized bed (macroscale level), or the use of computational fluid dynamics (CFD) (microscale)to model the detailed fluid flow in a complex vessel (macroscale).The macroscale “model”simply summarizes or interprets the detailed microscale results, usually by totalizing, averaging or otherwise correlating the microscale data. This is why the macroscale model in simultaneous integration could be better called the “macroscale function.” In the fluid bed example above, the macroscale function might estimate the average bed expansion or gross solids circulation rate, while in the CFD case, the macroscale function might calculate the residence time distribution. The microscale and macroscale models are decoupled in the sense that information is transferred in one direction only: from the microscale to the macroscale. Application

Simultaneous integration is used when: 0

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it is not possible to model any part of the system with sufficient accuracy at the macroscale level; some part of the region can be successfully modeled at the macrolevel, but the micro- and macroregions cannot be meshed together well enough (this is multidomain integration; see section 6.4.3.4); it is desired to view the system entirely at the microlevel.

It also serves as a baseline integration method against which other frameworks can be tested because it has “zero integration error” (Solomon 1995; Werner 1999). Advantages and Disadvantages

The advantages of simultaneous integration include the potentially high levels of detail, flexibility and accuracy. The main disadvantage is the very high computational burden, the highest of all the frameworks. This limits the size of the system and the length of time that can be simulated. Artificially reducing the system size risks failing to capture large-scale/long-timeeffects (McCarthyand Ottino 1998). Despite continuing growth in computer power, at least for the intermediate future, there will be problems of practical interest where microscale modeling of the entire system is impossible (Chan and Dill 1993). Simultaneous integration will also likely generate a large amount of detailed microscale data that is largely irrelevant to the modeling objective.

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improving the accuracy of the microscale model, because any microscale error will flow to the macroscale; increasing solution efficiency via improved numerical algorithms and distributed computing; recognising when simultaneous integration is truly necessary.

6.4.3.2 Serial Integration Framework

Description

There are three broad, partially overlapping possibilities in serial integration: simplification, transformation and one-way coupling. 1. Simplification The microscale model covers a small part of the system domain. Usually, it is this model that we “simplify” (Fig.6.6b). In this case, the microscale model is simplified by just “fitting a curve” to computed input-output data, by systematic order reduction methods, or by analytical solution if possible. The simplified or solved microscale model creates a relationship between macroscopic variables that is easier to evaluate than the complete solution of the original microscale model. We should perhaps refer to the microscale model as the “microscale function.” The macroscale model spans the system domain and calls the microscale function on demand. This kind of integration is used, for example, to link local and global scales in climate modeling. Mechanistic city-scale calculations relating atmospheric and urban variables to pollutant fluxes can be approximated by polynomials, which are then used in a global atmospheric chemistry model to predict climate change under different development scenarios. A more familiar example may be the analytical solution of the reaction-diffusionequations in a porous catalyst to yield a Thiele modulus-effectivenessfactor relationship that is then used in reactor scale modeling. In serial integration by simplification, information flows in both directions between the micro- and macroscales. However, the framework is decoupling in the sense that the solution process has two stages. First, the microscale model is first simplified to a “microscale function,” and then the macroscale model is solved, which involves calling the microscale function. Often the microscale model is “simplified”,but sometimes it is the macroscale one (Fig.G.6c). For example, if the focus of a modeling exercise were a particular unit in a process, then a model of the process excluding that unit could be built and then simplified, to provide a computationally cheap approximation to the operational environment of the unit. In this case, there is a “macroscalefunction” (the rest of the plant) that is called as required by the microscale model (the particular unit). 2. Transformation Another possibility in the serial integration framework is transformation. The microscale model describes a small part of the system domain. It is “formally

6.4 Multiscale Model Integration and Solution

transformed” into a macroscale model (Fig.G.Gd). This process is also called upscaling, coarse-graining, degree of freedom thinning, and constructing new effective theories or laws. Now, the microscale model is no longer needed and the system domain is described entirely by the new macroscale model. Many techniques are used for upscaling: volume averaging, renormalization and homogenization, among others. The three named methods have been used to upscale the equations for diffusion in porous media. No flow of information occurs between the microscale and macroscale models during solution because, in effect, the microscale model has been eliminated. Only the macroscale model must be solved. It is a decoupling framework in this sense. 3. One-way Coupling The third variation on the serial integration framework occurs when, because of the nature of the system, information flows naturally between the scales in one direction only (Fig.G.Ge,f).Or, that the approximation is close enough to satisfy the modeling goal. Physical vapor deposition (PVD) is an example. A vessel scale (macro) model predicts the average spatial distribution of metal sputtered onto a substrate in a PVD chamber. A “feature scale” (micro) model can then track the build up of the deposit layer on features, such as holes and trenches, on the substrate surface. Another example is the use of (microscale)molecular dynamics to calculate a diffusion coefficient, which is later used at the vessel (macro) scale via Fick‘s law. The information may flow either from the microscale to the macroscale, or vice versa. The independent model is solved first, and then the dependent one is solved. In this framework the solution of the models is decoupled in one direction. Application

The use of serial integration depends on the broad strategy chosen: 0

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Simplijcation.Virtually any microscale (or macroscale) model can be simplified by order reduction or approximation techniques; some models may be solved analytically under appropriate simplifying assumptions. Transformation. The ability to use transformation depends on the upscaling method chosen and the nature of the system. For example, sufficient “scale separation” is needed for homogenization (Auriault 1991). One-waycoupling. This method can be used when the nature of the system is such that one scale is dependent and the other is independent, or at least we can treat them as such. There is no feedback between the scales.

Many well-known relationships in science and engineering can be viewed as applications of serial multiscale integration by simplification or transformation. Examples include: equations of state derived from the kinetic theory of gases, the rheological equation for Newtonian fluids, Thiele modulus-effectiveness factor expressions for reaction-diffusion problems, and even, in some sense, Newton’s law of universal gravitation (Phillips 2001).

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The principal advantage of the serial approach is the elimination of the expensive, detailed microscale model. Consequently, serial integration can potentially produce the most efficient models among the five frameworks. Many powerful mathematical techniques for order reduction, analytical solution and upscaling can be applied to the microscale model. These techniques not only enhance calculation efficiency, but also, and perhaps more importantly, highlight the essential nature of the microscale model. They strip away unnecessary detail to reveal how (a few) key variables influence macroscale behavior. The &sadvantages include the restricted accuracy and flexibility of the approach. Considerable effort may be needed to apply the mathematical techniques referred to above. Challenges

The basic challenge is to find an appropriate balance between fidelity and computational cost through manipulation of the microscale model. This includes learning which mathematical techniques are best used to simplify or transform the microscale model, and knowing when one-way coupling is acceptable. A fitrther challenge is to provide mechanisms for revising the microscale representation as required (Oran and Boris 2001, p. 439). 6.4.3.3 Embedded Integration Framework

Description

The microscale model is “formally embedded” within the macroscale model in this framework (Fig.G.Gg), which was termed hierarchical by Pantelides (2001).The macroscale model spans the system domain, while the microscale model is local, restricted to a relatively small part of the domain. The microscale model calculates, on demand, a relationship between macroscale quantities. Hence, while its domain is small, the microscale model may be called (instantiated) at many points through the system. Ab initio molecular dynamics is an example of the embedded framework. In this application, the macroscale model is an MD simulation that tracks the motion of each molecule based on the forces that act upon it. The microscale model is an electron-atom scale computational chemistry method, such as density functional theory, that calculates the intermolecular force (potential)function on thejy as the MD simulation proceeds. The embedded approach is a true, interactive multiscale method because information is passed between two models that are actively being solved. Application

This framework is used when a suitable macroscale model exists but needs to be “informed” by localized microscale simulation, and the microscale model cannot be acceptably simplified. If a suitable simplification of the macroscale model were available, serial integration via simplification (Section 6.4.3.2) should be used to reduce computing demands.

6.4 Multiscale Model lntegration and Solution

Advantages and Disadvantages

Embedded integration has a natural appeal because of its orderly, hierarchical nature. It potentially has the flexibility and accuracy of simultaneous integration, but with a much lower computational load. The detailed, expensive microscale calculations are performed only where and when they are required. A disadvantage is the need to run the microscale model at all, because it may still consume the bulk of the computing resources. Because the micro-macro interface in embedded and serial (simplification)integration may be similar, it should be possible to swap these methods with little change needed in the macroscale model. Challenges

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finding the smallest domain and shortest time needed to simulate the microscale model to provide accurate results with a minimum amount of calculation; enhancing the micro-macro interface beyond that used in traditional serial (simplification) modeling. For example, Stefanovit and Pantelides (2000) show a new approach to linking molecular dynamics information with unit operation modeling.

6.4.3.4 Multidomain Integration Framework

Description

In the multidomain framework, the microscale and macroscale models describe separate but adjoining parts of the whole system (Fig.G.Gh). It is sometimes called “hybrid modeling” because it is often used to create hybrid, discrete-continuous models. There are many multidomain models in materials science. In investigating the fracture of brittle solids, for example, the region around the growing crack could be modeled with a discrete, atomistic, microscale method: molecular dynamics. Relatively far from the crack, the solid could be modeled at the macroscale level with a continuum technique, such as the finite element method. The interface between the micro- and macroscale domains may be either a point, line or surface, or it may be a buffer zone with a volume that is nonzero, but small compared to the size of the micro- and macrosimulated regions. The models in the two regions interact across an interface. This multidomain framework is a true, interactive multiscale method, with a two-way flow of information between the partial models via the interface. Application

Multidomain integration is used where some parts of the system can be adequately described at the macrolevel, while in other regions, only a microscale model will suffice. It is often used in models with heterogeneous media, for example, the gas and solid phases in chemical vapor deposition, or the catalyst and bulk phases in a packed bed reactor. In these applications, the microscale region is usually fKed in space. The other main application for multidomain models is materials science, par-

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ticularly the field of fracture mechanics. There, the microscale model is applied when the macroscale model fails some error criterion; the microscale region may change as the simulation proceeds. Advantages and Disadvantages

Like embedded integration, the multidomain method couples micro- and macroscale models to reduce the computational burden compared to simultaneous integration, while maintaining microscale realism where needed. The greatest disadvantage is the potential complexity of the micro-macro interface. It is important to guarantee the continuity of thermodynamic properties and transport fluxes across the interface, and to avoid unphysical wave reflections (Brenner and Ganesan 2000a,b; Curtin and Miller 2003). Challenges

The main challenge is to formulate a seamless interface between the microscale and macroscale regions. In some applications,techniques are also needed to move, grow and shrink the microscale region as the simulation proceeds to minimize computational requirements. 6.4.3.5 Parallel Integration Framework

Description

Both microscale and macroscale models cover the entire system domain in parallel integration (Fig.6.6i). However, the models are complementary in the detail with which they describe the important phenomena. There are currently few examples of parallel integration in chemical engineering. All combine a CFD model with a traditional unit operation model. For example, in a bubble column reactor, two phenomena are important: hydrodynamics and process chemistry. In a parallel framework, the microscale model might treat the fluid mechanics in detail via CFD, while the process chemistry could be represented in an abbreviated manner by a simple gas source (or sink) term. The macroscale model, on the other hand, could contain a comprehensive reaction scheme that was assumed to take place in simple fluid flow regime, such as a well-mixed or plug-flow region, or some combination of these. Current parallel models are solved by successive substitution. The macroscale model predicts some quantities that are inputs to the microscale model, while the micromodel outputs some variables that the macromodel requires. The models are run alternately until convergence. Parallel integration is an interactive multiscale method because both models are active and must be solved in concert. Application

To date, parallel CFD-unit operation models have been developed for a bubble column reactor (Bauer and Eigenberger 1999,2001),a batch stirred tank reactor (Bezzo, Macchietto and Pantelides 2000), an industrial crystallizer (Urban and Liberis 1999) and a high temperature electrochemical reactor (Gerogiorgisand Ydstie 2003). The

6.4 Multiscale Model Integration and Solution

parallel method is suitable where the important mechanisms in the system are weakly coupled (Pantelides 2001; Urban 2001). However, it is likely that the strength of the coupling that can be successfully accommodated by the parallel method would increase as the macroscale model is more finely discretized. Very strongly coupled systems may require embedded integration. Advantages and Disadvantages

The key advantage of parallel integration is the division of the system into two simpler problems. It is also a way to form a multiscale model from an existing software package that has limited interface options. The main disadvantage is the inherent approximation of the method and its consequent limitation to systems with weakly coupled active mechanisms. Challenges

One challenge for parallel CFD-unit operation models is to determine efficiently, perhaps automatically, an acceptable combination of ideal flow regions needed to approximate the CFD flow pattern. Another is to replace successive substitution with a more efficient and stable solution method. However, the most open challenge is to expand the range of applications beyond the CFD-unit operation examples reported so far. A general question is how best to partition the controlling phenomena between the parallel models. 6.4.3.6 Discussion of the Scheme

The extended classification scheme proposed for process modeling groups multiscale models according to how the microscale and macroscale models are linked. It helps in understanding the structure of multiscale models, but there are some open issues: 0

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The classification scheme does not provide formal definitions for the frameworks. A given multiscale model could potentially fall into more than one category. Conversely, applying a framework to given partial models does not guarantee a unique multiscale model; many variations are possible. While some qualitative properties of the frameworks are outlined, we lack comprehensive comparative information. There is little guidance and there are no quantitative tools available to help select the integration method. These would be especially useful early in the modeling process. It is unclear how, or indeed if, all frameworks could be applied to a given modeling problem. Some problems do not naturally seem to suit certain integration methods and the underlying reasons need to be understood. The way the integration framework depends on the partial models to be integrated, and conversely,how the partial models affect the properties of the integrating framework, also needs careful further investigation. The classification scheme considers the integration of two scales at a time. However, it can be applied to multiscale models with more than two scales. For example, in a three-scale system, the micro- and mesomodels could be linked with one

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framework, while the meso- and macroscales could be linked with another framework. This “painvise”scheme may not be adequate to describe all multiscale models with more than two scales. Some of these unresolved issues can be tackled through structural analysis techniques. For systems that may exist in more than one state or regime, there is an additional element needed in model integration. It is to define the criteria used to identify the operating regime or prevailing physical structure of the system. Li and Kwauk (2003) have developed a method for multiscale models.

6.4.4 Application of the Frameworks

So far, we have looked at some issues of conceptual modeling for multiscale process systems. There are also mathematical and software issues involved in implementing the models: the use of existing software, solution methods, and model testing. 6.4.4.1 The Use of Existing Sofiware

The chemical engineering community has already addressed many of the challenges in linking existing, previously incompatible process simulation software through the CAPE-OPEN project and its successors (Braunschweig,Pantelides, Britt, et al. 2000). CAPE-OPEN defines a set of open interface standards that allows parts of process simulators from different vendors to work together in a “plug-and-play’’manner. The standard is maintained by the CAPE-OPEN Laboratories Network, CO-LaN (http: www.colan.org). Their Web site contains a list of current projects and discusses issues in linking process engineering software. Work at the University of Aachen has been centered on the component-based hierarchical explorative open process simulator (CHEOPS) (http: www.lfpt.rwthaachen.de/Research/cheops.html)as an integrative modeling and solution environment. There are commercial examples of multiscale software integration. Some of these include linking a CFD package with another software type, for example, process modeling software (gPROMS with FLUENT and STAR-CD), a process simulator (ASPEN Plus with FLUENT), and a gas-phase/surface chemistry simulator (CHEMKIN With STAR-CD). A related development is underway in the field of medical research as part of the IUPS Physiome Project (Hunter and Borg 2003; Hunter, Robbins and Noble 2002). A series of XML-based markup languages is being developed to capture and exchange information on human physiology - from the scale of cells to the whole body.

6.4 Multiscale Model Integration and Solution

6.4.4.2 Solution Methods for Multiscale Models

Solution efficiency is important in multiscale modeling because the partial models may use computationallyintensive techniques, such as quantum chemistry, molecular simulation and so on. A key feature of multiscale modeling is transforming a problem that is intractable when viewed at the finest scale into a manageable one when considered at multiple scales. The modeling techniques used at each scale have established specialized solution methods. The challenge is to meld the scalespecific solution techniques into an efficient strategy for solving the multiscale problem. There is a broad and powerfd class of numerical methods known as multiscale (multigrid, multilevel, multiresolution, etc.) schemes (Brandt 2002). They seek to solve problems that are defined by equations at one scale, but have a multiscale character. This numerical approach complements the concept we have advanced in this work, namely defining separate models at each scale and linking them through a framework. The multiscale numerical method consists of recursively constructing a sequence of solutions to a fine-scale problem at increasingly coarse scales. Largescale behavior is effectively calculated on coarse grids, but it is based on information drawn from finer scales. Multigrid techniques have been applied to a vast range of problems. A few relevant to process engineering are: high efficiency methods for fluid dynamics, solution of partial differential equations in general, data assimilation and other inverse problems, optimal feedback control, efficient molecular dynamics and molecular modeling, and global optimization. See Brandt (2002) for a comprehensive review. Not included in that review is the recent "equation-free" (or "gaptooth/projective integration") method (Kevrekidis,Gear and Hummer 2004) that has been developed in the context of process systems. The solution of multiscale models constructed from partial models linked by a framework involves several issues, including: 0

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The efeect ofthefiarnework - In some frameworks, for example, the simultaneous and some serial ones, the solution process is decoupled. The best method at each scale can then be employed in isolation. The nature of the mathematical problem is different for each of the interactive frameworks. In multidomain integration, the models only interact through boundary conditions, while in embedded integration, information may also flow through the state variables and their derivatives at any location, through transport coefficients and fluxes, and source terms, for example. Thus far, only successive substitution has been used to solve parallel multiscale models. There are also fundamental questions, such as how the properties - differential index, stability, computational complexity, and so forth - of a multiscale model are determined by the properties of the partial models and the chosen framework. Time stepping - The partial models will usually have quite different time scales. In interactive frameworks, this results in stiff problems. Different microscale and macroscale time steps are often used. The microscale time step is usually much smaller than the macroscale one, but not always. When the microscale model is

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stochastic, temporal averaging is helpful in damping out microscale “noise,” which would otherwise propagate into the macroscale model (Raimondeau, Aghalayam, Katsoulakis, et al. 2001). Updating strategies - Partial models that have been simplified may need to be updated by referring to the original model, either periodically or when their estimated error becomes too high (Oran and Boris 2001). The accuracy required for the simplified model can be judged through error propagation methods, similar to those used in design sensitivity studies as presented, for example, in Xin and Whiting (2000). State transitions - For systems that may exist in different regimes, identification of the prevailing state is required. Li and Kwauk (2003) solved the multiscale model together with an optimization problem: minimization of a function (stability condition) that reflects the competing mechanisms that determines the system structure. Parallelization - Parallel processing fits naturally into some multiscale codes (Laso, Picasso and Ottinger 1997; Broughton, Abraham, Bernstein, et al. 1999).

6.4.4.3 Model Verification and Validation

The verification and validation (V&V) of models in science and engineering is a discipline in its own right (Oberkampf, Trucano and Hirsch 2003). The goal is to lend confidence (Kister 2002; Best 2003) to the use of the model. Pantelides (2001) identified the construction of validated models as a major challenge in chemical engineering. Here we touch on some aspects related to multiscale models. Verification is “the process of determining that a model implementation accurately represents the developer’s conceptual description of the model and the solution to the model,” that is, solving the equations correctly. It is a matter of software engineering and mathematics. There are two aspects: code verification and solution verification (Oberkampf,Trucano and Hirsch 2003). The former can be split into the verification of the numerical algorithm and software quality assurance. Solution verification involves checking for gross consistency (such as overall mass conservation), spatial and temporal convergence, and consistency with trusted solutions. Essentially, it is being confident that the numerical error in the predictions is acceptable and the qualitative behavior of the solution, for example the stability, corresponds to the developer’s expectations. In multiscale modeling, each partial model can be verified in isolation, using dummy functions or typical values for any variables that are shared between scales. The composite multiscale model should then be verified. This tests the integrating framework. Code and solution verification can be applied as before, but additional “trusted solutions” are potentially available. For each pair of linked scales, the chosen framework could be checked against simultaneous integration of those scales, since that has “zero integration error” (Solomon 1995; Werner 1999). The process is to compare predictions for the coarse scale variables using the chosen framework against the predictions of the same coarse variables derived from a complete simulation of the system at a finer scale. In practice, complete simulation of the system at

6.4 Multiscale Model Integration and Solution

the microscale may not be feasible, but selected microscale simulation of regions with typical or extreme behavior may be. Another possibility is to check the integration of three scales at a time against a two-scale model for the special case where the intermediate scale should not alter the solution (Gobbert, Merchant, Borucki, et al. 1997).Of course, this is most convenient when a suitable existing two-scale model is available, which might occur as part of the iterative model building process when it is decided that another scale is needed. Model validation is “the process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model”. It is a question of getting the actual physics of the system right. Ideally, validation consists of testing the model against experimental data drawn from a set of carefully targeted experiments (Pantelides 2001 ; Oberkampf, Trucano and Hirsch 2003). Both the data and the model will contain uncertainties. Two issues that are important for multiscale model validation are measuring data over a range of scales and estimating model parameters at different scales. If a model contains parameters that are unknown, that is, too uncertain, experiments can be used to estimate them. For hierarchical models, which include multiscale models, parameter estimation can be applied simultaneously at all scales, sequentially to the scales in some order, or independently at each scale, or a mixture of these (Robinson and Ek 2000). The ideal situation is to determine each parameter independently. Conversely, the simultaneous approach is seen as “bad empiricism” (Randall and Wielicki 1997), to be used only for parameters with mechanisms that are poorly understood. In climate science and meteorology, there are complex multiscale models with many poorly known parameters. Brandt (2002) discusses the use of (multiple)multiscale computational approaches to assimilating data on the fly into dynamic models of the atmosphere. Good data helps in both parameter estimation and model validation. For multiscale models, we would like data at each scale of interest. Different measurement techniques are used at different scales; see, for example, Balazs, Maxwell, deTeresa, et al. (2002); Gates and Hinkley (2003) for some techniques used in materials science. Like the various modeling techniques that describe the phenomena at different scales, we can locate the tools of measurement on the logarithmic time and length axes of a scale map (see Figs. 5 and 6 of Gates and Hinkley (2003)).However, there is a fundamental difference in the multiscale nature of models and measurements, at least at small scales. For models there is a rough positive correlation between time and space scales: small processes operate quickly and large processes slowly, in general. The opposite is true for measurement. There is an approximate negative correlation: it takes a long time to measure small things, and a short time to measure large ones. The consequence for multiscale modeling is that it is not possible to gather data to allow the direct validation or parameter estimation of some partial models. We have no choice but to view their effects through the filter of intermediate scales.

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Mdtiscale model integration refers to linking models at different scales together into a coherent whole. The constituent models may operate on vastly different characteristic time and length scales, and may be of dsparate kinds. There are many reported examples of model integration and we are beginning to understand its principles. Classification of the types of integration is helpful. One classification scheme proposed for process modeling identifies five broad frameworks for multiscale integration: simultaneous, serial, multidomain, embedded and parallel. There is some qualitative information available on their properties and when they can be applied. Aside from conceptual modeling concerns, there are issues of integrating existing software, solution of the model, and model validation. There is a good opportunity now, through using the large number of published models as examples, to improve our understanding of model integration. Key elements will be a clearer classification of integration methods and a suite of modeling tools to estimate the performance of a multiscale model - in terms of both dynamic and computational properties - at an early stage in the modeling process. As computing power increases and becomes more accessible, previously infeasible modeling techniques will become ripe for integration into multiscale models. New ideas will be needed to achieve tighter integration of different kinds of models.

6.5 Future Challenges

There is strong and increasing interest in the multiscale approach. Many examples of multiscale modeling are now scattered widely throughout the literature and there are the beginnings of a general “theory of multiscale modeling.” However, like any rapidly evolving field, the development is uneven. In specific areas, we can choose between several sophisticated multiscale techniques, while in other fields multiscale thinking has barely made an impact. We highlight here some future challenges in the multiscale modeling of process systems. Overall Strategy for Multiscale Modeling In Section 6.3 we outlined a general model building strategy and discussed its extension to multiscale modeling. Solomon (1995), Li and Kwauk (2003) and others pro-

vide alternative viewpoints. Are these strategies sufficient for the ejicient development of parsimonious multiscale models? One item that deserves more attention is the role of the modeling goal in multiscale systems. How can we use a statement of the modeling goal to drive model building towards : 0 0

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Identifying and selecting the scales to include in the model? Developing or choosing among alternative partial models at each scale? Guiding the integration of the partial models into a multiscale model?

6 5 Future Challenges

We need a better understanding of the decomposition of goal sets in multiscale problems: relating goals to scales and appreciating the filtering effect of the integrating framework. Rethinking the Partial Models

Pantelides (2001) expresses the point well: “it should not be taken for granted that techniques (e.g. for CFD or molecular modeling) that have evolved over long periods of time for ‘stand-alone’use automatically possess the properties that are necessary for their integration within wider computational schemes. Indeed, our experience has been that the contrary is often true.” We may need to reformulate the partial models to assist with mathematical aspects of “tighter”model integration: ensuring well-posedness, continuity, efficient Jacobian calculation, and so on. Stefanovie and Pantelides (2000) provide an example. Model Integration

Classification is the first step in generalizing our understanding of the options for linking multiscale models. Several classification schemes have been proposed (Sections 6.4.2 and 6.4.3),and there is some anecdotal information available on the properties of the classes. Are there more useful classification schemes? Attempting a formal definition of the classes, for example, Ingram, Cameron and Hangos (2004), may help improve upon current ideas. We also need to develop techniques to answer the question: how do the properties of the partial models and the nature of the integration method contribute to the properties of the resulting multiscale model? A suite of characterizing model metrics that can be applied at different stages of the modeling process will be of assistance here. Numerical Methods

Solving the partial models from some scales may entail a very high computational load. Indeed, the possibility of the repeated solution of such models in a multiscale simulation highlights the need for efficient numerical methods. Multiscale models might run across different platforms and processors. The partial models may be of different types (Section 6.3.2.3) and will almost certainly have very different time scales. SpeciaIist techniques, refined over time, are usually available for the models from different scales. We need to understand how the most suitable numerical methods for the partial models and the chosen integration scheme interact in order to develop efficient and robust solution methods for multiscale models. Multiscale Modeling Took

A multiscale perspective would enhance existing CAPE tools. To be effective aids for

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store validation data over the range of scales; archive the underlying assumptions of the partial models.

Extending the current work on open interface standards and heterogeneous simulation would facilitate these efforts. On a final note, we need to maintain a watch on how other disciplines are approaching the challenge of multiscale modeling and its application. There are interesting developments in materials science, ecology, climate studies, medicine and many other fields (Glimm and Sharp 1997; Li and Kwauk 2003).

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of Chemical Engineering. San Diego, CA, pp. 16-23. 2 Li, J., Kwauk, M. (2003) Chemical Engineering Science 58, 521-535. 3 Glimm,]., Sharp, D. H. (1997) SIAM News 30, 4,17 and 19. 4 Charpentier,J. C. (2002) Chemical Engineering Science 57, 4667-4690. 5 Cussler, E. L., Wei, J. (2003) AIChEJoumal49, 1072-1075. 6 Grossmann, I. E., Westerberg, A. W. (2000) AIChEJournal46,1700-1703. 7 Ingram, G. D., Cameron, I. T. (2002) In: APCChE 2002/Chemeca 2002, Christchurch, New Zealand, Proceedings CD-ROM, Paper #554. 8 Alkire, R., Verhofl M. (1994) Chemical Engineering Science 49, 4085-4093. 9 Maroudas, D. (2000) AIChEJournal46, 878-882. 10 Hangos, K. M., Cameron I. T. (2001a) Process modelling and model analysis. Academic Press, London. 11 Hangos, K. M., Cameron I. T. (2001b) Computers and Chemical Engineering 25, 237-255. 12 Robertson, G.A,, Cameron I. T. (1997a) Computers and Chemical Engineering 21,455-473. 13 Robertson, G.A,, Cameron I . T. (199713) Computers and Chemical Engineering 21,475-488. 14 Ingram, G. D. Cameron I. T., Hangos K. M. (2004) Chemical Engineering Science, in press. 15 Lerou, J. J., Ng, K. M. (1996) Chemical Engineering Science 51, 1595-1614. 16 Noble, D. (2002) Science 295, 1678-1682. 17Jensen, K. F., Rodgers, S. T., Venkataramani, R. (1998) Current Opinion in Solid State @ Materials Science 3, 562-569. 18 Pantelides, C. C. (2001) In: ESCAPE-11 (Eds: Gani, R., Jsrgensen, S. B.).Kolding, Denmark, pp. 15-26.

19 Stefanovit, J., Pantelides, C. C. (2000) In:

AIChE Symposium Series 96(323) FiJh International Conference on Foundations of Computer-Aided Process Design (Eds: Malone, M . F., Trainham, J. A,, Carnahan, B.). American Institute of Chemical Engineers, New York, pp. 236-249. 20 Phillips, R. (2001) Crystals, defects and microstructures: Modeling across scales. Cambridge University Press, New York. 21 Guo, M . , Li, J. (2001) Progress in Natural Science 11, 81-86. 22 Sobrnon, S. (1995) In: Annual Reviews ofCompukztional Physics 11 (Ed: Staufer, D.). World Scientific, pp. 243-294. 23 Werner, B. T. (1999) Science 284, 102-104. 24 McCarthy, J . J., Ottino, J. M. (1998) Powder Technology 97, 91-99. 25 Chan, H.S., Dill, K. A. (1993) Physics Today 46, 24-32. 26 Auriault, J. L. (1991) International Journal of Engineering Science 29, 785-795. 27 Oran, E. S. and Boris, J. P. (2001) Numerical simulation of reactivepow. Cambridge University Press, New York. 28 Brenner, H., Ganesan, V. (2000b) Physical Review E 62, 7544-7547. 29 Brenner, H., Ganesan, V. (2000a) Physical Review E 61, 6879-6897. 30 Curtin, W. A,, Miller, R. E. (2003) Modelling and Simulation in Materials Science and Engineering 11. R33-R68. 31 Bauer, M., Eigenberger, G. (1999) Chemical Engineering Science 54, 5109-5117. 32 Bauer, M., Eigenberger, G. (2001) Chemical Engineering Science 56, 1067-1074. 33 Bezzo, F., Macchietto, S., Pantelides, C. C. (2000) Computers @ Chemical Engineering 24, 653-658.

6 5 Future Challenges 34 Urban, Z., Liberis, L. (1999) Computers 99,

Dusseldorf, Germany. 35 Gerogiorgis, D. I., Ydstie, 8. E. (2003) In: Proceedings of Foundations of Computer-Aided Process Operations (FOCAPO 2003): A view to the fiture integration of RCD, manufacturing and the global supply chain, pp. 581-584. 36 Urban, Z. (2001) PSE User Group Meeting 2001. 37 Braunschweig, B. L., Pantelides, C. C., Britt, H. I., Sama, S. (2000) In: AIChE Symposium

Series 96(323) Fijh International Conference on Foundations of Computer-Aided Process Design (Eds: Malone, M. F., Trainham, /. A,, Carnahan, B.). American Institute of Chemical Engineers, New York, pp. 220-235. 38 Hunter, P. J., Borg, T. K. (2003) Nature 4, 237-243.

39 Hunter, P., Robbins, P., Noble, D. (2002) PFugers Archiv, European journal of Physiology 445, 1-9. 40 Brandt, A. (2002) In: Multiscale and multire-

solution methods: Theory and applications (Eds: Barth, T. J., Chan, T. F., Haimes, R.). Springer, Berlin, pp. 3-95. 41 Kevrekidis, I. G., Gear, C. W., Hummer, G. (2004) AICHE Journal 50, 1346-1355. 42 Raimondeau, S., Aghalayam, P., Katsoufukis, M. A,, Vlachos, D. G. (2001) In: Foundations of molecular modeling and simulation: Proceedings of thefirst International Conference on Molecular Modeling and Simulation, Keystone, Colo-

rado Cummings, P. T., Westmorland, P. R., Carnahan, B.). American Institute of Chemical Engineers, New York, pp. 155-158. 43 Xin, Y., Whiting, W. B. (2000) Industrial @ Engineering Chemistry Research 39,2998-3006. 44 Laso, M., Picasso, M., &linger, H. C. (1997) AIChEJournal 43,877-892. 45 Broughton,J. Q., Abraham, F. F., Bernstein, N., Kaxiras, E. (1999) Physical Review B 60, 2391-2403. 46 Oberkampf; W.L., Trucano, T. G., Hirsch, C. (2003) Technical Report SAND 2003-3769,

Sandia National Laboratories.

47 Kister, H. 2. (2002) Chemical Engineering Progress 98, 52-58. 48 Best, R. (2003) TCE 40-41. 49 Gobbert, M. K., Merchant, T. P.. Borucki, L. j., Cafe, T. S. (1997) Journal ofthe Electrochemical Society 144, 3945-3951. 50 Robinson, A. P., Ek, A. R. (2000) Canadian journal ofForest Research 30, 1837-1846. 51 Randall, D. A,, Wielicki, B. A. (1997) Bulletin ofthe American Meteorological Society 78, 399-406. 52 Balazs, B., Maxwell, R., deTeresa, S., Dinh, L.,

Gee, R. (2002) In: Materials Research Society Symposium Proceedings 731, 3-7. 53 Gates, T. S., Hinkley, J . A. (2003) In: Collection $Technical Papers A I M ASME ASCE A H S ASC Structures, Structural Dynamics and Materials Conference 2, 1233-1246.

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Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

7 Towards Understanding the Role and Function o f Regulatory Networks in Microorganisms Krkt V: Cernaey, Morten Lind, and Sten BayJirgensen

7.1 Introduction

Microbial function is carefully controlled through an intricate network of proteins and other signaling molecules, which enables microorganisms to react to changes in their environment. Thus microorganisms constitute examples of entire autonomous chemical plants, which are able to produce and reproduce despite a shortage of raw materials and energy supplies. Understanding the intracellular regulatory networks of microorganisms is important to process systems engineering for several reasons. One reason is that the microbial systems still constitute relatively simple biological systems, the study and understanding of which may provide a better understanding of higher biological systems such as human beings. Furthermore microbial systems are used, often following genetic manipulation, to produce relatively complex organic molecules in an energy-efficient manner. Understanding the regulatory networks in microorganisms, and especially understanding how to couple the microbial regulatory functions and higher level process and production control functions, is a prerequisite for process engineering. The focus of this chapter is discussing basic modeling problems when describing regulatory networks in microorganisms. In this introduction, we first present arguments to explain why researchers from so many different disciplines, but especially from the systems engineering field, are interested in gaining an increased understanding of the functioning and design principles of these regulatory networks. Second, fundamental modeling problems are highlighted. The introduction finishes with a statement of the purpose of this chapter and an overview of the remainder of the chapter.

Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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7.1.1 Why Gain an Understanding of Regulatory Network Function?

From an industrial point of view, a microorganism can be considered an autonomous plant suited for the production of complex biomolecules. Industrial production of chemicals such as food and cosmetics ingredients is, for example, increasingly based on biotransformation processes (Cheetham 2004), where the conversions of raw materials to useful products are catalyzed either by microorganisms or by enzymes obtained from microorganisms. On a macroscale, for example, in a bioreactor where millions of microorganisms reside, the conversion of raw materials to valuable products by the microorganisms has traditionally been monitored using probes for pH, dissolved oxygen, gas phase composition, and biomass concentration measurements. In recent years, however, interest in system-levelunderstanding of regulatory networks in biological systems, including microorganisms such as Escherichia coli (a prokaryotic organism) and Saccharomyces cerevisiae (a eukaryotic organism),has been an important research theme. This increasing interest in the microscale is, to a large extent, boosted by the fact that biology has evolved from being a data-poor science to a data-rich science, an evolution driven by progress in molecular biology, particularly in genome sequencing and high-throughput measurements (Kitano 2002; Vukmirovic and Tilghman 2000). Indeed, contrary to earlier efforts in developing system-level understanding of biological systems, it is now possible to collect informative system-wide data sets on protein-DNA interactions, protein-protein interactions, and increasingly smallmolecule interactions as well (Ideker and Lauffenburger 2003). An ever-increasing number of advanced analytical methodologies allow detailed monitoring of the dynamics of intracellular processes (e.g., Chassagnole et al. 2002). In the postgenomic era, the availability of genome sequence data of several organisms, including E. coli and S. cereuisiae, has already led to a focus shift from molecular characterization and sequence analysis to developing an understanding of functional activity and the interaction of genes and proteins in pathways (Salgado et al. 2004; Vukmirovic and Tilghman 2000; Wolkenhauer et al. 2003),a research area called functional genomics. In fact, microorganisms are networks of genes, which make networks of proteins, which regulate genes, and so on ad infinitum (Vukmirovicand Tilghman 2000). Gene expression and regulation, i.e., to understand the organization and dynamics of genetic, signaling, and metabolic pathways, is considered to be one of the main research challenges of the next 50 years (Wolkenhaueret al. 2003). Developing a system-level understanding of biological systems can be derived from insight into four key properties (Kitano 2002): (1)System structure, for example, the network of gene interactions; (2) System dynamics, for example, the dynamic response of a biological system to a change in the substrate concentration; (3) The control method, that is, understanding of the mechanisms that control the state of the cell; (4) The design principles of the cell, for example, simulations can support strategies to construct biological systems. Reaching a system-levelunderstanding of biological systems necessitates multidisciplinary research efforts to unravel the complexity of biological systems.

7.1 Introduction

One could, of course, wonder why not only biologists, but people coming from very different research fields, are involved and interested in developing an increased understanding of biological function. First of all, involving other research disciplines can be considered a necessity. Secondly,the versatility of microorganisms to produce industrially relevant chemicals, by expression of the appropriate gene(s), is an important factor promoting research aimed at gaining an increased understanding of biological function. Thirdly, the similarities between microorganisms and chemical plants, combined with increased data availability, almost naturally lead to an interest of systems engineering in understanding biological function. Each of these points will be presented in a little more detail below. Biology has grown to a scientific area that generates far more data than biologists are used to handle. The amount of complex data that are and will be generated with the technologies now available, and the need for modeling to understand the way networks function, requires - for efficiency reasons - that disciplines outside of traditional biology collaborate on the problem of understanding biological function (Vukmirovicand Tilghman 2000). The most obvious collaborators for this endeavor are systems theoreticians and engineers. The industrial interests in the understanding of biological function is illustrated by the tremendous and steadily growing list of products resulting from biotransformation processes mentioned in the review paper of Cheetham (2004).Clearly, improved understanding of the regulatory mechanisms responsible for the expression of the gene encoding a product of interest might lead to higher production rates (more product can be produced within an existing industrial facility), increased production yields (raw materials can be utilized more efficiently),and shorter time to market. Thus, for an industrial biotransformation process, the results of improved understanding of biological function are directly related to increased profit. The bacterium E. coli, to name one popular example, was called a “workhorse microorganism” for recombinant protein production and a fundamental understanding of intracellular processes, such as transcription, translation, and protein folding, make this microorganism even more valuable for the expression of recombinant proteins (Baneyx 1999).Knowledge of the mechanisms of complex regulatory networks involved in the transformation of extracellular signals into intracellular responses is important to improve the productivity of microorganisms. The E. coli lactose utilization (lac) operon, which will be used later in this chapter to illustrate the complexity of regulatory networks, has served as one of the paradigms of prokaryotic regulation, and therefore a considerable number of the promoters used to drive the transcription of heterologous genes (genes carrying the genetic code for a product of interest) have been constructed from lac-derived regulatory elements (Baneyx 1999; Makrides 199G). The interest of systems engineering groups in contributing to an improved understanding of microbial function becomes clear by considering the number of review and position papers that were published in recent years (e.g., de Jong 2002; Doyle 2004; Ferber 2004; Hasty et al. 2001; Ideker and Lauffenburger 2003; Kitano 2002; Smolen et al. 2000; Wolkenhauer et al. 2003). Microbial function is carefully controlled through an intricate network of proteins and other signaling molecules. Free-

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living bacteria have to maintain a constant monitoring of extracellular physicochemical conditions in order to respond and modify their gene expression patterns accordingly (Lengeler et al. 1999; Salgado et al. 2004). Microorganisms by themselves thus constitute examples of entire autonomous chemical plants, which are able to produce and reproduce despite a shortage of raw materials and energy supplies. Microorganisms can sense changes in the surrounding environment, and subsequently control the expression of genes in reaction to these changes. Such adaptation of the cell to changes in the environment is crucial for the survival of the cell, since it allows economical use of cellular resources (Lengeler et al. 1999), as a result of regulating the expression of all genes to produce the optimal amount of gene product at any given point in time. The energy consumption for protein synthesis and the relatively short half-life of the mRNA molecules are reasons for a cell to control both the types and amounts of each protein (Wolkenhauer et al. 2003). Making a link to chemical production plants, cell behavior can be compared with adjusting the production capacity and the operation strategy of a chemical plant to the availability of limiting amounts of raw materials, aiming at minimizing plant operating costs. In view of the similarities between the functioning of a microorganism and a chemical plant, it is not overly surprising that systems-engineering thinking is increasingly applied to these biological systems. Systems engineering has different applications. Reverse engineering, aimed at unraveling the functionality of regulatory networks, is one of the major goals in systems biology. However, more and more effort is also directed into forward engineering, aiming at the design of regulatory networks with a desired functionality (Elowitz and Leibler 2000; Ferber 2004; Hasty et al. 2001). This research area is also called synthetic biology, to distinguish it more clearly from the reverse engineering efforts in systems biology. Contrary to systems biologists, who analyze data on the activity of thousands of genes and proteins, synthetic biologists simplify and build. They create models of genetic circuits, build the circuits, see if they work, and adjust them if they don’t (Ferber 2004). In the synthetic biology field, one of the future visions is the construction of cells as small factories for complex chemical compounds such as pharmaceuticals.

7.1.2 Levels of Abstraction, Function, and Behavior

It is important to realize that models of biological systems play a central role in both reverse and forward engineering. However, a model of biological systems represents different types of knowledge and assumptions about the system depending on the problem to be solved. Thus, the aim of reverse engineering is to interpret the biological system in order to explain how its structure and behavior originate from interactions of its subsystems. The interpretation is based on a model of the expected structure and behavior. The model can be based on either previous experience or represent a purpose or design intention. In both cases, the aim is to test whether the model (the hypothesis)

7. J Introduction

is an adequate representation of empirical data. In contrast, the aim in forward engineering is to predict structure and behavior of a biological system from knowledge of the structure and behavior of its parts, and to test if the predictions match subsequent empirical observations or design intentions. In prediction, the model is assumed to be adequate and used to produce hypotheses about unobservable structure or behavior. Models have accordingly different roles in reverse and forward engineering of biological systems. A general problem in the modeling of dynamic systems is to determine a proper level of abstraction. Most natural and artificial systems can be modeled on a variety of levels but the choice of level is of particular importance for biological systems due to their extreme complexity. Unfortunately, levels can be defined relative to several dimensions in the modeling problem. For example, we can describe the spatial structure (the anatomy) on many part-whole levels, and we can also describe the behavior (dynamics) at several part-whole levels of temporal resolution. Another way to define levels in biological systems is to consider their functional organization. The idea here is to describe the biological system as a goal-directedsystem and to decompose the system into subsystems so that each subsystem serves the needs or provides the means for its superordinate system. The analysis that brings about this type of system information is usually called means-end analysis or functional modeling, and has been developed within cognitive science and artificial intelligence research. The use of means-end analysis to define levels of abstraction is a very powerful approach to handle the modeling of complex dynamic systems (Lind 1994). It is of particular importance for modeling systems with embedded control systems, such as biological systems. Control systems play a direct role in the constitution of functional levels (Lind 2004b) and their function can therefore not be described properly without means-end concepts. Note that when using concepts of means-end analysis we must distinguish carefully between the concepts of behavior and function. The two notions are often confused such that function is thought to be more or less synonymous with behavior. We stress the teleological meaning of function; it represents the role the system has in the fulfillment of a purpose or goal. Behavior refers to what happens when a system reacts to an intervention or a disturbance. Descriptions of behavior have accordingly no connotations to purposes or goals and are therefore distinct from functional descriptions. We will later return to a discussion of means-end analysis and functional concepts in modeling complex dynamic systems.

7.1.3 Overview of the Chapter

The main purpose of this chapter is to discuss basic modeling problems that arise when attempting to describe regulatory networks and their function in microorganisms. The focus is on the representation of the regulatory mechanisms in micro-

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organisms applied for production purposes. First, the central dogma of biology will be introduced briefly. The E. coli lac operon is subsequently used as an example of a regulatory network structure in microorganisms, illustrating the complexity of such networks. The lac operon example is followed by a discussion of the essential steps in the central dogma, identifylng possible sites for control actions. Formalisms to model the regulatory networks are then introduced briefly, and strategies developed to deal with the complexity of regulatory networks in microorganisms are highlighted. Finally, means-end analysis and functional modeling are presented as suitable methods to represent the complex interactions in regulatory networks, and their use is illustrated by means of the lac operon example.The chapter ends with a discussion and conclusions.

7.2 Central Dogma of Biology

According to the central dogma of biology, a term coined by Sir Francis Crick, three processes, illustrated in Fig. 7.1, are responsible for the conversion of genetic information. (1)DNA replication: a process involving several enzymes and duplicating a double stranded nucleic acid to give identical copies; (2) Transcription: a DNA segment constituting a gene or an operon is read and transcribed into a single stranded sequence of RNA, the messenger RNA (mRNA),by the RNA polymerase enzyme; (3) Translation: the mRNA sequence is translated into a sequence of amino acids, where the ribosome reads three bases (a codon) at one time from the mRNA, translates them into one amino acid, and subsequently joins the amino acids together in an amino acid chain (protein formation). The resulting proteins, depending on their structure, may function as transcription factors (or regulatory proteins) binding to regulatory sites of other genes, as enzymes catalyzing metabolic reactions, or as components of signal transduction pathways. In prokaryotic cells, transcription and translation take place simultaneously. In eukaryotic cells, the mRNA is formed in the cell nucleus, which is separated from the rest of the cell. The mRNA undergoes further processing and modifications before it is transported out of the nucleus, where the ribosomes take care of the translation. New research results appearing in the early 1970s meant that the basic principle of the central dogma, that information flows uniquely from DNA to RNA to protein, Transcription DNA

n

Translation

ReKtion

Figure 7.1

biology.

Schematic illustration of the central dogma of molecular

7.3 Complexity ofRegulatory Networks

needed adjustment. Indeed, with the discovery of reverse transcriptase in retroviruses, the central dogma (Fig.7.1) was extended to include the ability to convert RNA into DNA. Prions also form an exception to the original formulation of the central dogma, since these proteins can induce misfolding of other proteins.

7.3 Complexity of Regulatory Networks

Of course, Fig. 7.1 is a simplified representation of the complex processes taking place in, for example, a prokaryotic microorganism. In reality, the central dogma of molecular biology includes quite a number of possibilities for regulation of DNA replication and protein production processes, which in this chapter will be first illustrated with an example of the production of enzymes (proteins) in a prokaryotic organism.

7.3.1 An Example of Transcriptional Regulation: the lac Operon

The example focuses on transcriptional regulation, the intracellular control mechanisms that influence the rate of the process responsible for converting the genetic information contained in the DNA into mRNA. In prokaryotes, genes are grouped into operons (Fig. 7.2).An operon can thus consist of several structural genes, where each structural gene encodes a protein. The genetic information contained in the genes in one operon together provides the cell with the capability to perform a coordinated function, for example, the execution of one metabolic pathway to produce a specific amino acid, or the (partial) conversion or degradation of one specific substrate to a metabolic intermediate. All genes in an operon are transcribed at once, resulting in polycistronic messenger RNA (mRNA),that is, mRNA encoding for several proteins. An example of an operon, in this case the well-known E. coli lac operon, is provided in Fig. 7.2A. The regulation of the transcription of the lac operon is the result of a combination of different mechanisms. 7.3.1.1 Absence of Extracellular Glucose: Induction of the lac Operon Mechanism

The first mechanism that will play a role in the transcription of the lac operon is induction, which is schematically represented in Fig. 7.2. The foundation for the current level of understanding of this regulatory mechanism is the operon model formulated by Jacob and Monod (1961),where the clear distinction between structural and regulatory genes was introduced. The lac operon consists of three structural genes (Fig. 7.2), containing the genetic code for enzymes that will be responsible for the uptake and conversion of the sub-

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Figure 7.2. Induction o f the lac operon in the absence of glucose in the growth medium (based on the model o f Yildirim and Mackey (2003)) A Repressed lac operon, B Induced lac operon (A, B, P, L, L, and M, in italics in the figure, indicate the variables considered in the model of Yildirim and Mackey 2003) Lac1 = gene encoding for repressor protein, P = promoter region for repres

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sor protein, P = promoter region for structural genes, 0 = operator region for structural genes, LacZ = P-galactosidasegene, Lacy = P-galactoside permease gene, L a d = P-galactoside transacetylase gene, p-gal = P-galactosidase, per = p galactoside permease, transac = P-galactoside transacetylase

7.3 Complexity ofRegulatory Networks

strate lactose into its building blocks glucose and galactose. In the simplified representation in Fig. 7.2, the structural genes are preceded by one operator and one promoter. In the absence of extracellular glucose and lactose, the lac operon is repressed. The repression of the lac operon originates from the presence of a fourth gene, containing the genetic code for a repressor protein. This lac repressor gene, or regulatory gene, provides one of the keys for understanding the regulatory mechanism that allows E. coli bacteria to grow on lactose in the absence of glucose. The lac repressor gene has its own promoter (Pi in Fig. 7.2) allowing RNA polymerase to bind to Pi and to transcribe the lac repressor gene. The ribosomes translate the lac repressor mRNA, to form the lac repressor protein. In the absence of lactose, the lac operon is repressed, meaning that the lac repressor protein is bound to the operator region of the lac operon, preventing the RNA polymerase to bind to the promoter of the structural genes, and thus repressing the transcription of the structural genes (see Fig. 7.2A). Allolactose is the inducer of the lac operon and results from the intracellular conversion of lactose following uptake trough the cell membrane (Lengeler et al. 1999; Wong et al. 1997; Yildirim and Mackey 2003). Indeed, in the absence of extracellular glucose, and when lactose is present in the growth medium, lactose is transported into the cell by the P-galactoside permease (Fig. 7.2B). Intracellular lactose is subsequently converted into glucose, galactose, and allolactose. The lac repressor protein undergoes a conformational change after binding the inducer allolactose, and is then no longer capable of binding to the operator region of the structural genes (see Fig. 7.2B). RNA polymerase can now bind to the promoter of the structural genes and produce mRNA, which is subsequently converted into proteins (P-galactosidase, P-galactoside permease, and P-galactoside transacetylase) by the ribosomes. This induction mechanism of the lac operon is a positive feedback loop: increasing intracellular lactose concentrations will lead to an increase in the expression of the lac operon, and thus result in an increased production of, for example, permease enzyme molecules, which will again lead to increased intracellular lactose concentrations, until the maximum protein production rate is reached. Depletion of extracellular lactose will result in repression of the lac operon. A First Principles Model Example: Model of the lac Operon Induction

Modeling plays an important role in unraveling regulatory mechanisms. A model for the induction of the lac operon was proposed by Yildirim and Mackey (2003)and will be used here as an example. Since this model only considers the induction mechanism, the model is only valid in the absence of extracellular glucose. The model consists of five states (see Fig. 7.2B): intracellular lactose (L), allolactose (A), mRNA resulting from the transcription of the structural genes (M), P-galactosidase (B), and P-galactoside permease (P). The system is modeled with five nonlinear delay differential equations (DDEs) provided in Eqs. (1-s), and has two external inputs, the extracellular lactose concentration (LJ, which is assumed to be constant, and the growth rate (p). Note also that spontaneous mRNA generation has been omitted in Eq. (l),since its contribution could be neglected.

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dM dt

-=(YM'

+ K1 . (e(-pL.rM). A(t - t ~ )2 )- m . M - p . M K + K1 . (e(-fiL.rM) . A(t - t ~ ) ) 1

dB = (YB . e(-@"B) . M(t - tg) - y~ . B - p . B dt

Such a model is, of course, based on a number of model assumptions. In the case of this model, the delay times in the DDEs are assumed to be related to different biological phenomena. The delay in Eq. (1)represents the fact that there is a delay ZM between the start of transcription and the production of a complete mRNA. The delay tBin Eq. (2) represents the delay between the start of mRNA translation and the appearance of P-galactosidase, and tBthus corresponds to the time needed for translation. The delay tB+zP in Eq. (5) includes the assumption that P-galactosidase production needs to be finished before the production of the P-galactoside permease whereas tprepresents the time needed to produce P-galactoside can start (delay te), permease. The selection of DDEs with constant delays to model this regulatory mechanism actually includes the assumption that translational regulation does not influence the protein production rate. Indeed, translational regulation would lead to variations in the delays tBand zp. Furthermore, transcriptional control is only modeled as influencing transcription initiation. The constant delay tM in Eq. (1)includes the assumption that no regulatory mechanism influences transcriptional elongation and transcription termination. Besides the assumptions underlying the choice of the delay times in the model example, it is of utmost importance to have a proper understanding of the assumptions that were made when describing the transcriptional regulation mechanism of the lac operon. Actually, the model example in Eqs. (1-5) does not provide a detailed description of this regulatory mechanism (Santillin and Mackey 2004). Instead, the model example lumps the regulatory mechanism into one Hill-type equation, describing the production of mRNA as a function of the inducer, the allolactose concentration (Eq. (1)).The dynamics of the lac repressor protein, the lac repressor protein-allolactose complex and the RNA polymerase enzyme are not considered explicitly. The original paper by Yildirim and Mackey (2003)can be consulted for further detail on the kinetic expressions. A set of parameters, suitable initial conditions, and steady-state values obtained with these parameters can be found in Yildirim and Mackey (2003),as well as a demonstration of the capabilities of the model to describe the dynamics observed experi-

7.3 Cornphity ofRegulatory Networks

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Figure 7.3 Relative concentration dynamics of intracellular lactose (L), allolactose (A), mRNA (M), P-galactosidase (B), and P-galactoside permease (P) predicted by the model of Yildirim and Mackey (2003) for a step-change of glucose to lactose feeding at t = 0 (L= 0.08 mM. p = 0.0226 m i d ) . The initial values provided by Yildirim and Mackey (2003) were used.

mentally. Figure 7.3 provides the relative concentration dynamics predicted by the model for a step change in the feed from glucose to lactose at t = 0. For the figure, the data were scaled by dividing each concentration time series by its maximum value. After the appearance of lactose, the model predicts an increase of the allolactose concentration, resulting in induction of the lac operon and subsequent production of pgalactosidase. 7.3.1.2 Presence of Extracellular Glucose: Inducer Exclusion and Carbon Catabolite Repression

Glucose, not lactose, is the preferred carbon source of E. coli bacteria. In the presence of both glucose and lactose, Ecoli bacteria will first grow on glucose, and the enzymes encoded in the lac operon will not be produced. However, when all glucose is consumed, the presence of lactose will induce the production of the enzymes encoded in the lac operon, and thus provide E.coli bacteria with the capability of growing on lactose as an alternative substrate. When all lactose is consumed, the production of the enzymes encoded in the lac operon will be turned off again, thereby economizing on the cellular resources.

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The lac operon inducer mechanism (Fig.7.2) alone cannot explain why the lac operon is repressed when both extracellular glucose and lactose are present. Clearly, there must be additional regulatory mechanisms beside the lac operon inducer mechanism. Figure 7.4 provides a diagram of the main regulatory mechanisms of the lac operon included in the model of Wong et al. (1997),and indicates that the lac operon is indeed controlled by glucose at two levels (Lengeleret al. 1999; Wong et al. 1997): (1) Inducer exclusion; and (2) catabolite repression. Extracellular glucose, while being transported into the cell by the phosphoeno1pyruvate:sugar phosphotransferase system (PTS), an important transport system of E. coli bacteria, is converted to glucose 6-phosphate (GGP).The cell growth rate is assumed to depend on the G6P concentration. At this point, the model of Wong et al. (1997) assumes that the glucose uptake rate via the PTS is related to the external glucose concentration via Monod kinetics. It is important to realize, though, that the PTS itself is a protein complex consisting of several enzymes that will transfer a phosphate group from phosphoenolpyruvate (PEP) to glucose during glucose uptake, resulting in G6P (Postma et al. 1993). In the absence of extracellular glucose, cyclic adenosine monophosphate (CAMP)is synthesized by the adenylate cyclase (AC)enzyme, accumulates in the cell, and binds to the cAMP receptor protein (CRP). cAMP is considered an alarmone for carbon starvation of E. coli (Lengeler et al. 1999).Alarmones are molecules that signal stress conditions. The phosphorylated form of one of the enzymes of the PTS (HAG'') is an activator of AC (Postma et al. 1993).The cAMP:CRP complex binds to the CRP binding region, which is located near the lac promoter, and will enhance transcription initiation and also transcription of the structural genes by the RNA po1ymerase:a factor complex (RNAP:a in Fig. 7.4). In the presence of extracellular glucose, no CAMP will be generated since HAG'' is in its non-phosphorylated form, and no cAMP:CRP complex will be formed, resulting in catabolite repression, or the repression (inactivation)of certain sugar-metabolizing operons (such as the lac operon in this example) in favor of the utilization of an energetically more favorable carbon source (glucose in this example). In Fig. 7.4, the presence of extracellular glucose results in inhibition of the transport of lactose by the lac permease, a phenomenon known as inducer exclusion. It has been demonstrated that it is not extracellular glucose itself, but the non-phosphorylated form of the PTS enzyme IIAG" that inhibits the uptake of lactose by the lac permease (Postma et al. 1993). Several first-principles models have been formulated to describe the combined effects of inducer exclusion, carbon catabolite repression, and induction on the lac operon (e.g., Kremling et al. 2001; Santillin and Mackey 2004; Wong et al. 1997).An accurate description of the phenomena observed when E. coli bacteria grow on a mixture of glucose and lactose necessitates inclusion of the glucose effects in mathematical models of the lac operon, which results in rather complex models. In the model of Santillin and Mackey (2004),an additional layer of complexity in the regulation of the lac operon is considered explicitly by tahng into account in the model that the lac operon has three different operators, two different cAMP:CRP binding sites and two different promoters. Also, the model takes into account that DNA can fold in such a way that a single repressor molecule can bind to two different operators. Considering all possible interactions between the lac operon, the repressor, the cAMP:CRP com-

7.3 Complexity ofRegulatory Networks

A

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plex, and the RNAP:a complex results in 50 different binding states for the lac operon.

7.3.2 Potential Sites for Control Actions

The lac operon example is entirely focused on transcriptional regulation, more specifically transcription initiation, meaning that only part of the mechanisms that control the transcription of the structural genes, and thus the production of mRNA, are

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considered. In fact, the emphasis is on transcriptional regulation in the majority of the studies on regulatory networks in microorganisms. In a review on modeling transcriptional regulation, Smolen et al. (2000)explained this by the fact that two key approximations have historically been used to model genetic regulatory systems: (1) control is exercised at the transcriptional level, (2) the production of a protein product is a continuous process, with the rate determined by the balance between gene activation and gene repression. As a consequence, there are few or no studies that model both translational and transcriptional control in any specific genetic system (Smolen et al. 2000). However, prokaryotic cells are capable of rapidly adjusting to a wide range of environmental conditions (Lengeler et al. 1999), and this adjustment is achieved in two ways: (1) Instant responses involving a change in the activity of critical metabolic enzymes; and (2) Delayed, more long-term responses, involving positive or negative regulation of gene activity in a coordinated fashion. Transcriptional regulation is, of course, very important. However, Lengeler et al. (1999)provided examples that illustrated that regulation of protein synthesis not only takes place at the level of transcription initiation, that is, regulation of the binding of the RNA polymerase to the promoter, but also occurs at the levels of transcriptional elongation (i.e., during the formation of the mRNA chains) and termination (i.e., during the final stages of mRNA formation). Moreover, the mRNA is not a stable intermediate, and mRNA degradation provides a major control point of gene expression in virtually all organisms (Makrides 1996). Furthermore, Lengeler et al. (1999) indicated the importance of regulation during translation and mention protein stability as an additional factor that can be influenced by regulatory mechanisms. Finally, posttranslational modification of proteins is considered as a fine-tuning mechanism to adjust the activity of enzymes. Summarizing, transcriptional regulation alone provides only part of a more complicated picture. Information on mRNA-levels in the cell provides an indication of gene expression and transcriptional regulation, but should also be combined with protein measurements to track the final gene expression product.

7.4 Methods for Mapping the Complexity of Regulatory Networks

Models are ideally suited for the representation of complex regulatory networks. The lac operon example is first compared to the size of the genome to further illustrate the complexity. As mentioned above, most systems can be modeled on a variety of levels of abstraction. The role of modeling is discussed and illustrated with a design example. Current developments in the construction of high-level models will be illustrated with the search for network motifs in regulatory networks, which is a highlevel modeling example. A signal-orienteddetailed first principles modeling methodology will subsequently be introduced as an attractive example of a low-level modeling approach. Finally, high- and low-level modeling approaches will be contrasted, and the link between high- and low-level models will be explained.

7.4 Methods for Mapping the Complexity of Regulatory Networks

7.4.1 Complexity of Regulatory Networks

Regulatory networks are complex, which was illustrated using the lac operon example. There are two distinct characterizations of complexity that both apply to regulatory networks (Doyle 2004): (1) The classical notion of behavior associated with the mathematical properties of chaos and bifurcations (behavioral complexity); and (2) The descriptive or topological notion of a large number of constitutive elements with nontrivial connectivity (organizational complexity). Chaos, bifurcations, and the occurrence of multiple static or dynamic states in biological systems are beyond the scope of this chapter. Instead, we are more interested in the organizational complexity of regulatory networks, more specifically in methodologies that allow representation of the complex regulatory networks and its many elements in a systematic way. The lac operon example illustrates the degree of organizational complexity involved in the transcriptional regulation of a single prokaryotic operon, and gives an indication that proteins are the main catalysts, structural elements, signaling messengers, and molecular machines of living cells. The classical view of protein function focused on the local action of a single protein molecule, for example, the catalysis of one specific reaction in the metabolism of an organism. However, today there is a more expanded view of protein function, where a protein is defined as an element in the network of its interactions (Eisenberg et al. 2000). Each gene in the genome of an organism encodes for a protein. Thus, a first indicator of the overall organizational complexity of the regulatory networks is the number of genes in the genome. The genome of the well-studied prokaryote E. coli consists of 4408 genes with 179 transcriptional regulators (Salgado et al. 2004), whereas the genome of a typical eukaryote, S. cerevisae, consists of 6270 genes (Lee et al. 2002). The absolute numbers of genes might already provide an indication that the organizational complexity of eukaryotic organisms is higher compared to prokaryotic organisms. Most proteins interact with several other proteins, resulting in complicated protein-protein interaction networks. It is exactly these multiple simultaneous interactions of many proteins in the network that need to be understood and represented to understand the functioning of a living cell. As a reaction on sensing a change in the extracellular environment, the gene expression pattern will be modified. Contrary to prokaryotic cells, eukaryotic cells have a nucleus. For Eukaryotes provoking a change in the gene-expressionusually requires the movement of a protein from the body of the cell to the nucleus in response to the changes in the extracellular environment (Downward 2001). Thus, the cell compartmentalization will also necessitate the mapping and representation of transport processes between different cell compartments for eukaryotic cells, whereas such intracellular transport processes usually don’t need to be considered for prokaryotic cells.

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7.4.2 The Essential Role of Modeling

A conceptual problem arises of how to understand the operation of these complex systems. Positive and negative feedback within signaling pathways, crosstalk between pathways, time delays that may result from mRNA or protein transport, and nonlinear interactions all need to be considered to understand the operation of genetic regulatory systems (Smolen et al. 2000). Mathematical modeling of the dynamics of regulatory networks in microorganisms is therefore assumed to take on an essential role for a number of reasons (Mackey et al. 2004; Smolen et al. 2000): (1) Mathematical models can integrate biological facts and insights, that is, process knowledge on regulatory networks can be represented and summarized in a mathematical model; (2) Models can be helpful in identifying design principles for the regulatory networks; (3) Modeling can contribute to developing an understanding of the responses of both normal and mutant organisms to stimuli; (4) Model analysis can reveal potentially new dynamical behaviors that can then be searched for experimentally; and (5) Models can be used to verify the consistency and completeness of reaction sets hypothesized to describe specific systems. Failure of realistic mathematical models to explain experimentally observed behavior often points to the existence of unknown biological details, and can thereby also act as a guide for experimentalists. Many modeling formalisms have been applied to the description of regulatory networks and were reviewed in detail by de Jong (ZOOZ), including directed graphs, Bayesian networks, Boolean networks, nonlinear ordinary differential equations (ODEs), piecewise-linear differential equations, qualitative differential equations (QDEs), partial differential equations (PDEs), and stochastic equations and rulebased formalisms. Discussing the advantages and drawbacks of each modeling formalism is beyond the scope of this chapter. Instead, we will limit ourselves to highlighting positive and negative aspects related to applying the most widespread modeling formalism for the detailed representation of regulatory networks, nonlinear ODEs. Representing regulatory network dynamics with differential equations has certain advantages (Smolen et al. 2000; Hasty et al. 2001): (1)the model yields a continuous description allowing, in principle, for a more accurate physical representation of the system; (2) the models are supported by dynamical systems theory or, in other words, a large body of theory and methodology is available to characterize the dynamics produced by these models; (3) despite being computationally expensive, simulations with detailed models are still rapid compared to in viuo experimental work, allowing researchers to examine many hypotheses and concentrate experimental effort on the most promising of them. Using differential equations also has disadvantages (Alur et al. 2002; de Jong et al. 2002; Smolen et al. 2000; Stelling et al. 2002): (1) The approach is computationally more intensive than, for example, the Boolean approach, where discrete updating of model states is applied. (2) Differentialequation models require the assumption of a specific kinetic scheme, whereas the necessary mechanistic detail is in many cases not (yet) available. (3) There is often a lack of in uiuo or in vitro measurements of

7.4 Methods for Mapping the Complexity of Regu/a#ory Networks

kinetic parameters in the models. Parameter values are indeed only available for a limited number of well-studied systems such as the E. coli lac and trp operon. Application of system identification methods combined with the increasing availability of data might alleviate this problem. (4)Cell compartments modeled with differential equations are assumed to be spatially homogeneous. In some situations this assumption is not appropriate. (5) Differential equations do not yield a good description of systems where only a limited number of molecules are involved. For identical initial conditions, two regulatory systems may reach different steady states as a consequence of stochastic processes resulting from the low number of molecules involved. 7.4.2.1 A Differential Equation Modeling Example: the Repressilator

Simulations with detailed mathematical models are important tools to analyze or to predict the behavior of regulatory networks and to subsequently draw conclusions regarding their design principles (Hasty et al. 2001). The repressilator (Elowitz and Leibler 2000) is an example of a rather simple synthetic network consisting of three transcriptional repressor systems, each consisting of a repressor gene encoding for a repressor protein. The names of the specific proteins are not important in the frame of this paper, and will therefore be omitted. When the genes (e.g., gA) are transcribed to mRNA, which is subsequently translated, the result is the production of a repressor protein (e.g., PA). The repressilator is a synthetic network, and was designed such that a negative feedback loop was obtained: The first repressor protein (PA) inhibits the transcription of the second repressor gene (gB). The second repressor protein (pB) inhibits the transcription of the third repressor gene (gC). And finally, the third repressor protein (pC) inhibits the transcription of the first repressor gene (gA).This is schematically presented in Fig. 7.5. The repressilator example can be represented by a system of six coupled ODES (Elowitz and Leibler 2000), where mA, rnB,and mc represent the mRNA concentrations, and pA,pB,and p c represent the protein concentrations.

* dt

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lgal -\*/

Figure 7.5

Scheme of the repressilator

(Elowitz and Leibler 2000).

El The parameters a + ao,ao,n, and p in Eqs. (6-11) represent the mRNA production rate of the derepressed promoters, the mRNA production rate of the repressed promoters (due to the “leakiness”of the promoter), a Hill coefficient, and the ratio of the protein decay rate to the mRNA decay rate, respectively. Elowitz and Leibler (2000) demonstrated that depending on the selection of the model parameters, the system has a stable or unstable steady state. Both cases are illustrated in Fig. 7.6, and were obtained by simply varying the parameter a in the model. Note that both the protein concentrations and the time axis were normalized I

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7.4 Methods for Mapping the Complexity of Regulatory Networks

in Fig. 7.6 by dividing through the maximum protein concentration and the end time of the simulation, respectively. Based on the modeling work, it was concluded that oscillations are favored by the following cellular design principles : strong promoters coupled to efficient ribosome-binding sites, tight transcriptional regulation (low a,,), cooperative repression characteristics, and comparable protein and mRNA decay rates. These model-based design principles were subsequently used to construct an E. coli mutant showing oscillatory behavior in vitro (Elowitz and Leibler 2000).

7.4.3 Modularizing Complex Regulatory Networks

In the following, existing methods to decompose the complex network interactions into smaller elementary units will be highlighted. However, it should be mentioned explicitly that we did not attempt to make a complete overview of all available methodologies. Rather, we have chosen to present two approaches so that we could illustrate the difference between high-level and low-level modeling, There seems to be general agreement that suitable methodologies to represent the organizational complexity of regulatory networks should rely on hierarchical structures consisting of multiple modular elementary blocks. A module can generally be considered as a component or a subsystem of a larger system, and generally has some or all of the following properties (Csete and Doyle 2002): (1)identifiable interfaces; (2) can be modified and evolved somewhat independently; (3) facilitates simplified or abstract modeling; (4) maintains some identity when isolated or rearranged; and (5) derives additional identity from the rest of the system. However, what kind of modular structure should be selected for this purpose remains an open question and recent research has provided a number of attractive suggestions. The development of methodologies that allow a modular representation and simulation of large-scale dynamic systems is considered as one of the most important research topics in systems biology (Wolkenhauer etal. 2003). However, Csete and Doyle (2002) point out that the protocols (the rules that prescribe allowed interfaces between modules, permitting system functions that could not otherwise be achieved by isolated modules) are far more important to biological complexity than the modules. 7.4.3.1 Network Motifs

One way to deal with the organizational complexity of regulatory networks in microorganisms is the recognition of elementary modules, called network motifs. Such network motifs seem to be present in all kinds of complex networks (Milo et al. 2002) and can serve as elementary building blocks to reconstruct the connectivity in a regulatory network. For the prokaryote E. coli, Shen-Orr et al. (2002)extracted data from a database (Salgado et al. 2004) on direct transcriptional interactions between transcription factors and the operons they regulate, and augmented these data with a literature search, resulting in 141 transcription factors. A transcription factor, or a tran-

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scriptional regulator, is a protein that binds to regulatory regions of the DNA and helps control gene expression. The Lac1 gene encoding for the lac repressor protein is an example of a transcription factor (Fig. 7.2). Shen-Orr et al. (2002) found that a considerable part of the regulatory network of E.coli was composed of repeated appearances of only three elementary network motifs (Fig.7.7). In the feedforward loop network motif, a general transcription factor X regulates the expression of a second specific transcription factor Y, whereas both transcription factors jointly regulate the expression of a structural gene Z. Coherent and incoherent feedforward loops are distinguished. In a coherent feedforward loop the direct effect of the general transcription factor on the expression of the structural gene has the same sign as the net indirect effect through the specific transcription factor. In an incoherent feedforward loop the direct and indirect effect have opposite signs. The coherent feedforward loop, the most frequently occurring feedforward loop motif in E. coli (Mangan and Alon 2003), was originally thought to be designed to be sensitive to persistent, rather than short and fast, transient inputs (Shen-Orr et al. 2002), that is, as a circuit that can reject transient activation signals from the general transcription factor (X in Fig. 7.7). A more detailed mathematical analysis of the feedforward loop motif (Mangan and Alon 2003) indicated that coherent feedforward loops act as a sign-sensitive delay element, meaning that the coherent feedforward loop responds rapidly to step changes in the general regulator concentration X in one direction (e.g., OFF to ON), and with a considerable delay to step changes in the general regulator concentration X in the other direction (e.g., ON to OFF). The practical functioning of this coherent feedforward loop regulatory mechanism was demonstrated with the L-arabinose (am) utilization system in E. coli (Mangan et al. 2003). The influence of step changes in the global regulator cAMP on the expression of the L-arabinose system was investigated, and it was demonstrated that the ON response following a step increase of the cAMP concentration was indeed much slower compared to the OFF response (provokedwith the addition of glucose in the growth medium). It was concluded that E. coli might have an advantage in a rapidly varying environment with this type of asymmetric response. When glucose is suddenly present (corresponding to a CAMP OFF step) it is utilized immediately. However, when glucose is depleted from the growth medium (corresponding to a cAMP ON step), the cell can save on the energy spent for protein production by only responding to persistent CAMP ON stimuli. In a single input module network motif, a number of structural genes Z1, Z2,..., Z N are controlled by a single transcription factor X. The single input module can be compared to a single-input multiple-output (SIMO) block architecture in control (Doyle 2004), and is typically found in systems of genes that encode for a complete metabolic pathway. Shen-Orr et al. (2002) further indicate, based on mathematical analysis, that single input modules can show a detailed execution sequence of expression of the structural genes, resulting from differences in the activation thresholds of the different structural genes. In the dense overlapping regulons network motif there is a layer of overlapping interactions between a group of transcription factors X1, Xz,..., XN and a group of structural genes Z1, 2 2 , . .., ZN. In control terminology, dense overlapping regulons can be compared to a multiple-input multi-

7.4 Methodsfor Mapping the Complexity of Regulatory Networks

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Figure 7.7 Elementary network motifs found in the E. coli transcriptional regulation network (Shen-Orr et at. 2002).

ple-output (MIMO) block architecture (Doyle 2004). The dense overlapping regulons seem to group operons that share a common biological function (Shen-Orr et al. 2002). Shen-Orr et al. (2002)indeed illustrate that the motifs allow a representation of the E. coli transcriptional network in a compact, modular form. However, reality is more complex: the transcriptional network can be thought of as the “slow”part of the cellular regulation network (with a time scale of minutes). An additional layer of faster interactions, including interactions between proteins (often on a subsecond timescale) contributes to the full regulatory behavior and will probably introduce additional network motifs. This was confirmed by Yeger-Lotem et al. (2004), who extended the search algorithms for network motifs from genome-wide transcriptional regulatory network data to also include protein-protein interaction data (YegerLotem and Margalit 2003) and applied the methodology to S. cerevisiae. For a more complete review of recently developed methods to search for network motifs in highthroughput data see Wei et al. (2004). Again, eukaryotic organisms are more complex than prokaryotic organisms. In a study with S. cerevisiae on regulator-gene interactions, Lee et al. (2002)identified six frequently occurring network motifs, compared to only three for E. coli. Besides the Autoregulation

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Figure 7.8 Network motifs in the regulatory network o f 5. cerevisiae (Lee et al. 2002). Solid arrows indicate binding of a regulator to a promoter. Dotted arrows indicate links between regulators and genes encoding for a regulator. Capitals indicate genes (e.g., A), whereas normalfont (e.g., a) indicates proteins.

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feedfonvard loop and the single input module (single-input motif in Lee et al. 2002) found for E. coli (Shen-Orr et al. 2002), autoregulation, multicomponent loop, regulator chain, and multiple-input motif network motifs were identified (Fig.7.8). In an autoregulation motif, the regulator binds to the promoter region of its own gene. This mechanism was found for about 10% of the yeast genes encoding transcription factors. 7.4.3.2 A Signal-Oriented Modeling Approach to Modeling Regulatory Networks

Models are ideally suited to represent the knowledge about complex systems. Hierarchical modular modeling approaches are needed, since they lead to high model transparency at different levels of abstraction. Such model transparency is beneficial for engineers, but certainly also for biologists. Moreover, a modular structure contributes to allowing easy modification of the model by the model user. One can just modify one model module and subsequently plug the updated module into the overall model. A two-level hierarchical approach for modeling cell signaling mechanisms was proposed (Asthagiri and Lauffenburger 2000), where signaling modules would be defined as units whose underlying mechanisms can be studied first in isolation, and then integrated into a larger flow diagram of networked modules. Modules may be networked in a manner similar to the assembly of unit operations. Signaling outputs would be directed between different modules providing the interconnectivity and optimization; network performance can be assessed from a process systems perspective. According to Lengeler et al. (1999),cellular control is hierarchical, meaning that there are global control networks that are superimposed on the specific control systems, and that can overrule the specificcontrol systems. In prokaryotes, operons and regulons are at the lowest level of the control hierarchy as specific control systems. A regulon is a group of operons that are regulated by a common, but specific, regulator. The global control networks are coupled to complex signal transduction systems, which sense changes in the extracellular environment that require more drastic cellular adaptations than simply the expression or the repression of a few operons. Groups of operons and/or regulons controlled by such a global regulator are called a modulon. Finally, a stimulon represents groups of genes that will respond to the same stimulus. In the example of the lac operon, the repressor protein can be considered a local regulator, whereas CAMP can be considered a global regulator. In this cellular control hierarchy, functional units naturally appear by applying a set of three biological criteria (Kremling et al. 2000; Lengeler et al. 1999): (1)the presence of an enzymatic network with a common physiological task; (2) its control at the genetic level by a common regulatory network, corresponding to operons, regulons, and modulons; and (3) the coupling of this regulatory network to the environment through a signal transduction network. The prokaryotic cellular control hierarchy of Lengeler et al. (1999) is applied in the signal-oriented first principles modeling approach of Kremling et al. (2000), where complex metabolic and regulatory networks are decomposed into physiologically meaningful smaller functional units. Each functional unit is built up by combining

7.4 Methodsfor Mapping the Complexity ofRegulatory Networks

Substance storaee No genetic information:

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a number of elementary modeling objects (Fig.7.9). When building a mathematical model of a unit, each elementary model block in the representation of a regulatory network structure (see Fig.7.10 for an example) gets a mathematical equation assigned to it. As a result, functional units in complex networks are represented as mathematical modeling objects. The method was first applied to the modeling of the lac operon (Kremling et al. 2001). The development of the systems biology markup language (SBML), an XML-based language for representing models of systems of biochemical reactions, and for

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exchanging these models between simulation and model analysis tools (Hucka et al. 2003), is an important joint effort of a number of research teams. The use of SBML should facilitate the exchange of models between users of different software platforms. Indeed, instead of writing and validating model code for each software platform, a validated model in one software platform can be exported as an SBML model, which can subsequently be loaded by another software platform. The mere existence of the current SBML definition already contributes to modular representation of models, since a model for part of the processes in a cell can now be exchanged easily between researchers interested in cell modeling, and incorporated into other models and simulation software packages. Moreover, in the long term, it is envisaged that SBML will include the possibility of building large models by reusing a number of previously defined submodels (Finney and Hucka 2003). Clearly, this future SBML development is ideally suited for building large cell models with a modular structure. 7.4.3.3 Bridging the Gap between Network Motifs and the Signal-Oriented Modeling Approach

The network motifs (see Section 7.4.3.1) can be linked to the low-level modeling of regulatory networks (Doyle 2004), where the motifs represent modular components that recur across and within given organisms. One hierarchical modeling classification is proposed (Doyle 2004),where the top level corresponds to a network, which is comprised of interacting regulatory motifs. A module is at the lowest level in the hierarchy and describes transcriptional regulation. It is important to realize here that the network motifs are extracted from systemwide (genome-wide)molecular interaction datasets by applying statistical methods. They provide a general indication of the connectivity and the structure of the regulatory network, however, without any indication on the exact kinetics of each interaction. Network motifs might point in the direction of a model structure that can be applied to describe the connectivity in part of the network, but there are many model candidates that can correspond to each motif (Mangan and Alon 2003). However, development of a detailed (low-level) simulation model necessitates experimental data that can be used to discriminate between model candidates and to estimate kinetic model parameters (Mangan et al. 2003). The signal-oriented modeling approach (see Section 7.4.3.2) on the other hand, is based on detailed experimental work aimed at generating dynamic data for the key metabolites participating in the interactions related to a very small part of the genome. The signal-oriented modeling approach includes detailed mechanistic information on the kinetics of each interaction between model states, resulting in a detailed nonlinear ODE-basedmodel. Both approaches consider the regulatory network at a different abstraction level (Ideker and Lauffenburger 2003). The network motifs can be considered as high-level pathway models, whereas the signal-oriented modeling approach belongs to a class of extremely detailed low-level models. High-level and low-level models are of course connected. In fact, there are relatively few well-documented systems where detailed low-level modeling can be applied (de Jong 2002; Ideker and Lauffenburger 2003),whereas high-level informa-

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7.5 Towards Understanding the Complexity of Microbial Systems

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Illustration o f the use of different modeling formalisms to move from abstracted high-level models to specific low-level models (Ideker and Lauffenburger 2003).

tion on protein-DNA interactions and protein-protein interactions is available for an increasing number of microorganisms. Bridging the gap between the high-level and the low-level models or, in other words, increasing the throughput with which interesting and important biological problems can be brought from the high-level to the low-level modeling state is a major challenge for systems biology (Ideker and Lauffenburger 2003). Bridging the gap between high-level and low-level modeling might necessitate the sequential use of a hierarchy of modeling formalisms (see Fig. 7.11), where each formalism corresponds to an adequate description of a certain level of abstraction ofthe regulatory network (de Jong 2002; Ideker and Lauffenburger 2003). An example of a procedure to evolve from high-level to low-level models is provided in Ronen et al. (2002).

7.5 Towards Understanding the Complexity o f Microbial Systems

The models presented above represent selected but important aspects of microbial regulatory function as they can be expressed using dynamic systems concepts and theories. These theories have proven to be very powerful in dealing with analysis and design problems in control engineering and it is therefore natural to expect that similar successes can be obtained when they are applied to microbial systems. However, this expectation is based on the assumption that the complexity of engineering systems and microbial systems are comparable and measurable on the same scale. In that respect, Csete and Doyle (2002) indeed concluded that the complexity of engineering systems, taking a Boeing 777 with its more than 150,000subsystem modules as an example, is almost comparable to the complexity of biological systems. The modeling of microbial systems should therefore not represent fundamental new challenges, except maybe from the problem that the number of ODES required to

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describe their behavior will be significantly higher than for engineering systems, and that more nonlinear phenomena might be involved. This assumption however ignores basic interpretation problems in model building. These interpretation problems are tractable and their importance therefore remains unrecognized in most systems engineering problems, which becomes a major problem in modeling microbial systems.

7.5.1 The Interpretation Problem

The interpretation problem originates in the multifunctional nature of microbial systems. Where a subsystem in most engineering systems only serves one or a few functions, it may serve many interdependent functions in microbial systems. A function is not an inherent property of a subsystem, but is defined relative to other subsystems and by the purpose of the system of which it is a component. A protein may thus serve at least three different functions. It can serve as a substance (material or product, e.g., in protein degradation reactions) in a metabolic process, it can serve as an enzyme promoting another reaction, and it can act on the DNA for promoting or blocking the expression of genetic information (transcription factor).The complexity of microbial systems originates in this unique ability of proteins to enter into a multitude of functional relations. The identification of functions requires knowledge of how a subsystem contributes to the whole. This knowledge about the functional organization of the system is a prerequisite for the formulation of a set of ODES describing the system, because it determines the level of abstraction adopted and the system features to be included in the equations (Lind 2004b). As mentioned before, a distinction must be made between organizational (functional) complexity and behavioral complexity (Doyle 2004). Behavioral complexity can be expressed by ODES,but we need other concepts to model the organizational complexity. The purpose of a model of the organizational complexity is to define, in formalized language, the functional relations between subsystems and the biological system as a whole. Such a model comprises an abstract qualitative representation which can be used to communicate the understanding obtained for the biological system. Often, informal sketches or graphics are used to communicate functional knowledge. However, more formal concepts are required in order to ensure clear semantics and consistency of the models. A formalized model of the functional organization is therefore a complement to, and not merely a mediocre or less accurate version of, an ODE model. In the following we will discuss the interpretation problem in more detail in order to further motivate the application of functional concepts in the modeling of microbial systems, and to introduce and explain the basic concepts. We will subsequently present formalized generic concepts to model control (regulatory)functions. A key advantage of generic concepts is that they can be applied on an arbitrary level of abstraction and thus facilitate the modeling of complex control functions in microbial systems. Another advantage of the formalized concepts is their completeness.

7.5 Towards Undentanding the Complexity ofMicrobia/ Systems

7.5.1.1 Frameworks of Interpretation

In order to develop a deeper understanding of the problems in modeling complex systems, it is important to realize that modeling activity, in addition to mathematical aspects, involves a process of interpretation where the modeler makes sense of the events and phenomena in the problem under investigation. The interpretation problem is fundamental to humanities and social sciences but has thus far not been considered particularly relevant for the natural or the engineering sciences because interpretation is often considered in conflict with objectivity. However, when considering complex systems we must apply perspectives or make abstractions in order to handle the modeling problem at hand, and thus interpretations are unavoidable. However, interpretation of a phenomenon is always relative to a conceptual framework. According to Goffman (1974) we can distinguish between two so-called primary frameworks of interpretation. A framework of interpretation serves as a frame of reference and is seen as rendering what would otherwise be a meaningless and chaotic situation into something that is meaningful and with structure. The two primary frameworks are called the natural and the social frameworks, respectively, and are defined as follows: 0

0

...Natural frameworks identify occurrences seen as undirected, unoriented, unanimated, unguided, “purely physical.” Such unguided events are ones understood to be due totally, from start to finish, to “natural” determinants. It is seen that no willfid agency causally and intentionally interferes and that no actor continuously guides the outcome. Success or failure with regard to these events is not imaginable.. . ...Social frameworks, on the other hand, provide background understanding for events that incorporate the will, aim, and controlling effort of an intelligence, a live agency, the chief one being the human being. What the agent.. . does can be described as “guided doings.” These doings subject the doer to “standards,” to social appraisal of his action based on its.. . efficiency, economy, safety, etc.

Events and occurrences in engineering systems can clearly be interpreted within a natural framework. Engineering systems are, however, designed to exploit physical phenomena such that human purposes and aims can be fulfilled and therefore be understood within a social framework of interpretation. The natural and social frameworks are both broad categories. The natural frameworks include, for example, physics and chemistry, and similarly the social frameworks include several subcategories. Note that concepts of function and purpose belong to the social framework of interpretation. Habermas (1989) compared different approaches to functionalism within social science. His analysis identifies three approaches. Two of these are in Goffman’s sense of social frameworks for understanding the plan or intention of a system or an activity: 0

We can understand the plan teleologically, in which case it is based on the artisan model of instrumental action through which an end is reached through appropriate means.

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We can also conceive the plan dialectically, in which case it is based on the dramaturgic model of communicative action, in which an author makes an experience transparent through the role playing of actors.

Habermas’ analysis also indicates that we could define an additional framework of interpretation that could be called biological and was not considered by Goffman. This framework is characterized as follows: 0 We can also use a model borrowed from biology. According to this model, systems can be understood as organized unities that under changing circumstances maintain themselves in a specific state through self-regulation and reproduction. Accordingly, four different frameworks may be applied in the interpretation of an event or a phenomenon: 1. the natural framework, 2. the framework of instrumental action, 3. the framework of communicative action,

4. the biological framework.

Note that the four frameworks of interpretation should be seen as different ways to assign meaning to an observed event or phenomenon. Each framework defines a context for understanding the system according to a particular point of view. As mentioned below, the frameworks are often combined in the interpretation of complex systems. 7.5.1.2 Interpretation of Complex Biotechnological Systems

Interpreting complex biological systems will often require the application of more than one of the frameworks. For example, the behavior of a “dancing” bee can be described within a communicative framework by its communicative function. But it may also be described within a biological framework (at a higher abstraction level) by its function for the survival of the species. In order to understand the organization of bio(techno)logicalsystems it is necessary to apply the instrumental action, the biological, and possibly also the communicative framework, for example, when considering quorum sensing where a population of microorganisms are informed about a certain event. Since there is no blueprint (i.e., no designer of the cell), its regulatory function must be explained in evolutionary terms, where it must be seen as emerging from a selection process, leading to a competitive advantage for the cell. When the behavior of a cell population in a bioreactor is controlled from the outside it must be seen as an object of instrumental action. A major challenge in the interpretation of complex microbial systems is therefore to understand how to combine different interpretations of the same subsystem or how to combine the interpretations of subsystems that belong to different frameworks. As an example, the central dogma, which includes the transcription, translation, and expression of information in the DNA and RNA (communicative action), should be combined with the metabolic reactions (biological framework) and the control of cell population in a reactor (instrumental action).

7.5 Towards Undentanding the Complexity of Microbial Systems

7.5.2 Functional Analysis

The instrumental, communicative and biological frameworks support functional explanations, i.e., answers to a “why”question having the general form “in order to” (Achinstein 1983).The explanations are different, however. Within the framework of instrumental action the explanation of an event or object relates it to the intention of the actor. Within the communicative framework an occurrence is ascribed a communicative function (e.g., a message), and the occurrence is explained by its effect or role in an act of communication. Within the biological framework, observed events are seen as contributing to survival and adaptation of the system to its environment, e.g., an organ is ascribed a function in view of its contribution to the survival of the organism of which it is a part. Functional explanations express the reasons (not the causes) for the occurrence of an event and are therefore an integral part of means-end analysis. Means-end analysis is an old topic in philosophical logic with ancient roots in works of Aristotle but has more recently been developed within artificial intelligence (Simon 1981)and cognitive science research (Bratman 1987). Means-end analysis is the basis for multilevel flow modeling (MFM), which is a methodology for modeling complex industrial systems (Lind 1994) by integrating different frameworks. MFM is not intended to generate detailed dynamic models. Instead, it allows one to represent systems at different levels of abstraction and as such supports the building of detailed dynamic models in the conceptual phase (Gofuku and Lind 1994).MFM has an inherent logic that allows formal analysis of the organizational complexity, and is therefore also attractive for application to regulatory networks in microorganisms. 7.5.2.1 Formalization of the Concept of Function

One of the key research problems in means-end analysis is the formalization of the concept of hnction. Formalization is necessary in order to be able to build models of means-end relations in systems that are logically consistent and in order to be able to use the models for computational purposes. The formalization involves the solution of two problems. The first problem is to define a logic that can be used to make inferences about means-end relations. The other problem is to identify a basic set of socalled elementary functions, which can be used as generic modeling concepts. The question of means-end logic was addressed by Larsson (1996) for applications of MFM in fault diagnosis and by Larsen (1993) for problems of start-up planning. We will not consider these logics here, but instead focus on the problem of elementary functions, which is of particular interest in the present context of modeling regulatory networks in microbial systems. 7.5.2.2 Elementary Functions

Before we address the problem of elementary functions it should be mentioned that the concept of function actually has two core meanings. One meaning is related to

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the concept of action and the other is related to the concept of role. The first meaning is used when we define a function by what a system or actor is doing, and the second when we refer to the entities involved in the action. Consider the following example: the function of the pump is to “move the water.” Here the function of the pump in the first meaning is the intended result of the pump’s action, whereas the function in the second meaning is the role played by the pump in its interaction with the water, i.e., that it is the agent of the action. The function (role) of the water is similarly the object of action (the patient). By distinguishing between actions and roles we are accordingly able to define functions of systems more precisely. This clarification of the concept of function relies on the linguistic analyses of verb semantics (see, e.g., Lyons 1977).The solution to the problem of elementary functions depends therefore on whether we mean function in the sense of action or role. Elementary Roles and Embedding Relations

Elementary roles (such as agent, patient, instrument, etc.) have been defined by linguists, but some disagreements of minor importance for the discussion in this chapter still remain. Role relations are important for understanding system complexity. Thus, the same object or system could have several roles at the same time or different roles at different times. In this way system processes can be embedded into each other. With the roles mentioned above we have the following possibilities for role shifting: 0

0

0

An item may be the patient (product) of an action (transformation) and then become the agent (e.g., catalyst) of another action. An item may be the patient (product) of an action (transformation) and then become the patient (material) of another transformation. An item may be the agent of an action (transformation) and then become the patient (material) of another action (transformation).

An item may participate in this way in several processes at the same time provided it can play the roles simultaneously. Elementary Actions

The possibility of defining a set of elementary action types has been addressed by Von Wright (1963),and has been explored further for application in means-end analysis of complex dynamic systems by Lind (1994, 2002, 2004a,b). The elementary action types are actually derived from a set of corresponding elementary change types. The idea is that an action results in a change of state. Conceptually, the change caused by the action would not appear if the action was not done. The definition of an action therefore contains a reference to a hypothetical situation that is not realized because the action was done. Now, by defining a change as a transition between two states, we can define four so-called elementary changes shown in Table 7.1. Each change in the table is defined by both a linguistic description and a logic formula, which is composed of a proposition p representing the world state, a temporal operator T (Then) and one of the four change verbs “happens,” “remains,” “disappears,”and “remains absent.” In this way the formula -pTp (-p Then p) is a logic representation of the change described by “p happens.”

7.5 Towards Understanding the Complexity ofMicrobial Systems

We shall not go into details about the logic definitions here. However, it is notable that the list of elementary changes is a logically complete list so that all changes in the world can be defined provided we define the state in question by a proposition p. We will actually also need to combine elementary changes. Each elementary change has a corresponding elementary action type as indicated in Table 7.1. The action formula contains the temporal operator T (Then) and an additional operator 1 (Instead) used to indicate the hypothetical state. The logical formula -pTpI-p represents the action “produce p.” It is seen that if the action was not done the state of the world would be -p instead of p. The list of elementary actions can actually be expanded with four additional action types not shown in the table. These actions would correspond to actions where the agent refrains from intervening with the world. The total number of elementary actions is accordingly eight. The four (eight) elementary action types define a generic set of actions that have the great advantage of being defined on a logical basis. This means that the completeness of the action types is ensured. The elementary action types (Table 7.1) therefore form a very attractive basis for the definition concepts for modeling system functions. Note that the action types are generic because they are defined without specifying the proposition p. The action types can therefore be specialized to specific problem domains by proper specification of p. Another remarkable aspect of the action types is that they have a direct correspondence with the types of control functions used in control engineering. The correspondence is shown in Table 7.2. The completeness of the action types implies Table 7.1 The elementary action types of Von Wright (1963). p denotes a state, T denotes “Then,” and I denotes “Instead.” Types o f elementary change

I

Types of elementary action

Description

Formula

Description

Formula

p Happens

-PTP

Produce p

-PTPI-P

p Remains

PTP

Maintain p

pTp1-p

p Disappears

PT-P

Destroy p

PT-PIP

p Remains absent

-pT-p

Suppress p

-PT-PIP

Table 7.2 Correspondence between the elementary action types and control actions. Elementary action

I Control action

Produce

I Steer

Destroy

Trip

Suppress

I Interlock

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accordingly that any control function can be described by proper combinations of these four functions. Note that the descriptions of the controls do not represent the implementation of the controls. The descriptions only define the control purpose.

7.5.3 A Language for Modeling Functions of Microbial Systems

The elementary action types can be used as a systematic basis for the derivation of modeling concepts for a particular problem domain. As an example, MFM (Lind 1994) can be mentioned. A basic set of modeling concepts adapted to the domain of microbial systems is proposed in Fig. 7.12. Each of the actions shown can be defined formally as specific interpretations of the elementary action types or as compositions of two elementary actions (Lind 2004a).We will not provide all details here. Instead, we prefer to demonstrate with examples (see Section 7.5.4) how the modeling concepts can be used to represent the functional organization of microbial systems. The MFM modeling language (Fig.7.12) comprises three types of concepts. It contains a set of concepts for representing action (functions), concepts to represent goal states, and a set of concepts for representing means-end relations between actions, sets of actions, and goals. It should be stressed that MFM represents the actions or transformations done to material, energy, or information flows (fluxes) in a complex system. However, it does not represent the flows or fluxes themselves. This may seem disturbing, but the abstractions provided by MFM describe how the systems of

Actions

0 0 0 69 0 0

Q Q

Means-end relations Source Sink Transport

C I

Condition

T

Negated condition

-cI

Transcription I

Translation Storage Conversion

.p I

pp

Achievement

Producer-product

I

Separation

rlIn

Goal

t S

States

0

T

I

Mediation Figure 7.12

Steering

MFM

modeling concepts adapted to the microbial domain.

7.5 Towards Understanding the Complexity of Microbial Systems

transformation of the various substances (energy, material or information) are organized into means-end networks. The levels of abstraction can therefore not be defined without implicitly thinking in concepts of flows or fluxes. It should be noted that MFM also includes concepts to model part-whole relations, as well as concepts to model relations between functions and physical structures, but these relations are not used here (see Lind (1999) for more detail on these relations). The concepts in Fig. 7.12 will be explained briefly in the following. A deeper understanding can be obtained by studying the application examples presented in Section 7.5.4. 7.5.3.1 The Means-End Relations

Goals and functions can be connected by conditions, achievements, producerproduct, mediation, and steering relations. Each of the relations will be discussed separately. 0

0

0

0

0

0

The condition: a goal can define a condition that is necessary for the enablement of a function. This conditioning is expressed by a relation (C) between the objective and the function. The negated condition: a goal can define a condition that is necessary for the disablement of a function. This negated conditioning is expressed by a relation (-c) between the goal and the fimction. The achieve relation: goals are achieved by system functions. This relation is defined by the achieve relation (A). The (A) relation is a means-end relation where the goal is the end and the function or systems of functions are the means. The producer-product relation: functions can be related through a means-end relation called a producer-product (PP) relation. This relation is used when the interactions between a set of functions (an activity or process) result in a product that again serves another function in the system. The mediation relation: functions can also be related through a mediate (M) relation. This relation is used when a system has the role of being an intermediate between a system and another system that serves as an object of action. An example of such mediation could be the transportation of energy by the pumping of water. Here, there is a mediate relation between the pumping function and the transportation of energy. The steering relation: functions can also be related through a steering relation (S). This relation is used when the interactions between a set of functions (an activity or process modeled by a so-called flow structure) determine the state of another function. The connection relation: MFM also includes a so-called connection relation, which is not really a means-end relation, and is also not shown in Fig. 7.12. A connection is used to relate the functions (actions)into functional (flow)structures. A connection is symbolically represented as a line linking two functions and represents a contextual linkage of two functions. This means that they relate to the same goal perspective or that they share substances (change properties that belong to the

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same substance).The connection relation can be further specialized by taking into account causal directions in the interaction between functions. 7.5.3.2 The Flow Functions

MFM also defines a set of so-called flow functions (the actions), which are used in building models together with the relations described above. The symbols used for functions are shown in Fig. 7.12. Each ofthe functions represents an action on a substance that may be mass, energy, or momentum. The different substances are indicated by symbology. In action terms a source provides a substance, i.e., makes it available. Similarly a sink removes substances. A transport changes the spatial location of a substance, a conversion changes the composition of a material flow, and a separation separates a flow of material into its components. It is clear that some of the functions both apply to modeling material and energy flows. However, there are also functions that are dedicated to modeling transformation of, e.g., information flows. Two such functions are defined here for modeling the processing of genetic information in microbial networks, namely the transcription, and the translation functions. 7.5.4

Modeling Examples

In the following, the application of MFM to regulatory networks will be illustrated with examples that illustrate the capabilities of the methodology in decomposing the regulatory network into its elementary modules. Again, the lac operon will be used as an example since its regulation has already been described in detail (see Section 7.3.1). 7.5.4.1 Induction of the lac Operon

Figure 7.13 represents the induction mechanism of the lac operon (see also Fig. 7.2) using the symbols introduced in Fig. 7.12. The logic of the model can be explained by starting with the bottom part of Fig. 7.13, which represents the uptake of lactose and the conversion of lactose to allolactose, the inducer of the lac operon. The model shows in box I that the transport of lactose over the cell wall is carried out (“mediated”) by the f3-galactosidepermease, which is produced as the result of the translation of the mRNA (top part of the model). The subsequent conversion of lactose to allolactose is catalyzed (“enabled”) by the f3-galactosidase.The allolactose is afterwards assumed to be distributed in the cellular cytoplasm by diffusion (“transport function”). The functions described comprise the means for achieving (A in Fig. 7.13) the conditions for allolactose to be present in the cell. By changing perspective and thus moving upwards in the model we now consider the set of functions in boxes I1 and I11 in Fig. 7.13, describing how the state of the repressor (R) is influenced by the various functions of the microbial system. The

7.5 Towards Understanding the Complexity ofMicrobial Systems

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"beta galactosidase is present"

r

A

Y permease

F lac( Z,Y,A)

IV beta galacto

"no R a t IacZYA prornotor" A

I11

Figure 7.13. MFM representation o f the lac operon induction mechanism (see also Fig. 7.2).

allolactose

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principles to describe the production of repressor protein in detail (box 11) are similar to the flow model in box IV, representing the transcription and translation of the structural genes of the lac operon. The lac1 gene is transcribed and the resulting mRNA is translated into the repressor protein. Box I1 thus depicts the flow model for the transcription and translation part of the central dogma (Fig.7.1). The function in box I1 provides the presence of R. In other words, it provides the means for producing R. Box I1 represents the producer, whereas the fact that R is becoming present in box 111 is the product. The repressor protein can follow three possible paths, represented in box 111: (1)it can follow the bottom path where it binds with the operator of the lac operon, and will thus block transcription; (2) it can follow the middle path, where it binds with the inducer allolactose and undergoes a conformational change; and (3) it can follow the upper path, where the repressor protein is degraded (modeled as a "Sink). The presence of allolactose conditions two of the transports in box 111. When allolactose is present the repressor protein binds to the allolactose, whereas in the absence of allolactosethe repressor protein binds to the lac operon operator. In the first case, the functions described in box 111 comprise the means for achieving a de-repressed lac operon and thus transcription of the lac operon structural genes occurs. Again, we can now move upwards in the model. The transcription and translation of the lac operon is represented in box IV. The lac operon structural genes are transcribed into a polycistronic mRNA, and during the subsequent translation process the mRNA results in the different proteins (P-galactosidepermease and P-galactosidase).Conversion of the polycistronic mRNA to several proteins is modeled by combining the translation process with the subsequent separation function in Fig. 7.13. It is important to mention that the production of the third protein encoded in the lac operon, pgalactoside transacetylase, is not shown in Fig. 7.13, since that enzyme is assumed not to play any significant role in the induction mechanism. The difision of both proteins into the cytoplasm is finally illustrated in boxV and VI. Note that bgalactoside permease is a mediator (transport enzyme), whereas the role of galactosidase is an enabler (a catalyst). Now the loops to box I are closed.

0-

7.5.5 Inducer Exclusion and Carbon Catabolite Repression

Figure 7.14 is an extension of Fig. 7.13, including the regulatory effects of glucose on the lac operon, catabolite repression, and inducer exclusion (see also Fig. 7.4). Again, we start at the bottom of the figure to explain the modeled flows. The PTS system (box VII and VIII) is responsible for the uptake and the phosphorylation of glucose, resulting in GGP (for an explanation of the abbreviations, see Section 7.3.1.2). Note that both the energy level (box VIII: transfer of a phosphate group from PEP to GGP) and the component level (box VII) are represented, indicating the strength of MFM to represent a system at different abstraction levels. In the simplified schematic representation of the PTS for this purpose, it is assumed that a phosphate group is transferred from the phosphorylated PTS enzyme P-IIAG"to glucose during uptake

7.G Discussion and Conclusions

of glucose. The resulting PTS enzyme HAG'' again receives a phosphate group from PEP, i.e., it is involved in a loop. In fact, phosphorylation of IIAG1' takes place through a series of conversion steps, involving several of these loops. These conversions are not shown in detail since they do not contribute to the regulatory mechanisms, and are instead represented by one extra catalysis step between PEP and the catalysis of HAG1' to PIIAGICconversion. For the detailed mechanism of the PTS system, Lengeler et al. (1999) and Postma et al. (1993) should be consulted. With respect to the glucose effects on the regulation of the lac operon, the formation of IIAG" and PIIAG1'has been described in boxVII in all the detail needed. Again, we can change perspective and consider the functions where the presence of HAG" or PIIAG" will have an influence on the state of the system. The P-IIA'", the species that will be abundant in the absence of extracellular glucose, activates the conversion of ATP to cAMP by the adenylate cyclase (AC) in box X. Similarly to pgalactosidase, AC is modeled as an enabler (catalyst),but the transcription and translation processes leading to AC formation are not presented since these mechanisms were considered as not contributing substantially to the regulation of the lac operon. We have thus assumed that the enzyme AC is present ("Source" in box IX) and undergoes diffusion in the cytoplasm, and subsequently catalyzes the ATP to cAMP conversion in box X. The CAMP forms a complex with CRP, and this complex subsequently boosts the transcription of the structural genes of the lac operon (box IV), thereby releasing the catabolite repression of the lac operon. Thus, the catabolite repression mechanism has been described. Finally, in the presence of glucose IIAG1'will be abundant and will inhibit the uptake of lactose by the permease. Inducer exclusion is thus modeled as a negated condition, i.e., the absence of HAG'' will be the condition to reach full activity of the permease enzyme.

7.6 Discussion and Conclusions

To reveal and understand regulatory network mechanisms constitutes one of the most significant scientific challenges in the post-genomic era. Many researchers are devoted to uncovering these networks and utilize many different (novel) techniques to enable gene annotation, transcription factor identification, as well as characterization and representation of protein-DNA and protein-protein interactions. Behind these attempts remains a fundamental question of how to combine data from these many different sources for revealing the function of regulatory networks in microorganisms. The intuitive approach for a systems engineer is to generate a model of the system under study. However it is not always clear which modeling methods and what abstractions to apply. Therefore, this chapter first highlighted fundamental modeling problems in describing regulatory networks in microorganisms, and subsequently illustrated the potential of means-end analysis (also called functional modeling) to represent the functionality of complex regulatory networks at different levels of abstraction.

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f PP lac(2Y.A)

permease

sidase

"no Rat IacZYA promotor"

\ R-allolactose

R-IacZYA promotor

I mSSystem

~-

cAMP.CRP wmplex

"IIA'~Is present"

CRP

n

-T

ATF Gluwse

G6P

VII

X

cAMP.CRP

"adenylatecyclase IS presenl"

(k) adenylate cyclase

IX

Figure 7.14 MFM representation of the glucose effects on the lac operon (see also Fig. 7.4).

7. G Discussion and Conclusions

As mentioned above, microbial function is carefully controlled through an intricate network of proteins and other signaling molecules, which was demonstrated in a couple of examples mainly drawn form the lac operon. From a production process perspective, it is definitely an interesting question how systems engineers should couple the detailed description and understanding of the functioning of microorganisms (the microscale) to the higher process level descriptions (the macroscale). The proposed MFM-modeling-based methodology is especially suitable to support this coupling of the microbial regulatory functions and the higher level process and production control functions, since the same set of symbols might be used to represent the flows at the process as well as at the detailed (micro) level. This ability clearly distinguishes the proposed methodology from existing methods to represent regulatory network mechanisms: The network motifs (Lee et al. 2002; Shen-Orr et al. 2002) only represent the connectivity between system states, and do not allow a representation of the connectivity with higher process levels. The elementary modeling objects developed by Kremling et al. (2000) allow the representation of the regulatory networks at a very detailed level, and would probably also be suited to connect the regulatory network with higher process level functions. However, the representation with elementary modeling objects does not contain the degree of information available in the MFM models of regulatory networks, where the actions and means-end relation symbols (see Fig.7.12) provide a high degree of transparency on the way system states interact with each other. Thus, a first conclusion of this chapter is that the proposed representation of regulatory network systems, based on MFM, is ideally suited for supporting systems engineers in detailed model building in bioreactor systems. The applied modeling concept has been demonstrated to enable modeling the changes in qualitative behavior of microorganisms, and is as such able to summarize available process knowledge. If quantitative dynamic models were desired, then these could be developed within each region of qualitative behavior using the logic in the MFM model as a support in the generation of detailed mathematical descriptions. By providing a methodology to represent the regulatory networks at several abstraction levels, this chapter is of relevance to process systems engineering for several reasons. One reason is that microorganisms constitute relatively simple biological systems, and the study and understanding of these relatively simple biological systems may, with suitable extensions, enable better understanding of multicellular biological systems. Furthermore, microbial systems are increasingly used, often following genetic manipulation, to produce relatively complex organic molecules in an energy-efficient manner. Understanding the details of intracellular regulatory networks is a prerequisite for efficient coupling of microbial regulatory functions with higher-level process and production control functions. In other words, the final result of applying process engineering might be improved considerably when process-relevantparts of the intracellular regulatory networks are better understood, and the methodology proposed in this chapter can significantly contribute to represent and subsequently develop that understanding. Finally, and maybe most importantly, applying the MFM modeling method to regulatory networks in microorganisms almost naturally leads to modularizing the network into elemental

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building blocks that are understandable for systems engineers as well as biologists. Thus, the proposed modeling method could contribute substantially by providing a formalism that allows biologists and systems engineers to communicate efficiently about regulatory network functions. To reach a basic level of understanding of the function of these autonomous plants, which is what microorganisms are from an industrial point of view, a systematic description of fundamental regulatory and metabolic functions is proposed in this chapter: The proposed description, which is based on MFM (Lind 1994), might eventually lead to combining the basic understanding of microbial behavior with the semiotics of control. This combination leads to simple schematics for describing fundamental roles of molecules in cells, and their reactions for control and coordination of microbial behavior. In this respect, the flexibility of the MFM modeling formalism is especially noteworthy. In fact, in the lac operon example the lacI gene is expressed constitutively. This means that transcription and translation of the gene to the resulting repressor protein does not necessarily have to be modeled in detail in box I1 of Fig. 7.13. Indeed, since we assume that no regulatory mechanisms are involved in this process, the presence of the repressor protein could have been modeled by only including a “source” for the repressor protein in box 111, thereby omitting box I1 from the model. Thus, MFM models are flexible and can be extended easily. This is, for example, also illustrated by the straightforward extension of the lac operon induction mechanism (Fig.7.13) to also include the glucose effects (Fig.7.14). Clearly, when further building on existing MFM models, the “source” symbols are obvious candidates for extending these models, aimed at including more detail. An MFM representation of the DNA replication process could for example be coupled to the presence of the lacI and lac(Z,Y,A)genes in Figs. 7.13 and 7.14. One could also argue that this chapter has mainly addressed prokaryotic organisms, and that this will limit the applicability of the MFM modeling methodology severely. We claim that application of the proposed methodology to other organisms, for example, eukaryotic unicellular organisms, should be no problem except for obtaining the necessary fundamental knowledge. Again, an example will illustrate this. In eukaryotes, the mRNA might undergo several processing steps before it is transported out of the nucleus, where the ribosomes will finally take care of the translation of the mRNA to a protein. Applying the proposed MFM methodology to such eukaryotic systems, box IV in Fig. 7.13 (describingtranscription and translation of the prokaryotic lac operon) would definitely need several extensions to allow the detailed representation of similar eukaryotic mechanisms. This extension could be obtained by splitting up the prokaryotic version of box IV (Fig.7.13) into several boxes for the eukaryotic case, where each box represents a separate perspective. One box for transcription and its regulation, one box for the subsequent eukaryotic mRNA processing steps, one box for the transport of the processed mRNA out of the nucleus, and finally one box for the translation process. However, it is also evident from the examples in Figs. 7.13 and 7.14 that these extensions can be made in a Straightforward way by using the symbols and conventions provided in Fig. 7.12. Thus, a second conclusion of this chapter is that MFM modeling is highly flexible, allowing systems engineers to easily extend existing models (e.g., by adding flow

7.G Discussion and Conclusions

models related to the means for production of some “sources” in an existing model), and to transfer MFM modeling concepts to other (more complex) biological systems. Finally, it should also be pointed out that the proposed modeling methodology is not only useful in reverse engineering, where it could be applied to represent hypotheses on the operation of complex regulatory network systems. In our opinion MFM models could also be used in forward engineering to design regulatory network building blocks such as the repressilator (Elowitz and Leibler 2000) before developing a detailed mathematical description References 1 Achinstein P. (1983) The nature of explana-

tion. O x f r d University Press, Oxford, UK. 2 Alur R. et al. (2002) Modeling and analyzing biomolecular networks. Comput. Sci. Eng. 4 ()an/Feb),20-31. 3 Astltagiri A. R., Laufenburger D. A. (2000) Bioengineering models of cell signaling. Ann. Rev. Biomed. Eng. 2. 31-53. 4 Baneyx F. (1999) Recombinant protein expression in Escherichia coli. Curr. Opin. Biotechnol. 10,411-421. 5 Bratman M. E. (1987) Intention, plans and practical reason. Harvard University Press, Cambridge, MA. 6 Cltassagnole C., Noisommit-Rizzi N., Schmid J . W., Mauch K., Reuss M . (2002) Dynamic modeling of the central carbon metabolism of Escherichia coli. Biotechnol. Bioeng. 79, 53-73. 7 Cheetham P. S. ]. (2004) Bioprocesses for the manufacture of ingredients for foods and cosmetics. Adv. Biochem. Engin. Biotechnol. 86, 83-158. 8 Csete M. E., Doyle J . C. (2002) Reverse engineering of biological complexity. Science 295, 1664- 1669. 9 deJong H . (2002) Modeling and simulation of genetic regulatory systems: a literature review. J. Comp. Bid. 9, 67-103. 10 Doyle F.]. I l l (2004) A systems approach to modeling and analyzing biological regulation. Proceedings of the International Symposium on Advanced Control of Chemical Processes (ADCHEM2003). 11-14 January 2004, Hong Kong. 11 Downward /. (2001)The ins and outs of signalling. Nature 411, 759-762. 12 Eisenberg D., Marcotte E. M., Xenarios l., Yeates T. 0.(2000) Protein function in the postgenomic era. Nature 405, 823-826. 13 Elowitz M. B., Leibler S. (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403, 335-338.

14 Ferber D. (2004) Microbes made to order. Sci-

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15 Finney A,, Hucka M . (2003) Systems biology

markup language (SBML) level 2: structures and facilities for model definitions. http: www. sbml. org/. 16 G o f i a n E. (1974) Frame analysis. Penguin Books, London. 17 Gofuku A., Lind M . (1994) Combining multilevel flow modeling and hybrid phenomena theory for efficient design of engineering systems. Proc. Znd IFAC Workshop on Computer Software Structures Integrating AI/KBS Systems in Process Control. 18 Habemas J. (1989) On the logic of the social sciences. M I T Press, Cambridge, MA. 19 Hasty ]., McMillen D.,Isaacs F., Collins]. ]. (2001) Computational studies of gene regulatory networks: In numero molecular biology. Nature Rev. Gen. 2, 268-279. 20 Hucka M., Finney A,, Sauro H. M., Bolouri H. et al. (2003)The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 19, 524-531. 21 Ideker T., Lauffenburger D. (2003) Building with a scaffold: emerging strategies for highto low-level cellular modelling. Trends Biotechnol. 21, 255-262. 22 Jacob F., Monod]. (1961) Genetic regulatory mechanisms in the synthesis of proteins. J. Molec. Bid. 3, 318-356. 23 Kitano H. (2002) Systems biology: a brief overview. Science 295, 1662-1664. 24 Kremling A., Bettenbrock K., Laube B., Jahreis K., Lengeler]. W., Gilles E. D. (2001)The organization of metabolic reaction networks. 111. Application for diauxic growth on glucose and lactose. Metabol. Eng. 3, 362-379. 25 Kremling A , ,jahreis K., Lengefer]. W., Gilles E. D. (2000) The organization of metabolic reaction networks: a signal-oriented appro-

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7 Towards Undentanding the Role and Function ofRegulatory Networks in Microorganisms ach to cellular models. Mekzbol. Eng. 2, 190-200. 26 Lanen M. N. (1993) Deriving action sequences for start-up using multilevel flow models. PhD Thesis, Department of Automation, Technical University of Denmark. 27 Larsson J. E. (1996) Diagnosis based on explicit means-end models. Art$ Intell. 80, 29-93. 28 Lee T. I. et al. (2002)Transcriptional regulatory networks in Saccharomyces cerevisiae. Science 298, 799-804. 29 Lengeler]. W., Drews G., Schlegel H. G. (1999) Biology of the prokaryotes. Thieme Verlag, Stuttgart, Germany/Blackwell Science, Oxford, UK. 30 Lind M. (1994) Modeling goals and functions of complex industrial plant. Appl. Art$ Intell. 8, 259-283. 31 L i d M. (1999) Plant modeling for human supenisory control. Trans. Inst. Measure. Control 21, 171-180. 32 Lind M. (2002) Promoting and opposing. NKS-R-07 Project Report, Orsted DTU, Technical University of Denmark, Kongens Lyngby, Denmark. 33 Lind M . (2004a) Description of composite actions-towards a formalization of safety functions. NKS-R-07 Project Report, Orsted DTU, Technical University of Denmark, Kongens Lyngby, Denmark. 34 Lind M. (2004b) Means and ends of control. Proc. IEEE Conf Systems Man and Cybernetics, 10-13 October 2004, The Hague, Holland. 35 LyonsJ. (1977) Semantics 2. Cambridge University Press, Cambridge, MA. 36 Mackey M . C., Santillltn M., Yildirirn N. (2004) Modeling operon dynamics: the tryptophan and lactose operons as paradigms. C. R. Biolo. gies 327,211-224. 37 Makrides S. C. (1996) Strategies for achieving high-level expression of genes in Escherichia coli. Microbiol. Rev. 60, 512-538. 38 Mangan S., Alon U. (2003) Structure and function of the feed-forward loop network motif. Proc. Natl. Acad. Sci. 100, 11980-11985. 39 Mangan S., Zaslaver A,, Alon U.(2003) The coherent feedforward loop serves as a signsensitive delay element in transcription networks. J. Mol. Biol. 334, 197-204. 40 Milo R., Shen-Ov S., Itzkovitz S., Kashtan N., Shklovskii D.,Alon U. (2002) Network motifs: simple building blocks of complex networks. Science 298, 824-827. 41 Postma P. W., LengelerJ. W., Jacobson G. R. (1993) Phosphoeno1pyruvate:carbohydrate phosphotransferase systems of bacteria. Microbiol. Rev. 53, 543-594.

42 Ronen M., Rosenberg R., Shraiman B. I., Alon U. (2002)Assigning numbers to the arrows:

parameterizing a gene regulation network by using accurate expression kinetics. Proc. Natl. Acad. Sci. 99, 10555-10560. 43 Salgado H. et al. (2004) Regulon DB (version 4. 0): transcriptional regulation, operon organization and growth conditions in Escherichia coli K-12. Nucleic Acids Res. 32, D303-D306. 44 Santilldn M., Mackey M. C. (2004) Influence of catabolite repression and inducer exclusion on the bistable behavior of the lac operon. Bi~phys.J. 86, 1282-1292. 45 Shen-Ov S. S., Milo R., Mangan S., Alon U. (2002) Network motifs in the transcriptional regulation network of Escherichia coli. Nature Genetics 31, 64-68, 46 Simon H. A. (1981) The sciences of the artificial. MIT Press, Cambridge, MA. 47 Smolen P., Baxter D. A,, Byrne]. H. (2000) Modeling transcriptional control in gene networks-methods, recent results, and future directions. Bull. Math. Biol. 62, 247-292. 48 Stelling H., Klamt S., Bettenbrock K., Schuster S., Gilles E. D. (2002) Metabolic network structure determines key aspects of functionality and regulation. Nature 420, 190-193. 49 Von Wright G. H. (1963) Norm and action-a logical enquiry. Routledge and Kegan Paul, London. 50 Vukmirovic 0. G., Tilghman S. M. (2000) Exploring genome space. Nature 405,820-822. 51 Wei G.-H., Liu D.-P., Liang C.-C. (2004) Charting gene regulatory networks: strategies, challenges and perspectives. Biochem. J. 381, 1-12. 52 Wolkenhauer O., Kitano H., Cho K. H. (2003) Systems biology. IEEE Control Syst. Mag. 23, 38-48. 53 Wong P., Gladney S., KeaslingJ. D. (1997) Mathematical model of the lac operon: inducer exclusion, catabolite repression, and diauxic growth on glucose and lactose. Biotechnol. Prog. 13, 132-143. 54 Yeger-Lotem E. et al. (2004) Network motifs in integrated cellular networks of transcriptionregulation and protein-protein interaction. Proc. Natl. Acad. Sci. 101, 5934-5939. 55 Yeger-Lotem E., Margalit H. (2003) Detection of regulatory circuits by integrating the cellular networks of protein-protein interactions and transcription regulation. Nucleic Acids Res. 31, 6053-6061. 56 Yildirim N., Mackey M. C. (2003) Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data. Biophys. J. 84, 2841-2851.

Section 2 Computer-aided Process and Product Design

Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

Section 2 presents a state-of-the-art review on methods and computer tools currently available to support engineering design activities. The material i n this section is organized infive chapters covering the use of models is the development and improvement of processes and products. This activity relies on tools that integrate knowledgef r o m many disciplines, since it has to take care of all the constraints of process and product design, such as equipment and plantflexibility and operability, raw materials and energy usage, economy, health and safty. Feedstock and product purification are among the critical components o f a chemical process. This is why Chapter 1 addresses the synthesis of separation systems, with emphasis on distillation, one of the most energy-intensive unit operations in the chemical process industry. Besides the optimization of a single column, improved sequencing i n the case of multiple separations, heat integration and thermal coupling offer perspectives for significant energy savings and are reviewed. All potential solutions of the synthesis problem are considered by generating a superstructure to be optimized using mathematical programming techniques. Chapter 2 covers process intensification. The design of more escient and compact processing equipment, usually combining severalfunctions, has long been realized by drawing on intuition and expertise. Now systematic design procedures based on modeling thefundamental principles underlying the process intensification technologies are being developed. Significant achievements are reported in the design of single and multiphase reactors, of reaction-separation systems and of hybrid separation processes by using computer-aided methodsfor process intensification. The performance of a process is not only related to the proper design of the main equipment. All processes require utilities: water, solvents, waste treatment, and most of all, energy. Chapter 3 presents computer-aided methods for solving the optimal integration of processes with utility networks; it compares several formulations proposed to optimize the integration of different types of utility subsystems (e.g., combined heat and power, heat pumps and refigeration, water circuit). CAPE tools now allow the development, evaluation and optimization of new units and processes. Chapter 4 documentsfive industrial case studies that illustrate how a combination of rigorous models has been used to produce innovative designs meeting multiobjective targets besides economy: environment conservation, safty, operationalflexibility, controllability. They required the use ofcommercial tools but were enhanced with detailed models of the units that were not routinely available. An eficient design relies on accurate bench and pilot plant data combined with rigorous models based on thermodynamics, conservation laws, and accurate models of transport and fluid flow, with particular emphasis on dynamic behavior and uncertainty in market conditions.

The process industries undergo a movefrom the process-oriented to the product-centered businesses. A consequence of this switch is a growing interest in product development. Chapter 5 addresses this issue, with special emphasis on the product definition phase: how to translate customers’ needs into product properties and how to foster the generation of novel product concepts. Applications of computer-aided techniques, such as data mining or case-based reasoning, are also illustrated by practical examples.

Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

1 Synthesis of Separation Processes Petros Proios, Michael C. Georgiadis, and Efstratios N. Pistikopoulos

1.1 Introduction

Process synthesis has received considerable attention over the last 30 years and it continues to be an active research area in chemical engineering, with significant advances achieved in terms of developing synthesis methods for subsystems (reactor networks, separation systems, heat exchange networks, and mass exchange networks) and for total flow sheets. By definition, process synthesis is the determination of the topology of process units, as well as the type and design of the units within the flow sheet that will convert specified inputs into desired products. The synthesis task is usually driven by the optimization of a specific objective, typically based on economics. Systematic techniques for the synthesis and design of flow sheets are represented by two different approaches: hierarchical decomposition and mathematical programming. Reviews on early developments can be found in Hendry et al. (1973), Westerberg (1980),and Nishida et al. (1981).An overview of synthesis techniques is also available in the book by Biegler et al. (1997) and recent advances are presented in the excellent review article by Westerberg (2004). Early work on process synthesis appeared on developing insights into finding better separation sequences (using distillation) for separating mixtures of n components. The emphasis on investigating techniques for the synthesis of complex distillation networks can be mainly attributed to the role that distillation plays in the economy of the overall plant. Distillation is a highly utilized and at the same time one of the most energy intensive unit operations in the chemical process industry. Mix indicated in 1978 that distillation accounted for about 3% of the total US energy consumption and that a 10% savings in distillation energy could amount to a savings of about $500 million in national energy costs (Mixet al. 1978). Today, the expense of distillation-relatedenergy consumption has reached even higher levels, considering the expansion of the use of distillation in industry and the higher cost of utilities. These economic reasons have imposed energy efficiency as the main design target in distillation. Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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Due to its importance, distillation has received particular attention in the field of chemical engineering, with publications about the operation and design of countercurrent separation cascades dating as early as 1889 (Sore1 1889). Moreover, since multicomponent separations require the use of sequences of distillation columns, significant research efforts have concentrated on the synthesis of these systems aiming at energy efficiency. Research on this subject has been further powered by the fact that extensive energy savings can be achieved through the selection of the most energy efficient sequence, amongst a large number of available candidates. This is an explicit consequence of the dependence of the distillation systems’ energy consumption on the feed mixture and on the order in which its components are separated. Technological breakthroughs are constantly called in to propose new techniques for energy efficiency that would compensate for the ever increasing energy-related distillation expenses. Two of the most promising techniques are heat integration and the thermal coupling of distillation columns. The former is based on the energy savings that can be achieved by heat integrating two distillation columns, that is, by using the heat generated in a column’s condenser for the heat required in another column’s reboiler, while satisfylng appropriate temperature difference conditions. This technique can lead to substantial energy savings that can reach the order of 50% when compared to nonheat-integrated arrangements. Similar energy savings have been reported through the use of thermal coupling techniques in distillation, where heat units and their associated utilities are eliminated through the use of two-way liquid and vapor interconnections between columns, the latter being characterized as complex columns. These energy savings are the direct result of the elimination of heat units and the increase of thermodynamic efficiency, due to the minimization of remixing effects, which are generally associated with nonthermally-coupledarrangements. However, in order to apply the aforementioned synthesis techniques for energy efficiency, certain complicating issues need to be addressed, which are mainly of structural and physical nature. The structural complications are related to the large number of alternative arrangements that need to be considered. Even in the simplest case from a structural perspective, where sequences of simple columns are examined (columns with a single feed and two products), the extensive connectivity possibilities between columns lead to the generation of a large number of alternative column sequences, which increase with the number of components to be separated. Moreover, these structural complications become even more intense through the incorporation of structural possibilities associated with heat integration and thermal coupling. The physical complications are related to the complexity of the underlying physical phenomena, which involve simultaneous mass and heat exchange between liquid and vapor streams at the tray cascades. Furthermore, the physics of the problem are such that the choice of the optimal configuration is largely dependent on the feed mixture to be separated (its components’ relative volatilities and composition). It has been reported (Tedder and Rudd 1978; Agrawal and Fidkowski 1998) that for a particular separation, column configurations that are generally regarded as highly energy efficient (for instance, fully thermally coupled columns), can, in fact, have

1.7 Introduction

larger energy consumptions than sequences of more conventional columns. Consequently, in order to evaluate efficiently the energy consumption of a particular column sequence and the energy savings that can potentially be achieved through its heat integration or thermal coupling. The aforementioned physical phenomena need to be accurately captured. Summarizing, it is not an exaggeration to state that the economic importance and associated complications have made the distillation column sequencing problem for energy efficiency one of the most challenging synthesis problems in chemical engineering, with numerous approaches proposed for its solution. One of the earliest attempts was based on total enumeration. This approach is, however, limited to problems with only a few alternatives. Other main approaches are the heuristic and physical insight ones. The former relies on rules of thumb derived by engineering knowledge and/or by the use of shortcut models, while the latter is based on the exploitation of basic physical principles, which are also based, to a certain extent, on simplified models and on graphical representations of the problem. These approaches generally enable quick and inexpensive calculations for the alternatives’physical evaluation. However, the fact that they are derived based on simplifylng assumptions, which are valid only for certain cases, places a major limitation on their accuracy, validity, and applicability. Furthermore, more complications arise when the developed heuristics are conflicting, suggest more than one possible solution, or do not cover the details of the examined problem. The most recent approach addressing this problem is the mathematical programming (algorithmic) approach, where the synthesis of column sequences is formulated as an optimization problem. Based on mathematical programming, one of the most important systematic approaches that has been receiving increased attention over the last years is superstructure optimization. Superstructures are, in general, superset flow sheets incorporating every feasible realization of the process in question. The generation and evaluation of each alternative realization takes place with the solution of an optimization problem, which usually involves the use of continuous and binary (0-1) variables, rendering the problem a mixed-integer programming (MIP) problem. However, most of these methods either use sirnplifylng assumptions, limiting the validity and accuracy of the results, or treat the problem rigorously, but at an expense of computational effort. Section 1.2 will provide an overview of techniques for the synthesis of simple column sequences. The subsequent section will describe in a comprehensive way the synthesis problem of heat-integrated distillation trains. State-of-the-artmethodologies and algorithmic frameworks for the synthesis of complex distillation sequencing are critically discussed in Section 1.4.Finally, concluding remarks will be made in Section 1.5.

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1.2 Synthesis of Simple Distillation Column Sequences 1.2.1 Simple Distillation Column Sequencing

As already mentioned, in order to separate multicomponent mixtures into pure or

multicomponent product streams using distillation, more than one column needs to be employed, generating sequences of distillation columns. From chemical engineering knowledge it is known that for the same mixture separation, different distillation column sequences have different energy consumption levels, which can, in fact, be quite different from each other. Since distillation is an energy-intensive process widely used in the chemical industry, there is a substantial economic incentive in selecting the appropriate distillation column sequence for a particular separation. However, as already mentioned, two main complications make the distillation column sequencing problem one of the most challenging synthesis problems in chemical engineering, namely the increasing number of structural alternatives and the complexity of the underlying physical phenomena. From a structural point of view, developing a method that could incorporate all the alternatives of interest, without simplifying assumptions, such as sharp splits or product streams enriched in a particular component produced once in a column sequence, is not a trivial task after the first member of multicomponent separations. For the latter only three possible simple column sequences exist (Fig. 1.1). However, an illustration of the difficulty of the problem is realized when the next problem is considered, namely the quaternary separation. For this problem, if sharp split assumptions are used, only five different structures are possible (Thompson and King 1972). However, if the simplifying assumptions are removed then 22 possible alternative sequences have been identified (as will be shown later). It must be noted that these alternatives may include more sections than the minimum number required, however, this is desirable since when general design targets are optimized, such as the total annualized cost (TAC), these structures may potentially exhibit an optimal tradeoff between operating and capital cost in their additional sections. Therefore, a proposed sequencing method must also be able to incorporate and generate the structural alternatives systematically and automatically. From a physical representation point of view, the proposed method must be accompanied by a physical model that can capture the underlying phenomena accurately. Due to the complexity of the physical phenomena taking place within each distillation column, rigorous physical models are required for an accurate physical representation. However, the incorporation of rigorous models leads, by default, to the generation of nonconvex mathematical problems whose solution is quite involved and usually computationally expensive. In order to obtain easy and fast solutions, numerous methods have been proposed approximating the physical behavior of distillation columns with models based on a number of simplifying assumptions, thus trading accuracy and applicability for ease of calculation.

1.2 Synthesis ofsimple Distillation Column Sequences Figure 1.1 Ternary simple column sequencing alternatives.

(a) Direct Sequence

(b) Indirect Sequence

(c) _I

Finally, due to the aforementioned complexities, an efficient sequencing method must also be accompanied by an appropriate solution procedure. For the case of three or five structural alternatives, an implicit enumeration solution procedure is acceptable. However, even for the quaternary case including nonsharp separations, implicit enumeration is not the most efficient solution procedure, especially if rigorous models are used, due to the computational effort required. Due to the aforementioned importance and associated difficulties of the simple column sequencing problem, a number of methods have been proposed over the years for its solution, based on different approaches for process synthesis. These approaches involved heuristic methods (Seader and Westerberg 1977), evolutionary techniques (Stephanopoulos and Westerberg 1976), hierarchical (Douglas 1988), implicit enumeration (Fraga and McKinon 1995) and thermodynamic insights (BekPedersen 2003; Bek-Pedersen and Gani 2004). to name but a few. However, one approach that has received a lot of attention recently, is the superstructure optimization approach. In this approach a superstructure of the problem is generated that, according to Westerberg (Andrecovich and Westerberg 1985a, b), “Should contain all feasible distillation sequences and all feasible operating conditions for any column within the superstructure. ” Due to the importance of the superstructure optimization approach, particular attention will be given to the various superstructure optimization methods proposed for the synthesis of simple distillation column sequences.

1.2.2 Superstructure Methods for Simple Column Sequencing

An early systematic superstructure method for the simple column sequencing problem was proposed by Andrecovich and Westerberg (1985a,b).The proposed method addressed the separation problem of a multicomponent feed into pure product streams, generating a superstructure according to the separation tasks taking place in each column and to the column connectivities. Using the assumption of sharp splits in each distillation column, the generation of the superstructure was based on

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the list splitting technique of Hendry and Hughes (1972),ranking components in decreasing order of relative volatilities. The authors proposed two algorithms for generating systematicallythe column sequences. The synthesis problem was formulated as an MILP, with binary variables introduced for the presence of columns and with simple models based on total mass balances and feed split fractions. This work was further extended in the work of Wehe and Westerberg (1987),for the incorporation of bypass possibilities, addressing the more general problem, namely of distillation separation of multicomponent feed streams into multicomponent product streams. Simple sharp split column models were employed, which were generally linear, except for those of the superstructure splitters, where bilinearities were introduced (due to the absence of information about the composition of the splitter feed and the actual values of the splits obtained). In order to overcome this complication, the resulting nonlinear problem for each selected structure was relaxed through linearizations, generating a lower bound to the solution. For the sequence with the best lower bound, the nonlinear problem was solved to obtain an upper bound on the global optimum. Optimal column sequences were considered those that had similar upper and lower bounds, while the ones with lower bounds above the best upper bound were discarded. The general separation problem using superstructure optimization was also addressed by Floudas (1987). A superstructure method was developed for the sequencing of separators, without explicitly specifying the actual type of separators used. Distillation columns were also included as potential separator types. The objective was the minimization of a function related to the separation difficulty,which was calculated based on relative volatilities, for assuming distillation as the separation task. The method generated the superstructures based on the assumption of sharp (perfect) splits obtained in each separator. The number of separators in the superstructure was assumed fixed, thus eliminating the need for the introduction of binary variables. The optimal column sequences were derived by the selection of the column interconnections. In this method each separator was modeled using simple models of total and component mass balances, resulting in an NLP problem. Floudas and Anastasiadis (1988)addressed explicitly the synthesis of simple distillation column sequences for the general multicomponent feed and product problem. Their method, as the one of Floudas (1987), generated a superstructure containing series and/or parallel arrangements of distillation columns, along with stream splitting, mixing and bypassing. In order to construct the superstructure, systematic stream mixing and splitting rules were provided, based on the assumption of sharp split separations performed in each distillation column. Simple distillation column physical models were employed based on assumptions of isothermal and isobaric columns with equal condenser and reboiler heat duties and general total and component mass balances, where the component compositions did not participate as variables but as parameters derived a priori over a number of shortcut simulations. The objective to be minimized was the system’s total annualized cost (TAC), which was modeled as a function of the column feed flow rate. This was derived based on a linear approximationof the six-tenths factor rule used to scale equipment cost. The necessary coefficients for the cost functions were also based on simulations of shortcut

7.2 Synthesis ofSimple Distillation Column Sequences

distillation models at different feed flow rates. The problem was posed as an MILP with binary variables denoting the existence or not of distillation columns, to be determined by the optimization along with their continuous interconnections, thus generating the optimal sequence. In Aggarwal and Floudas (1990) the simple distillation column sequencing problem for general separations was revisited, relaxing the assumption of sharp separators. The latter were modeled based on the assumption of nondistributed nonkey components. The construction of the superstructures, and therefore the generation of the alternatives, was based on the assumption that in each column exist two adjacent (withrespect to their volatility) key components, which are allowed to distribute between the column distillate and bottoms (whose recoveries were considered optimization variables).All other nonkey components were only allowed to appear in the distillate or bottoms. Based on the fact that each column performs a separation of adjacent key components, a maximum of n-1 columns was set up for an n component feed. Binary variables were introduced for the presence of each potential distillation column. In the structural model of the method, possibilities of mixing, splitting and bypassing were explicitly incorporated. The objective function for the optimization problem was the minimization of the system's TAC. Cost models were generated through regression analysis, based on the solution of numerous shortcut models over a range of flow rates, compositions and key component recoveries, and the problem was modeled as an MINLP. Finally, the method was extended to incorporate cases of nonadjacent key components, with intermediate components allowed to distribute between the column product streams. The simple column sequencing problem of single feed to pure product streams was addressed in Novak et al. (1996)based on the generation of a network and a compact superstructure, which were viewed as combinations of interconnection nodes (mixers and splitters) and process units (distillation columns). The former were approximated using special linear constraints and the latter were based on the assumption of sharp splits. The compact superstructure was viewed as a simplification of the tree superstructure of Hendry and Hughes (1972) and of the network superstructure of Andrecovich and Westerberg (1985a,b) for sharp separations and was a modification of the superstructure provided by Floudas and Anastasiadis (1988),using less distillation columns. In the compact superstrucmre the number of columns was assumed fmed and binary variables were assigned only to the stream connectors. For an n component feed, the number of distinct sharp distillation sequences S (for all superstructure types employed) was equal to (Thompson and King 1972): S=

S ( n - l)! n!(n- l)!

The number of columns in the network and compact superstructures S" and 9, respectively, was found equal to:

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The distillation columns were modeled based on the Gilliland/Fenske/Undenvood method using assumptions of uniform relative volatilities per column and specifylng minimum recoveries of light and heavy key components. The objective to be minimized was set as the system’s TAC, employing the futed charge model of Grossmann (1985)for the capital cost calculations. The structural (pure integer) constraints were based on the topology of the superstructure. The generated MINLP problem was solved using the modified OA/ER algorithm (Kravanja and Grossmann 1994),implemented in the computerized synthesizer PROSYN-MINLP (Kravanja and Grossmann 1993). Smith (1996) addressed the simple column sequencing problem generating a superstructure for the structural alternatives using the state operator network (SON) representation. In this representation the separation tasks and equipment were defined while the assignment of tasks to the equipment had to be determined. Stream mixers and splitters were included before and after each distillation column to arrange all possible interconnections. The distillation columns were modeled using a slightly modified version of the rigorous tray-by-trayMESH distillation column model of Viswanathan and Grossmann (1993).The method did not employ any simplifying assumptions such as sharp splits or equimolar flow rates. Binary variables were used in order to determine the number of trays within each column (reflux tray location). In the SON method the number of columns was rationally fuced according to the number of components in the feed. The feed tray location and the column interconnections, defining the column sequences, were continuous decisions, determined by their flow rate values. The generated MINLP problems were solved in GAMS using the DICOPT MINLP solver (Kocis and Grossmann 1989).The solution procedure included the solution of the relaxed MINLP problem and then a branch-and-bound search over the nonzero relaxed binary variables. This procedure was followed since, as reported, the full MINLP problem failed to converge in DICOPT, due to scaling and domain errors in the set-up of the NLP subproblems. To overcome the implicit assignment of processing tasks to processing units, Papalexandri and Pistikopoulos (1996) developed a general synthesis framework based on a mass and heat transfer representation. Utilizing fundamental masslheat transfer principles within a superstructure environment, this building block synthesis method exploits modeling concepts from the well defined heat exchanger network (HEN)and mass exchange network (MEN) problem. Synthesis alternatives are not prepostulated, but instead embedded within a network of mass and heat exchanging modules that allow nonconventional hybrid systems to be revealed. In terms of total process flow sheet alternatives, the potential of this approach has been illustrated through application to ideal systems including separation, reaction, and heat integration operations. The approach has also been applied to the synthesis of flexible heat and mass exchange networks (Papalexandri and Pistikopoulos 1994). Ismail et al. (1999)modified and extended the masslheat transfer module to general multicomponent nonideal systems and reactive distillation systems (Ismail et al. 2001). Yeomans and Grossmann (1999a) proposed a systematic modeling framework based on superstructure optimization in which the sequencing of simple distillation

1.2 Synthesis ofSimple Distillation Column Sequences

columns was also addressed. The superstructures were obtained using the state task network (Sargent 1998) and the state equipment (operator) network representations (Smith 1996; Smith and Pantelides 1995).These two representations can be regarded as complimentary to each other. In the STN, tasks and states are defined while the equipment assignment is unknown, while in the SEN, as indicated above, the tasks and equipment are defined and the assignment of tasks to equipment must be determined. The synthesis problem was modeled with the generalized disjunctive programming (GDP) (Raman and Grossmann 1994). In order to use GDP, conditional constraints had to be identified from permanent ones, that is, constraints that held in all structural alternatives. Having identified the conditional constraints, these were then represented with disjunctions and assigned to Boolean (logical)variables representing their existence. For a Boolean variable of value equal to true, the conditional constraints (corresponding to a column physical model) became activated, otherwise all variables participating in these constraints were set to zero. The GDP models were then transformed to MILP problems using the convex hull formulation of the disjunctions (Balas 1985). The generation of the superstructures for the examined problem was based on sharp separations performed in the distillation columns, which were modeled using simple mass balances and recoveries, generating linear physical models. The above modeling framework was also implemented in Yeomans and Grossmann (1999b) for the synthesis of simple distillation column sequences employing nonlinear shortcut distillation column physical models. The latter were based on assumptions of sharp splits, where only the distribution of adjacent key components was allowed along with high recovery of the key components. The Fenske/Undenvood/Gilliland method was used for the calculation of the minimum number of trays and of the minimum reflux. The simple column sequencing problem using GDP was also addressed in Caballero and Grossmann (1999).The column superstructures were generated using the STN, SON and an intermediate representation. The latter was proposed based on the STN representation, however, allowing the columns the possibility of performing multiple tasks. The distillation column models were aggregated using mass transfer feasibility constraints at the section boundaries. These models bore many similarities with the ones used for the representation of simple distillation columns in the generalized modular framework (Papalexandri and Pistikopoulos 1996). However, the two models have many structural and physical modeling differences. Structurally, the building blocks of the superstructure are different. In Caballero and Grossmann (1999) each building block consisted of a set of two aggregated column sections with their heat units and a feed position mixer (with predefined interconnections between them). In the GMF each Mass/Heat (M/H) module and each Heat Exchange (He) module constitutes different building blocks. Moreover, as shown in Papalexandri and Pistikopoulos (1996),the GMF building blocks by definition have more extensive interconnection possibilities. From a physical modeling point of view, in Caballero and Grossmann (1999) simplift.lng assumptions of equimolar flow rates, isothermal operation and sharp splits were employed. Moreover, in order to provide valid mass transfer feasibility constraints, the mass transfer direction had to be known and postulated, which depended on the separation task taking place in

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a particular column section. The GMF mass transfer is arranged through driving force constraints, which are formulated in such a way that the mass transfer direction does not need to be prepostulated or known a priori. However, both models provide a lower bound on the energy consumption of distillation columns. In the proposed method the generated GDP problems were transformed into MINLP problems by assigning a binary variable to each Boolean variable and by transforming the disjunctions into big-M constraints. The most recent superstructure method for the simple column sequencing problem was proposed in Yeomans and Grossmann (2000a),using rigorous MESH models for the physical representation of distillation columns. The superstructure for the problem was generated using the SON representation, as implemented in Yeomans and Grossmann (1999a).The computational difficulties of the rigorous MESH models, associated with equations becoming singular, the solution of redundant equations, the need for good initialization procedures and the convergence difficulties due to nonexisting columns or flows were overcome using a GDP adaptation of the Viswanathan and Grossmann (1993)distillation column model. In the GDP model the feed, the condenser and reboiler stages were modeled as permanent and all the other stages as conditional, assigned to a Boolean variable. The disjunctions were modeled so that for existing stages (whose Boolean variables had a value of true), phase equilibrium constraints were enforced along with other MESH constraints, thus allowing mass and heat exchange between the contacting phases. For the opposite case, of nonexisting stages, the MESH constraints were applied without the phase equilibrium constraints and the input stream properties of the stage were set equal to those of the output stream, thus making the nonexisting stage a simple input-output stage. The translation of the GDP problems into MINLP problems was also done by transforming the disjunctions into big-M constraints. Moreover, appropriate initialization schemes were provided based on relaxed purity and recovery constraints. Table 1.1 presents an overview of superstructure methods for simple column sequencing. The synthesis of azeotropic separation processes has also received significant attention in the literature. Table 1.2 summarizes most of the developed approaches for the analysis, design, and synthesis of homogeneous azeotropic separation systems. Recently, Proios (2004) addressed the simple distillation column sequencing problem through the GMF synthesis model. From a structural point of view, a GMF structural model has been developed that can generate the feasible structural alternatives for the examined problems, aiming at addressing efficiently the first complication of the column sequencing problem. In the proposed GMF sequencing method the accompanying physical model is based on the GMF physical model originally introduced by Papalexandri and Pistikopoulos (1996) and extended by Ismail et al. (1999, 2001). This model, which is based on aggregation, can be viewed as an intermediate between the simplified and the rigorous models. On one hand it avoids potentially limiting simplifylng assumptions, such as equimolar flow rates, isothermal operation, sharp splits, etc., thus increasing the accuracy of the physical representation. On the other hand, due to aggregation it generates problems of smaller

Authors

Features

Simplified methods (sharp splits)

Andrecovich and Westerberg (1985)

I

Network superstructure, simple mass balances ~~

Wehe and Westerberg (1987) Floudas (1987)

I

General separation, nondistr/nonkey, mass balances General separation, mass balances

Floudas and Anastasiadis (1988)

General separation, mass balances

Agganval and Floudas (1990)

General separation, nondistr/nonkey. mass balances

Novak et al. (1996)

Network & compact superstructures, mass balances

Yeomans and Grossmann (1999a)

STN & SON superstructures, GDP, mass balances

Yeomans and Grossmann (1999b)

STN & SON superstructures, GDP, nonlinear shortcut models

Caballero and Grossmann (1999)

STN, SON & intermediate superstructures. aggregated models

Rigorous methods ( M E S H model, Vinuanathan and Grossmann 11993)) Smith (1996)

SON superstructure, MESH modified model

Yeomans and Grossmann (2000)

SON superstructure, GDP

size, which are easier to solve. Finally, the GMF was accompanied by a solution procedure, using formal MINLP solution techniques that find the optimal sequence without having to evaluate all structural alternatives. The overall synthesis framework and solution approach was demonstrated in several case studies (Proios 2004).

1.3 Synthesis of Heat-integrated Distillation Column Sequences 1.3.1 Heat-integrated Distillation Column Sequencing

As shown in the previous chapter, there is a substantial economic incentive for the development of methods for the simple distillation column sequencing problem, since significant energy savings can be achieved by finding the most energy efficient column sequence for a particular separation. However, even higher energy savings

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can be achieved in distillation sequences by incorporating techniques such as heat integration (HI) and thermal coupling. In this chapter, the former technique is employed, while the latter is examined in the following chapter. In general, distillation heat integration makes use of the heat generated in a column’s condenser in order to provide the heat required in another column’s reboiler (an illustration for the direct column sequence is shown in Fig. 1.2). The generated heat in the column’s condenser is produced by the condensation of the vapor effluent of the column’s top end, through heat exchange with a cold utility (coolingwater) due to a heat gradient. On the contrary, in a column’s reboiler, heat is required for the vaporization of the liquid effluent of a column’s bottom end, through heat exchange with a hot utility (steam). Structurally, two process streams exist: one vapor (hot) from a column’s top end that needs to be cooled and a liquid (cold)from a column’s bottom end that needs to be heated. Provided there is a sufficient heat gradient between those two streams the latter can be made to exchange heat with each other. In order to achieve the necessary heat gradients the column pressures are shifted appropriately, since the column pressure has a direct effect on the column temperature levels (for nonazeotropic and nonreacting mixtures, raising the column pressure raises the column temperature). The operating cost savings achieved through HI have been reported to reach 50% as compared with non-HI column sequences (Hostrup etal. 2001). The obtained energy savings compensate for a potential increase in the capital expenditure due to the use of pumps, which is usually considered negligible when compared to the overall column investment cost (Novak et al. 1996). Due to its economic importance, a large number of methods have been proposed for the synthesis of HI column sequences. Some characteristic methods proposed over the years included the dynamic programming algorithm of Rathore et al. (1974)

(a) Forward HI Match Figure 1.2

(b) Backward HI Match

Illustration of a ternary direct column sequence

Entrainer selection

Doherty and Caldarola (1985)

RCM, total reflux, ternary mixtures. Assumes linear distillation boundaries and no boundary crossing.

Stichlmair J., Fair, J. R., Bravo J. L. Chemical Engineering progress 85 (1989), p. 63

DLM, total reflux, linear distillation boundaries.

Foucher et al. (1991)

RCM, total reflux, ternary. Automatic procedure.

Laroche L., Bekiaris N., Andersen H. W., Morari M. AICHE Journal, 38 (1992), p. 1309

RCM, ternary. Analysis based on “equivolatilitycurves.”

~~

Column sequencing & bounding strategies

Laroche et al. (19926)

I RCM, all reflw, ternary. Separabilityin single feed columns.

WahnschafR et al. (1992)

RCM, all reflw, ternary mixtures, single feed columns. Bounds by feed pinch point trajectories.

Fidkowski et al. (1993)

DLM, all reflw. ternary, single feed. Introduces “distillation limit,” accounts for boundary crossing, algebraic-based.

Stichlmair and Herguijuela (1992)

DLM, all reflux, ternary, single feed. Accounts for curved distillation boundaries.

Jobson et al. (1995)

Achieves attainable product region based on simple distillation and mixing for ternary mixtures.

Safrit and Westerberg (1997)

RCM. Algorithm determining boundaries & distillation regions for n-component systems.

Rooks et al. (1998)

RCM. Equation-based approach to determine distillation region structures of multicomponent mixtures.

Boundary value design procedure Levy et al. (1985) Knight and Doherty (1986) Julka and Doherty (1990) Knapp and Doherty (1994) Fidkowski et al. (1991)

Ternary mixtures, single feed columns, CMO Ternary, single feed with heat effects. Multicomponent, single feed, CMO. Tracks fixed points ternary, double feed column, CMO. Calculates R,,, and R,,,.Up to four components, single feed, CMO. Algebraic continuation arc method to locate tangent pinch.

Stichlmair et al. (1993)

Ternary mixture, “pinch point” geometry.

Bauer and Stichlmair (1995)

Combines pinch point analysis with process MlNLP optimization for ternary mixtures in single-feed columns.

~

~~~

Castillo et al. (1998a)

Introduces the concept of staged leaves for ternary mixtures in single feed.

Thong et al. (2004)

Two-stage synthesis procedure for separating multicomponent azeotropic mixtures.

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the Branch-and-Bound techniques of Sophos etal. (1978) and Morari and Faith (1980),the thermodynamic insights (pinch) technique of Linnhoff et al. (1983), to mention but a few. However, focus will be, as elsewhere, on the superstructure optimization methods proposed for the solution of this problem. A distinct characteristic of the majority of these methods is that they are based on column sequencing methods, incorporating elements of heat exchanger network (HEN) synthesis. An overview of the superstructure HI column sequencing methods is provided next.

1.3.2 Superstructure Methods for HI Simple Column Sequencing

A number of superstructure methods have been reported in the open literature for the treatment of various facets of HI distillation sequencing problem. Andrecovich and Westerberg (1985a) incorporated HI possibilities in their MILP simple column sequencing method. This was realized by assuming instances of columns at different prespecified pressure levels, The pressure ranges were determined by the available hot and cold utility temperatures. The method constructed the HI sequencing schemes using two proposed algorithms. The hot and cold streams were the condenser, reboiler, and utility streams. Having specified the column pressures, the temperatures of the condensers and reboilers were also considered fxed. The heat duties of each condenser and reboiler at each column instance were assumed proportional to the feed flow rate. The HI possibilities were incorporated using HEN transportation (Cerda et al. 1983) and the transshipment (Papoulias and Grossmann 1983) problem formulations. However, in the above approach an invalid assumption was used, which assumed that the product of the column heat duty and the reboilercondenser temperature difference was constant in each column. This essentially meant that the reboiler and condenser heat duties and their temperature difference were assumed independent of the column temperature (or pressure) levels. In order to rectify this, Andrecovich and Westerberg (198513) removed this assumption. The results derived generated the same HI sequences as previously, but the calculated utility targets were improved. The HI column sequencing superstructure method of Andrecovich and Westerberg (1985b)was also used in Kakhu and Flower (1988),incorporating three types of complex columns (Petlyuk,side stripper, and side rectifier) to be heat integrated with other simple column sequences. The HI problem was addressed through the transshipment/transportation formulations. The distillation columns were modeled assuming sharp separations and total mass balances, generating a MILP problem, solved for the minimization of the TAC. The capital cost functions were based on a fEed-charge part and on a feed-flow-rate-dependantpart and the column heat duties were also given as simple functions of the feed-flow rate. Paules and Floudas (1988)addressed the HI simple column sequencing problem, generating a superstructure based on sharp split columns. The presence of each column was assigned to a binary variable, along with the HI possibilities between potential condenser and reboiler matches. In order to provide more realistic HI

7.3 Synthesis of Heat-integrated Distillation Column Sequences

schemes, the effects of pressure were taken into account implicitly, through relating the column pressure to the condenser temperature. The latter was included as a variable in the optimization problem. Through shortcut simulations at different pressure ranges and using regression analysis, the reboiler temperatures and the TAC expressions were formulated as functions of the condenser temperatures. The pressure ranges were determined by the available hot and cold utilities available. The overall problem was posed as an MINLP solved for the minimization of the TAC using the APROS methodology. Raman and Grossmann (1993) developed a model for the synthesis of HI distillation column sequences, where the sequencing problem without HI was based on the MILP method of Andrecovich and Westerberg (1985a). In the HI sequencing problem, columns of sharp splits were incorporated, which were represented in the HI problem through the temperatures of reboilers and condensers. In contrast with the HI method of Andrecovich and Westerberg (1985a),the reboiler and condenser temperatures were not considered constant, but were incorporated as variables. The columns were modeled operating at arbitrary pressures, since the authors did not incorporate pressure explicitly as a variable in their HI method. The columns’ capital cost was expressed in the fixed charge-variable charge (feed-flow-rate-dependant)model. In the operating cost model the heat loads of the reboilers and condensers were calculated by simple functions of the feed flow rate. The generated problem was solved using a proposed incorporation of logic in the branch-and-bound scheme. The simple column sequencing method of Novak et al. (1996) was also extended for the incorporation of HI possibilities, combining the compact and network column sequencing superstructures with the multistage HEN superstructure of Yee at al. (1990) (NLP) or Yee and Grossmann (1990) (MINLP in which binary variables were used for every potential heat match). The columns were modeled assuming sharp splits and using the Fenske-Gilliland-Underwoodshortcut model. The investment cost of pumps was found to be small compared to that of the columns and was therefore neglected. In the case studies, the MINLP form of the HEN problem was employed, along with a simple initialization and linearization scheme for overcoming convergence problems. In Grossmann et al. (1998)the above column sequencing superstructure method was coupled to the Duran and Grossmann (1986) HEN model and its disjunctive MILP reformulation. The latter was proposed in order to overcome difficulties experienced with the above model handling isothermal streams (due to its convergence to suboptimal solutions). In a revisited HI column sequencing problem from Novak et al. (1996), the Duran and Grossmann (1986) model and its disjunctive reformulation-generated HI structures were characterized by the same column sequencing, but with different HI schemes. The Duran and Grossmann (1986) model produced the worst solution, while its disjunctive reformulation and the Novak et al. (1996) solution produced the best results, which were quite similar. The HI column sequencing problem was also addressed as an extension of the simple column sequencing problem of Yeomans and Grossmann (1999a). However in order to produce more realistic HI results, energy balances were added in the simple mass balance models employed for the physical representation of the distillation

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columns. The HI synthesis procedure was based on the Raman and Grossmann (1993)model. The convex hull formulation was applied to the generated GDP problem for its translation to an MILP problem. For the incorporation of HI possibilities, the Raman and Grossmann (1993)HI model was also implemented in Yeomans and Grossmann (1999b), coupled to the proposed nonlinear GDP simple column sequencing model (involving nonlinear shortcut distillation column physical models). In the distillation column sequencing method of Caballero and Grossmann (1999) HI possibilities were incorporated based on the heat integration models of Paules and Floudas (1988)and Raman and Grossmann (1993).The heat loads of the isothermal reboilers and condensers were calculated as a function of their vapor flow rates and heat of vaporization,while simplified energy balances were used for the incorporation of HI possibilities. Heat exchange feasibility constraints were enforced between condenser and reboiler temperatures for their potential HI matches, which were assigned to a binary variable. The authors did not consider the incorporation of pressure in the proposed method. An objective function was used for the minimization of the sequences’ operating cost, which enabled the method to derive valid lower bounds of the represented systems’ energy consumption. Finally, a superstructure optimization method was proposed by Hostrup et al. (2001), incorporating thermodynamic insight principles. The main principle in this method was the use of thermodynamic insight techniques (Jaksland et al. 1995) in order to generate a superstructure of alternatives, to be further optimized using formal superstructure MINLP techniques. Three main steps were defined in this method: the problem formulation (identification of tasks and techniques), the flow sheet optimization, and the validation/analysis. A general separation problem posed in Agganval and Floudas (1990)was revisited also incorporating HI possibilities. In the first step of the method having provided the feed mixture to the ICAS synthesis toolbox (CAPEC 1999), based on thermodynamic insights two feasible separation techniques were identified (flash and distillation), which were eventually presented as a single technique, thus reducing the size of the generated MINLP problem. In the structural optimization step the compact superstructure of Novak et al. (1996) was employed and extended for the general separation case. HI was implemented in this model using the Yee and Grossmann (1990) HEN model. The columns were modeled using the Fenske/Gilliland/Undenvood shortcut model and the HI results indicated 50% lower operating costs than those of the non-HI case. Recently, Proios (2004) extended the GMF model for the simple column sequencing problem to account for heat integration opportunities. To this end, an HI block has been introduced, along with its structural and physical model components. A summary of superstructure-based methods for HI simple column sequencing is presented in Table 1.3.

1.4 Synthesis ofcomplex Disti//ation Column Sequences Table 1.3

Summary of superstructure methods for HI simple column

sequencing.

Authors

Features

Simplijed methods (sharp splits) Andrecovich and Westerberg (1985)

HEN transportation/transshipment, fxed P columns

Kakhu and Flower (1988)

Andrecovich and Westerberg (1985b) - three complex columns

Paules and Floudas (1988)

P via condenser temperature, regression analysis

Raman and Grossmann (1993)

Andrecovich and Westerberg/Paules and Floudas, Logic BB

Novak et al. (1996)

I HEN of Yee and Grossmann (1990)

~~

Grossmann et al. (1998)

HEN of Duran and Grossmann (1986) and its GDP MILP

Yeomans and Grossmann (1999a,b)

HI of Raman and Grossmann (1993), GDP

Caballero and Grossmann (1999)

HI of Raman and Grossmann (1993), GDP

Hostrup et al. (2001)

Thermodynamic insights and Novak et al. (1996)

Proios (2004)

Extension of the GMF to account for heat integration opportunities

1.4 Synthesis of Complex Distillation Column Sequences 1.4.1 Complex Distillation Column Sequencing

In Section 1.2 it was illustrated that choosing the most appropriate simple column sequence out of a set of possible alternatives can lead to substantial energy savings. In Section 1.3 these energy savings were extended by applying heat integration techniques on these simple column sequences. However, it has been reported, as will be shown below, that high energy savings can also be achieved through the incorporation of thermal coupling in complex columns (columns with multiple feeds and side streams). In this section, an overview of techniques for the complex distillation column sequencing problem (that is, the sequencing problem where both simple and complex columns are considered) is critically presented. An illustration of the complex distillation sequencing problem is given here for the ternary separation problem (the first member of the multicomponent separation). The alternative sequences for ternary distillation are shown in Fig. 1.3. From those alternatives, the most common in both literature and industrial practice are those consisting of simple distillation columns, which are shown in Figs. 1.3a-c. However, the number of alternatives increases considerably when complex columns are considered (Figs. 1.3d-k). All these complex columns are either partially or fully ther-

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(a) Direct Sequence

(b) Indirect Sequence

(c) 3-columns

(d) Prefractionator

(e) Petlyuk

(f) Dividing Wall Column

(h) Side Stripper

(i) Side Rectifier

Q

(g) Side Petlyuk

(i)RV Column

-;i 4-

(k) SL Column

Figure 1.3 Alternative configurations for the distillation o f a nonazeotropic ternary mixture.

mally coupled. From these, the fully thermally coupled (Petlyuk) column (Fig. 1.3e) has been receiving increasing attention, as it has been found that it can exhibit energy consumption reductions of up to 40% compared to conventional configura-

1.4 Synthesis ofcomplex Distillation Column Sequences

tions (Fidkowski and Krolikowski 1986; Glinos and Malone 1988; Triantafyllou and Smith 1992; Annakou and Mizsey 1996; Dunnebier and Pantelides 1999),which is due to its thermodynamic efficiency (minimization of losses due to unnecessary remixing). The dividing wall column (Fig. 1.30 is considered thermally equivalent to the Petlyuk column under the assumption of no heat transfer through the vertical wall. It also provides capital cost savings by including the main and the prefractionation column of the Petlyuk arrangement in the same shell. The two final configurations (RV and SL) are essentially a side rectifier column with a direct vapor connection and a side stripper column with a direct liquid connection, respectively (Agrawal and Fidkowski 1999a). However, some further column configurations have been reported in the literature (Fig.1.4) (Agrawal 2000b). As indicated in Fig. 1.4, these configurations are modifications of some of the complex columns, aiming at the improvement of the latter’s controllability.According to Agrawal (2000b),control difficulties can be encountered in complex columns, the source of which the author traced to the columns’ vapor interconnections. In the alternatives to Fig. 1.4, one or more “problematic” vapor interconnection streams have been replaced with additional rectifying sections and condensers or with stripping sections and reboilers. These configurations, which are assumed to be thermally equivalent to their “parent” configurations, increase the number of structural alternatives for ternary distillation to 18 versus the eight reported by Tedder and Rudd (1978). Therefore, the issue that needs to be formally addressed is to provide a method that systematically incorporates the complex structural possibilities of thermally coupled columns in a unified way with simple column sequences, while capturing accurately the underlying physical phenomena in order to obtain the most energy efficient distillation sequences.

1.4.2 Superstructure Methods for Complex Distillation Sequencing

The first systematic superstructure method for generating distillation column sequences, including complex arrangements, was the sequential column superstructure presented initiallyby Sargent and Gaminibandara (1976) and extended in Eliceche and Sargent (1981) (see also Fig. 1.5). The authors proposed a rigorous tray-by-tray model for the design of distillation columns with multiple feeds and side streams. The number of stages and the optimal feed and side stream locations in each column required the introduction of a large number of discrete variables, rendering the problem a large scale MINLP. The combinatorial complications were handled by treating the integers as continuous variables rounded off each time to the nearest integer variable, eventually reducing the problem to a nonlinear programming (NLP) problem. The work on the sequential column superstructure was continued in Mei (1995), by investigating the possibilities of aggregation and size reduction, by using as superstructure building blocks column segments (aggregations of trays) and connecting units, whose existence was denoted by binary variables. Using the fact that trays do not operate at equilibrium, their number in each column segment was not

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%-

(a) Modified Side Stripper

(b) Modified Side Rectifier

(c) Modified Petlyuk 1

(d) Modified Petlyuk 2

(e) Modified Petlyuk 3

(0 Modified Petlyuk 4

A?-

(g) Modified Petlyuk 5 Figure 1.4

Modified configurations for operability targets.

1.4 Synthesis ofComplex Distillation Column Sequences

specified as an integer variable. Moreover, each building block was modeled as a modular simulator using as inputs a guess of the number of trays, the flow rates, and the intensive variables characterizing each input stream and generating outputs of the calculated flow rates and intensive variables of each output stream to be included in the optimization. The case studies demonstrated the effect of the modular treatment of the problem, which led to mathematical models of less than 100 variables and equations, respectively. Agrawal (1996)proposed a satellite column superstructure method for the separation of near-ideal mixtures of four and more components (n)into pure (or near pure) products, generating more than one fully thermally coupled distillation column sequences, with a single reboiler and condenser. This superstructure involved n-2 columns as satellites around a central column, while generating an overall smaller minimum number of column sections than the sequential column superstructure (Sargent and Gaminibandara 1976) for n 2 4.The method was also extended to the generation of all the other feasible partially or nonthermally coupled column sequences for the examined separations through the addition of reboilers and condensers at the appropriate end of each column section of the satellite superstructure. A step-wise procedure was provided for the generation of all the FC structural alternatives. It was shown that although for n-3 the satellite and the sequential column superstructures are equivalent, for n 2 4 the former is more complete than the latter. However, no formal structural optimization model was provided by the authors for the automated generation of all the alternatives. The proposed satellite superstructure representation was extended in Agrawal(2000a),providing a systematic method to draw more alternative FC column sequences, which are topologically different and allegedly more operable than those presented in the previous publication. From a thermodynamic point of view, the additional structures were equivalent to those of the 1996 publication. A method was also proposed for the generation of feasible distillation column sequences was the state task network (Sargent 1998),which was analyzed more thoroughly in Doherty and Malone (2001). The method initially identified the network states (mixtures created by all the feasible separations) and then the separation tasks, which were assigned to a distillation column or, in the case of complex distillation columns, to a section of a distillation column. Having identified the possible states and tasks, a superstructure could be generated linking a distillation task between two consecutive states and by using logical constraints for the feasible sequencing of tasks. Since the STN is a one-to-one approach (one unit carries out only one task), this method generated a problem with a large number of units (contrary to the SON, which considered only a small number of multi-task units). The complex distillation column sequencing problem was also examined in Dunnebier and Pantelides (1999),based on the state operator network (SON) method (Smith 1996).In the proposed method the distillation column operators were physically modeled based on the rigorous tray-by-tray model of Viswanathan and Grossmann (1993). Binary variables denoted the existence of column operators, number of trays (reflux return tray location), and side stream locations (see Fig. 1.6). The authors included some very useful comments regarding the convergence complexities of the exam-

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ined problem, while stressing the importance of complex arrangements for energy efficiency and the use of accurate models for their representation. Yeomans and Grossmann extended their disjunctive model for the rigorous design of simple distillation columns (Yeomans and Grossmann 2000a) to the incorporation of complex arrangements (Yeomans and Grossmann 2000b) based on the sequential column superstructure representation of Sargent and Gaminibandara (1976). In both studies, once the superstructure for the column sequence had been postulated, the representation was translated into a GDP mathematical problem, which was then transformed into an MINLP problem. The column trays were arranged as permanent and conditional ones. The former were fured in the superstructure and corresponded to the feed, reflux, and boil-up trays. The conditional trays were assigned to a Boolean variable, which when false, turned these trays into simple input-output trays with no mass transfer. The authors concluded that their method avoided the numerical difficulties associated with including redundant equations and with singularities due to liquid and vapor flows taking a value of zero and improving the model's robustness. Caballero and Grossmann (2001) provided a superstructure method on the STN formalism of Yeomans and Grossmann (1999a)for the generation of all FC column sequences for n components from the satellite superstructure (Agrawal 1996). In conjunction with the guidelines provided in the latter publication, the superstructure was modified for the generation of all the partially and nonthermally coupled column sequences. The superstructure was modeled structurally using repositional logic expressions (Raman and Grossmann 1991) and each column was modeled physically

A=uFL. B

V Vk

L d l

B+C

1 +=

&

LC L

Figure 1.5 Sequential column superstructure (Eliceche and Sargent 1981).

7.4 Synthesis ofcomplex Distillation Column Sequences

Figure 1.6

Superstructure for ternary distillation (Dunnebier and Pantelides 1999).

using a simple shortcut distillation column models based on a modified version of Underwood’s equations (Carlbergand Westerberg 1989a,b)and using fixed costs for the heat units and column sections. The complete synthesis problem was modeled using GDP and solved using a modified version of the logic-based outer approximation algorithm. In a recent paper Agrawal (2003) proposed an improved systematic procedure for the generation of basic and thermally coupled column sequences. In both types of column sequences it was imposed that a product stream enriched in a particular component was produced only once. A systematic procedure including useful guidelines was provided initially for generating all the basic column sequences, followed by a procedure for the thermally coupled column case. The latter was based on the generation of thermally coupled columns by replacing reboilers and condensers of basic sequences with two-way interconnections between columns. The method that could be readily modified in order to incorporate other structural possibilities regarding the transfer of streams between columns can be viewed as a useful tool for the construction of a systematic superstructure optimization method for the automatic generation of the reported and of other “unknown” column sequences. Coupled to an efficient and appropriately designed physical model and solution procedure, solutions can be generated providing optimal designs from an energy and/or operability point of view.

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1 Synthesis of Separation Processes Table 1.4

Summary of superstructure methods for complex column sequencing. Features

Authors

Sargent and Gaminibandara (1976) ~~

Sequential column superstructure, MESH model

~

Eliceche and Sargent (1981)

Sequential column superstructure, M E S H Model

Mei (1995)

Sequential column superstructure, pseudo-aggregated model

Agrawal (1996, 2000, 2003)

Satellite column superstructure

Sargent (1998)

STN (Doherty and Malone (2001))

Dunnebier and Pantelides (1999)

I SON superstructure, MESH (Smith 1996)

Yeomans and Grossmann (2000b)

Sequential column superstrumre, MESH

Caballero and Grossmann (2001)

Satellite column superstructure, GDP, shortcut model

Shah and Kokossis (2002) Proios (2004)

I Synthesis of complex distillation sequences

Extension of the GMF for synthesis of complex column sequencing

Shah and Kokossis (2002) proposed a new representation for the synthesis and optimization of complex separation systems. This representation is based on tasks instead of units. The problem is formulated as an MILP and its efficiency was illustrated in several problems based on the literature as well as industrial problems.

1.5 Conclusions

A comprehensive review of state-of-the-arttechniques for the synthesis of energy efficient simple, heat integrated, and complex distillation column sequences, has been presented is this chapter. Emphasis was placed on systematic and rigorous approaches focusing on the systematic generation and evaluation of the alternative designs in a compact and unified way. The problem of significant economic and scientific interest required the tackling of underlying structural and physical complicating issues. These were inherently related to increasing the number of structural alternatives and the complexity of the underlying mass and heat exchange physical phenomena. The economic potential in conjunction with these complications has "compelled" virtually all major research groups, both in academia and industry, to propose methods for the distillation column sequencing problem, addressing it in its entirety or just certain components of it. The presented superstructure methods have contributed significantly to the efficient treatment of the distillation sequencing problem. It has been shown that the nonconvexities and discontinuities of the generated large scale MINLP problems, which have imposed a limitation of many rigorous approaches, have been overcome

References

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44 jobson, M., D. Hildebrandt, D. Glasser (1995) Attainable Regions for the Vapour-Liquid Separation of Homogeneous Ternary Mixtures. Chem. Eng.]. 59(1),51-70. 45 Julka, V., M. F. Dolterty (1990) Geometric Behaviour and Minimum Flows for Nonideal Multicomponent Distillation. Chem. Eng. Sci. 45, 1801-1822. 46 Kakhu, A. I., 1. R. Flower (1988) Synthesising Heat-Integrated Distillation Sequences Using Mixed Integer Programming. Chem. Eng. Res. Des. 66, 241-254. 47 Knight, /. R., M. F. Doherty (1986) Design and Synthesis of Homogeneous Azeotropic Distillations. 5. Columns with Nonnegligible Heat Effects. Ind. Eng. Chem. Fundam. 25, 279-289. 48 Knapp, J . P., M. F. Dohel?, (1994) Minimum Entrainer Flows for Extractive Distillation: a Bifurcation Theoretic Approach. AIChE J. 40(2),243-268. 49 Kocis, G. R., I. E. Grossmann (1989) Computational Experience with DICOPT Solving MINLP Problems in Process Systems Engineering. Comput. Chem. Eng. 13, 307-315. 50 Kravanja, Z., I. E. Grossmann (1993) PROSYN-An Automated Topology and Parameter Process Synthesizer. Comput. Chem. Eng. 17, S87-S94. 51 Kravanja, Z., I. E. Grossmann (1994) New Developments and Capabilities in PROSYNAn Automated Topology and Parameter Process Synthesizer. Comput. Chem. Eng. 18, 1097-1114. 52 Levy, S. G., D. B. Van Dongen, M. F. Doherty (1985) Design and Synthesis of Homogeneous Azeotropic Distillations. 2. Minimum Reflux Calculations for Nonideal and Azeotropic Columns. Ind. Eng. Chem. Fundam. 24, 463-474. 53 LinnhoJ B., H. Dunford, R. Smith (1983) Heat integration of distillation columns into overall processes. Chem. Eng. Sci. 35, 1175-1188. 54 Mei, D. (1995)An SQP Based MINLP Algorithm and its Application to Synthesis of Distillation Systems. Dissertation, Imperial College of Science, Technology and Medicine, London. 55 Mix, T. W., J . S.Dweck, M. Weiberg, R. C. Amstrong (1978) Energy Conservation in Distillation. Chem. Eng. Prog. 74(4), 49-55. 56 Morari, M., C. Faith (1980)The Synthesis of Distillation Trains with Heat Integration. AlChE]. 26,916-928.

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Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

2 Process Intensification Patrick Linke, Antonis Kokossis, and Alberto Alva-Argaez

2.1 Introduction

Since its emergence in the early 1980s, process intensification (PI) has received significant attention from the chemical and process engineering research community. PI promises novel, more efficient technologies with the potential to revolutionize the process industries. The term process intensification is associated mainly with more efficient and compact processing equipment that can potentially replace large and inefficient units commonly used in chemical processing but also includes methodologies, such as process synthesis methods, that enable the systematic development of efficient processing units. To date, numerous technologies have emerged from PI research activities and a number of commercial-scaleapplications have successfully been implemented. In a recent survey of industrial and academic experts, PI has recently been highlighted as one of the key enabling technologies required for the sustainable development of the chemical industry (Tsoka et al. 2004). One of the earliest and most prominent applications linked to process intensification has been the redesign of the Eastman Chemicals methyl acetate process (Siirola 1995). The original process was highly complex and consisted of more than twenty pieces of equipment. Through integration of reaction and separation processing tasks into multifunctional reaction/reactive separation equipment it was possible to devise an intensified process consisting of only three pieces of equipment. Since the success of the Eastman process, numerous other PI technologies have been developed and applied. Prominent examples (Stankiewicz and Drinkenburg 2004) include efficient reaction equipment such as spinning disk reactors and monolithic reactors, equipment for nonreactive operations such as compact heat exchangers, multifunctional reactor concepts such as reactive separations, hybrid separations such as membrane distillation, alternative energy sources such as ultrasound, and many other technologies. The numerous technological breakthroughs have led to comprehensive PI dissemination efforts for academic research as well as commercial applications including Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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international conferences, symposia, and networks. Most recently, a textbook by leading academic and industrial PI scientists and engineers has been published to provide a good overview over the technology developments to date (Stankiewicz and Moulijn 2004). PI developments have their origins in commodity chemicals due to the strength of the industry sector in the 1980s as well as the large economic impact of even small efficiencygains. As a result of the evolution of the chemical industries, PI has found applications in other sectors including fine chemicals and bioprocessing. This has resulted in a new definition of PI, which is now widely understood as the development of novel equipment, processing techniques, and process development methods for chemical and biochemical systems (Stankiewiczand Drinkenburg 2004). Process intensification plays an important role in the chemical industry’s attempt to develop into a sustainable industry that is both economically and ecologically viable. The following benefits associated with process intensification are important factors for sustainable development (Siirola 1995): 0

0

The development of more profitable processes, that are cheaper to run, require less energy, and produce less waste and by-products. Process intensification is linked to reduced costs for land, equipment, raw materials, utilities, and waste processing. The acceleration of the process development cycle. This is particularly important in industry sectors where time to market is a crucial factor, such as fine chemicals or pharmaceuticals. The development of greener and safer processes, which is closely linked with good company image. Such a positive image is crucial for enterprises in the chemical process industries in order to remain viable businesses.

Most of the reported developments have been realized on an ad hoc basis, drawing on design engineers’ intuition and expertise. Only more recently, the hndamental principles underlying process intensification technologies have been called upon in the form of systematic design procedures. The development of reactive distillation processes, such as the Eastman process mentioned above, is a typical example of an area where initial success stories have resulted from intuition and systematic design methods have subsequently been developed that can guide the designer in the development of such a system (Ciric and Gu 1994; Cardoso, Salcedo, Fey0 de Azevedo 2000; Hauan, Westerberg and Lien 2000; Hauan, Ciric and Westerberg 2000; Ismail, Pistikopoulos and Papalexandri 1999; Okasinski and Doherty 1998; Linke and Kokossis 2003a). Such methods are generally computer-aided design techniques that screen numerous novel and existing design options and assist the design engineer in the identification of promising process intensification routes. However, such methods are still emerging and very few tools are currently commercially available. Computer-aided tools have great potential to accelerate process intensification technology development. They enable the systematic screening and scoping of large numbers of alternative processing options and can identify novel options of phenomena exploitation that may lead to higher efficiencies. Such tools provide the basis for systematic approaches to novelty in process intensification and have the potential to

2.2 Process Intens9cation Technologies

identify processing options, which can easily be missed in design activities that rely on intuition and past experiences. The next section provides an overview over current process intensification technologies. In the remainder of this chapter we will present a number of recently developed systematic computer-aided process intensification methods.

2.2 Process Intensification Technologies

A large number of process intensification technologies have been developed. Accord-

ing to Stankiewicz and Drinkenburg (Stankiewicz and Drinkenburg 2004), the existing technologies can be broadly classified into process-intensifymgequipment (hardware) and process-intensifyingmethods (software).PI applications often result from a combination of equipment and methods. For instance, the application of methods often leads to the development of novel equipment. In the same vane, novel apparatuses often make use of new processing methods. The following sections will provide a brief overview of important process intensification methods and equipment and discuss industrial applications. A more detailed description of PI technologies can be found elsewhere (Stankiewicz and Moulijn 2004).

2.2.1 Process Intensification Equipment

Process intensifying equipment (hardware)can be broadly classified into equipment for reaction systems and equipment for nonreactive systems (Stankiewicz and Drinkenburg 2004). Examples of reaction equipment include spinning disk reactors, static mixer reactors, monolithic reactors, and microreactors. Static mixers, compact heat exchangers, rotating packed beds and centrifugal adsorbers are examples of equipment for nonreactive operations. Process intensifying equipment generally aims at improving crucial processing characteristics in terms of mixing, heat transfer, and mass transfer over those realizable in conventional equipment. Such equipment is generally smaller than their conventional counterparts, albeit offering improved processing performances. A number of PI equipment developments are illustrated below. Spinning Disk Reactors

The fluid dynamics of multiphase contact are dominated by surface forces so that very small interfacial areas are developed. This results in a lack of countercurrent interfacial motion in conventional multiphase contacting equipment, which in turn causes a low intensity operation in terms of reaction, mass, and heat transfer. The processing characteristics can be significantly enhanced through the application of a high-accelerationfield. In the spinning disc reactor (SDR) (Ramshaw 2004), such an

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acceleration field is established within a rotor that receives and discharges the working fluid. In general, the spinning disk device achieves highly efficient gas-liquid contacting from which numerous applications benefit, such as evaporators,aerators/ desorbers and reactors. The device is very compact and has been demonstrated to allow very good control over multiphase reactions, even in highly viscous systems. Multifunctional Heat Exchangers

Multifunctional heat exchangers combine heat transfer phenomena with other phenomena such as reaction, separation, or mixing in a single piece of equipment. A combination of heat transfer with one or more of these phenomena allows one to achieve better process performances or control for many systems. A reactor heat exchanger is a typical example for such a multifunctional heat exchanger (Thonon and Tochon 2004). Integrating the reactor within the heat exchangers enables better heat management that can significantly improve process yields and selectivities as well as reduce process energy requirements. Microreaction Technology

Microreaction technology (Ehrfeld 2004) achieves intense mixing and very high heat transfer rates by significantly decreasing the characteristic dimensions of a processing system to the scale of micrometers. The availability of high transfer rates for heat and mass in conjunction with very small material holdups allows good control over reaction systems, i.e., over reaction yields, selectivities and energy management. For instance, it is possible to operate highly exothermic reactions at isothermal conditions using miroreactors. Apart from more intense processing, microreaction technology offers advantages in process control, because the starting and boundary conditions for reactors and unit operations can be adjusted precisely and are easily scaled-up using parallelization. Structured Catalysts

Structured catalysts, such as monoliths, offer a number of advantages over nonstructured catalysts in terms of high rates, high selectivities,low energy consumption and easy scale-up (Moulijn, Kapteijn and Stankiewicz 2004). Monolithic catalysts are metallic or nonmetallic bodies that contain large numbers of channels of defined cross sectional shapes and sizes (Stankiewicz and Drinkenburg 2004). They cause very low pressure drops, offer high surface areas per reactor volume, allow the use of very small catalyst particles, such as zeolites, and exhibit very high catalytic activities due to short difision paths in the thin wash coat layer. The process-intensifymg benefits from a structured reactor results mainly from the possibility to decouple reaction kinetics, transport phenomena and hydrodynamics so that each can be optimized independently in order to achieve very good reactor performances (Moulijn, Kapteijn and Stankiewicz 2004).

2.2 Process Intensijcation Technologies

2.2.2 Process Intensification Methods

According to Stankiewicz and Drinkenburg (2004), process intensifylng methods (software) include multifunctional reactors, hybrid separations, the use of alternative energy sources and other computer-aided methods. A number of multifunctional reactor concepts have been developed including reactive separations, heat integrated reactors and fuel cells. Hybrid separations generally combine two or more types of unit operations into a single system and include processes such as membrane distillation and membrane adsorption amongst others. Alternative energy sources are used in process intensification to allow a better exploitation of chemical and physical phenomena. Such methods include the application of centrifugal fields, ultrasound, microwaves and electric fields. Other process intensifjing methods that do not fall into the above categories have been classed as “other methods” by Stankiewicz and Drinkenburg (2004). This class contains dynamic strategies for reactor operation, but also computer-aided design methods, such as process synthesis tools, that allow the systematic identification of promising design options that allow efficient processing. We explain a number of important process intensification methods below. Reactive Separations

Reactive separations are classes of reactors that facilitate component separations in the reaction zones with the aim to improve reaction yields and selectivities or to facilitate difficult separations. Examples of reactive separation processes include reactive extraction where a solvent is used to transfer components in and out of the reaction zone, reactive distillation where vapor is used as the separating agent, and membrane reactors, where the permeability and selectivity of a membrane is exploited to selectively add and remove components from the reaction zone. The most prominent success of reactive separations is the Eastman methyl acetate reactive distillation process mentioned in the introduction. Hybrid Separations

Hybrid separations are processing methods that exploit the synergies between different separation techniques in a single operation. Prominent examples of hybrid separations include extractive distillation, adsorptive distillation, membrane distillation, and membrane extraction. Hybrid separations are generally developed for systems where the performance of a unit operation is inefficient or problematic. For instance, distillation is an efficient process for the separation of close-boiling systems. The introduction of a suitable solvent into the column, i.e., extractive distillation, allows one to increase the driving forces in the column and thus improve the efficiency of the operation. The viability of a hybrid separation process depends strongly on the system under consideration. More information on the selection of hybrid separation processes can be found in Stankiewicz (2004).

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Process Synthesis

Process synthesis refers to the systematic development of process flow sheets through “the automatic generation of design alternatives and the selection of the better ones based on incomplete information” (Westerberg 1989). Process synthesis methods support the engineer in finding novel, improved solutions to process design problems. The synthesis objectives aim at finding those processing options that enable the production of desired chemicals in the most cost effective and environmentally benign manner possible. A number of process synthesis tools have been developed that can systematically determine the most promising process designs for a number of systems including reaction systems, reactive separations, hybrid separations, as well as water and energy management. Such methods will be described in detail later.

2.2.3 Process Intensification in Industrial Practice

There are a number of incentives for enterprises in the chemical industries to adopt process intensification technologies. Through the intensification of processes, stepby-step improvements can be realized that offer a strong possibility to fulfill current business requirements that are becoming more demanding in the current economic climate (Bakker 2004). A strong driver for the introduction of process intensification technologies is the need to remain competitive in expanding markets by achieving and maintaining the best low-cost position. Other strong drivers include the need to meet tightening legislative and environmental requirements. Process intensification technologies can positively respond to these drivers (Bakker 2004) as they offer numerous related advantages such as energy savings, reductions in space requirements, reductions in the number of required processing steps, reduced emissions and waste, and more flexible feedstock specifications. Bakker (2004)describes the possibilities for introducing process intensifylng technologies into an industrial process taking into account six largely sequential process development phases:

New ideas for a process. The development of a chemical path leading to a desired product defines the core of the process. There are typically a number of possible pathways that can be followed and it is important to consider PI technologies in order to determine the most intensified process option. This phase has great potential for introducing process intensification options because conceptual process changes are relatively cheap. At the end of this development phase, the chemical pathway is identified and a number of process parameters, such as concentrations and temperatures, have been roughly specified. Determining the process chemistry. This phase is concerned with identifjmg the optimal design and conditions for the reaction system so as to identify the intrinsically optimal route that produces the least by-products and waste. In this phase, the process design is also decided and care needs to be taken to ensure that the overall

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process design achieves optimal performance and can easily be implemented. Generally, the consideration of process changes in this phase is still relatively cheap. Pilot plant studies. This phase is typically concerned with optimizing performances on an equipment level in order to overcome heat and mass transfer limitations, for instance, by making use of process intensification equipment. The testing of PI options is still relatively cheap in this phase. Plant design. At this point, the process design is fixed and it is difficult to introduce major changes as this would require revisiting of the previous phases. It is important that the benefits of process intensification technologies have been identified before this phase. Start-up. At start-up, the process system is fully specified and there is no more room to introduce process intensification options. Debottlenecking and trouble-shooting. Once the process is operating, it is difficult to make significant changes as this would incur downtimes and loss of production. This is only viable if step changes in efficiencies can be gained from a process change. The application of process intensification technologies may offer such savings and their economic benefits should be identified in order to decide on high-impact retrofit projects.

Clearly, the quick evaluation of various processing scenarios is very important in order to identify the most intensified process for a given product. Without systematic methods, such a screening is likely to be incomplete and promising process options are likely to be missed. Computer-aided methods offer great potential to realize quick and systematic process screening so as to minimize the risk of choosing underperforming processing options.

2.3 Computer-Aided Methods for Process Intensification

The ability to process large amounts of data in modern computers has opened up the possibility of developing practical and user-friendly software tools to assist process engineers and process designers in their decision-making tasks. In the past, the lack of powerful computers and other data processing tools resulted in design tools that would heavily draw on past design experiences. Experience from previously successful designs, encapsulated in heuristics, were the basis for new process developments. However, the application of heuristics very often created contradictions and confusion and good, novel design options were often missed. With today’s vast computing power, it is possible to do away with heuristic rules and develop systematic methods that are not limited to the reproduction of past designs. For a widespread use of such computer-aided process intensification methods to be reached, the research community needs to deliver design methodologies that can be translated into usable software. In this section, we will describe a number of available methodologies and models that could become part of such a software toolbox.

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Systematic computer-aided process intensification methods are required to generate a set of feasible process design alternatives and to select the most efficient configurations from the set. Ideally, such decision support tools should allow a systematic determination of the most promising process designs, which closely approach the performance limits of the system and meet the business constraints, out of the set of all feasible alternative structural and operational process design options. Due to the complexity of the overall problem, applications of systematic decisionmaking technologies have addressed closely defined design subproblems, such as reaction, heat integration, and separation systems. These various subproblems exhibit a variety of challenging features. For instance, in reactor optimization these arise mainly from the highly complex chemical and physical models that need to be processed. In energy systems such as heat exchanger networks, on the other hand, the challenge is of combinatorial nature as vast numbers of different feasible design option exist. Separation system design presents examples of intermediate complexity in terms of model complexities. Despite the challenges in solving the individual design subproblems, the decomposition of the overall design problem into such subproblems makes numerous limiting assumptions that present opportunities for process choices. The conventional approach to process design has similarities with a bow tie model: one starts broad and collects all available process information. A structure is used, e.g., a tree diagram, to create a format for the required information. In this stage, all relevant factors that are anticipated to have an influence on the process are collected. Boundaries of the process are generally set early to reduce rework later on. Clearly, such an approach limits the degree of design novelty that can be achieved. It must be recognized that the system boundary is mostly an arbitrary construct defined by the interest and the level of ability and/or authority of the participating actors. From the above discussion, it is clear that process alternatives are commonly generated based on intuition and case-based reasoning. This provides a strong chance that promising design candidates are not arrived at and that novelty is not automatically accounted for in the design process. This risk of selecting underperforming designs can be reduced through systematic approaches that allow one to capture all possible design alternatives in a process representation and screen for the design that delivers the best possible performance for the specified performance measure. The following sections describe such methodologies for reaction system design, integrated reactionseparation systems, separation systems, solvent-process systems, and integrated water and wastewater systems.

2.3.1 Reaction Systems

The reactor is the part of the process where value is added by converting raw materials into valuable products. The intensification of this unit is crucial for profitability. Reactor design is particularly difficult since reaction, heat, and mass transfer phenomena need to be exploited simultaneously. Reaction and mass transfer models

2.3 Computer-Aided Methods for Process Intensification

tend to be highly complex and difficult to process numerically. There are a number of decisions that have to be made in reactor design such as the selection of appropriate mixing and contacting patterns, the choice of optimal reaction volumes, decisions on temperature policies, and the identification of the best feeding, recycling and bypassing strategies. In order to identify the reactor design options that achieve the best possible performance for a given system, this information needs to be analyzed simultaneously as one decision tends to impact the others. Such an activity is generally beyond the capabilities of a design engineer without the assistance of systematic design methods. As a result, the designer commonly employs textbook knowledge, heuristics, empiricism, past experience and qualitative reasoning on the basis of analogies with similar systems and case studies, which tend to be insufficient to guide the development of high-performance reactor designs for systems of industrial complexity. This design practice results in a lack of innovation, quality and efficiency in many industrial designs. The complexities of the reactor design task make the development of systematic computer-aided design methods a challenging task. From a practical viewpoint, such methods need to be applicable to general reaction systems and capable of supporting decision-making to enable the designer to quickly identify high-performance reactor designs. Over the past two decades, a number of such methods have evolved that can provide performance targets for the reaction system under consideration as well as guide design suggestions in terms of the best mixing, feeding, recycling, and operational policies at a conceptual level. In this section, we describe recent developments in reactor network synthesis that provide systematic decision support for various reactor design aspects such that the system’s performance is maximized with respect to the design objectives employed, Initial efforts in reactor network synthesis have focused on the development of systematic design methods for single-phase systems. More recent efforts have addressed multiphase systems. We will review the singlephase developments first, in order to give an overview over the different approaches followed. Approaches to conceptual reactor design can be broadly divided into optimizationbased and graphical methods. Optimization-based approaches make use of superstructure formulations. Reactor network superstructures include different possible structural design options in a single model, which is then searched in order to determine the optimal candidates. Achenie and Biegler (1986; 1988; 1990) were the first to propose the optimization of comprehensive reactor network superstructures combining options arising from combinations of axial dispersion models and recyclePFR representations. The structures were searched using deterministic optimization techniques in the form of nonlinear programming (NLP) methods to identif) the most promising design candidates embedded in the superstructures. Later, Kokossis and Floudas (1990; 1994) introduced the idea of a reactor network superstructure modeled on a mixed integer nonlinear programming (MINLP) formulation. They proposed superstructures to account for all possible interconnections amongst the generic structural units of ideal CSTRs and PFRs (represented by CSTR cascades) with the aim to screen for design options and estimate the limiting performance of the reaction system. Schweiger and Floudas (1999)later replaced the PFR represen-

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tation by rigorous DSR models in order to avoid the inaccuracies introduced by the use of CSTR cascades. The above methods all made use of deterministic optimization approaches to identify locally optimal designs. Marcoulaki and Kokossis (1999) have recently presented the application of a global search strategy in the form of simulated annealing. Their approach allows to establish solid performance limits for the reaction system and to systematically develop designs that can achieve the targets. We will explain this approach in detail later. In a parallel effort to the development of optimization-based approaches, Glasser, Hildebrandt and Crowe (1987)presented a graphical procedure, the attainable region (AR) method, which has its roots in the insighhl methods originally proposed by Horn (1964).This method allows one to identify the reactor network with maximum performance in terms of yield, selectivity or conversion, which can be located on the boundary of the AR in the form of DSR and CSTR cascades with bypasses. A number of extensions that generalize the AR method have been published (Hildebrandt, Glasser and Crowe 1990; Hildebrandt and Glasser 1990; Glasser, Hildebrandt and Glasser 1992; Glasser, Hildebrandt and Godorr 1994; Glasser and Hildebrandt 1997; Feinberg and Hildebrandt 1997). Though very useful for single-phase systems in lower dimensions, in higher dimensions the developments face both graphical and implementation problems and often lead to complex designs with multiple DSR and CSTR units and complex feeding and bypassing strategies. In an attempt to exploit insights gained from the AR concept to enhance superstructure optimization, Biegler and coworkers established rules for an efficient formulation and optimal search of superstructures. Balakrishna and Biegler (1992a,b) and Lakshmanan and Biegler (1996, 1997) presented NLP and MINLP formulations of this approach as well as a number of applications. From a practical viewpoint, a useful approach for reactor network synthesis should be able to identify the performance limits for the reaction system and to identify designs that can approach these limits. Targets can guide design evaluation in light of the ultimate performance possible for a given system and are also very useful to identify retrofit projects as they provide the incentives associated with such a project. Optimization-based methods can fulfill both of these requirements, provided confidence can be established in the optimization results. If local search strategies are deployed, there is no reason to be confident that the obtained solution can not be substantially improved. It is therefore important that global search strategies are adopted as in the method proposed by Marcoulaki and Kokossis (1999).Their application of stochastic optimization enables confidence in the optimization results, can afford particularly nonlinear reactor models, and is restricted neither by the dimensionality nor the size of the problem. Marcoulaki and Kokossis (1999) optimize the rich and inclusive superstructures formulated by Kokossis and Floudas (1990, 1994) to identify performance targets and to extract numerous design candidates that approach the targets. A reactor network superstructure containing three reactor units, each of which allows choices between CSTRs and PFRs with side feeding, is shown in Fig. 2.1. The superstructures generally feature complete connectivities between the reactor units, the feed streams and the reactor units, and the reactor units and the product. Structural restrictions that may result from practical consider-

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ations can be introduced as constraints for the optimal search. The stochastic search of the superstructures using the simulated annealing algorithm has been observed to systematically converge to the globally optimal domain, i.e., the performance targets of the system, and to produce numerous design alternatives with performances close to these targets for the numerous systems studied (Marcoulaki and Kokossis 1999). The stochastic optimization-basedreactor synthesis approach has been extended to multiphase systems to enhance its industrial relevance. Multiphase reactors are most frequently used in chemical processes besides fixed-bed reactors. A multiphase reactor network synthesis method needs to handle additional degrees of freedom that arise from the presence of multiple fluid phases in the system. When formulated as a superstructure optimization problem, the problem poses additional challenges to capture a significantly larger number of possible network configurations and to handle models for mass transfer and hydrodynamic effects. Mehta and Kokossis (1997) have proposed a compact representation of design options for multiphase reactors in the form of generic multiphase reactor units comprised by compartments and shadow compartments. A reactor module contains a single reactor compartment in each phase, which contains a single reactor unit, and can exist as either a CSTR or a PFR with side feeding. Each reactor compartment features diffusional mass transfer links with its shadow compartments in the different phases of the reactor unit. By combining different mixing and contacting patterns in compartments of different phases, a single multiphase reactor unit can represent all the conventional industrial reactors such as bubble columns, cocurrent and counter-current beds, and agitated reactors. By combining a number of such multiphase reactor units and their mixing and contacting options, it is possible to derive novel reactor configurations that can improve or enhance the performance of a multiphase reaction process. When embedded into a superstructure, a model can be formulated that contains all possible novel and conventional multiphase reactor design options from which the optimal candidates can be extracted using optimization techniques. Mehta and Kokossis (1997) adopted the stochastic optimization scheme proposed for single-phase systems (Marcoulaki and Kokossis 1999).The resulting methodology enables the development of performance targets and design options for multiphase systems.

Feed

Figure 2.1

RCTR 1 -

Single-phase reactor network superstructure representation.

RCTR 4

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Mehta and Kokosis (2000) have extended the approach to account for nonisotherma1 systems. Two approaches have been developed to handle temperature effects. Both approaches introduce elements associated with the manipulation of temperature changes. A conceptual approach (profile-basedapproach) imposes temperature profiles onto the reactor units, whereas a detailed approach (unit-based approach) introduces heaters and coolers into the reactor unit representations. The profile based approach allows one to determine the optimum temperature policies without considering the details of heat transfer mechanisms. As the profiles are imposed rather than simulated, the approach adds no hrther computational difficulties compared to the isothermal reactor network synthesis. The solutions obtained are easy to interpret and thus the approach helps in understanding of dominant tradeoffs in the problems. Results from the unit-based approach (UBS) provide the target that can be obtained from a network of adiabatic reactors with consideration of direct and indirect intermediate heat transfer options. The multiphase reactor network synthesis approach has been successfully applied to a number of industrial problems (Mehta 1998).For illustration purposes, consider the chlorination of butanoic acid as an example of a gas-liquid system. The chlorination of butanoic acid (BA) to monochlorobutanoic acid (MBA) and dichlorobutanoic acid (DBA) involves two reactions in the liquid phase.

+ Cl2 + MBA + HCl BA + 2 Cl2 + DBA + 2 HCl

BA

(1)

The reactions occur in the liquid phase and the gas phase contains chlorine feed and hydrogen chloride product. The modeling assumptions and problem data for this study are given in Mehta (1998). The study aimed at identifying reactor designs that can achieve the highest possible yield of MBA. The best conventional reactor design was observed to achieve a maximum yield of 74.4% (mechanically agitated vessel). Reactor network optimization of a superstructure identified a maximum yield of 99.6%. A number of designs were obtained from the optimization of a superstructure of three generic multiphase reactor units that achieve performances close to the target value. Designs range from simple designs employing only one reactor unit to designs featuring three units. Two simple designs are illustrated in Fig. 2.2. Despite their successes, current reactor network design methods still suffer from a number of shortcomings. As mentioned above, the attainable region is difficult to construct for problems containing a large number of reactions and components, local deterministic optimization-basedtechniques suffer from initialization and convergence problems, whereas stochastic optimization techniques such as the ones proposed by Marcoulaki and Kokossis (1999) and Mehta and Kokossis (1997) have proven to be fairly reliable but require long computational times to converge. Moreover, the optimization results are often incomprehensibly complex and impractical to implement, and the methods offer no insights into the key features that cause good performance in the designs identified. Research is in progress to address these issues. Most recently, a novel approach has been proposed that combines knowledge

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Gas Fe (Cf,)

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Liquid Product [BA+MBA+DBA+CI,+Cat)

Figure 2.2 Reactor designs for the chlorination o f butanoic acid (Mehta 1998).

acquisition tools with stochastic optimization methods to robustly and quickly address complex reactor design problems (Ashleyand Linke 2004). The method uses knowledge derived from reaction pathway information to gain an understanding of the system and devise a set of rules that are used to guide the optimal search. The approach was observed to consistently outperform the stochastic optimization-based methods in terms of search speed. Moreover, the knowledge gained during the search can be processed to provide insights into important features that determine a design with performances close to the targets of the system. 2.3.2 Reactive and Reaction-SeparationSystems

As indicated in Section 2.2, the combination of separation and reaction has given rise to a number of process intensification methods such as reactive distillation and reactive extraction. The appropriate integration of reaction and separation may lead to significant reductions in capital requirements, raw material waste and other operating costs. However, we would like to stress that the integration of these phenomena is by no means beneficial for all types of systems. There are strong and complex interactions between the various individual subsystems in a process flow sheet, in particular between the reaction and the separation systems that need to be considered in order to develop the most efficient processing options. The development of systematic synthesis tools for integrated reaction and separation systems is considerably more challenging than for the case where only one of these systems is considered as is the case in reactor network synthesis. As a result of these challenges, the

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area has received considerably less attention and the developments have mainly been confined to subproblems of integrated reaction and separation problems, such as reactor-separator-recycle, and specific reactive separation systems, such as reactive distillation column design. No general method capable of determining the most appropriate way of integrating reaction and separation is available, be it in a reactive separation process or in a process with separate reaction and separation units. We will review the existing approaches below before we describe in detail a method for integrated reactivelreaction-separation systems design. A first approach to reactor-separator-recycle process network synthesis was proposed by Kokossis and Floudas (1991),who extended their reactor superstructure formulations to account for the design of a separation train, in terms of determining separation task sequences, and the determination of recycle policies. Smith and Pantelides (1995)reformulated the problem to use more detailed separation process models in conjunction with process units that do not perform predefined tasks. An alternative approach was presented by Fraga (1996), who proposed a discretized reactor-separator synthesis problem that can be optimized using dynamic programming techniques. All presented approaches represent only systems with a single fluid phase, and can not account for reactive separation options. For the synthesis of specific reactive separation systems, a number of graphical techniques have been presented. Such methods aim at producing feasible rather than optimal design candidates and are available for reactive distillation, reactive extraction and reactive crystallization problems (Hauan, Westerberg and Lien 2000; Hauan, Ciric and Westerberg 2000; Okasinski and Doherty 1998; Barbosa and Doherty 1988; Ng and Berry 1997; Ng and Samant 1998; Nisoli, Malone and Doherty 1997). Ciric and Gu (1994) were the first to propose an optimization-based approach for reactive distillation column design with the number of stages and the feed locations being the structural decision variables. The problem was formulated as a MINLP and solved using a local deterministic optimization algorithm. Cardoso et d. (2000) later proposed the optimization of the same representation using stochastic techniques in the form of simulated annealing. A more general approach to reactive/reaction separation system synthesis has been presented by Papalexandri and Pistikopoulos (1994),who propose the optimization of superstructures of postulated units for reaction and separation using local deterministic optimization techniques. Additional applications in reactive and reactor/separation system synthesis using this method were reported in separate publications (Ismail, Papalexandri and Pistikopoulos 1999; Ismail, Papalexandri and Pistikopoulos 2001). The approach is limited by the multiphase reaction options it can handle and must cope with large MINLP formulations that include nonlinearities of the most general type. A general framework for the synthesis and optimization of processes involving reaction and separation was presented by Linke and Kokossis (2003a).The scheme employs rich and inclusive superstructure formulations of two different types of synthesis units and stream networks that allow for a conceptual as well as a rigorous representation of the chemical and physical phenomena encountered in general reaction and separation systems. Stochastic search techniques in the form of simulated

2.3 Computer-Aided Methods for Process lntensifcation Separation task feasibility constraints

* Mass exchange options (solvents, stripping agents,

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Figure 2.3 Strategy for reactive/reaction separation systems design (Linke 2001).

annealing and Tabu search can be applied to identify robust performance targets and a set of processing options from the superstructures that closely achieve the targets (Linke and Kokossis 2003b). The synthesis method is applied as part of an overall strategy (Linke 2001) as illustrated in Fig. 2.3. In the first instance, the available process design information for candidate reactive separation and mass exchange operations is incorporated into generic synthesis unit models. The proposed methodology utilizes two types of generic units, the reactor/mass exchanger (RMX) and the separation task (ST) unit, which are described below. Superstructures of these generic units are then formulated and the performance targets as well as a set of design candidates are obtained subsequently via robust stochastic optimization techniques. The overall synthesis strategy as well as the flexible process representation allows for iterations to incorporate the insight gained during the synthesis process. The two different types of synthesis units employed in the superstructures allow one to handle detailed information on reaction, mass and heat exchange (RMX unit) as well as conceptual information in the form of separation tasks (ST units). The generic RMX unit is a flexible and compact representation of the possibilities for phenomena exploitation in processing equipment. The RMX unit follows the shadow compartment concept described in Section 2.3.1 and consists of compartments in each phase present in the system. The streams processed in the different compartments of the generic unit can exchange mass across a physical boundary, which can either be a phase boundary or a diffusion barrier. Each compartment features a set of mutually exclusive mixing patterns through which a compact representation of all possible contacting and mixing pattern combinations between streams of different phases can be realized. With different combinations of mixing patterns in the compartments, the existence of mass transfer links between compartments, and decisions on the presence of catalysts, a variety of processing alternatives can be represented by a single generic RMX unit. For a vapor-liquid-liquidsystem featuring reactions in one heterogeneously catalyzed liquid phase, the six design alternatives in terms of phase interactions can be represented including a homogeneous reactor, a vapor-liquidand a liquid-liquid mass exchanger, a rectifier, a stripper, and gas-liquid-

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liquid reactors. Each of these instances represents a number of design alternatives resulting from different possible combinations of mixing patterns and flow directions in the compartments of the different phases. In contrast to the rigorous representation of reaction and mass transfer phenomena by RMX units, the ST units represent venues for composition manipulations of streams without the need for detailed physical models. In accordance with the purpose of any separation system, the separation task units generate a number of outlet streams of different compositions by distributing the components present in the inlet stream amongst the outlet streams. The ST units can accommodate for the different synthesis aims in the screening and design stages outlined in the previous section by different levels of component distribution and separation task constraints. Nonsharp separations arising from operational constraints on the separation tasks can easily be accomplished. The ST units can perform a set of feasible separation tasks according to the separation order to define a set of outlet streams. Depending upon the order in which the tasks are performed, a variety of processing alternatives exist for a single ST unit such as different distillation sequences performing the same separation tasks. The synthesis units are combined in network superstructures to provide for a framework that enables the simultaneous exploration of the functionalities of the different synthesis units along with all possible interactions amongst them. A number of synthesis units and a complete stream network are required to capture all different design options that exist for a process. Novelty is accounted for in the superstructures as the representations are not constrained to conventional process configurations but instead include all possible novel combinations of the synthesis units. The superstructure generation and the design instances that can be extracted from such superstructures are described in detail in Linke and Kokossis (2003a). The synthesis method has been applied to a number of case studies including reactivelreaction-distillation systems, reactive/reaction-extraction systems and bioreaction-separation systems (Linke 2001). For illustration purposes, we will discuss below an application in reactivelreaction-distillationand an application in the development of optimal wastewater treatment processes. The method was applied to synthesize process designs for ethylene glycol (C2H602)production from ethylene oxide and water (Linke and Kokossis 2003a). The model consists of two reactions as follows:

C2H40 f C2H602 +-C4H1003

(4)

The process goal is the production of 25 kmol/h of ethylene glycol with a minimum purity of 95 mol%. Ideal vapor and liquid phases are assumed (Ciric and Gu, 1994). The case study aimed at identifying the performance target and designs that minimize raw material and utility costs. Network superstmctures of four multiphase RMX units were generated accounting for vapor and liquid phases. The stream network includes complete intraphase as well as inter-phase connectivity amongst all compartments of the generic units present. Network optimization reveals a perform-

2.3 Computer-Aided Methodsfor Process lntensifcation

ance target of around 218Ok$/yr for the system, an improvement of 7% over the optimal reactive distillation column reported earlier (Ciric and Gu 1994). A number of designs were identified that exhibit performances close to the target. Figure 2.4 shows two such design alternatives of similar performance. In a different case study, the method was applied to design activated sludge processes (Rigopoulos and Linke 2002) using the comprehensive kinetic model available for activated sludge systems (Henze, Grady, Gujer et al. 1987). The study aimed at identifylng optimal schemes for combined oxidation/denitrification of wastewater that minimize both, the effluent COD and total nitrogen content (N). Superstructures were generated using RMX units to represent the possible single-phase and gas-liquid reactors and ST units to facilitate sludge separation. The study identified

Liquid

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b.l Figure 2.4 Strategy for reactive/reaction separation systems design (Linke 2001).

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Air f e d

Figure 2.5

Activated sludge process design.

designs that would achieve reductions by 97.4% in COD and by 84.9% in N. There is significant scope to improve the total nitrogen content reduction performances by adopting the novel designs delivered by this study. The new designs were observed to achieve improvements in reduction rates of total nitrogen content between 33% (Bardenpho process) and 80% (Ludzack-Ettinger process) over their conventional counterparts. A high performance design identified in this study is shown in Fig. 2.5. Details on the exploitation of the chemical and physical design insights are given in Rigopoulos and Linke (2002).

2.3.3 Membrane and Hybrid-Separation Systems

Membrane separation processes such as gas permeation, pervaporation, reverse osmosis, ultrafiltration, microfiltration, dialysis, and electrodialysis are frequently used in the chemical process industry. In addition, various membrane hybrid processes are used that couple membrane separation processes with other separation processes including adsorption or evaporation. Systematic design procedures for membrane separation systems have only emerged recently and it is common practice to compare a few known design alternatives in simulation studies. A large amount of work has been published in this area, a few of which are listed here. Stem, Perrin and Naimon (1984) and Stookey, Graham and Pope (1984) have analyzed the performance of single stage membrane permeators with recycle options. Mazur and Chan (1982) have studied multistage systems for natural gas processing and Kao (1988) have investigated recycle strippers and enrichers. Rautenbach and Dahm (1987) and Bhide and Stern (1991a,b) have performed economic feasibility studies for various membrane network configurations with and without recycle streams. Aganval and

2.3 Computer-Aided Methods for Process lntensifcation

Xu (1995, 1996) have provided broad guidelines based on process economics for two compressor cascades. Clearly, by their very nature, all of these approaches do not allow one to systematically identify the most efficient process option that may exist for a given membrane separation problem. Few approaches have considered the use of optimization technology to guide membrane network design selection. Tessendorf, Gani and Michelsen (1998) have presented various aspects of modeling, simulation, design and optimization of membrane-based separation systems modeled using orthogonal collocation. The proposed model can handle multicomponent mixtures and considers the effects of pressure drop and energy balances. Qi and Henson (1997, 2000) proposed an optimization-based approach that makes use of local deterministic optimization techniques to solve NLP and MINLP formulations for the design of gas permeation membrane networks using spiral wound permeators. Most recently, Marriott and Sorenson (2003a,b) have developed a general approach to modeling membrane modules considering rigorous mass, momentum and energy balances. Their approach constitutes a feed side flow model coupled with a permeate side flow model and a local transport model for the membrane system. They employ the detailed models in superstmcture formulations, which are optimized using genetic algorithms. Their application focuses on pervaporation systems and due to the complex models employed, the optimizations have proved to be extremely computationally demanding. A further membrane network synthesis approach has been proposed recently (Uppaluri, Linke and Kokossis 2004) that capitalizes on the developments in integrated reaction and separation process synthesis presented in the previous section (Linke and Kokossis 2003a) to develop a comprehensive gas permeation membrane network representation in the form of superstructures that make use of variants of the RMX unit presented previously. The superstructures capture all possible conventional and novel combinations of cocurrent, counter-current and cross flow gas permeation membrane units. The networks can be optimized using robust stochastic optimization techniques in the form of Simulated Annealing to extract those designs that exhibit the best economic performances. It overcomes major limitations of existing approaches as it allows one to quickly screen amongst all possible structural and operational process alternatives that may exist for gas permeation networks, to identify the performance limits of the system, and can accommodate for user preferences and problem specific modeling aspects. The approach can also be extended to account for hybrid separations by incorporating additional RMX and ST units in the superstructure formulations. The approach has been applied in a number of case studies in gas permeation network design as well as membrane hybrid network design. We will present two such applications below. In one case study, membrane networks are developed that exhibit minimum total cost for the recovery of hydrogen from synthesis gas using polysulfone membranes (Uppaluri, Linke and Kokossis 2004). High-purity hydrogen is required (99% H,) at high recovery. A number of optimal designs obtained from superstructure optimization for RMX units with different flow patterns are shown in Fig. 2.6. The designs using countercurrent membrane units exhibit the lowest network cost compared to

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networks with other flow patterns; however, the performance variations between various flow patterns is fairly small so that designs with cocurrent and cross flow patterns may be viable options if they are seen to offer operational advantages for this particular problem. In another case study, the superstructure optimization approach was applied to synthesize optimal hybrid adsorption/membrane process configurations for the

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sweetening of crude natural gas to meet pipeline specifications (Linke and Kokossis 2003a). Superstructures featuring three RMX units to represent the poly(etherurethane-urea) membrane units and three ST units to represent the irreversible fixed-bed adsorption process are optimized. The study sought to identify the most economic designs in terms of the total annualized network cost as well as to establish the influence of the hydrogen sulfide content in the feed gas on the performance targets of the membrane-absorption hybrid processes. The network optimization revealed a number of different processing options for different hydrogen sulfide feed concentrations. The optimal designs for the cases of lower hydrogen sulfide feed concentrations feature only networks of membrane units, whereas adsorptionmembrane hybrid systems were observed at higher concentrations. The dependence of the optimal total annualized process cost on the hydrogen sulfide feed concentrations is shown in Fig. 2.7. Details on the case study can be found in Linke and Kokossis (2003a).

2.3.4 Process-Solvent Systems

So far, we have presented a number of computer-aided methods for process intensification. These methods allow for the determination of the best processing schemes for a defined process system. Many processes make use of additional materials such as solvents to perform processing tasks. The process performance is strongly dependent on solvent properties and solvent selection is crucial to achieve optimal process intensification. Due to the strong interactions between the solvent and the process design, it is important to select the solvent and the process simultaneously. However, as process performance information is generally unavailable in the molecular design stage, available computer-aided solvent design methods (e.g., Marcoulaki and Kokossis 2000; Buxton, Livingston and Pistikopoulos 1999; Gani and Brignole 1983) design molecules for simultaneous optimality in a number of thermodynamic properties that are anticipated to have a significant effect on the process performance. A

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processing scheme is then designed in a subsequent stage for the solvent identified in computer-aided molecular design. Clearly, the success of such a design philosophy is highly sensitive to the formulation of the objective function and thermodynamic property constraints employed in solvent design and selection. The appropriate levels of importance of the thermodynamic properties used to judge the molecular performance are easily misrepresented without process performance feedback. If any important thermodynamic property effects are accidentally excluded, overestimated, or underestimated, the resulting process designs will inevitably underperform. As this is the case most of the time, there is a high risk of failure to provide optimal solutions to the overall design problem with such a sequential design strategy. Despite the potential for identification of improved designs, very few approaches have been reported that account for process performance feedback in molecular design. A possible design strategy has been proposed by Hostrup, Harper and Gani (1999). However, their method does not account for structural design interactions between computer-aided solvent and process design. This is a tribute to the combinatonal explosion problems faced when attempting to integrate process synthesis approaches with computer-aided solvent design approaches in a unified optimization model. A first approach to the simultaneous design process and solvent structures was proposed by Linke and Kokossis (2002).They developed a representation and optimization framework that exploits all possible molecular and process design options for solvent-processing reaction-separation systems. The proposed system representation takes the form of a process-molecule synthesis supermodel, which combines the superstructure based process design method (Linke and Kokossis 2003a) as presented in Section 2.3.2 to capture all possible processing options with a computer-aided design representation (Marcoulaki and Kokossis 2000) to capture all possible molecular structures of the solvent. The supermodels are then searched using simulated annealing to identify the optimal process-solvent system. Apart from the degrees of freedom associated with the process design options, the type and number of functional groups of the GC method that comprise the solvent molecules are treated as optimization variables during the search. The method was applied to the synthesis of reaction-extraction bioprocesses together with the corresponding solvents for (reactive) liquid-liquid extraction. The case study aimed at identifying optimal processes and solvents for the production of ethanol by fermentation as described by Fournier (1986).The objective for the study involved the identification of processing options that achieve maximum ethanol yields and glucose conversion while minimizing solvent flow rates. The objective function and problem data are given in Linke and Kokossis (2002). The classical fermentor design followed by a countercurrent liquid-liquid extractor using dodecanol as the solvent was taken as the reference design for the study. The performance target for this system was identified at an objective function value of 3.66 and the design achieved a glucose conversion into ethanol of 60.5%. By allowing process and solvent optimization, designs could be identified that achieved performance targets of 3135, a performance gain of three orders of magnitude. A number of process configurations together with different solvent molecules were revealed

2.3 Computer-Aided Methodsfor Process Intensification

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that allowed one to achieve near complete conversion of glucose (&Iu >w%)into ethanol at moderately increased solvent flows. The optimal process design options feature combinations of fermentor, reactive fermentor and extractors with recycles in the solvent as well as aqueous phases. More detail on the results of this study can be found in Linke and Kokossis (2002). Even though the results were impressive, serious numerical problems during the optimization were reported. The possible performance gains have stimulated additional research into a more numerically friendly method for the integrated design of processes and solvent. Such an approach was proposed recently by Papadopoulos and Linke (2004). They decomposed the design problem into a computer-aided solvent design and a process synthesis part. At the solvent design level, the design problem is reformulated as a multiobjective optimization problem in order to capture the relationships between physical properties and performance/environmental indices expected to impact on process performance and to extract solvent design candidates across the Paretooptimal front without having to make limiting assumptions prior to process design. The identified set of Pareto-optimal solvents is then introduced in the process synthesis task. A number of process synthesis tools can be employed in this approach, including the generic reaction-separationprocess synthesis scheme described in Section 2.3.1. The decomposition-based approach is illustrated in Fig. 2.8. Overall, this approach has been observed to be robust and quick for all cases studied, including liquid-liquid extraction, extractive distillation and reactivelreaction-extraction systems.

2.3.5 Water and Wastewater Systems

The previous sections have presented a number of computer-aided methods for the identification of process intensification options in reaction systems, reactive and reaction-separation systems, hybrid separation systems, and solvent-based process systems. Another area where computer-aided methods are useful for the identification of optimal processing schemes, is the design of industrial water-using systems.

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The design of such a system for the efficient use of the water resources is a complex problem involving a number of different tradeoffs. Apart from major process changes, e.g., replacing wet cooling towers by air-coolers, the efficiency of the water system can generally be significantly improved through a variety of practices that include water reuse, regeneration at different process stages, and recycling. The distinctive features of the wastewater minimization problem dictate the need for a focused rather than general purpose approaches. A major feature of the water system design problem in a typical industrial site is that the largest water users, e.g., cooling towers and steam system, are generally not mass transfer operations. Another advantage is the possibility that a limiting water profile can be constructed (Wang and Smith 1994),which allows the analysis of all water-using operations of water system on a common basis. This allows the combination of such insights gained from a graphical analysis of the problem with powerful optimization-based methods to systematically generate highly efficient water networks. In this section, we describe such an approach, which combines graphical water-pinch concepts with mathematical programming techniques in the form of superstructure models formulated as an MINLP problem (Alva-Argaez 1999). The optimization strategy employs a decomposition scheme and associates binary variables with the existence of different connections. The use of binary variables enables the approach to address complexity issues of the water network as it allows one to incorporate many practical constraints into the analysis, such as geographical constraints, flow rate constraints, and forbidden/compulsory matches. The formulation extends the domain of water pinch analysis with elements of capital cost to study the tradeoffs between freshwater costs, mass exchanger costs, and the cost of the required pipework. Consequently, the optimization results in cost-efficient networks rather than networks featuring minimum freshwater consumption. Former limitations of the water pinch method (Wang and Smith 1994) to address multiple contaminants are thus overcome since there are, in principle, no limits on the number of components or the number of freshwater sources. The design problem for a water-using system involves waterrelated elements in the form of: 0 0

a number of freshwater sources available to satisfy the demand water-using operations described by loads of contaminants and concentration levels.

The water system design problem with its sources and operations is sketched in Fig. 2.9. The design task is to find the network configuration to minimize the overall demand for freshwater (and consequently reduce wastewater volume) compatible with minimum total annual cost. The synthesis objective thus combines terms for low freshwater consumption, suitable network topology for water reuse, and low investment cost. The investment cost of the network includes piping costs and the approximate length of the pipe can be specified for each possible connection together with the materials of construction. The cost of mass exchange units assumes thermodynamic parameters and equilibrium relationships between the process streams and the water streams for the key contaminants, as well as the corresponding design equations and cost functions. The water-using system synthesis problem is initially

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presented based on the optimization of a superstructure representation. A natural decomposition of the problem is then developed based on the physical nature of the design problem as engineering insights gained through water pinch analysis. This allows the projection of the bilinear terms in the problem formulation following a recursive procedure which results in an efficient solution of the MINLP formulation. The method relies on the optimization of a superstructure model that facilitates all possible connections between freshwater sources and operations and different waterusing operations. The process streams in the mass exchange operations are considered implicitly in the superstructure model through the construction of the limiting water profiles. The number of units in the water network determines existing units and discrete options account for connections between sources and sinks of water. The superstructure is developed so that each freshwater stream entering the network is split amongst the water-using operations, each operation is preceded by a mixer fed by streams from the freshwater splitters and reuse streams emanating from the outlets of all other operations, and each operation is followed by a splitter that feeds the final mixer and the other operations in the system. The nonlinearities in the superstructure formulation are due to bilinear terms that appear in the mass balances (superstructure mixers and splitters), the nonlinear terms of the sizing equations, and the cost functions for the water-using operations. An example of a superstructure with a single freshwater source and two water-using operations is shown in Fig.2.10. Unlike some of the previous Mass Exchanger Network (MEN) developments (Papalexandri, Pistikopoulos and Floudas 1994), the superstructure provides options for water streams to merge. The minimum allowable composition difference used by mass exchanger network synthesis developments (Papalexandri,Pistikopou10s and Floudas 1994; El-Halwagiand Manousiouthakis 1989,1999a,b; Papalexandri and Pistikopoulos 1994)to avoid infinite sized mass exchangers is built into the limiting water profile data. In this approach, the given value is only a lower bound and it does not imply any preoptimized cost tradeoff between capital and operating costs. A customized solution strategy for the superstructure optimization problem was developed by making use of insights gained from water pinch analysis. From water pinch analysis it is known that for all water-using operations, at least one of the contaminants will reach its maximum permissible value in the optimal solution. If this was not the case, the flow rate through a particular unit could be decreased further

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Figure 2.10

Water network superstructure.

to decrease the water consumption, and disprove optimality. Since the mass load of an operation is fixed, the outlet concentrations must be maximized to reduce the water flow. Motivated by this observation and provided all outlet concentrations are set to their maximum levels in the water-using operations, an upper bound can systematically be obtained for the performance of the system. The observation is further exploited with the development of a “natural”decomposition strategy that is explained next. The original MINLP problem is decomposed into two related subproblems that are solved sequentially within an iterative procedure. A primal problem (PI) is developed in a first instance by projecting the nonlinear constraints onto the concentration space according to the observations in the previous section. The projection is consistent with the objective of minimum freshwater demands. Given that the “limiting” contaminants define the water demand in the operations, the fact of having a set of nonlimiting contaminants far away from the initial assumption does not detract from the quality of the solution obtained. Experience with the procedure shows that in numerous multicontaminant problems the minimum freshwater demand is identified under these conditions and the calculation of the outlet concentrations of nonlimiting contaminants is not significant. However, the projection is likely to correspond to an infeasible solution as not all contaminants will be able to reach their maximum level for a given mass load and water flow rate. Thus, there are limiting contaminants which achieve their maximum level and determine the water demand for the operation and nonlimiting contaminants that exit the operation at a lower concentration and do not affect the water consumption in the particular operation. The cost functions are linearized in problem (PI), which then becomes a mixedinteger linear programming (MILP) problem that can be solved to global optimality using any reliable branch and bound algorithm. Once the flow pattern corresponding to the optimal solution of problem (PI)is identified, a second problem (P2)can be solved to find the set of concentrations corresponding to the design obtained. Problem (P2) is formulated as an LP model that includes mass balance equations projected onto the flowrate space with respect to all flows in the network. The exit concentrations of all contaminants are the variables of the problem.

2.3 Computer-Aided Methodsfor Process Intensification

The overall procedure constitutes a robust methodology as will be illustrated below on the basis of an example in the design of a water system for an oil refinery. The mixed integer models allow one to consider the system-wide interactions between design decisions. In particular, the models can consider tradeoffs between various design alternatives, striking a balance between increased design expenditures and resulting improvements in the system's operation such as decreases in operating cost. The petroleum and petrochemical industry are heavy users of water. Oil refineries not only face the challenge of reducing the amount of waste generated, but also a fierce competitive market with other water users. Water reuse and recycling are widely practiced in petroleum refining operations; however, typically no systematic methods for exploring new design alternatives to make the most of water reuse and recycling are used. The presented methodology was applied in an oil refinery case study that considered a number of water users, namely an atmospheric distillation unit (CDU), vacuum distillation operation (VDU), hydrotreating (HDS), crude desalting (desalter), cooling tower (c.tower), boiler house, delayed coker, and other operations grouped together (others). The study considered three contaminants: hydrocarbons (HC), hydrogen sulfide (H2S)and salts (salts). The problem data, equipment cost correlations, and modeling assumptions are given elsewhere (AlvaArgaez 1999). The total annual cost of the design proposed without applying optimization techniques is 4,430k$/yr. This base case already accounts for some water reuse as the emuent from the CDU steam stripping feeds the desalter. The design corresponding to the base case is illustrated in Fig. 2.11. The problem was solved using the presented methodology considering simultaneously the freshwater cost, the cost associ-

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ated with the mass exchangers and the piping cost. The optimal design (Fig.2.12) was found after about 30 s of CPU time on an antiquated Pentium I11 processor. The optimized design incurs a total annualized cost of 4,040 k$/yr, corresponding to cost savings of about 10% as compared to the base case.

2.4 Concluding Remarks

There is constant pressure on the chemical process industry to improve processes and to create new process routes in order to remain competitive in a global market place and to comply with environmental regulations. This translates into a constant need to develop new, more efficient process technologies. Process intensification technologies have proven their value in improving process efficiencies. Most process intensification technologies, in terms of equipment and process concepts, have been invented through the ingenuity of a small number of individuals. Systematic computer-aided approaches have the capability of enhancing the innovation process significantly by offering valuable decision-support to the process designer. This chapter has presented important process intensification technologies as well as a number of the available computer-based methods that allow the systematic generation of intensified processes for reaction systems, reactive/reaction-separation systems, membrane (hybrid) separation systems, process-solvent systems and water and wastewater systems. A major advantage of all the presented methods is their ability to determine robust performance targets and to identify a variety of design

References I 3 2 5

options with similar close-to-targetperformances. By providing this information, the design tools offer invaluable decision support to the design engineer as it will allow the inspection of similarities and differences between high-performance candidates in order to select the most practical and novel process designs. Most of these methods have only recently been developed in academia and a close collaboration between academia and industry is essential to mature the technologies so that they can be used routinely by design engineers throughout the world. References 1 Achenie, L. E. K, Biegler, L. T. (1986) Jnd.

21 Feinberg, M., Hildebrandt, D.(1997) Chem.

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40 Kokossis,A. C., Floudas, C. A. (1991) Chem. Eng. Sci. 46, 1361. 41 Kokossis,A. C., Floudas, C. A. (1994) Chem. Eng. Sci. 49, 1037. 42 Lakshmanan, L., Biegler, L. T. (1996) Ind. Eng. Chem. Res. 35, 1344. 43 Lakshmanan, L., Biegler, L. T. (1997) Comp. Chem. Eng. 21, 785. 44 Linke, P., (2001) Dissertation, UMIST, UK. 45 Linke, P., Kokossis,A. C. (2002) Comput Aided Chem. Eng. 10, 115. 46 Linke, P., Kokossis,A. C. (2003a) AIChE]. 49(6), 1451. 47 Linke, P., Kokossis,A. C. (2003b) Comp. Chem. Eng. 27(5), 733. 48 Marcoulaki, E. C.. Kokossis,A. C. (1999) AIChE]. 45, 1977. 49 Marcoulaki, E. C., Kokossis,A. C. (2000) Chem. Eng. Sci. 55, 2547. 50 Marriott, ]., Ssrensen, E. (2003a) Chem. Eng. Sci. 58, 4975. 51 Marriott, I., Ssrensen, E. (2003b) Chem. Eng. Sci. 58, 4991. 52 Mazur, W. H., Chan, M. C. (1982) Chem. Eng. Prog. 78, 38. 53 Mehta, V. L., Kokossis,A. C. (1997) Comp. Chem. Eng. 21, S325. 54 Mehta, V. L., Kokossis, A. C. (2000) AIChEJ. 46, 2256. 55 Mehta, V. L. (1998) Dissertation, UMIST, U.K. 56 Moulijn,]. A,, Kapteijn, F., Stankiewicz,A. (2004) In: Stankievicz, A,, Moulijn, I. A. (eds.) Re-engineering the chemical processing plant-process intensification. New York: Marcel-Dekker, 191-226. 57 Ng, K. M., Berry, D. A. (1997) AIChE]. 43(7), 1737. 58 Ng, K. M., Samant, K. D. (1998) AIChE]. 44(6), 1363. 59 Nisoli, A., Malone, M . F., Doherty, M . F. (1997) AIChE]. 44,1363. 60 Okasinski, M.. Doherty, M . F. (1998) Ind. Eng. Chem. Res. 37, 2821. 61 Papadopoulos, A., Linke, P. (2004) Comput Aided Chem. Eng. 18, 259. 62 Papalexandri, K. P., Pistikopoulos, E. N. (1994) Comp. Chem. Eng. 18, 1125. 63 Papalexandri, K. P., Pistikopoulos, E. N.,Floudas, C. A. (1994) Chem. Eng. Res. Des. 72, 279.

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Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

3 Computer-aided Integration o f Utility Systems Franqois Marechal and Boris Kalitventzeff

3.1 Introduction

The integration of utility systems concerns the way energy entering a production plant will be transformed into useful energy for satisfymg the needs of the production processes. In order to identify the best options for the utility systems, one has to consider the chemical industrial plant as a system (Fig. 3.1) that converts raw materials into valuable products and by-products.The production is realized by a list of interconnected physical unit operations that will together form the chemical process. These transformations are made possible by the use of energy and support media, like solvent, water, and catalysts. The transformations are not perfect and

Raw materials 4

nWY roducts y-products

Waste heat Emissions

Water

Solids

Figure 3.1 The process as a system that converts raw materials into products and by products

ComputerAided Process and Product Enpneenng Edited by LUIS Puigjaner and Georges Heyen Copynght 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhelm ISBN 3-527 30804-0

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3 Computer-Aided Integration ofUtility Systems

therefore the process results in the production of wastes. Wastes are emitted in severa1 forms: 0 0 0

waste heat in the cooling water or the air; emissions as combustion or other gases (e.g., humid air, steam vent, etc.); liquid streams as polluted water or solvent; solids.

Emissions are regulated or taxed, so the waste emissions should reach levels that meet waste processing strategies. When performed on site, waste treatment integration offers opportunities for reducing the treatment cost by material recycling, waste energy recovery or by using process waste heat to improve the treatment efficiency. In this chapter, the utility system will be considered as being composed of different subsystems that serve the following production processes: 0

0 0 0 0 0 0

energy conversion; compressed gases: air, Oz, Nl, HI,etc.; cleaning in place; air conditioning and space heating; catalyst recovery; solvent recovery; water treatment; effluent treatment; etc.

We will, however, focus our presentation on energy conversion and water integration. Referring to Fig. 3.1, the goal of the engineer is to maximize the efficiency of the horizontal transformations from raw materials to products by minimizing the use of resources in the vertical transformations, leading by balance to the reduction of the emissions. The process integration techniques based on the pinch analysis (e.g., [I])have mainly focused on the definition of the minimum energy requirement (MER), expressed as the minimum (useful) heat requirement, and the definition of the optimal heat exchanger network (HEN) design that will realize the energy recovery between the hot and cold streams of the production processes at a minimum cost, achieving in this way the best tradeoff between the heat exchangers investment and the energy savings. In this approach, utilities are not really considered since they only appear as a way of supplying the minimum energy requirement. Using the analogy between temperature and concentration and between heat and flow rate, water usage may be tackled using a similar approach (e.g., [2]). We will therefore first present the methods for integrating the energy conversion and consider after the water usage aspects. The role of the energy conversion subsystem is to supply the energy requirement of the process at a minimum cost or with a maximum efficiency. This means converting the available energy resources into useful energy (energy services) and distributing it to the process operations. From the process synthesis perspective, the problem has five aspects:

Introduction I 3 2 9

0 0

the definition of the most appropriate energy resources; the selection of the conversion technologies; the definition of the most appropriate size of the equipment considered for the system (i.e., the investment); the definition of the most appropriate way of operating the conversion system; the definition of the utility-process heat exchanger network (energy distribution and interface between the utility system and the process).

In order to increase the efficiencyof the energy conversion system, the rational use of the energy resources will be obtained by the combined production of different services (polygeneration),for example, combined heat and power, combined production of hydrogen, and steam or refrigeration. In a process, the utilities are considered as a service provided to the production units. The control and the reliability issues are also part of the problem. For example, steam condensers are placed in such a way that these will control the target temperatures of the process streams. When the investment is made, the optimal management of the utility system will cover the aspects of exploiting market opportunities (e.g., electricity prices) and using at best the energy conversion equipment. The scheduling and the optimal management will then be an issue for the process. The role of the utilities becomes even more important when considering multiprocess plants being served by one centralized utility system. In this case, the utility system will transfer heat between processes. Considering the variations of demand over time, the definition of the optimal system becomes a multimodal and multiperiod problem where the best sizes of the equipment will have to be defined considering annualized value on a yearly production basis. In this perspective, another dimension to consider is the influence of the ambient conditions that will influence both the process demands and the conversion process efficiency (e.g., the influence of the ambient temperature on the refrigeration cycle coefficient of performance (COP)).

3.1.1 Defining the Process Requirements Using the Utility System

In an ideal situation, the definition of the hot and cold streams of a process is obtained combining the use of data validation (for existing processes) or simulation tools (for new processes). The data of an existing process are, however, not always available for the engineers doing the process integration study. This is particularly the case when the chemical production site utility system is managed by an energy service company that is different from the production companies on the site. Furthermore, if utility systems are easy to instrument, instrumenting processes are not always so easy, particularly when dealing with processes in the food industry. Taking advantage of the utility system instrumentation, it is possible to deduce the definition of the hot and cold streams of the process from the data collected on the utility system. Data validation and reconciliation tools (see Chapter 2 in Section 5 of this book) will be used to combine online measurements with other information

330

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3 Computer-Aided Integration ofutility Systems 0.4 r

Technology requirement

Thermodynamic

0

(1

SU

I1x)

requirement

150

100

250

QWW Figure 3.2 Dual representation of a heat requirement representing a stream heating by steam injection

from the specification sheets of the processes and obtain a coherent picture of the process. Correctly defining the temperatures and the heat loads of the hot and cold streams is essential for a proper process integration study. For this reason, the first step of the analysis is defining the operations required to transform raw materials into the desired products. The heating or cooling requirements are inferred from the operating conditions. In this respect, the MER may be computed in two different ways. The first (thermodynamic requirement) consists of determining the temperature profiles of the process streams that maximize the exergy supplied by the hot streams and minimizes the exergy required by the cold streams. The second (technological requirement) is to consider the equipment used to convert utility streams into useful process heat. Those two approaches produce the same overall energy balance but with a different temperature profile. The shape of the composite curve may differ from one representation to another. An example of this dual representation is shown in Fig. 3.2 for the case of water preheating by steam injection. The thermodynamic requirement corresponds to water preheating from its initial to its target state, while the technological requirement corresponds to the production of the injected steam. When using the Carnot Factor (1-TO/T) as the Y axis, the area between the two exergy composite curves corresponds to the ,,thermal" exergy losses due to the technological implementation of the operation. Following a systematic analysis [3], most of the process requirements may be defined from the knowledge of the process-utility interface.

3.2 Methodology for Designing Integrated Utility Systems

The process integration techniques aim at identifjmg the maximum energy recovery that could be obtained by counter-current heat exchange between the hot and cold

3.2 Methodologyfor Designing Integrated Utility Systems

600 550 SO0

450 eat recovery

400

/

350 300

250

0

5Mx)

IOOM)

15ooO

2OOOO

25000

3a700

Q(kW Figure 3.3

Hot and cold composite curves of the process

streams of the process. This technique, based on the assumption that a minimum temperature difference between the hot and the cold streams (ATmin), allows the calculation of the so-called MER target for the system. The identification of the pinch point, the point where the hot and cold composite curves of the process are the closest, is further used to design the heat recovery heat exchanger network structure. Using the concept of the hot and cold composite curves (Fig. 3.3), it is possible to compute the MER graphically. Mathematically, the minimum energy requirement is computed by solving the heat cascade model (3.1). This model is based on the definition of the corrected temperature list. The corrected temperatures are obtained by A Tmin and increasreducing the inlet and outlet temperature of the hot streams by 2 ing the temperature of the cold streams by - By assuming that the streams 2 have constant cp, the fluid phase changing streams are divided into stream segments with constant cp. In a more detailed study, the value of A Tmin will be related to the 2 heat film transfer coefficient, allowing for an account of a heat transfer resistance that depends on the fluid type considered. The heat cascade model (3.1) is a onedegree-of-freedomlinear programming problem that computes the energy required to balance the needs of the cold streams when recovering the maximum energy from the hot streams by counter-current heat exchange and cascading heat from the higher temperatures. The energy balance is written for each temperature interval. The grand composite curve (Fig. 3.4) is the plot of the heat cascaded as a function of the temperature:

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3 Computer-Aided Integration of Utility Systems

550

500

450 h

w

m

350 300 250

L

.

.....

_....-..... ....,-.... .................... 1......... ......... Cold utitizy :6948 kW ~

2ooo

0

6000

4000

QWW Figure 3.4

8 W

10W

12OOO

Grand composite curves of the process

subject to heat balance of the temperature intervals Vr=l,

kQ,,r+Rr+l-Rr=O

.... n,

(3.4

i=l

Rr

>0

V r = I,

.... n, + 1

(3.3)

where n is the number of specified process streams; n, is the number of temperature intervals; R, is the energy cascaded from the temperature interval r to the lower temperature intervals; Q,, is the heat load of the reference level of process stream i in the temperature interval r; Q, > 0 for hot streams and 5 0 for cold streams. The heat cascade constraints (3.2) is the equation system that is solved by the problem table method. An alternative set of equations (3.4) may be used to compute the heat cascade. This formulation has the advantage of involving only one R, per equation, with each equation being related to one temperature in the temperature list. From the analysis of the pinch point location, it may be demonstrated that the list of temperatures (and therefore of equations) may be reduced in this case to the list of inlet temperature conditions of all the streams:

2 (2

Q.k)

k=r

i=l

+ %,+I

- Rr = 0 V r = 1, .... n,

(3.4)

When considering the overall system (including the utility streams), it is necessary to define the complete list of streams to be considered in the system, including the hot

3.2 Methodologyfor Designing htegrated Utility Systems

and cold streams of the utility subsystem, prior to any heat exchanger network design. Compared to the conventional pinch analysis where hot and cold streams of the process have constant temperature and flow rates, the utility system integration has a much larger number of degrees of freedom since it requires the definition of the temperatures and flow rates of the utility streams that will minimize the cost of the energy conversion. This will be dealt with using modeling and optimization techniques. The modeling of energy conversion units will be used to determine the operating temperatures and compositions allowing the definition of the hot and cold streams of the utility subsystem. The flow rates will be determined by optimization in order to minimize the cost of the energy delivered. The constraints of the heat cascade will be considered in the problem and the solution will be characterized by a list of pinch points, one of these being the process pinch point, representing the maximum energy recovery between the process streams, the others corresponding to the maximum use of the cheapest utility. If, in the simplest cases, the calculation of the utility streams may be done graphically, it is more convenient to use optimization techniques to solve the problem, especially when cycles like steam network or refrigeration system integration is considered. There exist several ways of solving the optimal integration of the energy conversion system. All strategies are based on the definition of a utility system superstructure, which includes the possible conversion technologies that are envisaged. Although it is possible to set up a generic problem that would state and solve the problem in an automatic manner, it is more convenient to proceed by successive iterations, keeping in mind that learning from one step will result in new problem definitions and perhaps new ideas for the integration of alternative energy conversion technologies. This is particularly true because the problem definition is usually not known from the beginning and because the utility system integration may influence the process operating conditions. The philosophy behind the computer aided utility system integration is to have a method that supports an engineer’s creativity, helping him or her to identify the most promising options. Three major aspects have to be considered: 0

0

0

Technology data bases including thermoeconomic models of the different conversion technologies available. These models will be used to constitute the energy conversion system superstructure consistent with the technologies available on the market and the process requirement. An optimization framework for targeting the optimal integration of the utility system prior to any heat exchanger network design. The optimal utility system integration is by definition a mixed-integer nonlinear programming problem (MINLP), the integer variables being used to select in the superstructure the equipment to be used in the final configuration, while the continuous variables will be the operating conditions and the flow rates in the utility system. Graphical representations applying thermodynamic-based principles to assess, analyze, and understand the solutions obtained by optimization and to help in a

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3 Computer-Aided Integration ofUtility Systems

possibly new definition of the superstructure. Graphical representations are used to support the engineers when stating the problems and analyzing the results.

3.3 The Energy Conversion Technologies Database

When considering energy conversion technologies, we switch from the energy dimension to the thermoeconomic dimension, where cost of energy and the investments are considered simultaneously. It is therefore necessary to represent the market state by introducing market-related relationships between cost, sizes, and efficiencies. There is therefore a need to develop a technology database for the different conversion technologies. Today, Web-based techniques give access to the needed information [4].The required data are as follows: 0

Investment costs refer to the installed cost. It is computed by:

where C 1, C P, C C, C E, C G, C 0, 0

0

0

0

0 0 0

is the installed cost of the equipment e is the purchased cost of equipment e; is the cost of connections and piping; is the cost of engineering for equipment e; is the cost of civil engineering for equipment e; is the other costs like taxes, land, etc.

maintenance cost required to operate the technology on a yearly basis; operating costs that refer to the manpower and the consumables related to the use of the technology; The fuel consumption and the type of fuel concerned. This usually refers to the thermal efficiency computed by qh = useful heat (kJ). Ideally, efficiency should LHV (kj include partial load information. the electricity consumed or produced; the hot and cold streams that define the energy service delivery (process/utility heat exchange interface); the standard prescriptions for the technology specification; any information concerning the technology implementation; time to market information for emergent technologies; the list of suppliers.

From data collection using a technology market study, correlation equations are obtained. These give, as a function of the size parameters, the required values for the optimal integration models. For the efficiency correlation, it is important to analyze the degrees of freedom in order to avoid developing a correlation that would be inconsistent with the rules of thermodynamics. For this reason it is usually preferred to use simulation models for the technologies in which the model parameters are defined as correlation functions, e.g., isentropic efficiency as a function of the tur-

3.3 The Energy Conversion Technologies Database

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bine size. Two types of approach may be used. The first aims at representing the technology market by functions that correlate the model parameters with technology design parameters like temperature, pressure, or size. The model has the following generic form: Heat and mass balances B ( X , S, P ) = 0; Efficiency equations F ( X , S, P ) = 0; Cost correlations cr = a ( x ,s, PI; Correlations limits Smin I :S i Srnax; where X S

is the list of state variables characterizing the streams of the technology; is the set of sizing variables of the technology; is the set of characterizing parameters of the technology identified from the market database correlations.

P

When detailed thermoeconomic models are available, the efficiency equations will become more complex. Examples of such relationships may be found in [S],[GIor [7]. With this approach, it is assumed that the technology may be custom designed. It applies well, therefore, to steam turbines, heat pumps, or heat exchangers. The second approach considers that the technologies are available on the market with futed sizes and operating design conditions. This is the case, for example, for the gas turbines market in which a limited number of models are available. In both situations, the database data are used to calibrate the thermoeconomic models by computing the model parameters in the standard conditions in which the reference

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5 000 1

y = 11554x0

4 000

='' x

A

0

x +

1

10

100

1000

Electrical Power [kWe]

Figure 3.5 Comparison o f different cost correlations for diesel engines in cogeneration applications

ASCCC (cogen).MAN (C) Exsys (cogen) MTU (C) MTU (P) El-Nashar (mot)

FAA Chandalar Caddet (cogen) ListedLeroy Somer

10000

100000

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3 Computer-Aided Integration ofUtility Systems

data have been collected. The parameters are then used to compute the system performances, including part load efficiencies, in the operating conditions of the plant under study. Cost correlationshave to be used with caution considering the conditions, the area, the date, and the ranges for which they have been established. Consider for example the results of Pelet [8] shown in Fig. 3.5. The different correlations obtained from the literature and other market surveys are compared and show big differences. Tables 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7 give the values and correlation for the major energy conversion technologies. These data are representative of the European market in 2000. They have been gathered mainly by the partners of the European project EXSYS ([9],[4]). In order to update the cost given by the correlation, a plant or equipment index is used. The most important are the Marshall Swift index tables and the chemical engineering plant cost index (CEDCI),whose values are regularly updated in the Chemical Engineering journal. In the tables, all data are given with a CEPCI index of 382. For other equipment, usefil references like [lo] may be used. When data are available from other sources, e.g., from quotations, the effect of size is represented by the relation (3.7), the exponent a, may be obtained from different sources like [lo] or [11]:

is the installed cost of the equipment e of size S, in the year y . is the size of the equipment e. The size of the equipment is the relevant sizing parameter that mainly influences the cost of the technology. For this reason, when estimating the cost of a heat exchanger, the heat exchange area is preferred over its heat load. is the size of the known equipment similar to e. Se, is the cost of the known equipment. c Lf C E P C I ( y ) is the chemical engineering plant cost index for the year y. the year of the reference cost C res Yref

where C I, (S,,y) Se

Table 3.4

Gas engine, lean burn configurations. We: power in kW.

Gas engines, lean burn

Generator eff.

[-I

Mechanical eff. Engine cooling eff. @ 90'C Heat eff from combustion gases NO, emissions CO emissions Installed cost Catalyst cost Maintenance cost Lifetime

[-I [-I [-I

[mg/m3N] [mg/m'N] [El [El [E/kWhI

[hl

qgen= 0.015 . In(WJ + 0.8687, W,5 845 PW] qsn = 0.0013 . In(Wc)+ 0.9611, We> 845 [kW] I?= 0.2419 . ~ 0 6 " ql~,,@ = 0.7875 . W;o.'682 qth,=hap = 0.1556 . P I 3 250 650 C,",, = 0.026G . Wf+ 578.84 . We + 208174 85 @ Cm,,,= 0.0407 . W;0~2058+ 0.0034 48000

3.3 The Energy Conversion Technologies Database Table 3.5

Diesel engines. We electrical power in kW.

Diesel engines

Generator eff.

[-I

qS,,= 0.015 . ln(W,) + 0.8687, We5 845 [kW]

Mechanical eff. Engine cooling @ 90°C Heat eff. from combustion gases

[-I [-I [-I

qm

q f h , 4= 0.2875 . W g 0 O ” ’ l)h,chp = 0.5433 .

NO,

co

[mg/m3N] [mg/m3N]

100 400

Installed cost

[El

Catalyst Maintenance Lifetime

[El

C,,,, = -0.0266 (3.4. We)’ + 578.84 3 ~ 4We + 208174, We> 1000 C,,,, = 1147.62 . We> 1000 [kW] 136 . W, C,,,,, = 0.0407 . 48000

Table 3.6

q,, = 0.0013 . In(We)+ 0.9611, We> 845 [kW] = 0.0131 . h ( W J + 0.3452

[E/kWhI [hl

em, wss

Aeroderivative and heavy duty gas turbines.

Aeroderivative gas turbines: Weelectrical power in kW

[-I [-I [-I

[mg/m3 N] [mg/m’N]

[El Lifetime

[hl

‘lee.= 0.015 . In(W,) + 0.8687, W,5 845 IkW] qgen= 0.0013 . h ( W J + 0.9611, W,> 845 [kW] qmec= 0.0439 ln(W,) - 0.0684 q[h,&ap = 0.838 . ~ O S s 7 80 50 Cru,brnr = 1564. ’ i!4’”’, We < 50000 [kW] Ctu,brnr = 2977. ’ 77’1, W,> 50 000 [kW] 48 000

Heavy duty gas turbines: Weelectrical power in kW

vet,,= 0.015 . ln(W,) + 0.8687, We5 845 [kw]

[-I

[-I [-I

[mg/m’N] [mg/m3N]

[El Lifetime

[hl

Auxiliaries: Weelectrical power in kW,

Qth

qge.= 0.0013 . ln(We)+ 0.9611, W,> 845 [kW] q- = 0.0187 . ln(W,) + 0.1317 q t h a h p = 0.7058 . eo31r 50 50 ctu,blnr = 4786. . We < 50000 [kW] Crur(nne = 2977. . ~ 7 7 ’ ’ , We > 50000 IkW] 55 000

e7”’,

heat recovery boiler heat load in kW

Reference cost

IEI

c,, =

Recovery boiler

[El

Cb&r

cturbinc

(0.0503 . In =

Cboikr=

( L + 0.3208)+ ) 1000

(125.5 - 0.4 ’ )-( 0.2436 .

(Qfk

iv,

1000 [kw])”““’

. c,f

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3 Computer-Aided integration ofUtility Systems Table 3.6

(Continued)

Connection charges

C,

[El

=

C,,,= (0.0494 - 0.0047 . In

Instrumentation

(6)) . ,C,

. (L . C,ef )

Civil engineering

[El

CG.c= (0.1232 - 0.0005

Engineering

[El

Cc,= (0.1211 - 0.000735

[El

CdlV = (0.1403 - 0.0142 In

Whl

C,,,,int = 8.15

Maintenance €cts

. C,f

(0.09318 - 0.00011 . (*)

1000

. (A) . C, 1000

(F))w,

. C,I

. Wi0'3m1

Table 3.7 Thermoeconomic characteristics of industrial heat pump systems. The sizing parameter is the heat delivered Q h in kwth, the cost is expressed in €/kwh computed by Investment = a . (Qh)b, source

contribution of TNO in (9).

45 Electric compression heat pump Mechanical vapor recompression 30 Absorption heat pump NH3/H20 50 Absorption heat pump LiBr/H20 50 Absorption heat transformer LiBr/H20 50 Thermal vapor recompression 20

110 200 150 150 150 180

10- 3000 814 250-50 000 663.5 5-60000 810.2 5-60000 810.2 250- 4000 1164.8 15-50000 268.56

-0.327 -0.3925 -0.3154 -03154 -0.288 -0.4832

45% Carnot 45 % Carnot 1.4 (COP) 1.6 (COP) 0.45 (COP)

~~~

Table 3.8

Phosphoric acid fuel cells. PAFC

fld flfh

NO,

co

Installed cost Maintenance Lifetime

[-I [-I

0.35-0.4 0.25

1611

[PPml [PPml [WWI [€cts/kWh]

0

0 4000 1 40 000

1621 [621 [611 [611 ~ 3 1

-

I611

PI

Table 3.9 Solid oxide fuel cells. SOFC fld

7th

NO,

co

Installed cost Maintenance Lifetime

1-1 1-1 IPP4 [PPd F/kWI [€cts/kWh]

[hl

0.5 0.35 < 0.2 0 450 (long term) - 1500 1 > 20 000

~ 4 1 1641 [65] [611 1661

3.4 Graphical Representations Table 3.10

Proton exchange membranes.

PEMFC

-

9d 9th

1-1 [-I

0.3-0.4 0.5-0.45

NO,

[PPml IPPml [e/kWl [€cts/kWh] lhl

-

co

Installed cost

Maintenance Lifetime

[611

-

- 500 (long term) - 1000

1 87 600

1671, ~1 [611 1671

The total cost of the energy conversion system is given by (3.8):

where mf 0) Cf@)

is the flow rate of fuelfduring the period p [kg/s]; is the cost of fuelfduring the period p [Elkg]; W e l (p), Weel, @) is the electrical power imported (exported)during period p [kWl; C el (p), C el, (p) is the cost of electricity for import (export)during the period P [E/kJl; d @) is the duration of period p [slyear]; C me is the maintenance cost of equipment e [E/year]; ne is the number of equipment pieces; C re is the installed cost of equipment e [El; i is the interest rate for annualization; nP is the expected lifetime of the installed equipment e [year].

3.4 Graphical Representations 3.4.1 Hot and Cold Composite Curves

From the beginning, composite curves have been used to explain integration and identify energy savings opportunities. In most representations, temperature is a topological indicator in the sense that it allows one to pinpoint the process operations concerned with the pinch points or the pseudo-pinch points. The hot and cold composite curves mainly concern the process streams. They are used to quantify the possible energy recovery by exchanging heat between the hot and cold process streams. Four zones are of importance in the hot and cold compos-

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3 Computer-Aided integration of Utility Systems

ite curves (Fig. 3.3). On the right, we visualize the hot utility requirement. The heat recovery zone represents the possible heat recovery by an exchange of the hot and cold streams of the process. The remaining heat of the hot stream has to be evacuated using a cold utility. The left part of the graph therefore defines the cold utility requirement. The latter is divided between cooling requirements above the ambient temperature and the refrigeration requirements below the ambient temperature.

3.4.2 Grand Composite Curve

For analyzing the energy conversion system integration, the grand composite curve (Fig. 3.4) will be used. It describes as a function of the temperature, the way energy has to be supplied to or removed from the system. The process grand composite curve is divided into three zones. Above the process pinch point, the system is a heat sink to be supplied by a hot utility. Below the process pinch point and above the ambient temperature, the process is a heat source. The heat has to be evacuated from the process by a cold utility or used in another energy consuming subsystem like another process or a district heating system. Below the ambient temperature, the process requires refrigeration. The feasibility rule of the utility integration is that the grand composite curve of the utility system should envelop the process grand composite curve. Resulting from the linear nature of the composite curve calculation, the optimal integration of the utilities will result in the definition of utility pinch points (the intersection between the utility composite and the process composite). Each of them will correspond to the optimal use (maximum feasible flow rate) of the cheapest utility stream. Above the pinch point, the grand composite curve represents, as a function of the temperature, the heat that has to be supplied to the process by the hot utility. When ignoring the pockets in this curve, the process appears as a cold stream to be heated up by the hot utility. Knowing the temperature-enthalpy profile of the

Figure 3.6 Computing the flow rate of the hot utility

3.4 Graphical Representations

utility, it is possible to determine the flow rate of the utility stream by activating the appropriate pinch point, as illustrated in Fig. 3.6, for one hot utility whose inlet temperature is higher than the maximum temperature of the process streams. In this situation, the flow rate of the utility stream is computed by: (3.9)

where f w Ti,,,, T CPW

A Tmin/Zw Rk

is the flow rate of utility w [kg s-l]; are, respectively, the inlet and the outlet temperature of the utility stream w [“C]; is the specific heat of the utility w [kJ kg-’/’C] is the contribution to the minimum approach temperature of the utility stream w ; for k = 1, nk + 1, are the values of the heat cascade, and R,,k+lis the MER of the process.

It should be mentioned that in the example, the heat delivered by the hot utility is higher than the MER. The temperature Tk defining the intersection between the utility curve and the grand composite curve is called a utility pinch point. In the example, it differs from the process pinch point. Additional heat is therefore available from the hot utility and should be valorized. This situation often occurs in high temperature processes, like steam methane reforming processes, where the high temperature pinch point is activated by the integration of the combustion flue gases at the high temperature reforming reactor and where the heat excess available in the flue gases at lower temperatures is used to produce high-pressure steam that will be expanded in a condensing turbine to transform the excess heat into useful mechanical power by combined heat and power production. The same graphical representation applies for the cold utility and refrigeration requirements. The graphical definition of the utility flow rate has been widely used to define the flow rates in the steam system. The steam condensation defines a horizontal segment (constant temperature) whose length defines the steam flow rate. This approach cannot, however, be applied when utility streams interact as it is the case when steam is produced in a boiler, then expanded to produce mechanical power before being condensed to heat up process streams. The utility system is made of more than one stream and the flow rate of the one (e.g., the steam flow rate) will define the flow rate of the other (e.g., the fuel consumption in the boiler). Furthermore, since the heat of the flue gases is available for the process, nothing restricts its use for temperatures above the steam condensation temperature. As will be demonstrated later on, the problem has to be solved by optimization in this case. Nevertheless, the grand composite curve representation will be of prime importance to define the possible utilities that may be envisaged for the process under study. Another important aspect of the utility system integration is the combined production or consumption of mechanical power. Townsend and Linnhoff [12] have made

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a complete analysis on the use of the grand composite curves to analyze the optimal placement of the energy conversion systems and the combined heat and power production. Following the prescribed rules, the engineer is able to define a list of possible energy conversion technologies that may be envisaged for the process under study. The creativity of the trained engineers will unfortunately create a problem at this stage since they will usually be able to identify more than one technology per requirement.

3.4.3 Exergy Composite Curves

The exergy analysis is a thermodynamic-based useful concept that helps the understanding and analysis of efficiency in energy conversion systems. Exergy measures the thermodynamic value of energy. It defines the maximum work that could ideally be obtained from each thermal energy unit being transferred or stored using reversible cycles with the atmosphere being either the hot or cold energy source. The exergy approach (e.g., Kotas [13]) is used to represent in a coherent way both the quantity and the quality of the different forms of energy considered. The concept of exergy presents the major advantage of an efficiency definition compatible with all kinds of conversion of energy resources into useful energy services (heating and electricity, heating, cooling and electricity, refrigeration, heat pumping, etc.) and for all domains of energy use. While energy efficiencies are higher than 100% for heat pumping systems (because ambient energy is not accounted for), exergy efficiencies are lower than 100%. This gives an indication of how well the potential of an energy resource is exploited in the different technical concepts in competition. In the context of process integration analysis, the exergy concept is combined with the pinch analysis for reducing the energy requirement of the process [14,15] for optimizing the energy conversion system integration and for optimal combined heat and power production. The exergy composite curve concept has been introduced by Dhole [16] for this purpose. The exergy delivered (4by a stream exchanging heat with a constant cp from Tinto To,,is computed by: k = Q (1- -),TO

TJm

the logarithmic mean of temperatures computed by TJ,,,=

(a

where TJm is

a and To is the Tin In

(K’

ambient temperature (all temperatures are expressed in K). When representing the heat exchange in the temperature-enthalpy diagram, the exergy delivered may be represented by exchanging the temperature axis with the Carnot efficiency (1 - -). TO

T

The exergy then corresponds to the area between the exchange curve and the enthalpy axis (Fig. 3.7). When applied to the concept of the composite curves, the exergy composite curves (Fig. 3.8) represent the exergy lost in the heat exchange between the hot streams and

3.4 Graphical Representations

F

Fv

0.3

-

0.2

-

0.1

0

Figure 3.7

-

0

200

400

600

I000

8(KI

1200)

Q(kW

Exergy received by a cold stream heated from 350 to 500 K

0.6

0.5 Q.4 0.3

2 4

0.2 0.F

-0. I

xergy requirement

u

5000

loo00

1sm

2

m

25m

3oooo

QWW) Figure 3.8

Exergy composite curves defining the process requirements

the cold streams of the system. The exergy delivered by the hot streams (shaded area between the hot composite curve and the X axis) is deduced from the exergy required by the cold streams (shaded area between the cold composite curve and the X axis). As for the hot and cold composite curves, there are four zones to be considered in the representation. The hot utility requirement is defined by an area (below the cold composite curve) representing the corresponding exergy requirement. The area between the two curves represents the amount of exergy that will be lost in the heat recovery heat exchange network. This exergy loss may be recovered partly by properly

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3 Computer-Aided lntegration of Utility Systems

integrating combined heat and power devices. Therefore, the area between the curves and above the pinch point should be deduced from the hot utility exergy requirement to define the minimum exergy requirement of the process. After the heat recovery exchange, the remaining heat, to be evacuated from the system, is divided into two parts. Above the X axis (ambient temperature) the area between the hot composite curve and the X axis represents the exergy that could be obtained by integrating low temperature heat recovery devices like organic Rankine cycles. Below the X axis, the area represents the exergy required by the refrigeration system. Favrat and Staine [14]have added to this representation the exergy losses related to the compression work (as a function of the pressure drop) and the grey exergy. The grey exergy is the exergy required to construct the heat exchangers. It includes the raw materials exergy content as well as the exergy consumed in the construction process. This concept could be used to determine the threshold of the heat recovery effort; the exergy expenditure for the recovery equipment should not exceed the avoided exergy loss thanks to the heat recovery. This tradeoff is, of course, considered together with the economical tradeoff and should be used to define the appropriate value of the A Tmin.The grey exergy may become important when the process and the heat recovery exchangers are operated only part-time. When considering the exergy grand composite curve, the diagram represents the exergy required by the process. In this diagram, the special role of the self-sufficient pockets should be noted. This is the area representing the possible mechanical power recovery by the combined heat and power (Fig. 3.9). From this analysis, it is possible to identify the possible characteristics of the steam cycle as demonstrated by Marechal et al. [17]. This is done by integrating rectangles in the grand composite curve. The basis of any inserted rectangle being the vaporization and the condensation temperatures of the Rankine cycle. The use of the exergy concept allows one to

0.5

-

3.4 Graphical Representations

quantify the exergy required and therefore allows one to set a target for the energy conversion system. Because the grand composite curve is computed in the corrected temperature domain, we consider that there will be an exergy loss that is a priori accepted for limiting the heat exchangers investment and that is related to the definition of the A T,,,. The exergy analysis has been extended to account for the chemical reactions or the physical separations of the process operations [18]. In this represenAH-TOAS tation, the temperature axes is replaced by , which is an extension of AH the Carnot factor (1- -). TO

T

One heuristic rule resulting from the exergy analysis is to try to minimize the area between the hot and cold composite curves of the integrated systems.

3.4.4 Balanced Composite Curves

The optimization models presented hereafter allow one to overcome such difficulties by computing the optimal flow rates in the utility system in order to minimize the cost of supplying the energy requirement. The composite curves of the process, together with the utility system, can then be represented. They are known as the balanced composite curves. An example of such curves is given in Fig. 3.10. This representation is characterized by a number of pinch points, one being the process pinch point, the others corresponding to the maximum use of the cheapest utilities to satisfy the process requirement. In particular, we have a pinch point at the lowest and highest temperatures of the system, indicating that the energy requirement of the process is satisfied by the utility subsystem. These curves are, however, difficult to analyze and do not really help in improving the system efficiency. 3.4.5

Integrated Composite Curves

The integrated composite curves [19] will help analyzing the results of the optimization or understanding the integration of subsystems. This representation is based on the decomposition of the system into subsystems: processes, boiler house, refrigeration cycle, steam network, heat pump, utility system, etc., even very detailed subsystems may be considered like one or several existing heat exchangers. The integrated composite curves of a subsystem are obtained by subtracting from the grand composite curve of the overall system, the grand composite of the subsystem under study. The next step is to mirror the subsystem curve. The two curves intersect at the pinch points of the balanced composite curves. In order to locate the Y zero axis, it is convenient to consider the process pinch point location as a reference. From a mathematical point of view, the integrated composite curves are computed using the formulas below (Eqs. 3.10 and 3.11).

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3 Computer-Aided Integration of Utility Systems

Grand Composite curves of the system

Figure 3.10 Balanced composite curves of the process with boiler, steam, and cooling system

c

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 Q(kW

Composite curves of the system

0

5000 10000 1500020000250003000035000400004500050000

Q(kW

The set of streams is divided into two subsets. The first (set A) defines the subsystem whose integration should be checked, the second (set B) is formed by all the other streams. Set B will be referred to as being the reference set and will be represented by a curve (RBk,Tk), where Tk is the kth corrected temperature of the heat cascade and RBI, is computed by: (3.10)

3.4 Graphical Representations

where N B Q,

R,

is the number of streams in subsystem B; is the heat load of stream s in the temperature interval r; is the enthalpy reference that defines the position of the temperature axis (see below).

The opposite curve (RAk,Tk), corresponding to the subset A, is computed to make the balance of the first one. It defines the integration of the streams of set A to the others (reference set B):

r=k s=l

(3.11)

where NA is the number of streams in subset A; R,,k+l is the additional energy that can not be provided by the proposed utility set. The R,,+1 has been introduced to obtain a general definition. When the utilities are well integrated R,,+1 = 0. The value R, that appears in the definition of the two curves defines the position of the temperature axis on the energy axis. The value of R, is computed by considering that the curve of set B (the reference set) will intercept the temperature axis at the process pinch point temperature (Tk,).The latter being identified by computing the heat cascade where only the process streams are considered. If k,, refers to the process pinch point, R, is obtained by futing R Bkp= 0 in Eq. 3.10 giving: (3.12) r=k P s=l

Using this definition, the temperature axis divides the energy range into two parts: the positive values correspond to the energy concerned with the set B integration, while the negative values refer to the energy involved in the set A integration. The application of the integrated curves representation is described in more detail in Marechal and Kalitventzeff [191. Some applications of the integrated composite curves are given in the example below. Such representation is really useful when analyzing the integration of cycles. In this case, the difference between the hot and cold streams corresponds to the mechanical power that closes the energy balance. The representation will therefore be used to verify the integration of steam networks or refrigeration cycles and to confirm the appropriate placement of the cycles in the process integration. When the subsystem considered is a heat exchanger, the integrated composite curves of the heat exchanger are the graphical representation of the remaining problem analysis. When the two curves are separated by the temperature axis, no heat exchange is needed between the streams of the heat exchanger and the rest of the process, when this is not the case, the size of the heat exchange between the two curves will represent the energy penalty that would be associated to the heat exchanger under study. The graphical representation in this case also gives an indica-

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3 Computer-Aided Integration of Utility Systems

600 r 550

300

F

T

500 450

MER

4

TW)

z a l t y of heat exchanger

-

-

250 -1000 Figure 3.12 analysis

-500

I

0

,Q(kW)

500

1000

1500

2000

2500

3000

Integrated composite curves and remaining problem

tion on the temperature profile of the heat exchange required to reduce the penalty. An example is given in Fig. 3.12.

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

The analysis of the process requirement using the analysis of the grand composite curve and by applying the rules of optimal placement of Townsend et al. [12] and other rules for the optimal CHP schemes allows one to propose a list of energy conversion technologies able to supply the energy requirement of the process with a maximum of efficiency. The mathematical formulation of the targeting problem exploits the concept of effect modeling and optimization (EMO) [20], [21]. It will be used to select the equipment in a superstructure and determine their optimal operating flow rates in the integrated system. This approach assumes that the temperature and pressure levels are fixed, resulting from the analysis of the grand composite curve and the application of rules for the appropriate placement of utility streams [12]. The problem is then a mixed integer linear programming (MILP) formulation, where each technology in the utility system is defined for a nominal size and an unknown level of utilization to be determined in order to satisfy the heat cascade constraints, the mechanical power production balance, and additional modeling constraints. Two variables are associated with any utility technology w:the integer variables yw represents the presence of the technology w in the optimal configuration and fw its level of utilization. The objective function is the total cost including the operating costs and the annualized investment cost, both expressed in monetary units (MU) per year. Other objective functions like minimum operating cost or minimum of emissions may also be used. The annualizing factor is computed from the

3.5 Solving the Energy Convenion Problem Using Mathematical Programming

annualizing rate and the life time of the investment. In the equation system, we complete the model by an additional set of constraints (3.17) that are written in a generic form: (3.13)

w=l subject to: Heat balance of the temperature intervals w=l

nw

Cfwqw,r

w=l

+

2

+ Rr+l - Rr = 0

Q.r

i=l

Vr = 1, ..., nr

(3.14)

Electricity consumption

c

w=l

+ weli

fwww

-

wc ? 0

(3.15)

Electricity exportation nw

C f W W W + weli

-

We], - wc = 0

(3.16)

w=l

Other additional constraints (3.17)

k=l

w=l

%,ink

5

Xk

5

Vk = 1,. . . , n,

Xmaxk

(3.18)

Existence of operation w during the time period p: fminwyw ~ f ws f m a x W y w

Vw = 1 , .. . , nw,yw E 0, 1

(3.19)

Thermodynamic feasibility of the heat recovery and utility systems We1 2 0, R1

Wels L 0

= 0, Rnr+* = 0,

where yw c1w c2w

I Fw

(3.20) Rr

>0

Vr = 1 , .. . , nr+l

(3.21)

is the integer variable associated with the use of the technology w; is the fixed cost of using the technology w [€ year-']; is the proportional cost of using the technology w. This value is defined in [€-'I; is the fixed cost related to the investment of using technology w;I Fw is expressed in monetary units [€] and refers to the investment cost of the combustion and cogeneration equip-

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3 Computer-Aided Integration of Utility Systems

IPW t

ments as defined above as well as to the other equipment considered in the utility system (turbines, heat pumps, refrigeration systems, etc.); is the proportional investment cost of the technology w , I Pw [elallows one to account for size effect in the investment; is the annualizing factor of the investment. This value is used to express the investment of the energy conversion units in M U per year. = (1 1% ' ; - 1is the annualization factor +

i (1 +

4wr

fw ww

of the investment (years-') ment (years-') for a annualization interest i and an expected equipment life of nyean; is the number of technologies proposed in the superconfiguration of the utility system; is the heat load of the technology w in the temperature interval r for a given reference flow rate, qwr > ofor a hot stream [k w]; is the multiplication factor of the reference flow rate of the technology w in the optimal situation; is the mechanical power produced by the reference flow rate of technology w;w , < , 0 for a mechanical power consumer and > 0 for a producer [ k w]; is the selling price of electricity [Eper k W € kJ-'I; is the net production of electricity [kW]; is the electricity cost at import [€ kJ-'1; is the net import of electricity [kW]; is the total annual operation time [s year-']; is the overall mechanical power needs of the process; Wc < 0 if the overall balance corresponds to a mechanical power production [kW]; is the (n,) additional variables used in the additional equations of the technology models; are, respectively, the coefficients of the multiplication factor and the integer variables of technology w in the constraint i in the effect models; are, respectively, the coefficients of the additional variables and the independent term in the constraint i in the effect models; are, respectively, the minimum and maximum bounds of x,; is the minimum and maximum values accepted for fw. p @ ' s

The method presented may be applied to any kind of energy conversion technologies. It is based on the assumption that the operating conditions have been defined for each piece of equipment concerned and that only the flow rates are unknown. This is a limiting assumption but it allows one to solve most of the problems of energy conversion integration mainly because nonlinearities may usually be solved by discretizing the search space. The method has been further adapted to compute the optimal integration of steam networks [22], to incorporate restricted matches

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

constraints [23], to integrate refrigeration cycles [24] [25] and organic Rankine cycles[26]as well as heat pumps [27].It has been applied to integrate new technologies like the partial oxidation gas turbine [28],or to design new types of power plants by introducing the concept of isothermal gas turbines (291.

3.5.1 Gas Turbine and Combustion System

In order to demonstrate the ability of the formulation to tackle complex problems, the model for computing the integration of gas turbines and combustion will be given in more detail. The purpose is to explain how to formulate the problem as a linear problem even if the models appear to be nonlinear. The model represents the integration of the gas turbine including its partial load operation, the possible postcombustion of the gas turbine flue gas, the use of different fuels in the gas turbine and in the post combustion, and of course the integration of conventional combustion in a radiative furnace with possible air enrichment or air preheating. The post combustion and the partial load models are required because there is no possibility of identifylng a gas turbine model whose heat load will perfectly match the heat requirement of the process. The principle ofthe integration is illustrated in Fig. 3.21. The following integration constraints are added to the aforementioned problem. Hot stream corresponding to the flue gas of a gas turbine g is defined by: (3.22) where hg

@

CPf, TOT, TstackB

f, Yg

fgmin(-x’ ng

is the flue gas flow rate at the outlet of the gas turbine g in nominal conditions. These values result from the simulation of the gas turbine g is the total heat load of the flue gas from the gas turbine g is the mean cp of the flue gas at the outlet of the gas turbine g, is the temperature of the flue gas at the gas turbine g; is the stack target temperature accepted for the outlet of the gas turbine g after heat recovery; is the level of utilization of gas turbine g w i t h r . yg Sf,5 yg

. fgmaX;

is the integer variable representing the use or not (1,O) of the gas turbine g; is the minimum (maximum) level of utilization of the gas turbine g, is the number of gas turbines proposed in the utility system super configuration.

Hot stream corresponding to the post combustion (heat available for convective heat exchange) is given by:

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3 Computer-Aided Integration of Utility Systems

QF =fpc

. hg. cpfg . (Trad- TOTg) vg = 1, ng

(3.23)

is an arbitrary temperature used in the combustion model and representing the limit of the radiative exchange; is the fraction of the nominal gas turbine flue gas flow rate used for postcombustion; is the heat load supplied by the flow rate fraction of the flue gas flow rate of gas turbine g used in the post combustion device.

where Trad

J7 €T

Fuel consumption in the gas turbine g is as follows:

cJg . "cgt

LHV, - (yg . FCIg +fg . FCPg) = 0 Vg = 1, fig

c=l

(3.24)

is the number of fuels available for combustion in the gas turbines; the lower heating value of the fuel c; L H V, the flow rate of the he1 c in the gas turbine g; fi yg . FCI, +f, . FCP, is the linearized fuel consumption of gas turbine g as a function of its level of utilization.

where ncgt

Electricity production with the gas turbines W, is given by:

wgt

C (yg . Wlg +fg . WPg) = 0 %

-

(3.25)

g=l

where yg . W I, +f,

. W P,

is the linearized mechanical power production of the gas turbine g as a function of its level of utilization.

The parameters for the linearization are computed by simulation considering the partial load operation of the gas turbine. For each gas turbine g, the unknowns aref,, yg, andJP,t,while the other parameters are obtained from the thermoeconomic models. The quality of the linearization will mainly depend on the range in which the partial load operation is expected to happen in the optimal situation. The operating costs 0 C, and the investment costs I C, of the selected gas turbines are computed by:

C ( y g . ocr, +fg . omg) oc, n,

-

=o

(3.26)

g=l

%

Cyg.r a g

-

'C@ = 0

(3.27)

g=l

where yg . 0 C I, +fp . 0 C P, is the linearized maintenance cost of gas turbine g as a function of its level of utilization; is the investment cost of gas turbine g from the data Yg . I c 4 base catalog.

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

The fraction of the flue gas of the gas turbine used in the post combustion is limited to the level of utilization of the gas turbine g.

fgP C

Tfg vg= L n g

(3.28)

The combustion model is made up of different equations: (3.29) includes different terms representing the oxygen balance required by the combustion of the fuels and the oxygen supplied by air and post combustion flue gas. (3.29) a=l

J'Ln@"''x'

na

c= 1

is the oxygen content of the flue gas at the outlet of the gas turbine g is the oxygen content of the ambient air; is the amount of air used by the combustion in the system; its speis the flow rate of fuel c used in the combustion fC (fc""); cific cost is c,; is the fumes flow rate resulting from the combustion of one unit of fuel c; is the mean specific heat of the fumes resulting from combustion. This cp is considered between T rad and T stack; is the number of fuels that can be used in the system including those for firing the gas turbine (ncgt); is the oxygen requirement per unit of fuel c. For practical reasons, the oxygen requirement includes the minimum oxygen excess for this fuel; is the oxygen content of the enriched air stream leaving the air separation unit a; is the flow rate of enriched air leaving the air separation unit a under nominal conditions; is the level of utilization of the air separation unit a, w i t h r . ya 5 h 5 - r * y!i is the integer variable representing the use or not (1,O) of the air separation unit a is the minimum (maximum)level of utilization of the air separation unit a; is the number of air separation units considered in the system.

The fuel consumption balance of any fuel c that might be used either in a gas turbine or in standard combustion follows: (3.30) g=l

where fc

is the overall consumption of fuel c.

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High-temperature balance: radiative exchange model above T rad

Low-temperature balance: convective exchange below T rad:

cf. na

f

'

(3.32)

h a . cpa ' (Trad - Tstack) - Qcnv = 0

a=l

where

and is the total amount of heat available above T rad;

a,,"

L H V,

To Grh

cp,

is the total amount of heat available from T rud to T stuck; is the lower heating value of the fuel c. This value is the value computed by simulation of the combustion using the minimum accepted value of the oxygen content in the fumes; is the reference temperature used for computing the LHV is the heat load of air preheating, the existence of the air preheating equipment is defined by an integer variable Yprh and the following equation: Yprh&hrnin IG r h Iyprh&hmaX. The investment cost of the air preheating device is computed by linearizing the air preheater cost by 1 Cprh = 1 c Fprhyprh -I I c PprhGrh; is the mean specific heat of the enriched air leaving unit a at a temperature of

c.

Table 3.11 gives the values needed to compute the integration of some typical fuels used in the industry including some renewable fuels (wood and biogas). Table 3.11

Values for some typical fuels.

CH,

Natural gas Light fuel oil Heavy fuel oil Coal (lignite) Wood Biogas

LHV

b

Gad

(kllkg)

(K)

(K)

50000 39680 45316 44500 25450 18900 13358

2646.5 2270 2425 2423 2111 2185.43 2077

374 374 440 441 438 374 374

17.1 13.9 14.4 14.3 7.29 7.9 4.63

7.67 6.9 4.67 2.53

5 -

55 55 70 71 81 0 0

Natural gas composition: 87% Methane, 13% N2 Light fuel oil: '36.2% mass, H 12.4% mass, S 1.4% mass Heavy fuel oil: C-86.1% mass, H-11.8% mass, S-2.1% mass Lignite: C-56.52%, H-5.72%, 0-31.89%, N-0.81%, S-O.81%, Ash-4.25% Wood (wt) composition: C-49.5%, H-6%, 0 4 . 6 % + C1H1.400.7, C02 neutral 50% C 0 2 and 50% C H4, C 0 2 neutral Cost 2004, European market

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

Air Preheating: Outlet Temperature Calculation

When combustion is considered, air preheating plays the role of a chemical heat pump. It is used to pump waste heat available below the pinch point and make it available above the pinch point (by an increase of the adiabatic temperature of combustion). The effect of air preheating is, however, limited to the preheating of the stoichiometric air flow. When the flow rate is higher, the adiabatic temperature of combustion decreases and the benefit of air preheating is lost for the part corresponding to the excess air. When combined heat and power are considered using the steam network, the process pinch point no longer defines the maximum preheating temperature. In this case, the combustion air may be preheated up to the temperature corresponding to the highest condensation pressure of steam because this steam will produce mechanical power before being used as a preheating stream. The preheating heat load will become available at a higher temperature to produce an additional amount of steam at the highest pressure or to increase the superheating temperature. The air preheating temperature is therefore unknown and its optimal value has to be computed. When a heat cascade is considered, computing the optimal preheating temperature is a nontrivial task, mainly because the temperature is used to generate the list of the heat cascade constraints. This makes the problem nonlinear and discontinuous (i.e., according to the temperature, the stream will appear or not in a given heat cascade constraint). Some techniques have been proposed to solve this problem as a nonlinear programming (in our case mixed-integer) problem using smooth approximation techniques (e.g., [30]). This approach is explained in more detail in another chapter. A further approach consists of keeping the linear programming formulation by discretizing the temperature range in which the air preheating will take place in ni intervals of AT. The air preheating stream is to Tiri+l = T? + A T and by adding therefore defined by a list of cold streams from the following constraints: fair

vi = 1,.. . , ni;

>fai

(3.33a)

ni

Q r h = Cfaicpair,i(Ti+l - Ti)

(3.33b)

i=l

where fa; cp,ir,i

is the flow rate of air preheated from T, to Ti+l; the specific heat capacity of the air flow rate from Ti to Ti+l.

In the combustion model, the optimal temperature calculation model is also used to compute the outlet temperature of the air and enriched air preheating, fuel preheating as well as to compute the outlet temperature at the stack. This calculation is done in two steps: 1. Solve the model and compute the optimal flow rates in each interval cfai). 2. Compute the resulting temperature To,,, by solving from i = 1 to n,: Vai-, -fai) +fai . Ti+l

TO^

=

fai-1

with Too = Tain,which is the inlet temperature of the stream a.

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3 Computer-Aided Integration of Utility Systems

The precision of the model is related to the size of the discretizing temperature intervals. A compromise between the precision required for the equipment sizing and the number of variables is therefore required. A similar formulation is also used to compute the optimal temperature of the gas turbine flue gas after heat recovery. This systematic choice has been made to keep the robustness advantage of the MILP formulation. 3.5.2

Steam Network

The steam networks play a very important role in most industrial process plants. They are the major interface between the utilities and the process streams while allowing the combined production of heat and power. Furthermore, by transferring heat from one process stream to another, the steam network will be used to reduce the energy penalty resulting from restricted matches. The importance of steam networks is also present for site scale process integration since the steam network will be the method of transferring heat from one process to another. Targeting the integration of the steam network is an important part of the integration of the energy conversion technologies. In the first attempt to study the integration of energy conversion technologies, the idea was to consider steam as being a constant temperature stream that supplies or extracts heat from the process. This provided an easy way of understanding the multiple utilities integration from the grand composite curve analysis. When designing the site scale steam networks, Dhole and Linnhoff [lG]introduced the total site integration concept. They defined the total site composite curves to represent the integration of chemical plants that are composed of several processes and may be integrated. The purpose of these curves is to identify the way energy has to be transferred from one plant section to another using the steam network. This method assumes that heat recovery inside the process has already been performed before allowing for exchanges between processes. The construction of the total site composite curve is explained in Fig. 3.15 (top) for the integration of two processes whose grand composite curves are given on the left. After eliminating the pockets (self sufficient zones), the hot utility and cold utility profiles are composed to build the hot and cold site profiles. The exchange between the processes is then realized using the steam network as a heat transfer medium. If this approach is convenient from the graphical point of view, it can not be applied when considering the integration of steam networks in practice. Two major defaults should be removed. First, the pockets can not be ignored because these may hide heat exchange potentials. This is demonstrated in the center of the figure, where the integrated composite curves of process 1 versus process 2 shows the energy saving that could be obtained when exchanging heat between the two processes. In this example, the energy saving mainly results from the integration inside the pockets. In reality, the hot and cold composite curves of the whole system should be considered. The second drawback of that total site approach is that it ignores the combined production of mechanical power in the steam network. This is shown in the bottom of

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

"1r

'I

T

Total site representation

Process 1

Process 2

-'

.

.

,

..j

n g through process/process i n t e g r a t i o n Energy s a v ing

Process 1 Figure 3.15

Coolkng syste

Process 2

Total site integration and steam network

the figure where steam is produced at high pressure in process 2 to be used at high pressure in process 1 and low pressure steam produced in process 1 is used at a lower pressure in process 2. In between, the steam is expanded in a back pressure turbine to produce mechanical power. From the exergy analysis, the potential for combined production of mechanical power is proportional to the size of the pockets

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in the exergy grand composite curve, These can not therefore be ignored from the heat integration and CHP perspective. In real steam networks, the steam production and condensation cannot be considered at a constant temperature. One should consider the preheating and the superheating for the steam production and the desuperheating, condensation, and liquid undercooling for the steam condensation. Furthermore, the maximization of the mechanical power production will be obtained by optimizing the heat exchanges within the steam network, e.g., by condensing lowpressure steam to preheat high-pressure water of the steam network. The MILP formulation presented above may be extended to define a more precise model of the steam network. The formulation has been given in [22]. It is based on the definition of a steam network superstructure (Fig. 3.13). This superstructure was first proposed by Papoulias and Grossmann [31]. It has been adapted to account for the temperature-enthalpy profiles of the steam production (i.e., preheating, vaporization, and superheating) and consumption (de-superheating, condensation, and undercooling). One of the difficulties of this formulation has been to guarantee a coherency between the heat and the mass balances while using linear equations. This has been obtained by a special formulation of the mass balances of the steam network headers. Hot and cold streams of the steam network are considered in the heat cascade model and the contributions of the steam expansion in turbines are added in the mechanical power balances. It should be mentioned that using this

t, DT13

Dfn

SI

DC14

Df42

2 ’

DL43

L2

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DC24

3’

Figure 3.13 Superstructure of a steam network including three production/usage levels and one condensing level (deaerator)

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

model, the optimal flow rate in the steam network will be determined also by considering the possibility of exchanging heat between the streams of the steam network. This leads to an optimized steam network configuration where steam draw offs are used to preheat the high pressure water. The model may therefore be used as a tool for a rapid prototyping of complex steam cycles in conventional power plants [29]. The integrated composite curves of the steam network (e.g., in Fig. 3.20) is used to represent the results of its integration. In the figure, the overall mechanical power production corresponds to the balance between the hot and cold streams of the system. It is made of the contribution of the expansions between the different pressure levels in the superstructure. A post processing analysis will be required in order to decide the best configuration of the steam turbine(s): single turbine with multiple draw off or multiple back-pressure turbines. The use of the process pinch point as a reference to locate the zero heat (temperature axis) is used to verify the appropriate placement of the steam headers and the combined production of mechanical power. Combined heat and power is well located when it takes heat above the pinch point and sends it above or when it takes heat below and sends it to the cold utility [12]. When the combined heat and power production satisfies the rules for appropriate placement above the pinch point, the part of the integrated composite curve of the steam network that will appear in the left side of the temperature axis should correspond to the mechanical power production. This would indicate that the additional energy required by the system is equal to the mechanical power produced. If not, the rules for appropriate placement are not satisfied and the reason for this penalty should be investigated.

3.5.3 Refrigeration Cycles

Refrigeration cycles are used as a cold utility below the ambient temperature. The major principle of the refrigeration cycle is to use the compression power to change the temperature level of the streams. A simple refrigeration cycle is presented in Fig. 3.14, it is composed of one compressor, one evaporator (at low temperature), one condenser (at higher temperature) and a valve. From the process integration point of view, the refrigeration cycle is defined by one hot and one cold stream and by the corresponding mechanical power consumption. The optimal flow rate will be determined by the MILP formulation presented above. The temperature levels obtained from the grand composite curve analysis will usually define the type of fluid to be used but other considerations have to be taken into account like the environmental aspects (CFC refrigerants) or safety (flammability).The use of fluids already in use in the process plant is also an important criterion. The efficiency of the integration of one refrigeration cycle depends on its compression ratio and on the flow rate. It will depend also on the structure of the cycle and on the possibility of combining cycles with different refrigerants or with different pressure levels. The problem is therefore highly combinatorial since refrigerants, structures, pressure levels and flow rates have to be optimized. A graphical approach based on the exergy analysis has been

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340

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-lo00

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lo00

QW) Figure 3.14 cycle

Integrated composite curve o f a single stage refrigeration

proposed in [32]. This approach illustrates the methodology of the integration of complex refrigeration systems. A nonlinear programming model has been proposed in [33]. The method presented by Marechal and Kalitventzeff [24] shows the extension of the MILP formulation to tackle this complex problem. The method first identifies the most important temperature levels in the grand composite curve using an MILP formulation. The systematic integration of the cycles with the possible refrigerants is made by applying heuristic rules. From this first selection, the remaining cycles for which the refrigerants, the configuration, the temperature levels, and the mechanical power are known are added in the energy conversion system superstructure and the best configurations are sorted out by solving the MILP problem. When several cycles compete, integer cut constraints are added to the problem to systematically generate an ordered set of solutions. The integer cut constraint is used to avoid the generation of an already known solution when solving the MILP problem. The restriction of the kthsolution is obtained by adding the following constraint:

\w=1

w=l

where yk n,,l

/

(3.34)

is the value of yw in the solution k is the number of solutions so far.

The use of an integer cut constraint is an important tool when solving utility system integration. The systematic generation of multiple solutions allows the comparison of the proposed utility system configurations using different criteria (not accounted in the definition of objective function) and to perform a sensitivity analysis to uncertain problem parameters like the cost of energy or the investment.

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

3.5.4 Heat Pumps

When appropriately placed heat pumping systems should drive heat from below to above the pinch point. Several types of heat pumping systems may be considered (mechanical vapor recompression, mechanically driven heat pumps, or absorption heat pumps). Table 3.7 gives some useful thermoeconomical values for the evaluation of the industrial heat pumping systems. The optimal integration is determined by first identifying the streams or the temperature levels for which heat pumping is envisaged. The simulation of the heat pump system using process modeling tools defines the hot and cold streams characteristics that are added to the system integration model and the optimal flow rate for each of the levels will be computed by solving the MILP problem. When mechanical vapor recompression (MVR) is used, a hot stream at lower temperature (rnvrLo,,,) is replaced by another hot stream (mvwh;&)with a higher temperature and a mechanical power consumption W,,,,,. From the grand composite curve analysis, it may happen that only part of the stream will be recompressed. This situation will be represented by considering as variable the fraction used in the two streams (resp.finvrL,, andfinvrhigh) and by adding the following constraints (3.35): (3.35)

The inequality represents the situation presented in Fig. 3.17 where the stream is partly recompressed and the remaining part is cooled to lower temperature. It should be noted that the heat load of the high temperature stream is higher than the one of the colder stream because it includes the mechanical power of compression. All the streams have the same target temperature. The adiabatic valve being considered after liquid subcooling. The useful heat of the recompressed stream is therefore lower than the total heat load since part of the heat remains available below the pinch point temperature. When the recompressed stream may be vented (e.g., in the case of evaporation stream), the constraint becomes an inequality. In this case, there are in fact three options: (1)venting the stream and the heat is lost to the atmosphere, (2) condensing the stream and use it because the heat is needed below the pinch point, and (3) recompressing it and using the recompressed stream as a hot utility. The optimal flow rate in each of the options will be obtained from the optimization and will take into account the constraints of the heat cascade. As for the refrigeration cycles, multiple options corresponding to different technologies with different levels of pressure and temperature may be considered and will be handled by the optimization. The results of the heat pump integration correspond to the activation of multiple pinch points. The integrated composite curve of the heat pumping system (Fig. 3.23) is a useful tool to verify the appropriate integration of the system. 3.5.5 Handling Nonlinear Cost Functions

In the problem formulation, linear costs are needed. When the whole range of sizes is covered by the model, a piecewise linearization technique may be used to re-

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Size Figure 3.16

Piece-wise linearization o f the cost function (exponent 0.75)

present a nonlinear cost function. The generic investment cost function C (S) = C,f.

S (q)’ will be approximated by a set of segments (Fig. 3.16) defined by the

following set of constraints (3.36) in the linear optimization problem definition:

(3.36)

where C (S) is the installed cost of the equipment of size S; STa is the maximum size in segment i; is the size of the equipment in segment i; Spi is the integer variable used to select the segment i. yi The linearization by segments is also applicable for performances indicators of technologies like power and efficiency of a gas turbine. The linear formulation may in this case be extended to account for piecewise linearizations of nonlinear functions.

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

3.5.6 Using the Exergy Losses as an Objective Function

Due to the linear nature of the problem, the use of the energy cost as an objective function may reveal some difficulties [27]. When the cost of fuel and electricity is such that the electrical efficiency of a cogeneration unit is attractive without the use we' is greater than of heat (i.e., when the electrical efficiency of the unit qel= ___ L h V CLHV (e there is an economical interest to produce electricity even without Cd (ekJ-') cogeneration). In this case, the linear programming procedure leads to a situation where the cogeneration unit is used at its maximum. This situation usually does not occur when the investment costs are properly considered or when the costs of the different forms of energy are coherent with respect to the electrical efficiency. Nevertheless, the relative price of the different forms of energy will influence the technology selection and their level of usage in the integrated solution. When the target is the maximization of the system efficiency, alternative formulations that take into account the value of energy in the objective functions have to be considered. The minimization of the exergy losses (3.37) is an attractive way of formulating the problem:

(3.37) where AExw is the exergy consumed to produce the hot and cold streams and the electricity of equipment w ; A exwk is the exergy supplied by the conversion unit w in the temperature interval k,

ATmin/2s is the contribution to the ATminof the stream s; ATmin/2,2 0 for hot streams and 5 0 for cold streams; qsk is the heat load of the stream s in the temperature interval k computed for the nominal conditions of the related equipment; is the number of streams of conversion unit w. Using this formulation it is possible to define the set of energy conversion technologies that minimize the exergy losses of the system. It is even possible to introduce the aspects related to the investment by adding the grey exergy into the exergy consumption of the conversion technologies. 3.5.7 Handling Restricted Matches

The use of mixed-integer linear programming in process integration was first proposed to solve the problem of the heat exchanger network design [34]. The design

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problem is presented as a transportation problem by Cerda et al. [35] or as a transshipment problem [3G]with a significant reduction of the number of variables. Such methods can easily be extended to account for restricted matches by adding constraints to the MILP formulation. The draw back of such approaches is that they are designed to find a feasible heat load distribution that satisfies the MER target. In this case, the restricted matches are an advantage because they reduce the search space of the possible connections. When there is no feasible solution, these methods give the energy penalty of the restricted matches. Unfortunately, none of these methods search for ways of reducing the penalty. The integration of energy conversion technologies allows one to go one step further in the analysis because the utility can be used as an intermediate heat transfer fluid to reduce the restricted matches penalty. Steam or hot water may be produced in one section and condensed or cooled down in another section of the plant. Another fluid (e.g., Dowtherm) can be used as a ,,heat belt". The computer aided design methodology will help in defining the characteristics of the streams to be considered as an intermediate heat transfer fluid to indirectly exchange heat between the streams of the penalizing restricted matches. The formulation proposed by Marechal and Kalitventzeff [23]uses a MILP formulation of the restricted matches to target the energy penalty of the restricted matches. The penalty is divided into two parts according to the position of the process pinch point. From the optimization results, hot and cold restricted matches penalty composite curves are computed (Fig. 3.22). The hot curve represents as a function of the temperature the heat that can not be removed from hot streams involved in restricted matches by the allowed process streams. This heat has to be received by the cold streams of the heat transfer fluid system. The cold composite represents the temperature enthalpy profile of the heat that has to be sent back to the process by the heat transfer fluid in order to avoid the restricted matches energy penalty. The two composite curves have a pinch point temperature that is identical to the one from the MER calculation since no heat can be exchanged through the pinch point in the MER situation. Any combination of hot and cold streams of the utility system that is ill allow one to eliminate the penalty, provided that the framed by the two curves w temperature difference will be sufficient. The use of intermediate streams imposes the use of two heat exchangers instead of one, thus doubling the A T min value. One should note that the restricted matches composite curves are designed in order to preserve the possible combined heat and power production (i.e., producing highpressure steam by the hot streams and sending it back to the process at a lower pressure after expansion in a turbine). Once the utility streams characteristics are identified, restricted matches constraints are added to the optimal energy conversion system integration problem. The constraint formulation is an extension of the heat load distribution formulation [37] that complements the MILP problem definition. As a result, the complete list of hot and cold streams to be considered for the heat exchanger network design will be obtained and we know that there exists at least one feasible heat exchanger network configuration that satisfies the restricted matches constraints and the minimum cost of the energy target.

3.5 Solving the Energy Conversion Problem Using Mathematical Programming

3.5.8 Nonlinear Optimization Strategies

The analysis of the integrated composite curve helps in interpreting the results of the optimization and verifjrlng (or optimizing) the choice of the utility system operating conditions (temperature or pressure levels) that were supposed to be “well chosen” when stating the MILP problem. In order to solve the nonlinear problems with a linear programming formulation, three strategies may be used (1)the appropriate formulation of the problem constraints as it is explained for the gas turbine integration, (2) the piecewise linearization, and (3) the continuous search space discretization. The latter consists of defining different operating conditions of a given technology as an option among which the optimization will select, using integer variables. The method presented here has tackled the problem of nonlinearities by discretization of the continuous variables search space. Other authors have directly tackled the MINLP problem. These formulations use the alternative heat cascade formulation (3.4) because in this formulation each constraint is associated with one key temperature (stream inlet temperature) of the system. In this case, when the inlet temperature changes, the definition of the equation is not changed, but it may happen that some of the streams that were above the key temperature will now be below. This creates a discontinuity that is usually not acceptable for most nonlinear programming solvers. In order to smooth the discontinuities, Duran and Grossmann [30] have defined a temperature enthalpy diagram of the streams that is continuous for the whole temperature range. For a cold stream, the enthalpy profile (Fig. 3.11) of a stream is defined by (3.39).

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Temperature-enthalpy profile of a stream in the process

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3 Computer-Aided Integration ofUtility Systems

Heat supplement from W

Heat below the pinch Figure 3.17

Mechanical vapor recompression

(3.39a) (3.39b) (3.39c) A smoothing technique is then used to round the corners of the enthalpy profile. The difficulty being to tune the smoothing parameters to be compatible with the nonlinear solver criteria without introducing infeasibilities in the heat integration results. This technique was first proposed by Duran and Grossmann. It has been used and extended by other authors [38],[39] for simultaneously optimizing process performances and heat integration problems. When isothermal streams are considered in the problem, the smoothing technique to smooth the discontinuities is not valid anymore from the mathematical (numerical) point of view. In this case, it is necessary to use integer variables that will represent the contribution of the isothermal stream i in the heat balance above the temperature j constraint. The expressions y i j = 0 if T j < Ti and yii = 1 if 2 Ti are represented by the following equations:

(3.40a) (3.40b) This formulation is generic and may be used to represent all situations even the situations solved by the smoothing approximation technique. The resulting MINLP problem will have a huge number of integer variables and will be solved using an outer-approximation algorithm or using disjunctive optimization techniques.

3. G Solving Multiperiod Problems

I

3.6 Solving Multiperiod Problems

Even when the processes may be considered as being stationary, it is often necessary to consider multiperiod operations where the requirements of the processes are considered to be constant during a given period. The utility system serves the different processes of the plant. It therefore has to answer the varying demands of the different processes. Two different problems have to be addressed in this case: the optimal design of the system and the optimal operating strategy.

3.6.1 Optimal Design

In multiperiod problems, the goal of the optimal design task is to determine the best investment to be made in terms of energy conversion equipment considering the varying demands of the processes. This implies taking into account the partial load efficiencies of the equipment, but also to determine in each period the best way to operate the system. The use of MI(N)LP methods for solving multiperiod process synthesis problems has been reviewed by Grossmann et al. [40] and [41]. Multiperiod optimization is a well known problem that has been considered mainly in heat exchanger network design: e.g., [42],[43],[37].The generic problem is formulated as follows (3.41): (3.41)

vt = 1, ..., nt V t = 1, . . . ) nt Vt = 1, . . . , nt Vt = 1, . . . , nt

Yt S

Y

(3.42a) (3.4213) (3.42~) (3.42d)

is the set of modeling constraints during the period t ; is the set of inequality constraints during the period t; is an array representing the operating conditions of the equipments during time period t ; is an integer variable representing the use or not of an equipment during time period t ; is the array of the sizing parameters of the equipment sets, once an equipment is selected (see the value of y), it is used throughout all operating periods; is the array of the integer variables representing the global selection of the equipment sets (i.e., the decision to invest or not);

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c (xt,s) I (y, s) tt

is the operating cost during the operating period i; is the total annualized cost of the equipments of size s; is the operation time of period t.

In the general formulation, the set of constraints may be linear or nonlinear and it is assumed that the operating scenari in each period are independent and without heat exchange between periods. If this is not the case, the problem becomes an even more complex batch process synthesis problem. By generating the heat cascade constraints in each period, the model for integrating the energy conversion units presented above may be adapted for solving multiperiod problems [27]. The partial load operation of some of the units has been introduced in the model. Shang et al. [44] have demonstrated that a transshipment model can account for part-load efficiency of expansion turbines and boilers. When using heat cascade constraints, it is assumed that it will be possible to make the heat exchanges required by the heat cascade representation in each time period. If this appears to be a restriction with respect to the heat exchange network structure, its impact may be limited by considering the utility system that will offer a greater flexibility in terms of process utility interface. The multiperiod design formulation implicitly accounts for the optimal operating scenario. Iyers and Grossmann [45] proposed a MILP formulation for multiperiod problems where the design problem includes operational planning of the system in which the limited availability of the units is accounted for. To overcome the difficulties related to the size of the problem, the authors propose a bilevel decomposition of the problem that reduces the size of the MILP problems and considerably reduces the computing time required.

3.6.2 Optimal Operating Strategy

Once the utility system structure is defined, an optimal operating strategy has to be established. Some authors ([46], [47]) propose a linear programming approach to solve this multiperiod problem in order to calculate the optimal scheduling of the system, considering the start up and shut down time as well as the unavailability of the equipment during the maintenance period. The optimization of the operating conditions of a utility system has been solved by Kalitventzeff et al. [48], [49] as a mixed integer nonlinear programming problem. The nonlinear models allow one to account for effects of pressure, temperature, and flow rates, and to consider the available heat exchanger areas. To solve the MINLP problem, the outer-approximationalgorithm was applied [50]. Combined with data reconciliation techniques for process performance follow-up, this method has been applied with success for the optimal operation of complex industrial utility systems. The MINLP problem formulation assumed that each operating scenario may be optimized independently of the history. It has been demonstrated [49] that this MINLP formulation may be used to make the simultaneous optimization of heat exchanger

3.7 Example

network integrated with a complex utility system. The method has also been applied to solve utility system retrofit problems where available equipment (e.g., an exchanger) is attributed to a specific operation (e.g., a heat exchange). Papalexandri et al. [51] proposed a MINLP strategy that solves the multiperiod optimization problem and accounts for uncertainties in certain parameters.

3.7 Example

Let us consider the system requirements defined in Fig. 3.4 that result from the hot and cold composite curves of Fig. 3.3. For the calculations,we assumed that all possible process improvements were already implemented before analyzing the grand composite curve for the energy conversion technologies integration. In terms of energy, the requirements are given in Table 3.12. From the grand composite curve, several utilities may be proposed. The simplest solution is to integrate a boiler house using natural gas (with a LHV of 44495 kJ kg-’) and to cool the process with cooling water. The refrigeration needs will be supplied with a refrigeration cycle using ammonia (R717). The operating conditions of the refrigeration cycle (Table 3.13) have been obtained by simulation considering the temperature levels in the composite curve and the A T min to be reached in the heat exchangers. The results are presented on table 3.5 and the integrated composite curves presenting the results of the optimization are presented in Fig. 3.18. The refrigeration cycle consumption is 314 kW. It should be noted that the energy consumption is higher than the MER due to the losses at the boiler house stack (398 K). The solution accounts for the possibility of air preheating to valorize the energy excess available in the process. The heat load of air preheating is 131 kW. Table 3.12

Minimum energy requirements for the process

Heating requirement Cooling requirement Refrigeration requirement Table 3.1 3

6854 kW 6948 kW 1709 kW

Above 365 K Between 365 and 298 K Below 298 K (lowest T = 267 K)

Refrigeration cycle characteristics. Refrigerant Reference flow rate Mechanical power P

(bar)

R717 0.1 394

Ammonia kmol/s kW

T,. ( K)

TO”,

Q

A T minp

(K)

(kW)

(K)

340 264

304 264

2274 1880

2 2

~~

Hot stream Cold stream

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3 Computer-Aided Integration of Utility Systems Table 3.14

Steam cycle characteristics.

P

HP2 HP1 HPU MPU LPU LPU2 LPU3 DEA Table 3.15

T

Comment

(bar)

(rc)

92 39 32 7.66 4.28 2.59 1.29 1.15

793 707 510 442 419 402 380 377

superheated superheated condensation condensation condensation condensation condensation deaeration

Results of the energy conversion system integration for different options.

~

Option

Boiler Boiler + steam GT + steam Boiler + heat pump Boiler + steam + heat pump

7071 10086 16961

-

666

-

-

5427

2957 2262

-

-

8979 9006 9160 2800 2713

738

-

-

485 496

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In order to valorize the exergy potential (Fig. 3.8), a steam network has been integrated. The steam network characteristics are given in Table 3.14. The grand composite curve obtained by the integration of the steam network is given in Fig. 3.10. This figure is not readable and even the integrated composite curves of the system

3.7 Example

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-

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-

300 200 -2000

, 0

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QWW) Figure 3.19 Integrated composite curve o f the utility system: boiler, steam, refrigeration, and cooling water

(Fig. 3.19) becomes difficult to read. Based on the same results. The integrated composite curves of the steam network (Fig. 3.20) offer a better visualization of the steam network integration. It should be mentioned that the choice of the process pinch point location as a reference for locating the temperature axis allows one to verify the

Q v

I-

700 600

500 400 300

200 -4000

-2000

0

2000

4000

6000

QWW) Figure 3.20 Integrated composite curve o f the steam network: boiler, steam network, refrigeration, and cooling system

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3 Computer-Aided Integration of Utility Systems

appropriate choice of the steam pressure levels. The energy balance of the hot and cold streams of the steam network is the net mechanical power production. When the steam levels are appropriately placed, the mechanical power production corresponds to a supplement of energy to be supplied to the process. The fact that the mechanical power production appears on the left of the temperature axis proves that the steam network characteristics are appropriate for the optimal production of mechanical power. The area between the two curves gives an indication of the quality of the exergy valorization. Applying the rules of the appropriate placement of heat pumping devices, three heat pumping cycles have been proposed and simulated (Table 3.17). The high values of the coefficient of performance (COP) are explained by the very small temperature raise to be obtained from the heat pump when considering small A T,in/2 values for the heat exchangers. Using the optimization tool, the optimal flow rates in the three cycles have been computed together with the new value of the fuel in the boiler house. In the example considered,this leads to a situation where the whole heat requirement may be provided by the heat pumps. The integrated curves of the heat pump system integration are given in Fig. 3.23. When the steam network is considered the results are slightly different since in this case, an additional amount of energy is supplied to the system to produce mechanical power by expansion in the steam network. The solution of heat pumping is then compared with a combined heat power production using a gas turbine. The summary of the energy conversion integration target is given in Table 3.16. It is shown that a MER of 6854 kW for the heating requirement and of 1709 kW for the refrigeration requirement is finally supplied with an equivalent of 893 kW of fuel when considering the possibility of heat pumping and when valorizing the exergy content of the process streams. Compared to the boiler house solution, the new situTable 3.16

Overall energy consumption of the different options based on 55 % fuel equivalent for elec-

tricity. Option

Boiler Boiler + steam GT + steam Boiler + heat pump Boiler + steam + heat pump Table 3.17

Cycle 3 Cycle 2 Cycle 0

Fuel

Net electricity

Total consumption

(kwd

(kWe)

(~WLHV)

7071 10 086 16 961

371 -2481 -7195 832 125

7746 5575 3879 1513 893

-

666

Characteristics of the heat pump system, based on R123 as working fluid.

5 6 6

354 361 361

7.5 10 7.5

371 384 371

15 12 28

130 323 34

Figure 3.21 Integration of a gas turbine with postcombustion process.

ation corresponds to a fuel consumption reduction by a factor of eight. These data have been computed by considering a fuel equivalence of 55 % for electricity production. The method applied here allows one to quickly evaluate energy conversion alternatives and to quickly assess the impact of process modifications on the processes. Using the targeting method based on optimization, the major advantage of the approach stands in the accounting of the energy savings in terms of cost of energy or in terms of exergy losses rather than in terms of energy. The method allows one to make a first selection of energy conversion options to be further analyzed in more detail using thermoeconomic optimization tools. In this analysis, the cost of the energy conversion system should then be considered together with the cost of the heat exchanger network in order to assess the profitability of the solutions. I

1

500

460 460

440 420 400

380 360

f i

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Figure 3.22

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Restricted matches penalty composite curves.

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-2000

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8000

10000

12000

Q(kW Figure 3.23 Integrated composite curve o f the heat pump system: boiler, refrigeration, and cooling system

3.7.1 Rational Use of Water in Industry

In chemical process systems, water is often used as processing support. The optimization of the water usage concerns two aspects: water savings (i.e., resource usage minimization) and the emissions (i.e., the level of contaminant in the waste water). Resulting from the mass balance and the process requirements, both objectives are 1200 1100 1000 -

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,

I (

antagonistic since the mass of contaminant to be extracted from the process is constant: if the flow rate decreases, the concentration should increase. El-halwagi and Manousiouthakis [52] presented the analogy between the energy pinch analysis and the mass exchange networks: the concentration replaces the temperature and the water flow replaces the heat load. The method uses mathematical programming techniques where a constraints set defines the water cascade. Wang and Smith [2] proposed a graphical method based on the same analogy. In their approach, each unit utilizing water is represented by a limiting water profile defined as a concentration/mass of contaminant profile (Fig. 3.25). This profile represents the worst conditions (in terms of flow rate and concentration) that should have the support water entering and leaving the unit in order to realize the mass transfer required. This allows one to draw a composite curve that may be assimilated to the utility profile in the energy domain and to which the fresh water curve (utility) will be integrated. The utility curve starts with a mass of contaminant of zero. Its slope corresponds to the fresh water flow rate required for the system. The minimum flow rate is computed by activating the pinch point between the utility curve and the limiting profile. By balance, the maximum concentration of water is determined. From this targeting procedure, it is then possible to design a water exchange network using a procedure similar to the pinch design method for the heat exchangers. The proposed method suffers however from the drawbacks of being based on the limiting profiles of the mass transfer that does not allow for generic representation of the water usage in the chemical industry and on the difficulty of handling multiple contaminants even if the authors have proposed an adaptation of the method [53] [54]. Dhole et al. [55]proposed another representation of the water cascade considering the water units as being source and demand of water with a given level of purity (Fig. 3.26). The advantage of this representation is the generalization of the approach to represent all types of water usage units and not only the one based on the mass transfer profiles. This allows one to draw the hot (source) and cold (demand) composite 250 200

150 h

E,

uva

100

50

.,

n

Figure 3.25

0

50

150 200 250 300 Mass load of contaminant (kg/h)

100

Water limiting composite curve

350

400

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3 Computer-Aided Integration of Utility Systems

1.oo

0.9990 0.9980 h

'8

a

Demand composite Source composite ------

r : '

r

le recovery by mixing

-

0.9970

0.9950

-

0.9940

-

0.9960

Figure 3.26 Water sourcedemand profiles

curves, and the corresponding grand composite curve, and to identify the pinch of the system. This approach, however, suffers from the problem of not being able to identify the purity changes that will result from the mixing of a high purity stream (above the pinch point) with a low purity stream (below the pinch point) to produce water with a medium purity above the pinch point. This has some energy analogy with heat pumping by absorption heat pumps. Based on this representation, Hallale [SG]proposed an algorithm that computes the possible recovery by mixing and that is based on the water surplus profiles (Fig. 3.27). By analogy with the process integration techniques, the methods based on the graphical representations help to identify the possible integration of water regeneration equipment. The rules are similar to the one of the heat pumps integration.

.

0

with water k l o w the pinch

0.5

Figure 3.27 Water surplus composite

1 1.5 Water flow (t/h)

2

2.5

The graphical representations are attractive to solve and explain water usage targeting. These methods become difficult to apply, however, when multiple contaminants are concerned. In this case, the use of mathematical programming appears to be more convenient. Several formulations have been used: following the formulation of El-halwagi (521, Aha-Argaez et al. [57] proposed a transshipment formulation of the problem that allows one to introduce multi contaminants constraints and balances in the system. Such methods simultaneously solve the water target and the mass exchange network. Like in the case of energy conversion integration, the water usage minimization will not only concern the minimization of the water usage but also the integration of water treatment or purification units that will concentrate the effluent and perhaps transfer the contaminant to another phase (e.g., solid). In this case there is an interest to consider the combined integration of the water usage and the energy. This would become especially true in situations where waste water will be treated by concentration (cold stream) followed by biomethanation to produce biogas (fuel) or in the case of thermal water desalting systems. Furthermore, when analyzing the grand composite curve of the water requirement profile (water surplus composite), one may suggest introducing water regeneration units (e.g., filtration) that will change the shape of the water profile and will allow one to further reduce water consumption. This approach will therefore be similar to the one used in the energy conversion technology integration. The mathematical programming approach [58]presented below is based on a multiple contaminants transshipment model. It may be combined with heat integration by summing up MILP problems. The link between the two models will be the value of the level of utilization fw,which is related to water usage and to energy effects: (3.43)

subject to: Vd = 1, ..., nd s=l

Vs = 1, ..., ns

c . c2 . ns

s= 1 as

s= 1

h sd

d=l

xs,,

h d <.f,(d) xd

(1 - Xs)

.x z y

V ' = 1, ..., n,

h d <.fw(d) . (1 - Xd) xd

Vd = 1, ..., n d

(3.44) Vd = I, ...,

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3 Computer-Aided Integration of Utility Systems

where

mi,d

r

Cs.d

is the flow rate exchanged from source s to demand d is the purity required by demand d is the purity of source s; is the level of usage of unit w corresponding to demand d (respectfully,source s); is the flow rate in the nominal conditions of demand d is the fraction of impurity j in source s; is the maximum allowable fraction of impurity j in demand d; are, respectllly, the fixed and the proportional cost of the unit w expressed as a function of its nominal flow rate; is the cost of exchanging one unit of flow rate from source s to demand is the number of utility units in the system; is the number of process units in the system; is the number of demands (sources); is the total number of impurities to be considered in the problem.

By analogy with the energy integration technique, the use of graphical representations allows a representation of the quality of the integration and suggests further process modifications. It should be mentioned that the approach suggested for water usage may be implemented by analogy to design and retrofit of the hydrogen networks in refineries. Both graphical techniques [59]and mathematical programming techniques [60] are transposed from water minimization to the refinery hydrogen management domain. In this context, it will be important to consider simultaneously the energy conversion integration techniques, since hydrogen recovery from the hydrogen network will have to be accounted for by its energy content value. The energy consumption used to produce pure hydrogen and the possible combined production of mechanical power must be considered, with the latter being used to drive the recycling hydrogen compressors. 3.8 Conclusions

In process design, the optimal integration of the utility system allows one to transform energy minimization problems into energy cost or exergy loss minimization. It offers a way of considering the energetic problem as a whole, adopting a system vision for the use of energy in the process. The optimal integration of the utility system defines the complete list of streams to be considered in the heat exchanger network design. It is therefore an important step in any process integration study. The computer-aided methodology for integrating the energy conversion system (utility) in chemical production sites combines the use of graphical techniques and mixed integer linear optimization. Considering that the problem formulation is not always known from the beginning, it is important to use methods that support an

References 1379

engineer’s creativity rather than a push-button method. The MILP technique is a robust problem formulation and solving method for process engineers. Many complex problems in process design and operation can be formulated and efficient solutions may be identified. Obviously, the utility integration study, as presented in this chapter, represents only one step of the problem, but it allows one to capture, in a quick and easy way, the major aspects of the integration and to identify the most important options while eliminating the less attractive ones. An important aspect of the methodology presented is the fact that it does not need the definition of the utility system structure. That will be done only when the best options are identified and together with the heat exchanger network design and optimization.

1 Boland P. Hewitt G. C. Thomas B. E. A. Guy

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A. R. Marsland R. H . LinnhofB. Townsend D. W.A user guide on process integration for the efficient use of energy. The Institution of Chemical Engineers, 1982 Smith R. Wang Y. P. Wastewater minimisation. Chem. Eng. Sci. 49(7) (1994) p. 1981 - 1006 Paris J. Brown D. Markchal F. A dual representation for targeting process retrofit, application to a pulp and paper process. Appl. Therm. Eng. 25(7) (2005) p. 1067-1082 Markcha1 F. KalitventzefB. Optimal insertion of energy saving technologies in industrial processes: a web based tool helps in develop ments and coordination of a European r&d project. Appl. Therm. Eng. 20 (2000) p. 1347-1364 Pelster S. Environomic modelling and optimisation of advanced combined cycle cogeneration power plants including C02 separation options. Dissertation, LENI, EPFL, 1998 Von Spakovsky M. R. Pelster S. Favrat D. The thermoeconomic and environomic modeling and optimization of the synthesis, design, and operation of combined cycles with advanced options. J. Eng. Gas Turbines Power 123(4) (2001) p. 717-726 Frangopoulos C. A. Comparison of thermoeconomic and thermodynamic optimal designs of a combined-cycle plant. In: Rakopoulos C. D., Kouremenos D. A., Tsatsaronis G. (eds.) International Conference on the Analysis of thermal and energy systems, pp. 305-318, Athens, Greece, 1991 Pelet X. Optimisation de systPmes energetiques integrks pour des sites isoles en con-

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siderant les parametres economiques, d’kmissions gazeuses, de bruit et de cycles de vie. Dissertation, Laboratory for Industrial Energy Systems, Swiss Federal Institute of Technology Lausanne, 2004 Arpentinier et al. Exsys ii: an expert system for optimal insertion of intensified energy saving technologies (iest) in the industrial processes. Publishable final report of project joe3-ct97-0070. Technical report, EU commission, 2001 Whiting W. Shaeiwitz]. Turton R. Bailie R. Analysis, Synthesis and Design of chemical processes. Prentice-Hall, Englewood Cliffs, NJ, 1998 Barthel Y.Rairnbault C. ArlieJ. P. Chauvel A. Leprince P. Manuel d’kvaluation economique des procCdes, avant-projets en rafinage et petrochimie, nouvelle edition revue et augmentee. Ed. Technip. 2001 LinnhoflB. Townsend D. W. Heat and power networks in process design. Part 1: Criteria for placement of heat engines and heat pumps in process networks. AIChE J. 29(5) (1983) p. 742-748 Kotas T. J . The Exergy Method of Thermal Plant Analysis. Krieger, Melbourne, FL, 1995 Favrat D. Staine F. Energy integration of industrial processes based on the pinch analysis method extended to include exergy factors. Appl. Therm. Eng. 16 (1996) p. 497-507 Lavric V. Baetens D. Plesu V. De Ruyck]. Broadening the capabilities of pinch analysis through virtual heat exchanger networks. Energy Convers. Manage. 44(14) (2003) p. 2321-2329

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3 Computer-Aided lntegration of Utility Systems 16 L i n n h o f B . Dhole V. R. Total site targets for

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fuel, co-generation emissions, and cooling. Comput. Chem. Eng. 17(161)(1992)p. s l 014 0 9 KalitventzefB. Marbchal F. Identify the optimal pressure levels in steam networks using integrated combined heat and power method. Chem. Eng. Sci. 52(17) (1996) p. 2977-2989 Ohba T. Ishida M. A new approach to energy-utilization diagrams for evaluation of energy of chemical systems. In: Pulido R. Tsatsaronis G. Rivero R., Monroy L. (eds.) Energy-Efficient. Cost Effective, and Environmentally-SustainableSystems and Processes, pp. 845-852. Instituto Mexican0 del Petroleo, 2004 KalitventzefB. Marbchal F. Targeting the minimum cost of energy requirements: a new graphical technique for evaluating the integration of utility systems. Comput. Chem. Eng. 20 (1996)p. S225-S230 KalitventzefB. Marbchal F. Heat and power integration: a milp approach for optimal integration of utility systems. Proceedings of the 22nd Symposium of the working party on use of computers in chemical engineering, COPE’91 (Barcelona) 1991 Kalituentzef B. Marbchal F. Process integration: Selection of the optimal utility system. Comput. Chem. Eng. 22(200) (1998)p. S149-S156 KalitventzefB. Marbchal F. Targeting the optimal integration of steam networks. Comput. Chem. Eng. 23: (1999)p. s133-sl36 Kalitventzef B. Marbchal F. Restricted matches and minimum cost of energy echal requirements: tools and methodology for producing practical solutions. PRES’99: 2nd conference on process integration, modelling and optimisation for energy saving and pollution reduction (1999)p. 433-438 KalitventzefB., Marbchal F. A tool for optimal synthesis of industrial refrigeration systems. Comput. Aided Chem. Eng. 90 (2001) p. 457-463 KalitventzefB. Pierucci S. Closon H . Marbchal F. Energy integration: Application to an olefins plant. ICHEAP 4: Fourth Italian Conference on chemical and process engineering 277 (1999) p. 131-134 KalitventzefB. Marbchal F. Favrat D. A methodology for the optimal insertion of organic rankine cycles in industrial processes. 2nd International Symposium on Process Integration, Halifax (2001) 2001.

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Kalitventzef B. Marbchal F. Targeting the integration of multi-period utility echal F. systems for site scale process integration. Appl. Therm. Eng. 23 (2003)p. 1763-1784 Kalituentzef B.,Marbchal F. Study of the insertion of the partial oxidation gas turbine to satisfy high temperature requirements of industrial processes using energy integration techniques. Proceedings of ESCAPE 10, Elsevier Science, Amsterdam (2000) pp. 679-684 Marbchal F. KnlitventzefB. Dumont M . N. Process integration techniques in the development of new energy technologies: application to the isothermal gas turbine. Keynote lecture, paper F3.1, Proceedings of CHISA 98, 13th International Congress of Chemical and Process Engineering, Prague, 23-28 August 1998, p. 205, 1998 Duran M. A. Grossmann I. E. Simultaneous optimization and heat integration of chemical processes. AIChE J. 32(1) (1986a)p. 55 Grossmann I. E. Papoulias S. A. A structural optimization approach in process synthesis: i. utility systems. Comput. Chem. Eng. 7(6) (1983a)p. 695-706 Dhole V. R. L i n n h o f B . Shaft work targeting for subambient plants. Aiche Spring Meeting, Houston, Paper 34d, April 1989 Seider W. D. Colmenares T. R. Synthesis of cascade refrigeration systems integrated with chemical processes. Comput. Chem. Eng. 13(3) (1989) p. 247 Grossmann I. E. Sargent R. W. H. Optimum design of heat exchanger networks. Comput. Chem. Eng. 2 (1978)p. 1-7 Mason D. Linnhof B. Cerda /. Westerberg A. W . Minimum utility usage in heat exchanger network synthesis. A transportation problem. Chem. Eng. Sci. 38(3) (1983) p. 378-387 Grossmann I. E. Papoulias S. A. A structural optimization approach in process synthesis -ii. Heat recovery networks. Comput. Chem. Eng. 7(6) 1983b) p. 707-721 Kalitventzef B. Marbchal F. Synepl: a methodology for energy integration and optimal heat exchanger network synthesis. Comput. Chem. Eng. 13(4/5) (1989) Glavic P. Kravanja 2. Cost targeting for hen through simultaneous optimization approach: a unified pinch technology and mathematical programming design of large hen. Comput. Chem. Eng. 21 (1997) p. 833-853 Kravanja Z. Sorsak A. Grossmann 1. Hostrup M . Gani R. Integration of thermodynamic

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insights and minlp optimization for the synthesis, design and analysis of process flow sheets. Comput. Chem. Eng. 25 (2001) p. 73-8 3. Santibanezj. Grossmann I. E. Applications of mixed-integer linear programming in process synthesis. Comput. Chem. Eng. 4(49) (1980) p. 205-214 Grossmann I . E. Mixed integer programming approach for the synthesis of integrated process flowsheets. Comput. Chem. Eng. 9(5) (1985) p. 463-482 Grossmann I. E. Floudas C. A. Ciric A. R. Automatic synthesis of optimum heat exchanger network configuration. AIChE J. 32(2) (1986) p. 276-290 Grossmunn 1. E. Floudas C. A. Automatic generation of multiperiod heat exchanger network configuration. Comput. Chem. Eng. 11(2) (1987) p. 123-142 Kokossis A. Shang 2. A transhipment model for the optimisation of steam levels of total site utility systems for multiperiod operation. Comput. Chem. Eng. 28 (2004) p. 1673-1688 Iyer R. R. Grossmann I . E. Synthesis and operational planning of utility systems for multiperiod operation. Comput. Chem. Eng. 22(7-8) (1998) p. 979-993 Grossmann I. E. Iyer R. R. Optimal multiperiod operational planning for utility systems. Comput. Chem. Eng (8) (1997) p. 787-800 Hui C.-W.Cheung K.-Y. Total-site scheduling for better energy utilisation. J. Cleaner Prod. 12(2) (2004) p. 171-184 KalituentzefB. Mixed integer non linear programming and its application to the management of utility networks. Eng. Optim. 18 (1991) p. 183-207 Markchal F. Methode danalyse et de synthese energetiques des procedes industriels. PhD thesis, Laboratoire danalyse et de synthese des systemes chimiques, Facult6 des Sciences appliquees, Collection des publications no. 164, Universite de Liege, 1995 Grossman 1. E. Duran M. A. A mixed-integer nonlinear programming algorithm for process systems synthesis. AIChE J. 32(4) (1986b) p. 592-606 KalitventzefB. Dumont M . N . Papalexandri K. P. Pistikopoulos E. N. Operation of a steam production network with variable demands, modelling and optimisation under uncertainty. Comput. Chem. Eng. 20(207) (1996) p. S763-S768 Manousiouthakis V. I!-halwagi M . A. Automatic synthesis of mass exchange networks with single component targets. Chem. Eng. Sci. 45(9) (1990) p. 2813-2831

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Smith R. Kuo W.-C./. Effluent treatment system design. Chem. Eng. Sci. 52(230) (1997) p. 4273-4290 Smith R. Doyle S. /. Targeting water reuse with multiple contaminants. Process Safty and Environmental Protection: Transactions of the Institution of Chemical Engineers 75(3) (1997) p . 181-189 Tainsh R. A. Wasilewski M. Dhole V. R. Ramchandani N. Make your process water pay for itself. Chem. Eng. 103(1) (1996) p. 100-103 Hallale N. A new graphical targeting method for water minimisation. Adv. Environ. Res. 6 (2002) p. 377-390 Kokossis A. Alua-ArgaezA. Vallianatos A. A multi-contaminant transhipment model for mass exchange networks and wastewater minimisation problems. Comput. Chem. Eng. 23(10) (1999) p. 1439-1453 Pans /. Brown D. Marechal F. Marechal F. Mass integration of a deinking mill. pretires 90ieme conf. ann. PAPTAC, 2004 Liu F. Hallale N. Refinery hydrogen management for clean fuels production. Adu. Enuiron. Res. 6 (2001) p. 81-98 Towler G. P. Zhangj. Zhu X.X . A simultaneous optimisation strategy for overall integration in refinery planning. Ind. Eng. Chem. Res. 40(12) (2001) pp. 2640-2653 California Energy Commission: California Distributed energy resource guide (visited 2005) www.energy.ca.gov. Taylor W . R. Holcomb F. H . Binder M . /. Reduced emissions from cogeneration, applications of dod fuel cell power plant fleet. DoD fuel cell, www,dodfuelcell.com, cited 29 June 2004, 2004 Cropper M. Why is interest in phosphoric acid fuel cells falling? Fuel Cell Today, 8 October 2003 Crawley G. Operational experience with Siemens-Westinghouse cogeneration experience. Fuel Cell Today, 17 October 2001 Teagan P. Yang Y. Carlson E. /. Srimamulu S. Cost modelling of SOFC technology. 1st International conference on fuel cell development and deployment, www.~~kell.uconn.edu/f~c/pdf/fcic-pro~amoral-4a.2.pdf; 2004 Colson-lnam S. Solid oxide fuel cells - ready to market? Fuel Cell Today,7/anuary 2004 n o m a s C. E. Cost analysis of stationary fuel cell systems including hydrogen cogeneration. Technical report, Directed Technologies, December 1999, www.directedtechnologies.com, 1999

Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein I 3 8 3 Copyright 02006 WILEY-VCH Verlag GmbH 8

4 Equipment and Process Design 1. David L. Bogle and 6.Eric Ydstie

Abstract

The chapter introduces the computational basis for equipment and process design in the chemical manufacturing industries. Problems, discussed through the use of case studies, range from modeling, simulation and optimization of existing, and proposed processes. These case studies include minimization of exergy losses in distillation through design and control, conceptual design of a complex, multiphase system with fluid flow from pilot plant scale-up data, handling uncertainty in the design of a multiphase reactor, a biochemical process design, and control system design for a fluid bed reactor modeled using a population balance approach. In all of these studies we focus on exploiting a very rich mathematical structure defined by conservation laws and the second law of thermodynamics. The basic objectives will be briefly outlined dealing with the fundamentals of modeling systems of considerable complexity. The hierarchical modeling approach will lead to classes of systems that are suitable for the use of optimization, control design, and the study of the interaction between these for both unit operation and flowsheet design. Design objectives continue to include profitability, but are increasingly directed towards flexibility and robustness to allow for greater attention to responsiveness to market demands, and for improved environmental Performance. Controllability is also currently a focus of attention since flexibility implies dynamic performance objectives that include change of product and feedstock as markets change and environmental constraints become more important. The work we describe in the case studies represent recent developments and trends in the industry in the area of computer-aided process engineering (CAPE). The applications focus on the use of optimization techniques for obtaining optimal designs and better approaches for controlling the processes close to or at the optimal point of operation. Designs use rigorous models based on thermodynamics, conservation laws, and accurate models of transport and fluid flow with particular emphasis on dynamic behaviour and market condition uncertainties.

Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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4.1 Introduction

Engineering design combines the need to specify a unit or system that can manufacture a product to meet output specifications while creatively developing new approaches with the potential to improve the performance of existing units. In many parts of the process industries this is now done routinely using computer-aided process engineering (CAPE)tools. Such tools mean that many alternative designs can be developed and evaluated and that designs can be optimized to obtain the best performance under a wide range of market conditions. Only a few years ago it was impossible to carry out such studies for all but the simplest processes. However, very significant advances have been made in optimization theory, modeling complex systems, and nonlinear control over the last decade. These new methods take advantage of the rapid increase in computational speed, distributed and Web-based computing, flexibility in software development, and graphical interfaces to a degree not imagined a few years ago. This conflux of trends of ideas will necessarily lead to opportunities to better the design of, optimize, and operate chemical processes, as we are able to analyze and integrate a much broader range of physical scales and physical phenomena with CAPE tools. In this chapter we will outline some recent trends in equipment and process design through a series of case studies. We have chosen to focus on cases where the use of simulation tools was significantly challenged and where opportunities for new areas of application, as well as new research, are present. When new processes are being developed there is an important requirement to obtain accurate property data on which to base the design. This means that for process development there is a strong need to tie the design process with bench and pilot plant data to attempt to reduce uncertainty. Of course, CAPE tools allow us to do this more efficiently. In several case studies this has been achieved. A second trend is the need to obtain more accuracy on spatial distributions as the production of by-products, for example toxic wastes or unsafe local transients, need to be reduced or eliminated. This can now be done with Computational Fluid Dynamics (CFD) calculations and one of the case studies uses this capability. One of the key aspects of design is the design of objective specifications. These are needed, in the first instance, to make a product to a minimum specification and quantity at least cost. Increasingly other objectives are becoming more important such as: environmental (least waste, least water usage, least COz generation), safety (least toxic by-products),flexibility (maximizing the window of operation), controllability (metrics based on closed loop responses to disturbances), and uncertainty (based on expected distributions of key variables). 4.2 The Structure o f Process Models

In this section we will very briefly review the basis for process modeling, a field whose aim it is to develop the physical relationships that will allow us to make pre-

4.2 The Structure of Process Models

dictions, using mathematical models, of how the choice of design and control variables impact the quality and quantity of a product we make. The field of modeling, of course, is extremely broad. However, a fact that is often overlooked is that thermodynamics, most notably the second law, provides us with a very rich framework for analysis of the topological structure of vector fields that define the static and dynamic behaviour of that class of systems we are interested in discussing. For example, classical thermodynamics defines the relationships amongst the extensive variables used to define conservation laws (conservation of energy, mass, charge, and momentum) and the intensive variables (temperature, pressure, chemical potential, voltage, and stress) that need to be controlled in order to maintain quality. Process design and control need to be concerned with both types of variables. The topological structures of these vector spaces are very different, however, and it is important to keep this fact in mind. The space of intensive variables is usually quite smooth since the intensive variables provide driving forces for flow and sharp gradients in temperature, pressure, electric field, stress, and chemical potential are difficult to maintain. The space of extensive variables can be quite irregular since the energy, composition, charge, density, etc., are discontinuous across phase boundaries. This has consequences for how we solve the conservation laws since these are stated in terms of the extensive variables. A thorough understanding of the underlying topology of the problem at hand can provide insights that can be used to obtain better solutions more rapidly, and solution structures may be obtained that are the result of discontinuous jumps in logic that are normally difficult to encapsulate in stand-alone computer programs. All systems we deal with satisfy to very high degree of accuracy the basic axioms of equilibrium thermodynamics : 0

0

0 0

0

The state of a fluidlsolid element is represented by the (n+2) dimensional vector Z = ( U,V, M ~.., ., M , ) ~of extensive variables. u is the internal energy, v the volume, and Mi is the amount (mass or moles) of chemical component i. This state can be augmented to include charge, momentum, and indeed any other extensive property. There exists a C2 function S(Z), called the entropy, which is first order homogeneous so that for any positive constant h we have S(h, z) = AS(z). The entropy of an isolated system increases (second law of thermodynamics). The energy of the system is conserved (first law of thermodynamics). 8 S / 8 U > 0 (the temperature is positive).

The vector Z of extensive variables belongs to a convex subset of Rn+2,which we denote by Z . Based on these concepts and the conservation laws derived therefrom we develop optimization and control models for almost all unit processes of chemical process systems, the structure of phase and reaction equilibrium, and the potentials (pressure, temperature, chemical potential) used to derive transport relations. In equilibrium thermodynamics we can combine subregions of a system while retaining the idea that the extensive variables describe the state of the system. This means that two fluid elements with states Z1 and Z2,respectively, give a new element when combined with the state Z3=Z1+Z2.The second law of thermodynamics can now be stated in the following manner:

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S(Z3) z S @ l )

+ S(Z2)

(1)

In other words, the entropy of combined systems is never smaller than the sum of the entropies of its subsystems. It follows that for any positive constant A:

+

S(hZl+ (1 - h)Z2) L S(hZ1) S((1 - h P 2 )

Using the homogeneity gives the relationship:

This expression shows that the entropy is concave. Concavity of the entropy function and dissipation by entropy production lies at the heart of the beautiful structures that emerge as we simulate the static and dynamic behavior of process systems. Consider for example phase equilibrium. When Gibbs developed the tangent plane criteria for phase stability he used concavity of the entropy as a basis for defining the condition for phase equilibrium as shown in Fig.4.1. The slope of the supporting hyper-plane indicated satisfies: C * t W ) = sup ( S ( 2 ) - WTZ) ZE

t l

c

(4)

/’_

Figure4.1 Illustration (a) shows a projection o f an entropy function and a unique point of stability indicated at the point Zl. The slope ofthe tangent line, w1 defines the intensive variables at that point. Illustration (b) shows a projection, as it might arise in a cubic equation o f state like the van der Waals equation. In this case the EOS

gives three points where the slope, and hence the intensive variables, are the same. The actual entropy is defined to be the smallest concave envelope as indicated by the straight line segment. Along such segments the entropy is concave but not strictly concave. Entropy functions below the line are in violation of the second law.

4 2 The Structure of Process Models

Intensive Variables

Extensive Variables

Figure 4.2 The mapping w = PZ from the space o f extensive to the space of intensive variables is not one to one The line Z , , corresponding to a uniform scaling o f all extensive variables maps to the same point w, in Z’ indicating that pressure, temperature and chemical potential remain invariant Subregions with positive measure (regions in Z with phase or reaction equilibriumZ) map to points with zero measure (in 2”).

The vector w is called the vector of intensive variables in the entropy formulation. These variables belong to the convex set: C * ( W ) = sup (S(Z) - W T Z ) ZE c

The sets Z and Z” are subsets of Rn+2 and they are dual in sense of LegendreFenchel. It is critical to note that the mapping between these spaces is not one to one. The lack of invertibility between extensive and intensive parameters occurs on the subsets of Z where the entropy, although concave, is not strictly concave as illustrated in Fig. 4.2. If the entropy is linear it follows that it is merely concave in some directions. Along these directions we can scale the extensive variables without changing the intensive variables. We may also have discontinuities in the space of extensive variables (Callen 1985). The noninvertibility between the space of extensive and intensive variables is critical for process design. It allows the scaling of the size of the system along lines in the space of extensive variables without changing the intensive variables. We can therefore independently control the “size of the systems” (amount present in each phase) without changing the quality (pressure, temperature, chemical potential). In more practical terms, we can also change the hold-up without changing the properties of phase or reaction equilibrium. These scale invariant properties are so common that we use them always without thinking in thermodynamics. In transport we find similar scaling laws centered on the ideas of dimensionless numbers like the Reynolds, Rayleigh, Nusselt, and Damkohler numbers. These numbers include transport properties and estimates of the size of the domains. The relationship between size and shape, being the extensive properties that define the domain of interest, and the intensive properties is much more complex. It is especially difficult to unravel in systems that integrate fluid flow, chemical reactions and multiple phases. In chemical engineering we need to develop models to deal with systems that are not at equilibrium and have variations in space and in time. By far the most common approach is to develop a lumped description (a network model) of the process. In this

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approach we use nodes distributed in space that are interconnected by steady or unsteady flow of material and energy. These nodes may represent a single phase in a separation system, a unit process, or an entire plant. Irrespective of size we have additivity satisfied so that we can write conservation equations: $inventory = inlets - outlets + generation dt Inventory is a macroscopic measure of the total internal energy, volume, number of moles of each species respectively so that

-

where the subscript t refers to total. The balances can be written on the form: dv dt

- = @ ( x 9m, 4

Nedlow to system

+ p\-.--/( x , m, 4

Net Generation

The variables x, rn, d represent the state, manipulated and disturbance variables needed to define the flow and generation variables. We make a distinction between the extensive variables Z and the inventories v. The extensive variables were assumed to represent the state of the system with some degree of accuracy. The inventory v does not represent the state unless the system under consideration is spatially uniform in each phase. For example, consider a distillation column with benzene as one component. The macroscopic inventory balance can be applied to the entire system with generation equal to 0. But the inventory of benzene does not represent the state of the system since it can be distributed in an infinite number of different ways throughout the column. In order to have a unique (state)description we need to look at a more refined system. In a stage to stage model of the system we find that the inventory in fact represents the state to a good degree of approximation. Thus we find that uniformity can be approached asymptoticallyby tessellating the physical space finer and finer. In the limit we define the point densities:

M

Z and z = lim -

p = lim -, v-0

v

where M =

M-0

n

M

(9)

M~ is the total mass or moles. These limits exist if the spatial varia-

i=l

tions of Z(x) are sufficiently regular. Local pressure, temperature and chemical potential is defined as before.’ The inventory vi is related to the density of the extensive variables z according to:

1 This Assumption is called “the principle of local

action” or the hypothesis oflocal equilibrium. The latter term can be misleading since it does not at all

mean that the TRP system is at equilibrium: there may be variations is space and time.

4.2 The Structure ofprocess Models

And for a system with one spatial dimension partial differential equations of the form: aPz af -+-=a at ax t is the time, x is position, pz defines the local state vector,f= Cfi,...,fn+2,Tis the transport velocity and u refers to the production rate of property z. These equations are easy in principle to extend to include the momentum transfer and the equations then include the Navier-Stokesequations as a special case. By applying the ideas above to the entropy we find that we can write the entropy balance of the form

*+ at

ah

- = a, where a, 1 0 ax

where s is the entropy density,fr represents the entropy flu through a point and os is the entropy production. The inequality states that the production of entropy is always positive. In a nonequilibrium system the rate of entropy production represents the dissipated energy and an important design objective therefore is the minimization of entropy production. In irreversible thermodynamics the entropy production is given by the expression a, = X T L X

where the vector X represents the thermodynamic driving forces (these are gradients in pressure, temperature and chemical potential) and L is a positive definite matrix. It is quite straightforward to show that the entropy production is minimized when the forces are kept constant and equal in certain staged systems (Farschman et al. 1998). This leads to the important idea of equipartition of forces as a criterion for energy efficient process design (Sauar et al. 1996). The assumptions made up to this point are well tested and hold at very small scales and sharp gradients. Only at extremely small scales and extreme gradients is it necessary to resort to other techniques like lattice Boltzmann methods to give better approximations to physical behaviors. The field of process control has typically concerned itself with how to adjust the flows in a network with fixed network topology. The field of process design has typically been concerned with how to design the network topology. The distinction between the two fields is becoming more and more artificial however since the topology and flows are very closely interlinked. In a high performance system it is not possible to separate the two and equipment and process design should be considered in conjunction with the process control.

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4.3 Model Development

The book by Hangos and Cameron (2001) sets out systematic approaches to model development and analysis breaking down the steps as shown in Fig. 4.3. If undertaking a first principles modeling task, it is important first to establish the need for the model which define the “goalset definition” in their methodology. These should be specific and quantifiable and a list of common objectives is listed in the first section of this chapter. Models for controllability for example will be different from those required for capital cost minimization. As outlined in the previous section conservation relations can be developed for mass, component mass, energy, momentum, or population of a species. The choice of which relations are required will depend on the nature of the system. For example the population balance relations are only required if there are discrete elements whose properties affect the purpose of the system being analyzed. These may be solid particles, or micro-organisms, or bubbles in liquid or vapour phase. Further constitutive relations will be required to model transfer rates, reaction rates, property relations, and equipment and control constraints, as appropriate. Increasingly there is a need in process development and design to develop more detailed models of parts of the process. This is done for a variety of reasons. More confidence may be required in a new design of unit operation, particularly if the unit has only been used in the prototype. Dynamic simulation requires a more detailed model of complex units so that time-dependent elements, especially those where the time delays are related to transport effects, can be predicted with some degree of accuracy. Contractors are increasingly being required to provide dynamic simulations with the final delivery of a process design. Safety and environmental expectations of customers and local residents now can require more detailed models of plant items. All of these require sophisticated models.

4.4 Computer-aided Process Modeling and Design Tools

Most modeling and design is now done using CAPE tools. Bogle and Cameron (2002) recently reviewed the tools available and how they are used throughout the design and development lifecycle. Sequential modular packages commonly used in industry are based on calculating the sequence of unit operations in the order in which they appear in the flowsheet, using modules for each unit of the flowsheet. If the flowsheet contains one or more recycles a stream must be guessed (“torn”)and iterative methods are used to converge the calculated values to the guessed values. Most modules are written in FORTRAN, a high-level computer language widely used in the engineering and technical world. Examples are AspenPlus and ProII. The calculation procedure is at the heart of the computational procedure used by these packages to solve the problem. However, there are many other parts which make the whole package a convenient tool for the design engineer. The following are

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r_ Model Solution

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-

L - _ _ - -

i-

- - - -- --- -

-j

W I

Model Calibration and Validation

I

~

Figure4.3 The systematic approach to model building (Hangos and Cameron 2001).

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the most common elements: input interpreter, unit operation subroutines, physical property prediction subroutines, algorithms to select torn stream, recycle convergence block, costing databank, output post-processor. Equation-oriented packages use a matrix representation of material and energy balance problems. The set of modeling equations is assembled and can be solved numerically. From the equation set an occurrence matrix is assembled. This dictates the number of degrees of freedom, which in turn dictates the number of specifications that can be made. Once a square system is obtained the design equations can be solved giving the solution to the design problem. gPROMS (Oh and Pantelides 1994) and Aspen Custom Modeler are the most prominent examples currently. These systems have advantages over modular systems in that it is easier to set up and solve other types of problems such as dynamic simulation, optimization, and parameter estimation. In both cases models of each unit operation are required. In the design tools simple models are usually used so that overall designs for process flowsheets can be assembled quickly to be used as targets for more detailed design of individual units. There are many software programs for doing detailed design calculations obtaining internal flows and configurations such as the number of trays in a column or the size of a reactor’s packed bed. These calculations are often done using specialist programs which require a more detailed level of modeling. Eggersmann et al. (2002) reviewed the current state of such programs. Network representations use diagrams and pictures and show in a graphical manner the topological properties of a system as they are distributed in space and/or time. Examples of systems that can be effectively represented as networks include a flowsheet of a chemical process, chemical reaction networks, metabolic pathways, decision trees and transportation systems. An example of such a network is shown in Fig. 4.4 on the left. Each node represents an activity where information/material is processed, whereas each vertex can represent flow of information, material or energy. We note that several activities can be lumped together as indicated by activity A+ We can therefore decompose a process into a multiscale hierarchy ranging from the molecular to the macrolayer where economic decision making takes place (Grossman and Westerberg 2000; Ingram et al. 2004). These decompositions have motivated very active research which very recently has led to significant breakthroughs in our ability to model very complex systems with integrated fluid flow, multiple phases and population balances to express the dependency of physical properties on particulate matter. It is important to note that the additivity of the extensive variables allows us to combine such nodes very easily and we therefore have computational architectures with integrating scaling (Farschman et al. 1998; Mangold et al. 2002). These architectures are flexible and amenable to parallel computing using highly distributed Web-based systems and or cluster computer. Such systems are becoming increasingly easy to define and maintain due to interface standards like TCP/IP and CAPE-OPEN (www.colan.org). In this paper a series of case studies is presented where a sophisticated level of modeling was required. This required the use of commercial tools but enhanced with detailed models of the units that were not routinely available in the design sys-

4.5 Introduction t o the Case Studies

1

Physical network

login machine,

batch server.

Distributed Computer Representation

Figure 4.4 The topology o f the process system can be mapped onto a distributed cluster of fast computers interconnected by a high bandwidth communication for parallel processing.

tems on the market. This trend seems set to expand - that customers will require high-fidelity models of critical units often integrated into the design systems that they currently use so that the systems can be used for troubleshooting, startup and shutdown, and for plant enhancements. A closer relationship is required with existing plant, pilot plant, and bench-scale experimental data to improve the fidelity of the models. A rigorous thermodynamic basis is also necessary. This constitutes a development from the use of design systems as balance tools setting targets for detailed design to a staged use of the tools with increasingly sophisticated models used to achieve different objectives. These case studies highlight these more challenging objectives .

4.5 Introduction to the Case Studies

In the following sections we give an overview of five case studies that describe how CAPE tools have been used to improve process design and control for existing and conceptual processes. The first two case studies, of distillation and of a complex mul-

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tiphase reactor, involve detailed and thermodynamically rigorous models, which are used to ensure the design will meet the productivity targets and reduce or minimize the energy consumption and hence improve the environmental performance. The third case study also considers a multiphase reactor and shows how uncertainty can be systematically considered in the design of the unit. The final two case studies, of a biochemical reactor and separation system and of a fluidized bed reactor, both involve particulates, which requires the use of population balance techniques to model the fluid particle systems. Each case represents a unique challenge and opportunity for modeling and optimization.

4.5.1 Case Study I: A Binary Separation Case Study

Distillation column design is a well established activity where the use of CAPE tools is routine. However design is usually based around well established procedures that are now being challenged with increasing need for design for environmental performance or for better closed loop performance. The CAPE tools allow engineers much more scope for developing designs to meet these new objectives. In this case study a column for the separation of vinyl chloride monomer was designed to meet environmental targets and determining the effect on closed loop dynamic behavior. While the conventional wisdom is that optimizing environmental performance results in a degradation of performance by other measures, this case study shows that is not necessarily the case. For this design a column with total condenser and reboiler was required for the purification of vinyl chloride monomer (VCM). The product quality is fxed at both ends of the column which in turn fmes the product withdrawal at both ends of the column. The feed location is an independent variable and was to be optimized. For binary mixtures there will be one feed plate where the feed stream and the internal column flowrates match. For an optimized column the feed always enters the column on a single plate and is not split over a range of plates. The pressure drop between the top plate and the reboiler is approximately 0.165 bar and depends on the number of plates and the size of the internal flow rates. The objective was to develop a column design which utilized the energy most efficiently by trying to avoid irreversible losses. A diabatic distillation column is a column with intermediate sideheating and cooling on individual plates. Parts of the heating and cooling utilities are re-directed from the bottom reboiler and the top condenser of the adiabatic column to individual plates, generally around the feed plate. The advantage of such a configuration is that heating and cooling can be supplied at alternative temperature levels and the column profiles can be moved towards the reversible column profiles, as discussed by Kaibel et al. (1989, 1990).A n operating curve close and parallel to the equilibrium line is a thermally optimal column configuration.

4.5 Introduction to the Case Studies

4.5.1.1 An Environmental Design Problem: Minimizing Exergy Loss

The concept of equipartition of the driving forces as an optimization criterion to minimize the exergy losses was first introduced by Ratkje et al. (1995).Earlier, Tondeur and Kvaalen (1987)proposed the equipartition of the entropy production (EDF) criterion while Sauar et al. (1996) showed that a better approach was to design for uniform driving forces for different paths over a given transfer area will yield minimum entropy production. The criterion has been further discussed by Kjelstrup et al. (1996) and Sauar et al. (1997). Similar techniques based on thermodynamics have also been used to find the optimal temperature profile in equilibrium chemical reactors (Sauar and Ydstie 1999). The method used here is to directly optimize the exergy losses to minimize the production of entropy, the ExL criterion. Entropy is produced because of the existence of driving forces. To make uniform these driving forces the separation exergy has to be redistributed and this should be an automatic result of minimizing exergy loss. The advantage of the ExL criterion over the EDF criterion is that it considers all possible sources of entropy production without the need for additional constraints. Kaibel (1990) pointed out that the main exergy losses occur on the feed plate due to (1)the feed condition and the adiabatic column design and (2) due to heat transfer, and pressure losses and are necessary because of there being a finite number of plates. The latter (2) can only be reduced by substantial capital investment. The first (1)can be improved by minimal capital investment, i.e., specifylng the feed condition correctly and a re-distribution of the heating and cooling utilities. Hence, the degrees of freedom are the heating/cooling utilities on each plate, the feed location, and the feed condition. The feed location and composition have been determined by traditional means. Heating and cooling requirements were calculated for exergy optimized 23, 25, and 35-plateVCM columns (Table 4.1).For the 25-plate column, seven side-reboilers and two side-coolers (Hagemann et al. 2001) would be required to minimize exergy 02.5

om a2 0 175 0.15

a075

00s

UOX 0

Total Condenser Figure4.5

Plates

Exergy loss profiles for the VCM column (ExL Criterion).

Reboiler

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4 Equipment and Process Design Table4.1

~~

Exergy comparison of the diabatic with the adiabatic column [CJh-’1.

~~

Adiabatic - Diabatic

22 Plates

25 Plates

35 Plates

Difference in total cooling duty

-0.285 (+31.6%)

-0.182 (+20.7%)

-0.123 (+14.9%)

Difference in total heating duty Difference in total exergy losses, including losses due to intermediate heating or cooling

1-0.341 (-9.8%)

I

-0.552 (-25.1%)

1-0.385 (-11.5%)

I

-0.684 (-37.3%)

1-0.34 (-10.8%)

I

-0.592 (-35.4%)

loss. Seven reboilers is not a practical option but we will see later how much of this benefit can be achieved by a more practical solution. The exergy loss profiles of the VCM column are shown in Fig. 4.5. The profiles for the nonoptimized adiabatic column show very large exergy losses within the stripping section caused by large vapor phase gradients. In contrast, the optimized column shows an almost uniform profile over the mid section of the column. This is directly related to the re-location of heating and cooling utilities. The overall exergy losses are substantiallylower for the optimized column and decrease further with an increase in the number of plates. A dip in the exergy losses at the feed plate can be explained with the composition of the entering feed flow rates. The flow rates (vapor + liquid) closely match the outlet feed plate compositions and, hence, do not create exergy losses. 4.5.1.2 A Practical Interpretation

For the 25-plate column in Table 4.1 the thermally optimal design requires sideheating utilities on plates 14 to 20 and side-cooling on plates 12 and 13. For industrial application an additional seven side-heating units and two cooling units is clearly not a practical solution. However the intermediate cooling utilities can be neglected as these represent only 5.7% of the total required cooling utilities and the seven side-heating units can be replaced by one side-reboiler with very little effect on the results. This single side reboiler, located on plate 15, provides 74% of the side-reboiler utilities of the diabatic thermally optimal design, with an increase in exergy losses of only 11%. This side reboiler is located in an area of low driving forces. Figure 4.6 shows the temperature profiles for the adiabatic, diabatic thermally optimal, and diabatic one side-reboiler VCM column design. For the diabatic thermally optimal column design, the temperature and concentration (not shown) profiles are almost linear between the pinch points at both ends of the column. For the diabatic column with one side-reboilerthe profiles are also closer to being linear than for the adiabatic design. The critical factors for obtaining such “near linear” profiles are the location and size of the side-reboileras well as the condition and location of the feed stream. A shifting of the side-reboilerinto a region of already high-driving forces, or using the side-reboiler to dump available heating utilities, will increase the exergy

4.5 Introduction to the Case Studies

Figure 4.6

Temperature profiles for adiabatic and optimized 25-plate

col u m n.

losses but, perhaps more importantly for the control system design, will alter the column profiles away from near linear profiles. This optimized column design with one side-reboiler column configuration has been used to study a possible control system design and compare it with an adiabatic 25-plate VCM column configuration. 4.5.1.3 Effects on the Controllability of the Process

The new process design has a different dynamic response to feed disturbances which in turn can provide equivalent process stability and/or less capital investment. The design for both columns is based on the widely used LV column control configuration (Luyben 1990).The operating objectives for the VCM column are high product recovery and product purity. The selection of the LV configuration has the advantage that it has a direct effect on the product compositions and is weakly dependent on the level control scheme (Skogestad 1997).The controllers were tuned using the ZeiglerNichols tuning method. Results for 3% feed flow rate and 3 % feed composition disturbances and for a 1"C feed temperature disturbance are tabulated in Table 4.2. The response time, defined as time taken for the output variable to return within a fured distance from the new or old stationary value, has been used as a measure of the quality and stability of the process and control system design. Table 4.2 gives a summary of the response times, in hours, after the feed disturbances at time 0.05 h. It is assumed that the product compositions have reached a new stationary operating condition if the differential values are within (and remain within) *lo% of their final values. The results show that especially for the bottom composition the response times of the diabatic column are much shorter. In all of the feed disturbances reported here, except one, the dia-

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4 Equipment and Process Design Table 4.2

Summary of response times after the feed step disturbances.

Distillate Temperature (Composition) Control ~~

Response time [h]

Disturbance at time 0.05 [h]

Adiabatic

Diabatic

Flow rate: [+3.0%]

0.68

0.53

VCM composition: [+3.0%]

0.72

0.81

Temperature: [+l.O"C]

0.69

0.54

Bottom temperature (composition) control Response time [h]

Disturbance at time 0.05 [h] ~~

~~~~

Adiabatic

Diabatic

Flow rate [+3.0%]

>0.95

0.37

VCM composition: [+3.0%]

>0.95

0.58

Temperature: [+1.0 (C]

>0.95

0.56

batic column shows better disturbance rejection. The exception is the VCM composition of the distillate after a 3.0% VCM feed composition increase. The reason for this result is the fixed side-reboiler duty. Most of the additional VCM entering the column is vapor. Hence, after the disturbance the side-reboilerevaporates a higher fraction of ethylene dichloride (EDC)until the main reboiler is adjusted to the new operating condition. These results are perhaps surprising because the side-reboiler for the diabatic design introduces strong cross-coupling effects causing considerable oscillations within the column. However the improved controlled response is due to the removal of the sharp nonlinearity in the temperature profile (Fig.4.6).Shifts in the internal temperature, used as a measured variable in the control scheme, are less drastic producing a more even closed loop response. This design study was undertaken using a tray to tray column model with rigorous thermodynamics using SPEEDUP, now a part of the Aspen Custom Modeler system. The ability to develop a rigorous model was essential here in order to achieve the two goals, that of improving environmental performance and analyzing the controllability. Ideally the ability to make structural changes to the process, in this case by adding and removing side-heaters and coolers, could be done automatically but this challenge still remains. The use of integer programming techniques would permit this but they are not yet available in modeling systems.

4.5 Introduction to the Case Studies

4.5.2 Case Study 2: Conceptual Design of Multiphase Reactor

In this case study we review very briefly a challenging program for developing a new process for producing primary aluminum (Johansen et al. 2000; Johansen and Aune 2002).The aim of this joint research program amongst ALCOA, ELKEM and Carnegie Mellon is to develop a process for production of primary aluminum which reduces capital, energy and environmental costs relative to the Hall process with a significant amount (Motzfeldt et al. 1989). The process under consideration, carbothermic reduction, uses electric heating, but a carbothermic process is more energy efficient and has very high volumetric productivity leading to better economies of scale. 4.5.2.1 Carbothermic Reactor Modeling

Carbothermic reduction can be realized in a two-stage process with additional steps needed to purify the product, recover aluminum from off-gases, and recover heat. A diagram of the process is shown in Fig. 4.7 and it is based on a reversible, multiphase and multispecies chemical reaction phenomenon. The kinetics of the reaction mechanism has not been completely finalized, but ionic species identification has been conducted and there is a reaction rate model proposed by Frank et al. (1989);the reactants are aluminum oxide (A1203)and carbon (C), key reaction intermediates are aluminum suboxide (A120)and aluminum carbide (A&), and the end products are molten aluminum (Al) and carbon monoxide (CO).The complexity is illustrated by this simultaneous coexistence of solids and gases in the multicomponent reactive slag.

n

Figure 4.7

Diagram of the carbothermic aluminum process.

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4 Equipment and Process Design

1. The first stage is the pre-reduction smelting zone. Carbon and aluminum oxide pellets are continuously fed to a submerged arc smelter, melt, react and form a molten slag contained in an inert-atmosphere, oil-cooled reactor. The reaction of aluminum oxide with excess of carbon to form the AI4C3-richslag of the first stage is (T >1900(,))

2. The second stage is the high-temperature reaction zone: the first-stage molten slag

flows into this core stage (a multielectrode, high-temperature submerged arc reactor); the slag is heated to a higher temperature, avoiding severe local surface superheating caused in open arc reactors. Liquid A1 droplets and CO bubbles are rapidly generated, with concurrent A14C3injection to avoid carbon depletion. The decomposition of the first-stage, A14C3-richslag to form the second-stage, Al-rich phase is (T > 2000°C):

3. The third stage is a vapor recovery reactor (VRR), where A1 and A120 vapors react with C to form A14C3. Vaporization occurs as CO vapors sweep the second stage: unless Al,,) is recovered counter-current to incoming solid feed, metal losses are catastrophic for process economics. Undesirable vaporization is thus reduced by staging and feeding the first and second stage gas streams to the VRR. The recovered A& (recycle stream) is re-injected into the reactor, minimizing A1 emissions and maximizing yield. Counter-current flow exceeds incoming reactant preheating needs, thus allowing for energy recovery via cogeneration. 4. The fourth stage of the process is the purijcation zone: liquid aluminum flows through an overflow weir to a flotation and skimming unit, where entrained A14(& dissolved C,,, can be removed via proprietary technology. The goal of our research is to develop models for preliminary design, pilot plant scale-up, optimization of the process and process control. To make best use of energy aluminum vapors and their energy content must be recovered, preferably as aluminum carbide in the VRR. The approach we investigate solves the problem by feeding a carbon material at the top of the counter-current VRR. The carbon reacts with the aluminum compounds in a series of heterogeneous noncatalytic reactions forming solid and gas products. Liquid products, like molten slag, can be avoided by running the VRR at high temperature. The most important products are aluminum carbide, which is needed in the smelting stage of the aluminum process, and hot carbon monoxide gas, which can be used for co-generation of electricity. Garcia-Osorio and Ydstie (2004) developed an unsteady state reactor model which incorporated a shrinking core, mass transfer limited reaction separate mass and energy balances for the vapor, liquid and solid phases and dynamics distributed in space and time. The model equations take the form of conservation equations. The A120and A1 reactants take the form:

4.5 Introduction to the Case Studies

i = A120, Al Similar expressions were developed for the product species. Mass transfer limited diffusion was assumed for the solid gas reactions. All liquid phase reactions were assumed to be in equilibrium. The final model was solved using the method of lines and a stiff (index 1) DAE solver was used to solve the final large-scale system of differential algebraic equations. The model was matched to small-scale experiments and interfaced with proprietary thermodynamics describing the Al-0-C system for temperatures in the range 800 to 2050°C. It was found that the model predicted pilot plant results within 10% and that it could be effectively used for sensitivity studies and scale-up. An example of a sensitivity study is reported in Table 4.3. F, is the dimensionless feedrate of carbon with the range [SO-lOO%].

FC*

AI,O %

A1 %

x,

0.5

96

16

59

0.7

I 98

I 33

I 43

98

47

32

1

%

Gerogiorgis (2002, 2003a,b) developed a number of different CFD models to study design of the aluminum producing stage (stage 2). The objective was to solve the steady state PDE problems for the potential (V), field intensity (9,temperature (IT), velocity ( U,, U,)and pressure (P) to obtain reliable starting points so as to solve molar balances for species concentration profiles in a complete model, reliable for performance evaluation. Constant thermophysical properties were assumed in many studies, although a temperature-dependent electric conductivity has been used to illustrate the strong coupling between the charge balance and the Joule heat generation term. The k - E standard model of turbulence was used in the momentum balance in order to analyze the turbulent slag flow in the reactor. We also developed a two phase flow model to study how the gas generation impacts the fluid flow and mixing characteristics. The finite element CFD model of the reactor has been solved for different geometries with quadratic finite element basis functions on a fine unstructured triangular grid (12,124elements), using commercial simulation software (FEMLAB v. 2.3). Different FEMLAB multiphysics modules have been used and integrated for the simulations (Conductive Media DC, Convection and Conduction, k-E turbulence model and a two-phase model based on constant relative slip velocity). For the present case, nominal Reynolds numbers indicate slag flow well within the turbulent regime (Re -30,000). The steady state CFD problem with turbulence considers three PDE balances and the corresponding partial differential equations on a two-dimensional domain. The first part is the steady state electric charge balance:

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v2v = vxx+ vw = 0

(17)

The second part is the steady state heat balance:

);;-(

v . (kVT - pCpTU) + o ( V V ) ~- ko exp

- AH = 0

The third part is the steady state momentum balance, which comprises the continuity and velocity PDE system: v.u=o p ( U . VU)

-

v.

[(. +

p -C-, ) k2 ak

. (vu + (VUf)

6

1

= -VP

complemented with the two standard k-E turbulence model: p ( U . Vk)

-

V

[

. (w

+p

s ok

c) E

Vk] = pC, k2 (VU

+ (VU) T ) 2

&

- PE

(20)

The imposed voltage on all electrode tips (Vi, i = 1-6) is set, zero voltage is used on both long horizontal domain sides to approximate the potential in the third lateral dimension, and zero gradient (VV= 0) is used on all other wall sides to account for the insulating behavior of solidified slag. Inlet slag (2173 K) and wall (473 K) temperatures are set, and insulation (VT= 0) is assumed at all six electrode tips. An inlet vertical slag velocity is assumed ( Uo= 0.01 m s-', with a suitable wall function (Gerogiorgis and Ydstie 2003a) on the walls and all six tips. A slip boundary condition is used for the slag-free surface and zero pressure has been assumed at the reactor outlet. A typical result for a two-dimensional simulation showing constant velocity contours is shown in Figs. 4.8 and 4.9. The simulation revealed several important aspects about the process that play a role in understanding the underlying physics and impact design and scale-up. The simulation showed that short, submerged electrodes were preferable to long ones in order to get better energy distribution and flexibility for controlling the process by

.. ...

Figure 4.8 Single-phase flow model with constant velocity contours for steady state multiphysics CFD model. The slag moves under the electrodes and avoids the primary reaction zone. This problem can be solved by raising the floor of the reactor.

4.5 lntroduction to the Case Studies I403

Figure 4.9 Two-phase flow model with major flows superimposed on small-scale movements leading to coarser scale representation of the system, with each zone being represented by a uniform mixing tank with inter-stage flow.

moving the electrodes. Our current studies focus on three-dimensional multiphase simulation to study the issues of back-mixing between stage 1 and 2 and the feasibility of controlling the flow between the stages. 4.5.2.2 MINLP Model for Electrode Heating System Design

One design challenge is to optimize electrode positions and the imposed voltage profile so as to achieve the reaction advance without unnecessary reactor space or energy use. Obviously, dense electrode placement and high voltage result in excessive superheating, causing aluminum evaporation, major yield reduction and losses, while sparse electrode placement and low voltage fail to achieve adequate slag heating resulting in limited conversion and low productivity. In order to address some of these issues Gerogiorgis and Ydstie (2003b) developed a Mixed Integer Nonlinear Programming (MINLP) finite volume model of stage 2. The goal here is to perform electrode placement as well as imposed voltage profile optimization for maximization of A1 production under mass, heat and molar species balance constraints. The mathematical formulation is based on a CSTR series steady state process model; each finite volume is assumed a CSTR with perfect separation of reactants and products. The advance of the reversible reduction towards A1 (liquid) and CO (gas) is considered governed by the overall reaction proposed in a recent kinetic study (Frank et al. 1989). A typical result from the optimization procedure is shown in Fig. 4.10. The long term objective of the research is to develop a multiscale hierarchy of models of the aluminum process, which can be used for pilot plant design, scale-up and optimization of the final process, including control studies. Towards this end we have developed a new approach for Web-based distributed simulation (Garcia-Osorio and Ydstie 2003), which will allow the solution of modules in a network representation of the process over the web.

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4 Equipment and Process Design

IMPOSED ELECTRODE VOLTAGE

I

2bU0

F I I I ' I E VOLI

J

M E I! (i)

h

ABSOLUTE TEMPERATURE

2400 2(lll

2ouo

5 L

1500

1600 !JNl

I2UO 1000

I

I

-I

6

FENITE V O L U M E # (i) Figure 4.10 MINLP model comparing a reactor with three electrodes (gray) and one with six electrodes. The optimal imposed voltage is shown on the left while the resulting temperature profile is shown in the figure on the right.

4.5.3 Case Study 3: Uncertainty Analysis in a Multiphase Reactor

Risk analysis methods for the quantification of uncertainty and identification and ranking of major contributors can be integrated as part of the design process using CAPE tools. The risk analysis problem is a stochastic optimization of process performance under uncertainty and tolerance to operating policy variable error. This case study shows how this is applied to a multiphase reaction process where there is considerable uncertainty in the parameters. The use of stochastic optimization using complex models is a very computationallydemanding problem but is now achievable on the desktop.

4.5 Introduction to the Case Studies

I

405

The general problem for design under uncertainty was stated by Swaney and Grossmann (1985) and has been solved by determining the largest possible operating window based on conservative assumptions or using scenario-basedapproaches. The flexibility analysis assumes a “wait and see” approach where it is assumed that operating variables can be used to achieve any possible scenario. This requires a twostage optimization. The “here and now” approach takes design and operating variables together in a single optimization. A design was required for the production of a pharmaceutical intermediate, formed from the amination of a bromopropyl compound produced in an exothermic multiphase reactor. Sano et al. (1998) developed a kinetic model based on reaction calorimetry data obtained under laboratory conditions in order to determine the optimum feasible and safe operating policy. Solid particles of the active pharmaceutical ingredient (API) bromopropyl feed compound (A) reside in an organic solvent (methanol) inside the reaction vessel. A fuced volume of a 50 wt% aqueous dimethylamine reagent (B) is added to the vessel at a constant flow rate under continuous agitation. The solids gradually dissolve and react with the dimethylamine. The exact physicochemical phenomena for this process are not known. The reaction consists of a parallel-series reaction in which the dimethylamine reacts with the dissolved API feed to form the desired intermediate (C),which in turn reacts with the active feed (A) to form a dimeric by-product (D) in parallel: A + B --f C A C --f D

+

Main reaction Subreaction

D is very difficult to remove in the downstream purification stages. The model assumes intrinsic first order reaction kinetics. An initial rate limiting period due to the dissolution of solids (B) was observed to be independent of solvent concentration and agitation speed within the range of conditions approved. A crude approximation of first order kinetics is assumed in the model for this dissolution controlled period. This period was observed to last until approximately 55% conversion of A for all the conditions considered, at which point the reaction appeared to be limited by the intrinsic reaction kinetics. The kinetic model was integrated with a standard semibatch reactor model with constant volume addition (of reagent B). Consideration of the cooling capacity of the reactor resulted in a limiting relationship between the operating policy variables of feed B addition time, tad& and isothermal temperature, Tiso.For the purposes of this study, this relationship is well-approximated with Tisoas a quadratic function of tadd since data regarding the energy balance is unavailable. The model equations used to describe this process are given in Johnson (2002). 4.5.3.1 Nominal Optimization

Sano et al. (1998) state that one of the objectives for the development of the model was to help determine the best operating conditions for maximum product yield, Yc.

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4 Equipment and Process Design

A reaction time, tf, of less than 8 h (terminated when the rate of conversion of A falls below 0.1%) and a final yield in the impurity, YD,of below 2% must be maintained. The optimization problem is:

max YC

Tiso,tadd

subject to: batch reactor model equations

tf58 YD 5 2% 288 5 Tiso 5 313 0.5 5 tadd 5 3.0

The optimal results the authors determined through repeated simulation of their model are given in Table 4.4. Optimization of an equivalent model for maximum product yield subject to the stated constraints using a nonlinear constrained Sequential Quadratic Programming (SQP) optimization software, are shown to compare reasonably well (Table 4.4). 4.5.3.2

Consideration of Uncertainty in the Stochastic Model

Uncertainty in the model parameters could have a large effect on any results predicted by the model. This may be of particular importance regarding the optimal operating policy determined subject to the safety constraint and the desired limits on process performance. The uncertain parameters which have a nonnegligible influence on yield of C, Yc, yield of D, YD and the final time, tf, are the kinetic rate law parameters, the transition point from dissolution controlled kinetics to intrinsically controlled kinetics, GiSs, and the isothermal temperature, Tiso.These parameters are assumed to be uncorrelated. The main results from sensitivity analysis of the sample generated under the SQP nominal optimum conditions are shown in Table 4.5. The results indicate that the Table 4.4

Comparison between the optimal literature results (determined through repeated simulation) and optimal results obtained with SQP optimization.

I

Sano et al. (1998) Not given

I

~~

97.1 1.4

6.7 298.0

~~

SQP optimization

296.8 1.79

Ea 1 A, Ea2 A2

Yield C

Yield D

Final time

-0.62 0.02 0.74 -0.03

0.60 -0.02 -0.76 0.03

0.99 -0.03 0.03 -0.00

activation energy parameters in the intrinsic reaction rate laws (Eal and Eaz)are the most strongly related to the observed uncertainty in the output criteria. The optimization problem under uncertainty aims to maximize the expected product yield ( Yc)with the expected violation in both the impurity yield ( Yo)and the final time ( t f ) . The formulation of the stochastic optimization problem is:

subject to the process model equations with stochastic parameters. The operating policy decisions, tadd and TI,,, are scenario independent, assuming the a priori here and now mode of control where knowledge of particular realizations of the uncertainties is not assumed in the optimal solution. One-sided stochastic constraints are incorporated to maintain the desired limitations on the impurity yield (YDless than 2.0%) and the final time (tf less than 8 h). Since certain realizations within the uncertainty space may result in values of the endpoint impurity yield and final time exceeding the desired limits, expected violations of these limits of 1.5% and 1 h, respectively, are permitted. This allows some tolerance to the desired limits (E,,I the summed extent of violation of those observations failing divided by the total number of observations), which permits the determination of optimal solutions which are not overly conservative. A stochastic optimization algorithm is used to solve the problem, where a Hammersley sequence sampling scheme is used to place observations in the uncertainty space. A reduced convergence criterion of i 2 % deviation in the output distribution parameters is permitted to reduce the number of observations per objective function evaluation. The results for the optimizations under uncertainty in the key parameters are given in Table 4.6, where the value for the product yield objective mean-variance weight ( x ) is 0.5. It is clear that under the nominal optimal operating policy decisions, the expected violation of the final time, Ev,,l { t f } ,is significantly greater (4.49h) than the desired limit (1h). In addition, the expected yield of impurity, E{ Y,} at 2.75% with an expected violation, Emo1{ Y,}, of 1.39% is not satisfactory. Comparing the results obtained when uncertainty was considered, shown in Table 4.6, it can be immediately seen that an improvement in the Ycmean-variance objective is achieved (largely due to the 47% reduction in the variance) with maintenance of both the stochastic constraints for En,, { tf } and E,,,I { Y,}. Huge reductions in the both the expected final time (2.35 h) and the 5-95% fractile width (5.17h) are

Criteria

Nominal optimal operation

Uncertain optimal operation

Scenarios

456

418

24.97

0.234 [94.30,26.91 2.77 2.35 15.92 7.55 5.17

[0.59, 0.591 [1.39. 4.491

[0.53, 0.981 [1.25, 0.051

1.79 296.8

1.12 312.4

(h) TI, (K)

#add

observed, with no significant loss in the expected product yield or increase in the impurity yield. Full statistical information can be obtained from the results of the stochastic optimization (Johnson and Bogle 2006). The use of this approach allows the designers to explore the effects of uncertain parameters on the design of unit operations, and ofwhole processes (Johnson 2002). In spite of the considerable computational time required this can be achieved with relatively simple models. The approach can also highlight which parameters are the most sensitive to the design and therefore may require better prediction or measurement to be able to have confidence in the design.

4.5.4

Case Study 4: Biochemical Process Design

Much biochemical process development is done using simple tools such as BioProdesigner (Petrides 1994). This tool accounts for material and energy flows and in particular is useful in tracking the large amounts of water that are used. However the incorporation of detailed models for fermentation is difficult particularly if structured models are used where the organism which produces the product organism is modeled with different compartments inside the cell. This is necessary if there are separate phases in the cell as is the situation in this case study where the product forms as a solid granule or inclusion body (IB) inside the cell. In order to optimize the process it is necessary to consider this solid phase since it will significantly affect the primary separation. Biochemical processes and some batch chemical plants consist of cascades of complex unit operations, which process a physical stream to produce valuable products.

4.5 Introduction to the Case Studies

The product is produced in a fermenter, which is usually designed to maximize the biological yield of product. However, it is not always true that maximal biochemical yield will necessarily produce the most efficient yield of purified product since effective separation also depends on the production of by-products that must be separated, and also sometimes on the physical form of the product. To consider these aspects it is again necessary to consider the fermenter design in tandem with the rest of the process. This can only be done effectively with CAPE tools. In this case study the product forms as an inclusion body within the cells so the physical form of the product affects the efficiency of the process. The models must include the size of the inclusion bodies as they grow and are separated and so this study includes population balance techniques for modeling particle size. The process produces a protein which forms as inclusion bodies in recombinant E.coli cells in a fermenter operated in fed-batch mode and purified by a chain of downstream processing units (Bogle,Hounslow and Middelberg 1991).This process is typical for the production of animal hormones such as bovine somatotropin (BST) and porcine somatotropin (pST),for which much of this work was done, and for synthetic insulin. A culture of genetically engineered cells is fully grown in a fermenter in a solution of various salts using glucose as a carbon source. After initial growth phase the E. coli cells are induced either chemically or thermally after which the protein is produced and grows as inclusion bodies within the cell. The inclusion bodies continue to grow in size until the cell population is killed by raising the temperature yielding a slurry of cells each containing an inclusion body. Once the fermentation has been stopped the cells within the broth must be concentrated in a filtration stage, in which some of the water is removed, and the cells disrupted in a high pressure homogenizer to release the intracellular material which includes the solid inclusion bodies. The homogenate is then diluted with water to improve its handling characteristics prior to centrifugation. In the centrifuge the dense inclusion bodies are collected as the concentrate while the less dense cell debris is lost in the supernatant. Naturally, some inclusion bodies are lost and some debris will appear in the concentrate. Further purification stages are required prior to final formulation. The optimal operating conditions for a plant will depend on the characteristics of the product, in particular number and size of inclusion bodies. These characteristics can be measured after a fermentation and this information used to decide optimal conditions. Simulations can be used to determine timings for key events for optimal operation (Bogle et al. 1995). Here we investigate the optimization of the process given a known set of fermenter outlet conditions. 4.5.4.1 The Optimization Problem

A key tradeoff occurs in this process: inclusion body recovery and product purity vary in a conflicting manner when using the operation of the centrifuge as a manipulated variable to improve performance (see Fig. 4.11).This results in an optimization problem: only one of the objectives can be optimized at a time with the manipulation of the centrifuge variables yielding low values for the other objectives. This indicates

I

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410

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4 Equipment and Process Design 1.3

too 15

Figure 4.11 Tradeoffs between product purity, recovery and productivity as a function of one operating variable-centrifuge throughput.

1.1

Overall IB

Productivity

Recovey 50

or Purity

(%I

0.9

25

c

0.7 0

2500

5000

7500

Centrifuge throughput (Ifh) +Purity

&Overall

IB Recovery -A-

Productivity

that no overall process optimum with acceptable values for each objective can be attained with these specified variables, although both are of practical importance. However, it is important that process interactions are not to be neglected when an overall process optimum is intended, and it is of particular interest to ensure that acceptable levels of recovery, purity, and productivity can be achieved when the entire process is considered simultaneously The objectives can be optimized separately or the whole process can be considered and all values of the objectives examined. The objective functions for the optimization of the BST production process are IB recovery, product purity, and process productivity.The time invariant parameters and control variables and their limits are described in the next section. More details can be found in Graf (1996). In this work we chose to focus on the following operational case: a fermentation has taken place and an engineer wishes to determine the best operating conditions economically. The fermentation time was fmed at 14 h. The boundaries of the optimization variables are either due to practical limitations, e.g., for the settling area and the throughputs, or due to the lack of further experimental data, e.g., for the number of homogenizer passes. 4.5.4.2 Maximization of the Inclusion Body Recovery

The results of the optimization of the overall inclusion body recovery are given in Table 4.7. The results show that a maximum of 95% of all inclusion bodies can be recovered by manipulating the specified variables. The results from the simulation (Fig.4.11) show the contrasting behavior of IB recovery and purity when variables influence the separator-centrifugeefficiency. This feature is supported by the optimization results. The purity is low with 73.8% when the IB recovery is optimized. A productivity of 1.021 kg BST h-' can be achieved due to the high recovered mass of product. Each optimization variable has taken the value of one of its limits except the separator-centrifuge settling area. It has been found during the simulation that the

Overall IB recovery

("/.I 9s

Harvester settling area

Purity

(%I

Productivity (kg BST h-')

Overall process time

Final process volume (L)

(h)

73.8

Recovered mass of BST (kg)

1.021

37.3

1304

38.1

Harvester

Number of

Dilution rate

Separator

Separator

passes

(m3

throughput (L h")

(-1

settling area (m7

throughput (L h-')

250000

1200

3

4

105837

1200

(-1

IB recovery increases with the number of homogenizer passes and a higher dilution rate. The upper limits of these variables are thus the expected values. The IB recovery is maximized when the losses in the centrifuges are minimized suggesting that the settling areas should take the values of the upper limits and the throughputs of the lower limits. In the case of the separator-centrifuge settling area, the critical IB separation diameter is already so small that an increase of the settling area cannot influence the IB recovery any more, an effect which is based on the discretization of the IB size distribution: the critical IB separation diameter is already smaller than the smallest diameter of the discretized IB size distribution. Hence the rate of change of the IB recovery with the further increase of the settling area has fallen below the sensitivity ofthe gOPT optimization procedure, and the optimization has stopped, showing values for the separator-centrifuge settling area which are not bounds. However, an IB recovery of 100% could not be achieved due to losses in the first centrifuge and the nonideal separation efficiencies.

4.5.4.3

Maximization of the Product Purity

The values of the objectives and the optimization variables for maximizing the product purity are listed in Table 4.8.A product purity of 92.1% in the separatorcentrifuge sediment can be achieved with a concomitant low overall IB recovery of 12.7%, and a productivity of 0.163 kg BST h-'. These results prove again the main feature of this type of process: the higher the purity, the lower the IB recovery and productivity. The values of the optimization variables show that the loss of cells in the harvester-centrifuge is minimized by the highest possible settling area and the lowest throughput. Three homogenizer passes and a dilution rate of 1.1 are expected values from the simulation whereas the values for the separator-centrifugeare not on the bounds. As has been earlier found with the discretized IB size distribution, any further change of the separator-centrifuge settling area and the throughput would not increase the purity, since the critical diameter for cell debris is already bigger than the biggest discretized diameter of the cell debris size distribution.

412

I

4 Equipment and Process Design Table 4.8 Results for the maximization o f the purity. The table shows the values ofthe objectives and the optimization variables determined for the maximum. Overall process time

Productivity (kg BST h”)

recovery

Final process volume (L)

Recovered mass of BST

(kg)

(h) ~~

12.7

Harvester settling area (m2)

92.1

1 Harvester

throughput (L h-’)

1200

250000

0.163

1 Number of 3

30.6

366

1 Dilution rate ISeparator

settling area

1.1

92062

5.1

ISeparator

throughput (L h-’)

3203

4.5.4.4 Maximizationof the Process Productivity Table 4.9 shows the results for maximizing the productivity. In this optimization, an

overall productivity of 1.16 kg BST and recovery of 93.5% is yielded whereas the purity is low at 55.7%. Again, the discrepancy between the IB recovery, productivity, and the purity is demonstrated: a high productivity and recovery together with a low purity. These results clearly prove that only one objective can be maximized at a time using the specified variables. The settling areas of both centrifuges are pushed to the upper bounds. This is expected since the best recovery of IBs will increase the productivity,and the settling areas do not influence the overall processing time. Furthermore, the throughput of the first centrifuge is 1200 L h-’ due to the best cell and thus IB recovery. The values for the harvester-centrifuge settling area and its throughput have been the same for all three maximizations suggesting that the influence of the Table 4.9 Results for the maximization o f the productivity. The table shows the values o f the objectives and the optimization variables determined for the maximum. Overall IB recovery

(W 93.5 Harvester settling area

I Purity

I

I Productivity I Overall pro- I Final process volume (L)

(kg)

155.7

11.16

132.7

I815

Harvester throughput

1200

37.9 Separator throughput (L h-‘)

(m9

250000

Recovered mass of BST

2

2.5

250000

1811

4.5 Introduction to the Case Studies

harvester-centrifuge on the objectives is negligible as long as its efficiency is maximized. The values for the number of homogenizer passes, the dilution rate, and the separator-centrifuge throughput are the values previously found as suboptimal for the productivity and were thus expected. The objectives IB recovery, purity, and productivity have been maximized separately. A maximum I B recovery of 95%, a purity of 92.1%, and a productivity of 1.16 kg BST h-' processing time can be achieved in separate optimizations of the specified variables. The optimization results confirm the previously found feature of this type of biochemical process: the contrasting behavior of IB recovery and purity. It has been demonstrated that only one of these objectives can be optimized with concomitant low values of the other objectives when operational and design variables are used. We have found that higher levels of each objective can be found if structural changes are made, in particular by adding a recycle around the centrifuge. An acceptable compromise has thus been found for high values of the objectives. These results have been obtained assuming a set of experimental data from one of many fermentations undertaken. To achieve the best process design the ideal would be to operate the fermenter in such a way as to allow the inclusion bodies to grow as much as possible and then obtain the cleanest split possible between inclusion bodies and cell debris. Models have been developed for the nucleation and growth of the inclusion bodies using population balance models (Graf) as well as for the growth of the host organism. These have also been implemented but a formal optimization is not valuable. It will inevitably result in requesting the arbitrary upper limit on the fermentation time since the stopping of the growth of the inclusion bodies has not been modeled and is not understood. The unit operation and process design can be achieved by the use of CAPE tools but this case study shows how development of designs in many new areas needs to be done in close conjunction with laboratory and pilot plant. This is especially true when designs are being optimized since they seem very often to predict designs that are on the boundary which is least understood physically. This then should lead to a planned and directed new experimental program to help in understanding the key process bottlenecks.

4.5.5 Case Study 5: Design of Fluid Bed Reactor

The micro-electronic and photovoltaic (PV) industries require high-purity silicon feedstock for production of integrated circuits and solar cells. Unfortunately, the cost of producing high-purity silicon, around $40-$65 kg-', is too high for the photovoltaic industry. Since their silicon purity requirements are not as restrictive, PV manufacturers have so far been able to use silicon that does not meet quality specifications as feedstock for solar cell production. Yet, this source of feedstock is diminishing. The annual rate of growth of total installed capacity of PV systems varied between 20% and 40% from 1992 to 2001 (Ydstie 2004), but the growth in the electronics

I

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4 Equipment and Process Design

industry is 10% or less. Continued growth in the PV market is therefore contingent on new sources of solar-grade silicon. A promising method of affordable solar-gradesilicon production involves thermal decomposition of SiH4 (silane)inside a fluid bed reactor (Lord and Milligan 1998).A diagram of the reactor is shown in Fig. 4.12. The silane feed is preheated and delivered to the reactor in a single jet or multiple jets through the primary injector at the bottom of the reactor. A secondary injector is used to stabilize the jet. Resistance wall heaters are used to maintain reactor temperature. Hydrogen, a product of reaction, and the fluidizing gas are vented out the top. Silicon particles can be removed as product from an outlet close to the bottom of the reactor. The objective of our study is to develop a model suitable for control systems design, scale-up and process optimization. 4.5.5.1 Model Development

The fluid bed population balance model (White and Ydstie 2004) assumes that Silane decomposition in the fluid bed reactor occurs according to the overall reaction SiH4+ Si + 2 H2

(25)

The analysis of solid-phase behavior is presented first, and the new kinetic expressions are highlighted. To predict changes in silicon particle size, the model tracks particle movement to and from different intervals. This is accomplished by using mass and number balances over each size interval with constitutive equations to define particle growth. A representation of particle movement between size intervals is shown in Fig. 4.13.

4.5 Introduction to the Case Studies Interval i ,

'rxn,

Intervalj

:

i. .................................

*

. :

:...............

.............................

' 4 k!:l ...................................... i

Figure 4.13 Progression of silicon particles through size intervals.

Changes in the particle size distribution can be described with a mass balance over each size interval. Ignoring the potential for particle aggregation initially allows the change in number of moles of silicon in an interval (Mi) to be defined as: d Mi dt

-= j - 1

-$

+ rxn;.

So to predict the size distribution, it is necessary to define molar flow due to particle growth from one interval to the next If;) as well as the number of moles added to each size interval due to silane decomposition (rxni).This equation is then combined with a number balance and combined with rate expressions for chemical reaction. Since reaction occurs through heterogeneous and homogeneous pathways, the overall reaction rate is considered to be the sum of the rate expressions for both pathways. r = rhet - rhom

(27)

Rate expressions for both the heterogeneous and homogeneous reactions can be obtained from Furusawa et al. (1988). The reaction rate expressions require knowledge of silane and hydrogen concentrations, which can be determined from a gas phase model of the bed zone of the reactor which behaves as a CSTR so that we can write:

where V, is the volume of gas in reactor (L); C,,is the concentration of silane leaving reactor (mol L-'); cSi is the concentration of silane into reactor (mol L-'); C h o is the concentration of hydrogen leaving reactor (mol L-I); Fin is the flow into reactor (L s 8 ) ; and F,,, is the flow leaving reactor (L s-l). The model has been further modified to account for particle aggregation using binary interactions so that:

The value of ki,ican be estimated by comparing model predictions with experimental data. A typical model result is shown in Fig. 4.14.

I

415

416

4,

I

4 Equipment and Process Design

experiment

1i

i

i

-0- model -c

2-:.........

0 ...........

0

4

~

3

'7

; . wi , I i j

5

..-- 1

2

~

.

.....

0.5 ~

1

i

0

1

0.5

10'~

Radius, x ~ o - 3 ...............,

0

0.5

0.5

1

4 !-.

1

0

1

0

0

0.5

lo9 4

....

.......................

0.5

0.5

1

.

,

1

0

lo9 Figure 4.14 Experimental ( x ) and modeled no aggregation or loss of powder.

~~

............

j

0

0.5

0.5

x 10'~ ( 0 ) size

distribution with

4.6 Conclusions

The five case studies have shown the way in which CAPE tools can be used to design thermodynamically and physically complex industrial units and flowsheets. Increasingly this level of detail is being demanded by designers especially where tight quality, safety and environmental requirements are being expected. In all cases there has been a clear need for sophisticated modeling tied closely to bench-scale or pilot plant experimental systems to ensure that the right data is obtained to the right degree of accuracy. Some of this work has been done with commercial tools and some with bespoke programs. But it is clear that more sophisticated systems are required to facilitate rapid development of such models involving complex thermodynamics, irregular geometries, and particulates. These challenges remain. Systems which guarantee thermodynamic consistency are required as well, and systems incorporating the sys-

References

tematic and efficient assessment and minimization of uncertainty and its use to guide experiments remains a challenge to the developers of CAPE tools.

References 1 Bogle I. D. L., Cockshott A. R., Bulmer M.,

2

3

4

5

6

7

8

9

10

Thornhill N., Gregory M., and Deghani M. (1995)A Process Systems Engineering View of Biochemical Process Operations. Comput. Chem. Eng. 20(6/7),943-949. Bogle I. D. L., Hounslow M. I. and ,Middelberg A. P. J . (1991) Modelling of inclusion body formation for optimisation and recovery in a biochemical process. In Proceedings of 4th International Conference on Process Systems Engineering, Montebello, Canada. Bogle I. D. L. and Cameron D. (2002) CAPE tools for off-line simulation, design and analysis. In B. L. Braunschweig and R. A. Gani (eds.) Software Architectures and Toolsfor Computer-aided Process Engineering, Elsevier, Amsterdam. Callen H. B. (1985) Thermodynamics and an Introduction to Thermostatics, Wiley, New York. Eggersmann M., Hackenberg]., Marquardt W., and Cameron I. T. (2002)Applications of modelling - a case study from process design. In B. L. Braunschweig and R. A. Gani (eds.) Software Architectures and Toolsfor Computer-Aided Process Engineering, Elsevier, Amsterdam. Farschman C. A,, Viwanath, K. P., and Ydstie B. E. (1998) Process Systems and Inventory Control. AIChEj. 44(8). 184-1855. Frank R. A,, Finn C. W. and ElliottJ. F. (1989) Physical Chemistry of the Carbothermic Reduction of Alumina in the Presence of a Metallic Solvent: 2. Measurements of Kinetics of Reaction. Metall. Mater. Trans. B 20(2), 161. Furusawa T., Kojima T. and Hiroha H. (1988) Chemical Vapor Deposition and Homogeneous Nucleation in Monosilane Pyrolysis within Interparticle Spaces: Application of Fines Formation Analysis to Fluidized Bed CVD. Chem. Eng. Sci. 43(8), 2037-2042. Garcia-Osorio V. and Ydstie B. E. (2003) Distributed, Asynchronous and Hybrid Simulation of Process Networks Using Recording Controllers. rat. J . Robust Nonlinear Control 14, 227-248. Garcia-Osorio V. and Ydstie B. E. (2004) Vapor Recovery Reactor in Carbothermic

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Aluminum Production. Chem. Eng. Sci 59(10), 2053-2064. Gerogiorgis D. I., Ydstie B. E. and Seetharaman S. S. (2002)A steady state electrothermic simulation analysis of a carbothermic reduction reactor for the production of aluminium. In Cross M., Evans ]. W. and Bailey C. (eds.) Computational Modeling of Materials, Minerals, and Metals Processing: 273; Warrendale, PA (TMS). Gerogiorgis D. I. and Ydstie B. E. (2003a) A finite element CFD sensitivity analysis for the conceptual design of a carbothermic aluminum reactor. In Crepeau P. (ed.), Light Metals 2003: 407; Warrendale, PA (TMS). Gerogiorgis D. I., et al. (2003) Process Systems Tools for Design and Optimization of Carbothermic Reduction Processes. In Das S . K. (ed.) Aluminum 2003, Warrendale, PA (TMS) (in press). Gerogiorgis D. I. and Ydstie B. E. (2003b) An MINLP model for conceptual design of a carbothermic aluminum reactor. In Proceedings of European Symposium on ComputerAided Process Engineering (ESCAPE-13), Lappeeranta, Finland, pp. 131-136. GrafH. (1996) Modelling and Optimisation of an Inclusion Body Type Biochemical Process. Externe Diplomarbeit (Universitat Dortmund) Project report, Dept of Chemical Engineering, University College, London. GrafH. and Bogle I. D. L. (1997) Simulation as a tool in at-line prediction and control of biochemical processes. In Proceedings ofthe 1st European Conference on Chemical Engineering, ECCEl Florence, Italy. Grossmann 1. E., Westerberg A. W. (2000), Research challenges in process systems engineering. AIChEJ. 46, 1700-1703. HagemannJ., Fraga E. S. and Bogk 1. D. L. (2001) Distillation column design for improved exergy utilisation and its impact on closed loop performance. In Proceedings of the World Congress of Chemical Engineering, Melbourne, Australia, Sept 2001. Hangos R and Cameron I . T. (2001) Process Modelling and Model Analysis. Academic Press, New York.

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(2004) Classification and Analysis of Integrating Frameworks in Multiscale Modelling. Chem. Eng. Sci 59(11), 2171-2187. Johansen K., et al. (2000) Carbothermic aluminum. In Proceedings ofthe Sixth Intemational Conference on Molten Slags, Fluxes and Salts (paper #192), Stockholm, Sweden (CD edition). /ohansen K. and Aune J. A. (2002) US Patent 6,440,193 to Alcoa Inc. & Ekem ASA. Johnson D. B. (2002) Integrated design under uncertainty for pharmaceutical processes. PhD Dissertation, University of London. Johnson D. B. and Bogle I . D. L. (2006) An approach for integrated design under uncertainty of pharmaceutical processes. Reliable Computing (in press). Kaibal G. (1990) Energieintegration in der thermische verfahrenstechnik. Chem. Ing. Tech. 62(2), 99-106. Kaibel G., Blass E., and Kohler]. (1989) Gestaltung destillativer Trennungen unter Einbeziehung thermodynamischer Gesichtspunkte. Chem. Ing. Tech. 61(1), 16-251. Kjelstrup S. and Hafskjold B. (1996) Nonequilibrium molecular dynamic simulations of steady state heat and mass transport in distillation. I d . Eng. Chem. Res. 35, 4203-4213. Lord S. M. and Milligan R. J. (1998) Method for Silicon Deposition, US Patent 5,798,137. Luyben W. (1990) Process Modelling Simulation and Controlfor Chemical Engineers. McGraw-Hill, Boston. Mangold M.,Motz S., Gilles E. D. (2002) Network theory for the structured modelling of chemical processes. Chem. Eng. Sci. 57, 4099-4116. Motzfeldt K., Kvande H., Schei A., Grjotheim K. (1989) Carbothermal Production of Aluminum - Chemistry and Technology. A1 Verlag, Dusseldorf, Germany. O h M. and Pantelides C. C. (1994) A Modelling and Simulation Language for Combined Lumped and Distributed Parameter Systems. In Proceedings ofthe 5th International Conference on Process Systems Engineering, Kyongju, Korea 1. 37-44.

33 Petrides D. P. (1994) Biopro designer - an

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advanced computing environment for modelling and design of integrated biochemical processes. Comput. Chem. Eng. 18, S621-S625. Ratkje S. K. E., Hansen E. M., Lien K. M . and Hafskjold B. (1995) Analysis of entropy production rates for design of distillation columns. Ind. Eng. Chem. Res. 34(9). 3001-3007. Sano T., Sugaya T. and Kasai M . (1998) Process improvement in the production of a pharmaceutical intermediate using a reaction calorimeter for studies of the fraction kinetics of amination of a bromopropyl compound. Organic Res. Develop. 2, 169-174. Sauar E., Ratkje S. K. and Lien K. M. (1996) Equipartition of forces - a new principle for process design and optimization. Ind. Eng. Chem. Res. 35(11), 4147-4153. Sauar E., Rivero R., Kjelstrup S. and Lien K. M.(1997) Diabatic column optimization compared to isoforce columns. Energy Consew. Mgmt. 38(15), 1777-1783. Sauar E. and Ydstie B. E. (1998) The temperatures of the maximum reaction rate and their relation to the equilibrium temperatures. /. Phys. Chem. A 102,8860-8864. Skogestad S. (1997) Dynamics and control of distillation columns. Trans. IChemE Part A 75, 539-562. Swaney R. E. and Grossmann I. E. (1985) An index for operational flexibility in chemical process design. Part 1 Formulation and theO T ~ AIChEJ. . 31(4), 621-630. Tondeur D. and Kuaalen E. (1987) Equipartition of entropy production - an optimality criterion for transfer and separation processes. Ind. Eng. Chem. Res. 26(1), 50-56. White C. and Ydstie B. E. (2004) Modeling the Fluid Decomposition of Silane to form Photovoltaic Grade Silicon in a Fluid Bed Reactor, Technical Report, Carnegie Mellon, Chemical Engineering. Ydstie B. E. (2004) Decision making in complex organizations: the Adaptive Enterprise. Comput. Chem. Eng. (in press).

Computer Aided Process and Product Engineering Luis Puigianer and Georges Heyen . Co. KGaA, Weinhein Copyright 02006 WILEY-VCH Verlag GmbH 8

5 Product Development Andrzej Kraslawski

5.1 Background

Over the past 20 years, the changing political situation and the advent of new technologies have had a profound impact on societies in the vast majority of countries all over the world. This influence has been manifested, among other things, by the growing demand for totally new products and services. This trend has resulted in mergers, and acquisitions of many businesses at a scale not previously observed. The changes in the organization of the businesses have very deeply influenced the manufacturing and processing industries. The main consequences have been the lowering of profit margins of commodities, growing pressure on the environmental aspects of production and conservation of raw materials, very high costs of the research and development (R&D),significance of low-tonnage and high-added value products, growing importance of the customer-oriented products, reduction of the time between development and bringing the product to the market, shortening of the life cycle of the products, and issues related to intellectual property rights. All of these factors have caused a visible change in processing industries. It was manifested by the move from process-oriented to product-centered businesses. The consequence of this switch is the emergence of interest in product development. The activity has been known to the industry for many years. However, its importance and the need for a more systematic approach have only been realized in the last five to eight years. It is obvious that the driving force for the generation of new chemical products is higher satisfaction of existing needs or fulfillment of new ones. However, it is hard to imagine even a fraction of those needs when realizing that the largest substance database, the Chemical Abstracts Service (CAS),contains around seven million commercially available chemicals, and around 4000 new substances are added each day. The generalization of the product concept is captured by the multilevel approach proposed by Kotler (1989).The product is composed of core, tangible and augmented parts as presented in Fig. 5.1. The core product assures the minimal fulfillment of Computer Aided Process and Product Engineering. Edited by Luis Puigjaner and Georges Heyen Copyright 0 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30804-0

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Figure 5.1

The concept of product (Kotler 1989)

the customer’s needs. It is a physical component of the product, e.g., paint or adhesive. The tangible and augmented part of the product is composed of physical as well as the immaterial elements, e.g., services. The activity of CAPE specialists related to product development concentrates on the analysis of the function-property-compositionrelations (Villadsen 1997; Rowe and Roberts 1998a; Cussler and Moggridge 2001; Gani and O’Connell2001; Wesseling 2001; Charpentier 2002). However, when looking at Kotler’s concept of the product it is clearly visible that such an approach corresponds exclusively to the core product and still there is a lot to be done by the CAPE community with respect to tangible and augmented elements of the product. The consequence for the process engineer is a need to look not exclusively at the composition or form of the core product but also at its function in a more general context, e.g., chemical sensors in the package that inform about the state of the core product or degradability facilitating the elimination of the product. The behavior of the product at the market is described by the concept of product life time. The concept introduced by Buzzell (1966)and later much criticized (Dhalla and Yuspeh 1976)is still a good tool for the determination of where the product is in its development path and the consequences for its engineering. The generalized product life time is presented in Fig. 5.2. The analysis of a product life cycle shows that the involvement of process engineers changes during its duration. The most intensive design activities of process engineers at the product development phase are later replaced by the involvement of the production phase and incremental changes of the product. The last phase, elimi-

5.1 Background I421

quantity

I) Figure 5.2

time

Product life cycle (Baker and Hurt 1999).

nation, is also a place where the process engineers have to be involved to ensure the required environmental, legal and technological compromise. The first phase of the product life cycle, new product development (NPD),is a type of activity where process engineers have always been active. However, the structural changes of many industries, as mentioned at the beginning of this chapter, imposed a need for more comprehensive participation of the process engineers in NPD. The involvement of process engineers will be easier to structure by analyzing the theories of new product development. Saren (1984)has introduced four types of NPD: departmental stage model, activity-based approach, decision stage model and conversion process. The activity-basedapproach developed by Booz, Allen and Hamilton (1982) seems to be the most adequate for the illustration of the involvement of process engineers in product development. The modification of activity-based model presented by Ulrich and Eppinger (2000) is the starting point for the analysis of the methods and tools to be used by process engineers in NPD. The generic activity-based model of product development is composed of the phases presented in Fig. 5.3. Chemical product development is a process composed of the definition phase and product design, as presented in Fig. 5.3. The very essence of the product development is the identification of how the needs could be satisfied by the chemical and physical interaction of the substances and the chemical composition and structure of those substances. It means the determination of the needed (expected) function of the product, translation of the required function (adhesiveness, chemical impact, smell, taste, elasticity, etc.) into the properties like viscosity, density, color, smell, microstructure, etc., and finally the identification of the single chemical component (its molecular structure) or mixture possessing those properties. In the definition phase a specialized business strategy and market research tools are used. Business strategy tools are applied in the planning phase to outline what should be produced and for whom.

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Chemical Product Development

Design Phase (implementation Product Concept TestingProduct

\ Figure 5.3

Chemical product development (Ulrich and Eppinger 2000).

Market research tools are used to identify the needs and requirements concerning the pre-defined class of the products and the market segment. In the CAPE community, product developmentwas traditionally reduced to product design and, due to the character of the chemical industry, product design was equivalent to formulation. Formulation is defined as the blendinglmixing and processing of ingredients in order to obtain a product characterized by the precisely determined properties (specifications).The main design problem in formulation is finding the relation between the composition and structure of the mixture as well as the type and conditions of processing it and the required final properties of the product. The great variety of products is a source of the fundamental difficulties in the generalization of the product formulation. Usually the know-how in formulation is restricted to a narrow class of the products (e.g., specialized soaps or rubbers). Knowledge of formulation for one type of the product is not easily transferable to the other types of products. The high specialization and the very complicated physical and chemical phenomena related to the processing of the ingredients, as well as their interactions, resulted in characteristicdevelopment of methods and tools used in formulation and more generally in product design. There are three main classes of methods applicable to product design: 0 0 0

experimental design knowledge-based methods computer-aided molecular design.

5.1 Background I423

Historically, the oldest method of product design was experimental design that evolved into sophisticated statistical methods, capable of handling very complicated mixing and processing problems. The interest of the CAPE community has been recently attracted to the statistical analysis of processes that can lead to the identification of new products. The attractiveness of experimental design has been strongly limited by the considerable costs and time needed for the realization of the experiments and the subsequent analysis of data. The trends for limiting the cost and time of development, the existence of a huge amount of data, and information related to the development of new products and stored by the companies were the main factors promoting the application of knowledge-based methods. The common feature of these methods is the use of historic data and information and the generation, on this basis, of new knowledge applicable to the actual product design problems. The proposed methods and tools are: rule-based systems, neural networks, genetic algorithms, case-based reasoning systems, TRIZ, and data mining. The last group of methods is related to computer-based molecular design. This approach has been dynamically developing over recent years. An important contribution of the CAPE community in this field was recently reported by Achenie, Gani, and Venkatasubramanian (2003). There are two approaches to product design (Fig.5.4). Product design can be formulated as a forward problem if the design starts from the given structure of a molecule or composition of the material, and aims at determining the properties of this material. The second approach, considering design as a reverse problem, starts with the given properties of a material and finds the molecular composition fulfilling the requirements. Experimental methods are an example of the forward problem formulation. The knowledge-based methods as well as computer-based design could be used in forward or reverse problem formulation. This chapter is limited to the description of experimental and knowledge-based methods. It covers the last 10 years and does not include the review of computerbased molecular design. The relatively new subjects to the CAPE community, such as quality function deployment, case-based reasoning and TRIZ method, are described in more details. The application of other methods and tools is reviewed in the context of the designed product type, i.e., catalysts, dyes, rubbers, etc.

Forward problem

problem

structure Figure 5.4 problem.

Forward and reverse formulation of the product design

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5.2 Definition Phase

Identification of consumer needs is a field where market research is given the last word. Often the comments found in process engineering literature about the “identification of the consumer voice by the use of questionnaire” are misleading and in fact quite different questions should be asked. Although direct contact with the customer is not a core speciality of process engineers, a basic knowledge of the methods and the applicable tools is very important. Attention should be focused on two types of tools. The first group is composed of the tools enabling not only identification of the actually expressed requirements but also their translation into engineering characteristics of the materials. The second group focuses on the identification of future needs, which are not yet foreseen by the great majority of consumers. Both activities are aimed at positioning the future product with respect to customers, competitors, and regulators.

5.2.1 Translation of the Requirements into the Parameters

The capturing of the requirements of a new product is composed of information gathering, information transformation, and finally generation of the requirements. The specialized techniques are the domain of market research and many noteable books exist dealing with the subject, e.g., Bruce and Cooper (2000).There are several methods and tools related with the quantification of the product attributes expressed by the customers and their translation into the engineering variables of the product. The most common are: conjoint analysis and quality function deployment. Conjoint analysis is used to determine the desired level of the product attributes and their influence on various business decisions. It enables futing of the tradeoffs between the requirements. The detailed description is presented in Baker and Hurt (1999). Quality function deployment is a more popular method enabling determination of the engineering variable values of the product, identification of the attributes and variables to be improved, and positioning of the product with respect to the competitors. Quality Function Deployment

The quality function deployment (QFD) method enables the conversion of the needs of the customers into the product design variables (ReVelle et al. 1998).QFD is composed of four steps: 0

identification of the attributes essential for the consumer when evaluating the product, and determination of the relative importance of the attributes; determination of connections between the attributes identified by the consumer and design variables;

5.2 Definition Phase I425

estimation of the target values of the design Variables satisfylng the needs of the consumers; assessment of the degree of satisfaction with the existing products (designs). QFD is realized by the tool called House of Quality (Hauser and Clawing 1988),presented in Fig. 5.5. The basic form of House of Quality is used not only for the translation of the customer needs into the product design requirements but is also applied to the systematic identification of product features. This evaluation could be based on the assessment of the competitiveness of the given product with regard to the other products on the market or by the identification of tradeoffs using a correlation matrix. An application of the House of Quality for the chocolate couverture is given by Viaene and Januszewska (1999) (Fig. 5.6). The analysis, conducted by the use of the questionnaires, has revealed that the most important characteristics applied by the customers in the evaluation of the chocolate couverture are: flavor, appearance, and texture. Those attributes were introduced into the “what”part of House of Quality (Fig. 5.6). There are the parameters of the couverture that are measured by the instrumental analysis (acidity level, sugar content, fat content, hardness, particle size, adhesiveness). Moreover there is a group of sensory variables: color intensity, color brightness, texture on surface, texture on snap, melting in hand, aroma, acidity, bitterness, cocoa body, fruitiness, sweetness, smokiness, first bite, oily mouth coating, aftertaste, adhesiveness, smoothness, melting in mouth. Both variables determined by the instrumental anal-

A Correlation matrix

Customer needs (product attributes)

Figure 5.5

The House of Quality.

Product positioning with regard to comoetitors

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.......

..................... 9 ........... 4

StrongPos @ Strong @ Positive Medium0 Negative Weak A Strong Neg %

0 x

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3 1

t

........... 0 I

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Figure 5.6 House of Quality for the chocolate couverture (Viaene and Januszewska 1999).

ysis and sensory analysis are introduced into the “how“part. There were several samples evaluated by the customers. Every sample had the different values of the variables related to the instrumental and sensory analyses. In the next step the relation between the attributes proposed by the customers and the variables from the “how” part are determined using the statistical analysis of the results of the questionnaire. The relations between the attributes and the variables are expressed as strong positive, positive, neutral, negative, strong negative. They are symbolically introduced into the central part of the House of Quality. Next, based on the importance of the attributes given by the customers and the technical knowledge obtained from the analysis of the relations between the attributes and variables, the target values for the variables are determined and introduced at the bottom of the House. The roof of the House represents the interrelations between the design variables. It allows the study of the tradeoffs between them. On the right side of the House, the positioning of the

5.2 Definition Phase I 4 2 7

given product with respect to the products of the competitors is given. It allows for the identification of the weakest attributes of “our” product with respect to the competitors. The detailed information on the application of QFD to the product development is given in ReVelle et al. (1998).

5.2.2 Forecasting Methods for New Product

The forecasting of new products is an activity where the various methods are applied, e.g., expert judgment; brainstorming, one-to-one interviews with customers and salespeople, surveys of customers, idea generation, etc. The complete review is given by members of TFAM working group (2004). The methods of the new product forecasting could be divided as market and product-oriented. The market-oriented methods deal with the information collected from the analysis of the consumers’ behavior, their needs and economic factors related to the existing products. They use complicated statistical tools, mostly penetration models, however new approaches emerge, e.g., chaos theory (Phillips and Kim 1996). The market-oriented methods of new products forecasting are mainly the domain of market analysts. However, the obtained results are not encouraging as the estimations of the sales of the new products have errors reaching SO-60% (Kahn 2002). 5.2.2.1 Idea Generation

The phase of idea generation is a key point for engineers interested in the enhancement of their creativity.There are two major groups of methods supporting idea generation: intuitive and analytical. The most common, unstructured intuitive methods applied to the engineering problems are explained below. Brainstorming

Brainstorming is the most popular creativity enhancement method. Originally introduced by Osborn (1953), it is based on four rules: (1)evaluation of ideas must be done later; (2) the quantity of the generated ideas is most important; (3) encouragement of strange and “wild”proposals, (4) improvement and combination of the generated ideas is welcomed. There are many types of brainstorming: stop and go, Gordon-Littlevariation, trigger method, etc. The detailed description is given in Proctor (2002). Synectics

The objective of the method developed by Gordon (1961) is to look at the problem from a different perspective (to make familiar strange and vise versa). To achieve it, a set of four analogies (personal, direct, symbolic and fantastic) and metaphors is used. The detailed description of the method is presented in Proctor (2002).

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Lateral Thinking

Lateral thining is the group of methods introduced by De Bono in 1970 based on the unconventional perception of the problem. The main factors enabling the lateral thinking are: identification of the dominant ideas, searching for new ways of looking at the problem, relaxation of the rigid thinking process, and the use of chance to encourage the emergence of the other ideas. The most common analytical methods applied to the engineering problems are as follows. Morphological Analysis

This method was introduced by Zwicky in 1948. It is based on the combination of the attributes of the product or process (like properties, functions, etc.) with the elements of the product or the process. It can generate all possible combinations of the attributes, however, its applicability is practically limited to three-dimensional analysis of the attributes. There exist many variations of the method, such as attribute listing and SCIMTAR (Proctor 2002). Analogies

Analogies is a group of methods with case-based reasoning as the most useful for engineering applications. They exploit the similarities between the features of the existing problems and the features of the problems or design known from the past. A good survey of case-based methods is given by Leake (1996). The special class of analogy-based methods is an approach exploiting biomimetics, i.e., analogies with living systems (French 1994). TRIZ

TRIZ is a popular method of systematic creativity support introduced by Altshuller in 1984. There are many books that describe TRIZ (theory of inventive problem solving), e.g., Savransky (ZOOO), Salamatov (1999). However, the book by Mann D. (2002) seems to be the best introduction. The main findings of TRIZ are: 0

0 0 0

All innovations start with the application of a small number of inventive principles. Trends exist in the evolution of technologies. The best solutions transform the harmful effects into useful ones. The best solutions remove the conflicts existing in the system.

Those findings are translated into the basic principles of TRIZ. Contradictions There are two types of contradictions. A technical contradiction takes place when there are two parameters of the system in conflict, i.e., the improvement in the value of one parameter lessens the value of the other one. The technical contradictions are solved by applying a contradiction matrix. A physical contradiction takes place when

5.2 Definition Phase I429

the parameter should simultaneously have two different values. The physical contradictions are removed applying the principles of separation in time and space. Ideality There is a tendency in the evolution of the systems that they always change towards the state where all benefits of their function are realized at no cost or harm. Functionality Every system has its main useful function and all its elements have to fulfill this function. Otherwise they are under-used and could be a source of conflict. The notion of functionality allows for the generalization of various aspects of the system functioning resulting in the possibility of transferring know-how between the various fields (technical, medical, biological, etc.). Use of Resources Any physical element of the system or phenomena accompanying its functioning has to be used to maximize the system functionality. The application of the basic principles is realized as shown in Fig. 5.7. It is based on the translation of the actual problem into the general one (identified by Altshuller), finding the general solution to this problem (also identified by Altshuller), and then transforming it into the solution of the actual problem. The basic principles are implemented in several tools (inventive principles, Sfields, contradiction matrix, separation principles, knowledge effects, trends, etc.) (Mann 2002). An example of application of the modified TRIZ to product design, is given by Nakagawa (1999). The problem was the insufficient quality of the porous polymer sheet. The reason was a foam ratio in the process of forming a polymer sheet that was too low. The gas dissolved in the molten polymer was escaping during the process through the surface of the sheet, which resulted in the creation of small, unequally distributed bubbles. The application of USTI resulted in several alternatives for concerning the material itself as well as its manufacturing. The proposed solution dealing directly with the polymer composition was the addition of the solid powdcr to the molten polymer to act as secds for thc bubbles.

Problem

Solution

Specific

Specific Solution

Figure 5.7 The general principle of application of TRIZ (Mann 2002)

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5.2.2.2 Creativity Templates

The analysis of the market by interviewing the customers and salespeople can lead to incrementally innovative products. However, as was shown by Goldenberg and Mazursky (2002),it will not introduce a breakthrough product earlier than the competition. The reason is that the saturation of the market with the new-needs-aware customers is so low that their identification by the market research is practically impossible early on in the process. There are practically very few customers able to express the new breakthrough need in terms of the existing products. With time the awareness of the consumers grows and it becomes easier to reach new-needs-aware persons, but the competitors have the same chance as well and the competitive advantage of our product diminishes (Fig. 5.8). Goldenberg and Mazursky (2002) have suggested an introduction of creativity templates to generate new products. This idea was inspired by the TRIZ method (Orloff 2003). The first step consists of listing the attributes of the existing product. Next the attributes are manipulated

Number of customers

A

:>

c;' 0

000

0

b

0

0

0

unaware customer

b

Time

.

0 . 0

0

a o

C

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new-needs- aware customer

Figure 5.8 The change of the new-needs awareness o f customers with time. (a) Early period o f the need recognition-very few aware consumers. (b) Mature period of the need recognition-aware consumers are numerous. (c) Saturation period of the awareness-the need is obvious for the considerable majority o f the customers.

5.3 Product Design I 4 3 1

using one or more templates (subtraction of the attributes, their multiplication, division, task unification or attribute dependency change). The examples of the method application are given in Goldenberg et al. (2003). 5.2.2.3 Product-oriented Methods

The product-oriented methods could be of special interest for the CAPE community. They concentrate mainly on the technical aspect of the product and try to forecast future trends of the technology development. The most common method is technology/product roadmapping (Petrick and Echols 2004). An example of a roadmap for technology/product planning is presented by Phaal et al. (2004). The forecasting of technological trends and products based on the use of TRIZ is presented by Mann (2003).He cited the generic trends of technology evolution: 0 0 0 0 0

0 0 0 0 0

systems with increasing benefits and decreasing cost and harm; increased dynamization within systems; increased space segmentation; increased surface segmentation; increased controllability: increased complexity followed by reduced complexity; use of all available physical dimensions within a system; decreased number of energy conversions; increased rhythm coordination; increased action coordination.

Mann (2003) also presented the application of trends to the forecasting of the product development.

5.3 Product Design

The concept of design is very difficult to define due to the fact that design activity could lead to extremely diverse products and consequently a broad spectrum of activities is possible. A survey of chemical product design could be performed from the following various perspectives: 0

0

classes of the products (e.g., basic chemicals, specialty chemicals, pharmaceuticals, crop protection, consumer products); development phases of the project: - development of the product concepts (determination of its form, features and draft specifications); - detailed design (identification of the complete specification and corresponding composition and structure); - product testing; - final design;

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- The specificity of the product development has to be taken into account, e.g.,

0

0

pharmaceuticals the steps involved are discovery, initial tests, clinical trials, legal approval: degree of innovativeness: new class of products, derivatives of the existing group of products, incrementally improved and breakthrough of new products; used tools, etc.

The next subchapter focuses on the methods and tools used in chemical product design.

5.3.1 Experimental Design

Experimental design is one of the oldest engineering product design methods. Traditionally, it can be divided into the studies of the influence of process variables (temperature, pressure, etc.) and analysis of the properties of the materials by changing their composition. The first type of experiment is called experimental process design and the second is mixture design. There is also a third type called combined experimental design, when the characteristic of the materials is studied as a hnction of mixture composition and process conditions. 5.3.1.1 Planned Experimental Design

The mixture design experiments are carried out to precisely determine the composition of the components. The usual experimental mixture design deals with three to ten components. The existence of the various types of the component constraints (relational constraints, constraints resulting from the interaction of components) is typical for the mixture design. The studied attribute (taste, viscosity, stability, etc.) is recorded for every experiment. Depending on the number of components and the specific composition constraints, a precise number of experiments are conducted according to the selected experimentation plan. A pliori determined compositions and experimentally obtained values of the attributes are used for obtaining linear, cubic or higher-order regression equations. A good introduction to mixture design is given by Eriksson et al. (1998). The most popular fields of application of mixture design are pharmaceuticals, food products, polymers and paints. A few of the most typical are: 0

0

Blending: Olive oils were blended to ensure the best sensory quality. There were eighteen experiments carried out to find the optimal composition, according to eight sensory criteria, of four olive oil mixtures (Vojnovic et al. 1995). Formulation: A mixture of dispersants was designed for use in oil spill cleanup. In order to evaluate a mixture of three different dispersants, ten experiments were performed consisting of the measurement of the effectiveness of oil dispersion (percentage of oil removed from the water surface) as a function of the dispersant mixture composition (Brandvik 1998).

5.3 Product Design 1433 0

Pharmaceuticalfomtulation: The objective was to find the optimal composition of the tablet coating ensuring the required release rate of the active ingredient from inside of the tablet. The coating was composed of three components and the release rate was determined by the measurements of the diffusion coefficient in 13 experiments with different mixture compositions (Bodea and Leucuta 1997).

The combined experimental design is an active field of research allowing for the prediction of the product properties not only as a function of the composition but also of processing mode. The application fields are the same as for the mixture design: 0 Blending: In a process consisting of the blending of three different flours, the product attribute was baking quality. Sixty-sixexperiments were performed to identify the polynomial that correlates the composition of the flour mixture and mixing time with the baking quality of bread (Naes et al. 1999). 0 Phamaceuticalformulation: The studied property of the material was its viscosity as a function of its composition (active substance: tolfenamic acid, block copolymer, ethanol and buffer) and one process variable (Cafaggi et al. 2003). Mixture design is a very well-established method within product design. However, changes in the business environment, as mentioned at the beginning of this chapter, have caused long time periods and relatively high costs, which are the main factors limiting the attractiveness of experimental mixture design. 5.3.1.2 High-Throughput Experimentation

The estimated number of inorganic compounds is around 10’ (only for materials composed of five elements). The number of drug-like structures is projected to be as high as loG3(Cawse 2002). Consequently, new methods of research are needed to identify, with relatively high probability, new promising materials in such a huge space. High-throughput experimentation (HTE)is used to tackle the problems where the parameter space is too large to be handled efficiently using conventional approaches. HTE consists of the use of miniaturized laboratory equipment, robotics, screening apparata and computers. The main field of HTE application is the determination of the composition of drugs, multifunctional materials, coatings, and catalysts as well as the determination of their formulation. A good introduction to the subject is given by Cawse (2002). The main problems related to the application of CAPE to HTE consist of handling the complexity resulting from the amount of data and highly complicated phenomena under consideration. As a consequence, the research interest concentrates on the design of databases, data mining, integration and representation, conversion of data to knowledge, experimental control systems, and decision-supporting systems to facilitate “hit-to-lead”process aiming at the maximization of the number of the successful designs. The successful application of HTE to products design are numerous, e.g., homogeneous and heterogeneous catalysts (Murphy et al. 2003). The review of HTE applications to materials design is presented by McFarland and Weinberg (1999). A few examples given below focus on CAPE-related problems in THE.

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Catalyst design: Caruthers etal. (2003) have outlined a procedure for the more comprehensive use of data from HTE in catalyst design. The final objective of the presented research is to achieve a balance between the speed of the data generation and abilities of their transformation into knowledge. The proposed process of “knowledge extraction” consists of planning HTE in a way that allows for the discrimination of the models of catalybc reaction models, determination of the kinetic constants and relating them to the catalyst microstructure. The proposed forward modeling is realized by the application of the rules capturing the human expertise, neural network (NN), and genetic algorithm (GA). The proposed system is not yet fully automatic. Phamaceuticals design: The acute problem facing the pharmaceutical industry is growing discrepancy between the R&D investment and a number of newly registered drugs (Anonymous 2004). The declining number of new drugs is due, among other factors, to the concentration of efforts in HTE technology on the generation of active pharmaceutical ingredients (API) - selectivity and potency at the target are the mainly studied features - and neglect of the studies of the API form (salt, polymorphic, hydrates, solvates, etc.). Those forms determine the properties of the API, such as solubility, stability, biocompatibility, etc. Those properties, on the other hand, determine the metabolism, toxicity, and formulation design of the pharmaceuticals. A comprehensive description of the problem is presented by Gardner et al. (2004).

An interesting application of CAPE tools to HTE is the use of case-based reasoning for identifylng the required conditions for protein crystallization (Jurisica et al. 2001).

5.3.2 Knowledge-based Tools 5.3.2.1 Case-based Reasoning

Case-based reasoning (CBR) consists of solving new problems using the solutions of old problems. The central notion of CBR is a case. The main role of a case is to record a single past event where a problem was