Advances in Industrial Control

Series editors

Michael J. Grimble, Glasgow, UKMichael A. Johnson, Kidlington, UK

More information about this series at http://www.springer.com/series/1412

Radoslav Paulen • Miroslav Fikar

Optimal Operationof Batch MembraneProcesses

123

Radoslav PaulenDepartment of Biochemical and ChemicalEngineering

Technische Universität DortmundDortmundGermany

Miroslav FikarInstitute of Information Engineering,Automation and Mechanics

Slovak University of Technologyin Bratislava

BratislavaSlovakia

MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 3 Apple HillDrive, Natick, MA 01760-2098, USA, http://www.mathworks.com

ISSN 1430-9491 ISSN 2193-1577 (electronic)Advances in Industrial ControlISBN 978-3-319-20474-1 ISBN 978-3-319-20475-8 (eBook)DOI 10.1007/978-3-319-20475-8

Library of Congress Control Number: 2015942552

Mathematics Subject Classification: 49J15, 34K35, 93C15

Springer Cham Heidelberg New York Dordrecht London© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

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Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technologytransfer in control engineering. The rapid development of control technology has animpact on all areas of the control discipline. New theory, new controllers, actuators,sensors, new industrial processes, computer methods, new applications, newphilosophies…, new challenges. Much of this development work resides in industrialreports, feasibility study papers and the reports of advanced collaborative projects.The series offers an opportunity for researchers to present an extended exposition ofsuch new work in all aspects of industrial control for wider and rapid dissemination.

This Advances in Industrial Control series monograph, Optimal Operation ofBatch Membrane Processes by Radoslav Paulen and Miroslav Fikar is the firstmonograph in the field of membrane processes to appear in the series. The use ofmembrane technology to separate desirable species from the undesirable ones, or toremove pollutants and produce a purified gas or liquid is of critical importance insome fields of process control, biotechnology and medical engineering, just to listthree application areas.

Many in the control community are not familiar with membrane technologies,and the authors open the monograph with an educational chapter of introductorymaterial on the field. Once past the introductory sections, the authors declare thatthe focus of the monograph will be on diafiltration membrane problems. In asentence, diafiltration uses membrane technology to separate two or more solutesfrom a solution, thereby increasing the concentration of a desired product anddecreasing the concentration of impurities in the solution. The authors cite someinteresting medical and food industry applications of the method.

By way of contrast, some workers in the chemical engineering field may not befamiliar with the use of generalised mathematical models, and classical optimalcontrol theory that the authors utilise in their reported studies. In Chap. 2,the authors present the main results of the optimal control theory applied. Since thetechnical framework is one of nonlinear processes the authors devote Chap. 3 to thevarious analytical and numerical solution methods for such problems.

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The main chapters of the monograph (Chaps. 4–7) present a variety ofscenario-based studies of diafiltration membrane problems. Chapter 8 presents ageneral membrane model and also reports two short industrial application casestudies. A chapter of conclusions and future possible research directions closes themonograph.

This is a scholarly monograph presentation that will appeal to different reader-ships. For the control community the attraction is the opportunity to gain insightinto an important industrial area that is not often given monograph length treatmentin the control literature. Within the engineering disciplines of chemical engineering,biotechnology, medical engineering and similar fields, the monograph demonstrateswhat the application of optimal control has to offer for nonlinear process problems.It is a very welcome multidisciplinary addition to the Advances in IndustrialControl monograph series.

Glasgow M.J. GrimbleScotland, UK M.A. Johnson

vi Series Editors’ Foreword

Preface

The principles of optimality govern our everyday life. Any natural or artificialsystem, which surrounds us or influences our closest vicinity, tends to operateoptimally, i.e. it tries to maximise or minimise some given function under presentconstraints. This can be seen even in such microscopic phenomena as the bondingof atoms to form molecules in order to minimise the overall potential energy(function). The primary neural network in humans is the brain, in which the neuronsuse the minimum of wiring material (neuronal connections) necessary to complywith the constrain imposed by the amount and rate of information transfer. Seeds insunflowers are collocated in order to maximise their number subject to the givenarea and the seed shape. In these cases, it is nature which decides, using evolu-tionary (trial and error, survival of the fittest) principles, about the optimal designof the systems.

Artificial systems, such as traffic, electricity, or logistic networks, are designedby engineers who express the objective and constraints in mathematical form offunctions and equations. If only this were true—just look at the traffic congestionaround oxford. Using such a mathematical model of reality, actual design problemscan be then solved by exploiting tools (e.g. algorithms) provided by mathematicsand computer science. The solution is then given by a set of discrete values ofdecision (optimisation) variables. Once this is done, we are sure that nothing betteris possible to achieve for the actual form of objective and constraint functions andfor the actual state of the system.

But what if the system state or any of these functions involved in the optimi-sation problem are changing over time? Then, we obviously need to repeat thewhole optimal design procedure at each time instant. From the practical point ofview, we no longer speak about discrete decision (control) actions but we considerthe corresponding time-dependent trajectories. We attribute all dynamic changeshappening at the observed system to an entity that we call process. Again, since ourgoal is the optimisation of the system, we need to devise a mathematical modelof the reality, a dynamic (process) model.

It is interesting to note that a variety of problems of the design of optimal processoperation (i.e. optimal process control) arise in fields of engineering (chemical,

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mechanical, electrical,…), computer science, economics, finance, operationsresearch and management science, space exploration, physics, structural andmolecular biology, medicine and material science. Such problems include findingof an optimal control strategy which minimises energy or raw material consumptionduring the production processes, maximises production profit, or leads to optimalprocess model identification (optimal experiment design, parameter estimation).Although similar concepts from static design apply for this optimisation problem,the situation is complicated by the presence of dynamic forms of system state andobjective and constraint functions.

Tools for solving static optimal design problems date back to the end of the firsthalf of the last century. Development of linear programing methods, followed soonby nonlinear programming (NLP) ones, enabled effective computer solution ofvarious engineering problems arising in many fields. Dynamic optimisation(DO) represents a mathematical approach for solving problems of open-loopoptimal process control.

The techniques utilised to solve DO problems in the class of deterministicapproaches fall under two broad frameworks: variational (indirect) methods anddiscretisation (direct) methods. Variational methods address the DO problem in itsoriginal infinite-dimensional form exploiting the classical calculus of variationstogether with dynamic programming or Pontryagin’s maximum/minimum princi-ple. A big advantage of this is that we look for an exact solution to the problemwithout any transformations. On the other hand, use of these approaches canbecome difficult if we want to solve DO problem for more complicated systems.Then discretisation plays an important role since the original infinite-dimensionalproblem is transformed to a nonlinear programing problem. Once transformed intostatic form, the DO problem can be solved approximately by means of staticoptimisation just as in static optimal design. It is then only a matter of utiliseddegree and form of discretisation as to how close the obtained solution will be to theoriginal problem. Discretisation methods can be subdivided into two broad clas-sifications known as simultaneous and sequential.

The simultaneous method is a complete discretisation of both state and controlvariables often achieved via collocation. While completely transforming a dynamicsystem into a system of algebraic equations eliminates the problem of optimising inan infinite-dimensional space, simultaneous discretisation has the unfortunate effectof generating a multitude of additional variables yielding large, unwieldy NLPs thatare often impractical to solve numerically.

Sequential discretisation, usually achieved via control parameterisation, is adiscretisation approach in which the control variable profiles are approximated by asum of basis functions in terms of a finite set of real parameters. These parametersthen become the decision variables in a dynamic embedded NLP. Function(functional) evaluations are provided to this NLP via numerical solution of a fullydetermined initial value problem (IVP), which is given by fixing the control pro-files. This method has the advantage of yielding a relatively small NLP andexploiting the robustness and efficiency of modern IVP and sensitivity solvers.

viii Preface

In this monograph, we deal with membrane processes which stand for anemerging technology in the chemical and bioprocess industry, used both in pro-duction and downstream processing. Membrane processes, such as membranedistillation, pervaporation, membrane purification, diafiltration, and processesexploiting membrane-equipped reactors, are receiving growing attention mainlydue to reduced energy demands and higher efficiency of the achieved separation orprocessing goals. These systems, however, did not receive much attention from theprocess optimisation community and that is why they provide many opportunities,for example, the development of optimal operation design.

Purification of a solution can be achieved by employing a semi-permeablemembrane which retains or concentrates (in) valuable species. A diafiltration pro-cess combines two possible ways of treating a solution to concentrate its valuablecomponents and to dilute (dispose of) present impurities. It can be performedcontinuously or discontinuously. This depends on several physical factors and onproperties of initial solution as well as the final product. The process can be con-trolled, either in continuous or batch set-up, by influencing concentrations throughan addition of solute-free solvent (diluant). Utilisation of this diafiltration buffer canbe dynamically adjusted to optimise the process performance, e.g. minimum time orminimum diluant operation can be attained.

This monograph concentrates on finding a general optimal operation strategy forbatch diafiltration processes which are a particular class of membraneseparation/purification processes. The existing operating practice is explored andimproved operation, based on the optimal control theory, is provided. The resultspresented summarise the research outcomes of our group since 2009, which havebeen published in journals and at various IFAC, IEEE, and membrane-process-oriented conferences.

The monograph is organised as follows. The first part (Chaps. 1–3) introducesthe theory of membrane processes, optimal control, and dynamic optimisation in away to provide tools that are exploited in the second part for finding an optimaloperation of batch diafiltration processes. The theory of membrane processesincludes the definition of separation problems, the derivation of dynamic mathe-matical models of batch membrane processes, and the introduction of typical costspecifications. The part on control theory involves an introduction to the problemsof dynamic optimisation mainly from a chemical engineering point of view. It isfollowed by an explanation of methods (analytical and numerical) that can beexploited to treat the problems of optimal control of membrane processes.

The second part (Chaps. 4–8) then builds upon the theoretical basis and uses it toestablish a solution to treated problems. It is divided into sections treating mem-brane models with increasing complexity. First, the limiting flux model is treated.The next chapter deals with perfect separation of solutes with arbitrary flux models.A further generalisation is studied when the macro-solute is perfectly rejected or ifboth rejection coefficients are constant. Finally, Chap. 8 discusses the most generalmodel. Each chapter starts with a derivation of optimal operation and continues

Preface ix

with selected case studies that present various aspects of considered optimal controlproblems and discuss the possible advantages and drawbacks of real implementa-tion of optimal operation of diafiltration processes.

The objectives of the monograph can be summarised as follows:

• to introduce the reader to the field of dynamic optimisation and optimal controlof (chemical) processes,

• to survey analytical and numerical methods for solving problems of optimalcontrol,

• to present a study of optimal control for general diafiltration processes,• to derive analytical solution for most common classes of batch diafiltration

processes,• to propose a simple numerical approach to treat the general case of optimal

operation for batch diafiltration processes,• to present a comparison of the resulting optimal operation with the standard

industrial control techniques and discussion of advantages of optimal operationand of future challenges for optimal operation of diafiltration processes.

Some of the programs and figures of the examples presented in the monographare freely available at the web page:

http://www.kirp.chtf.stuba.sk/*fikar/books/mem/index.htm.

x Preface

Acknowledgments

We would like to express our sincere gratitude to Prof. Abderrazak Latifi forlong-lasting cooperation in the area of optimal chemical process control with hisgroup at Ecole Nationale Supérieure des Industries Chimiques (ENSIC) in Nancy.

Ján Mikleš, Ján Dvoran, Alojz Mészáros, Monika Bakošová, Michal Kvasnica,Anna Kolesárová and Vladimír Baláž from our home Institute of InformationEngineering, Automation, and Mathematics (IAM) at the Slovak University ofTechnology in Bratislava and Boris Rohál-Il’kiv, Ivan Taufer, Jozef Markoš, ŠtefanSchlosser, François Lesage, Michael Daroux, Jean-Pierre Corriou, SigurdSkogestad, Tor Arne Johansen, Greg Foley and Benoît Chachuat are also amongthose we want to mention in order to thank them for their unreserved aid wheneverwe needed it.

Our research has been supported by several granting agencies over the past sixyears. Slovak Research and Development Agency provided support for projectsAPVV 0551-11 (Advanced and effective methods of optimal process control) andAPVV 0029-07 (Algorithms for optimal control of heat and mass transfer processeswith hybrid dynamics). Scientific Grant Agency of the Slovak Republic financedprojects 1/0053/13 (Optimal process control) and 1/0071/09 (Advanced methods ofoptimal control of chemical and biochemical processes). Finally, the project ITMS26240220084 (University Scientific Park STU in Bratislava) has been supported bythe Research 7 Development Operational Programme funded by the ERDF.

The preliminary version of this document was reviewed by Alena Kozáková andAntonín Víteček, to whom we are greatly indebted. At this place, we would alsolike to thank Mario E. Villanueva, Martin Jelemenský, and Ayush Sharma forproofreading.

Martin Jelemenský and Ayush Sharma joined our group working on optimalcontrol of membrane processes. Martin works on membranes and concentrates onfouling phenomenon. He helped us with case studies and coauthored several worksdealing with multiobjective optimal control. Ayush concentrates on experimentalwork, verification, and improvements at laboratory scale.

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There are few other people at IAM we would like to thank: Andrea Kalmárováand Monika Mojžišová for being helpful in processing paper and administrativeworks; Katarína Macušková also for drawing some schemes; Michal Čižniar,Martin Herceg, Ivana Rauová, and Richard Valo, Stano Vagač, L’uboš Čirka,Katarína Matejíčková, Jana Závacká, Lenka Blahová, Marian Podmajerský, MartinKalúz, Jana Kmet’ová, Juraj Oravec, and Alexander Szűcs as Ph.D. students andmembers of our department who made our life much more pleasant.

Special thanks are devoted to our friend, Dr. Zoltán Kovács, who introduced usto the world of membrane filtration. Henrik Manum, Johanes Jäschke, MagnusJacobsen, Ramprasad Yelchuru, Brahim Benyahia, Shahid Ayoub, Stephane Quino,Ahmed Maidi, Ali Assaf, Salim Zodi, Amine Bouarab, Huan Dinh Nguyen, DinhNang Le, Aziz Assad, Billy Homeky, Juan Lizardo, Ivan Gil, Minghai Lei, LeiWang, Mylène Detchebbery, Lívia Petáková, Mario E. Villanueva, Channarong“K” Puchongkawarin, Jai Rajyaguru, Boris Houska, Andreas Nikolaou, andCheng S. Khor are the last people mentioned to whom Rado is indebted for makinghis research abroad stays pleasant and inspirational.

Our final thanks go to our families that have supported us and have made ourlives much more pleasant.

Dortmund Radoslav PaulenBratislava Miroslav FikarMay 2015

xii Acknowledgments

Contents

1 Membrane Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Membrane Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Pore Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Operation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.4 Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.5 Fouling of Membranes . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Mathematical Modelling of Membrane Processes . . . . . . . . . . . . 111.3 Diafiltration Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 Process Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Fouling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.3 Operational Modes of Diafiltration. . . . . . . . . . . . . . . . . 191.3.4 Optimisation of Diafiltration Process . . . . . . . . . . . . . . . 20

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1 Objective Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 Typical Optimal Control Tasks . . . . . . . . . . . . . . . . . . . 282.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3 Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Linear Time-Invariant System . . . . . . . . . . . . . . . . . . . . 332.3.2 Input Affine System. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Summary of Problem Definition . . . . . . . . . . . . . . . . . . . . . . . 34References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Solution of Optimal Control Problems. . . . . . . . . . . . . . . . . . . . . . 373.1 Necessary Conditions for Optimality . . . . . . . . . . . . . . . . . . . . 373.2 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.2 Dynamic Programming. . . . . . . . . . . . . . . . . . . . . . . . . 423.2.3 Pontryagin’s Minimum Principle . . . . . . . . . . . . . . . . . . 42

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3.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.1 Control Vector Iteration . . . . . . . . . . . . . . . . . . . . . . . . 453.3.2 Boundary Condition Iteration . . . . . . . . . . . . . . . . . . . . 453.3.3 Complete Discretisation . . . . . . . . . . . . . . . . . . . . . . . . 463.3.4 Control Vector Parametrisation . . . . . . . . . . . . . . . . . . . 473.3.5 Direct Multiple Shooting . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Methods for Computing Gradients . . . . . . . . . . . . . . . . . . . . . . 503.5 Feedback Strategies for Optimal Control . . . . . . . . . . . . . . . . . . 52

3.5.1 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . 53References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Operation at Limiting Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.1 Process Model and Definition of Optimisation Problem . . . . . . . 57

4.1.1 Filtration Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.2 Optimisation Problem. . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Optimal Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.2 Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.2 Separation of Pectin from Sugar . . . . . . . . . . . . . . . . . . 714.3.3 Purification of Soybean Water Extracts . . . . . . . . . . . . . 74

4.4 Models Derived from Limiting Flux. . . . . . . . . . . . . . . . . . . . . 794.4.1 Viscosity Dependent Mass Transfer Coefficient . . . . . . . . 794.4.2 Osmotic Pressure Model. . . . . . . . . . . . . . . . . . . . . . . . 80

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Perfect Rejection of Both Solutes . . . . . . . . . . . . . . . . . . . . . . . . . 835.1 Optimal Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.1 Separation of Lactose from Proteins . . . . . . . . . . . . . . . . 855.2.2 Albumin–Ethanol Separation . . . . . . . . . . . . . . . . . . . . . 95

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Perfect Rejection of Macro-Solute . . . . . . . . . . . . . . . . . . . . . . . . . 1096.1 Optimal Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.1 Dye–Salt Separation. . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2.2 Radiopaque–Ethylene Glycol Separation. . . . . . . . . . . . . 1176.2.3 Sucrose–Sodium Chloride Separation . . . . . . . . . . . . . . . 120

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

xiv Contents

7 Constant Incomplete Rejection of Solutes . . . . . . . . . . . . . . . . . . . 1297.1 Optimal Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2.1 Extended Limiting Flux Model . . . . . . . . . . . . . . . . . . . 1327.2.2 Three Component Separation . . . . . . . . . . . . . . . . . . . . 137

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8 General Membrane Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.1 Optimal Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.1.1 Singular Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.2.1 Radiopaque–Ethylene Glycol Separation. . . . . . . . . . . . . 1478.2.2 Separation of Peptide from Trifluoroacetic Acid . . . . . . . 148

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 1539.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Contents xv

List of Figures

Figure 1.1 Classification of membranes with regard to poresize and filterable/retained components . . . . . . . . . . . . . . . . 2

Figure 1.2 Dead-end membrane filtration . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.3 Cross-flow membrane filtration . . . . . . . . . . . . . . . . . . . . . 4Figure 1.4 Hollow fibre membrane . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.5 Flat plate membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.6 Spiral membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.7 Batch membrane system with full recycle . . . . . . . . . . . . . . 7Figure 1.8 Batch membrane system with partial recycle . . . . . . . . . . . . 8Figure 1.9 Continuous membrane system . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.10 Continuous membrane system with recycle . . . . . . . . . . . . . 9Figure 1.11 Multistage series membrane system . . . . . . . . . . . . . . . . . . 9Figure 1.12 Multistage single pass membrane system

“Christmas tree” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 1.13 Concentration polarisation. . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 1.14 Cross-flow membrane module . . . . . . . . . . . . . . . . . . . . . . 13Figure 1.15 Generalised scheme of continuous diafiltration process . . . . . 15Figure 1.16 Schematic representation of a generalised batch

diafiltration process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 1.17 Standard fouling models. a Complete pore blocking

model. b Intermediate pore blocking model. c Standardpore blocking model. d Cake formation model . . . . . . . . . . 18

Figure 1.18 Representation of classical three-step processing(C-CVD-C), pre-concentration combinedwith variable-volume diafiltration (C-VVD),and variable-volume diafiltration (VVD) operationin terms of the α function . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 2.1 Typical optimal control tasks. a Fixed terminal timeand terminal state. b Free terminal time, fixed terminalstate. c Fixed terminal time, free terminal state. d Freeterminal time and terminal state . . . . . . . . . . . . . . . . . . . . 29

xvii

Figure 3.1 Schematic representation of the orthogonal collocationon finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Figure 3.2 Control vector parametrisation algorithm. . . . . . . . . . . . . . . 49Figure 3.3 Principle of direct multiple shooting . . . . . . . . . . . . . . . . . 50Figure 4.1 Optimal trajectories for αðtÞ for x0 ¼ 20, cs0=csf ¼ 5,

and chosen values of xf . a xf ¼ 1:5, b xf ¼ 2, c xf ¼ e,d xf ¼ 4, e xf ¼ 7, f xf ¼ 10 . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 4.2 Optimal operations of diafiltration with limiting fluxmodel in concentration diagram with one initial point (�)and two endpoints (�) . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 4.3 Optimal control of diafiltration with limiting flux modelfor minimum time, minimum diluant, and multi-objectiveoptimal operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 4.4 Pareto front under limiting flux conditions . . . . . . . . . . . . . 71Figure 4.5 Optimal macro-solute concentration during CVD step

for different values of ratio wT=wD . . . . . . . . . . . . . . . . . . 72Figure 4.6 Optimal values of processing time and diluant

consumption for different values of weightcoefficients wT ; wD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Figure 4.7 Optimal macro-solute concentration during CVD stepfor different values of wT=wD for soybean extracts . . . . . . . . 75

Figure 4.8 Economically optimal, minimum time, and minimumdiluant strategies for purification of soybean extractsto the prescribed final purity in state diagramof concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 4.9 Optimal values of processing time and diluantconsumption for different values of weightcoefficients wT ; wD in case of fixed initial and freefinal conditions by cheese whey extracts . . . . . . . . . . . . . . . 76

Figure 4.10 Various control strategies for purification of soybeanextracts with fixed final concentrations(denoted by �). Top plot—state diagram,bottom plot—optimal control. . . . . . . . . . . . . . . . . . . . . . . 78

Figure 5.1 Separation of lactose from proteins: comparisonof minimum time and C-CVD control strategyin concentration diagram (top plot) and correspondingcontrol (bottom plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Figure 5.2 Separation of lactose from proteins: analytical minimumtime control in concentration diagram. . . . . . . . . . . . . . . . . 88

Figure 5.3 Numerical solution of the minimum time problemfor separation of lactose from proteins: volumeof consumed water, concentration diagram,and control variable α during the simulation . . . . . . . . . . . . 92

xviii List of Figures

Figure 5.4 Concentration state diagram for multi-objective optimal,minimum time, and minimum diluant strategiesfor separation of lactose from proteins. Singularsurfaces for particular operations are denoted by circles . . . . 93

Figure 5.5 Optimal control for multi-objective optimal, minimumtime, and minimum diluant strategies for separationof lactose from proteins . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Figure 5.6 Pareto front for separation of lactose from proteins . . . . . . . 94Figure 5.7 Analytical and numerical minimum time control

for Case 1. Top plot—optimal concentrations diagram,bottom plot—optimal αðtÞ . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 5.8 Analytical and numerical minimum time controlfor Case 2. Top plot—optimal concentrations diagram,bottom plot—optimal αðtÞ . . . . . . . . . . . . . . . . . . . . . . . . . 100

Figure 5.9 Analytical and numerical minimum time controlfor Case 3. Top plot—optimal concentrations diagram,bottom plot—optimal αðtÞ . . . . . . . . . . . . . . . . . . . . . . . . . 101

Figure 5.10 Economically optimal, minimum time, and minimumdiluant strategies for albumin/ethanol separationfor Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 5.11 Economically optimal, minimum time, and minimumdiluant strategies for albumin/ethanol separationfor Case 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 5.12 Pareto front of optimal values of processing timeand of diluant consumption for albumin and ethanolseparation (Case 9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 6.1 Dye–salt separation: optimal operation. Top plot–optimalconcentrations diagram with different values of αmax,bottom plot—optimal αðtÞ . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 6.2 Optimal macro-solute concentration during CVD stepfor different values of wT=wD for the separationof radiopaque and ethylene glycol . . . . . . . . . . . . . . . . . . . 118

Figure 6.3 Optimal values of processing time and diluantconsumption for different values of weight coefficientswT ; wD in case of radiopaque and ethylene glycolseparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 6.4 Economically optimal, minimum time, and minimumdiluant strategies for radiopaque and ethylene glycolseparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 6.5 Case A: optimal control (top left), optimal α (top right),concentrations (bottom left), and volume (bottom right)as functions of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

List of Figures xix

Figure 6.6 Analytical optimal control of sucrose—sodium chlorideseparation. Upper plot—optimal concentrations diagram,lower plot—optimal αðtÞ s . . . . . . . . . . . . . . . . . . . . . . . . 126

Figure 6.7 Analytical time-optimal control of sucrose—sodiumchloride separation, Case A . . . . . . . . . . . . . . . . . . . . . . . 127

Figure 7.1 Dependence of optimal singular concentrationon rejection coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 7.2 Minimum time operation with R1 ¼ 1, R2 ¼ 0and C/CVD optimal modes . . . . . . . . . . . . . . . . . . . . . . . . 135

Figure 7.3 Minimum time operation with C/VVD optimal modes . . . . . 135Figure 7.4 Operation with C/CVD modes. . . . . . . . . . . . . . . . . . . . . . 137Figure 7.5 Operation with constant concentration modes . . . . . . . . . . . 138Figure 7.6 Comparison of different control strategies (top—state

space, bottom—control profiles). . . . . . . . . . . . . . . . . . . . . 140Figure 8.1 Minimum time control of radiopaque–ethylene glycol

separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure 8.2 Concentration state diagram for economically optimal,

minimum time, minimum diluant, and minimumproduct loss strategies for separation of peptidefrom trifluoroacetic acid . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Figure 8.3 Optimal control for economically optimal, minimum time,minimum diluant, and minimum product loss strategiesfor separation of peptide from trifluoroacetic acid . . . . . . . . 150

xx List of Figures

List of Tables

Table 1.1 Typically applied pressures and pore sizes for differenttypes of pressure-driven membrane processes . . . . . . . . . . . . 3

Table 1.2 Residence times and membrane areas for differentconfigurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Table 4.1 Multi-objective optimal operation of under limitingflux conditions compared with minimum time,minimum diluant, and traditionally used operations . . . . . . . . 71

Table 4.2 Economically optimal operation of apple juice underlimiting flux conditions compared with minimum time,minimum diluant, and traditionally used operations . . . . . . . . 74

Table 4.3 Economically optimal operation of soybean extractfiltration with the prescribed purity of the productcompared to other control strategies . . . . . . . . . . . . . . . . . . 77

Table 4.4 Economically optimal operation of soybean extractfiltration with prescribed final concentrations comparedto other control strategies. . . . . . . . . . . . . . . . . . . . . . . . . . 79

Table 5.1 Multi-objective optimal membrane operation for separationof a lactose from proteins compared with minimum time,minimum diluant, and traditionally used operations . . . . . . . . 94

Table 5.2 Permeate volumetric flowrate relation constants(Data taken from [5]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Table 5.3 Initial and final conditions on macro-/micro-soluteconcentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Table 5.4 Minimum operation times and diluant consumptionsfor different N�PWC αðtÞ . . . . . . . . . . . . . . . . . . . . . . . . . 96

Table 5.5 Operation times and minimum diluant consumptionsfor different N�PWC αðtÞ . . . . . . . . . . . . . . . . . . . . . . . . . 97

Table 5.6 Comparison of control strategies of batch DFprocess for albumin/ethanol separation (Case 3) . . . . . . . . . . 104

Table 5.7 Comparison of control strategies of batch DFprocess for albumin/ethanol separation (Case 9) . . . . . . . . . . 106

xxi

Table 6.1 Design factors and their levels . . . . . . . . . . . . . . . . . . . . . . 114Table 6.2 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Table 6.3 Comparison of time optimality loss (δ) between

optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Table 6.4 Comparison of control strategies of batch DF process

for radiopaque and ethylene glycol separation. . . . . . . . . . . . 119Table 6.5 Experimentally obtained coefficient values

for R1; R2, and q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Table 6.6 Comparison of optimality loss (δ) between optimal

control and traditionally used strategies . . . . . . . . . . . . . . . . 127Table 8.1 Comparison of control strategies of batch DF process

for separation of peptide from trifluoroacetic acid . . . . . . . . . 150

xxii List of Tables

Nomenclature

ΔP Transmembrane pressurebuðtÞ Vector of polynomial approximations of control variablesbxðtÞ Vector of polynomial approximations of state variables0 All-zeros vector1 All-ones vectorQ State weighting matrix of dimension nx � nxR Control weighting matrix of dimension nu � nuJ� Value function, cost-to-goFð�Þ Integrand in objective functionalGð�Þ Non-integral part of objective functionalJ Objective functionalp Vector of parametersu(t) Vector of control variablesx(t) Vector of state variablesy Vector of optimised parametersA (Effective) membrane areac ConcentrationD Diffusivity coefficientH Hamiltonian functionk Mass transfer coefficientnp Dimension of the vector of parametersnu Dimension of the vector of state variablesnx Dimension of the vector of state variablesq Volumetric flux, liquid flow through the membraneR Rejection coefficientRm Membrane resistancet Independent time variabletshift Shift durationu(t) Flowrate of pure diluant into the feed tankV Retentate volumeV(t) Volume of the solution inside the feed tank

xxiii

Vp Volume of permeatewD Unit price of the diluantwM Unit price of productwP Unit price of product relative to its puritywT Unit price of processing time

Greek Symbols

α Control variable of diafiltration processμ Viscosityπ Osmotic pressureλ Vector of adjoint variables

Mathematical Notation

A Vector space, setR Real-vector spaceS Space of symmetric matricesA matrixA functionala vector or vector functiona;A scalar or scalar function

Subscripts

0 Initialsing Singularf Finalmax Maximal value, upper boundmin Minimal value, lower bound1 Macro-solute2 Micro-soluteg Gellim Limitingp Permeatew Wall

Abbreviations

AV Adjoint VariablesBCI Boundary Condition IterationC Concentration (mode)

xxiv Nomenclature

CVD Constant-Volume DiafiltrationCVI Control Vector IterationCVP Control Vector ParametrisationD Dilution (mode)DF DiafiltrationDO Dynamic OptimisationDVD Dynamic-Volume DiafiltrationFD Finite DifferencesIP Interior-PointIVP Initial Value ProblemMF MicrofiltrationMPC Model Predictive ControlNCO Necessary Conditions for OptimalityNF NanofiltrationNLP Nonlinear ProgrammingOC Orthogonal CollocationOCP Optimal Control ProblemODE Ordinary Differential EquationPMP Pontryagin’s Minimum PrinciplePWC Piece-Wise ConstantRO Reverse OsmosisSE Sensitivity EquationsSOC Self Optimising ControlSQP Sequential Quadratic ProgrammingTPBVP Two-Point Boundary Value ProblemUF UltrafiltrationVVD Variable-Volume Diafiltration

Nomenclature xxv