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ATTAINABLE REGION THEORY

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ATTAINABLE REGION THEORY

An Introduction to Choosing an Optimal Reactor

By

DAVID MINGUniversity of the Witwatersrand, Johannesburg

DAVID GLASSER and DIANE HILDEBRANDTMaterial and Process Synthesis research unit, University of South Africa

BENJAMIN GLASSERRutgers, The State University of New Jersey

MATTHEW METZGERMerck & Co., Inc

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Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior writtenpermission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive,Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should beaddressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online athttp://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations orwarranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability orfitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategiescontained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall beliable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at(800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For moreinformation about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Names: Ming, David, 1985- author. | Glasser, David, 1936- author. | Hildebrandt, Diane, 1961- author.| Glasser, Benjamin John, 1968- author. | Metzger, Matthew, 1983- author.

Title: Attainable region theory : an introduction to choosing an optimalreactor / by David Ming, David Glasser, Diane Hildebrandt,Benjamin Glasser, Matthew Metzger.

Description: Hoboken, New Jersey : John Wiley & Sons, 2016. | Includes index.Identifiers: LCCN 2016022700 (print) | LCCN 2016025203 (ebook) | ISBN

9781119157885 (cloth) | ISBN 9781119244714 (pdf) | ISBN 9781119244707(epub)

Subjects: LCSH: Chemical reactors–Design and construction. | Statisticaltolerance regions.

Classification: LCC TP157 .M528 2016 (print) | LCC TP157 (ebook) | DDC660/.2832–dc23

LC record available at https://lccn.loc.gov/2016022700

Cover Image Credit: Courtesy of the Author

Typeset in 10/12pt TimesLTStd-Roman by SPi Global, Chennai, India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

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CONTENTS

Preface xi

Acknowledgments xiii

Prior Knowledge xiv

How this book is Structured xv

Software and Companion Website xvii

Nomenclature xix

SECTION I BASIC THEORY 1

1 Introduction 3

1.1 Introduction, 31.2 Motivation, 3

1.2.1 Toluene Production as a Case Study, 31.2.2 Part One: Initial Investigations, 41.2.3 Part Two: Iterative Improvement, 41.2.4 Part Three: Coffee, 51.2.5 Part Four: Additional Improvements, 61.2.6 What this Book is About, 7

1.3 Reactor Network Synthesis, 81.4 Solving the Reactor Network Synthesis Problem, 12

1.4.1 Reactor Superstructures, 121.4.2 AR Theory, 141.4.3 Attainability Problems Outside of Reactor Design, 15

1.5 Chapter Review, 16References, 17

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vi CONTENTS

2 Concentration and Mixing 19

2.1 Introduction, 192.1.1 Review, 192.1.2 Revisualizing Concentration Data, 19

2.2 Concentration Vectors and Dimension, 232.2.1 Moving on a Line: Table Salt and Water, 232.2.2 Moving Freely Through Space, 25

2.3 Mixing, 282.3.1 Introduction, 282.3.2 Additional Insights, 332.3.3 Different Ways of Synthesizing CC, 352.3.4 Mixing and Attainability, 372.3.5 n-Dimensional Mixing and Convex Hulls, 41

2.4 Chapter Review, 47References, 47

3 The Attainable Region 49

3.1 Introduction, 493.2 A Mixing and Reaction Game, 49

3.2.1 Introduction and Rules of the Game, 493.2.2 Filling in the Region, 493.2.3 Scenario 1: Selecting Points from Region A, 503.2.4 Scenario 2: Selecting Points from Region B1, 513.2.5 Further Improvements, 53

3.3 The AR, 573.3.1 Ten Experiments, 573.3.2 The Limit of Infinitely Many Batches, 57

3.4 Elementary Properties of the AR, 583.5 Chapter Review, 61

References, 61

4 Reaction 63

4.1 Introduction, 634.2 Reaction Rates and Stoichiometry, 63

4.2.1 Benzene Reaction Rate, 634.2.2 Toluene Reaction Rate, 64

4.3 Reaction from a Geometric Viewpoint, 664.3.1 The Rate Vector, 664.3.2 Rate Fields, 69

4.4 Three Fundamental Continuous Reactor Types, 734.4.1 Motivation, 734.4.2 The Plug Flow Reactor, 734.4.3 The Continuous-Flow Stirred Tank Reactor, 824.4.4 The Differential Sidestream Reactor, 95

4.5 Summary, 1024.6 Mixing Temperatures, 102

4.6.1 Motivation, 1024.6.2 Adiabatic Energy Balance, 1024.6.3 Non-adiabatic Energy Balance, 104

4.7 Additional Properties of the AR, 1054.8 Chapter Review, 106

References, 107

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CONTENTS vii

5 Two-Dimensional Constructions 109

5.1 Introduction, 1095.2 A Framework for Tackling AR Problems, 1095.3 Two-Dimensional Van De Vusse Kinetics, 110

5.3.1 Introduction, 1105.3.2 Scenario 1: a1 = a2, 1115.3.3 Scenario 2: a1 < a2, 1145.3.4 Scenario 3: a1 > a2, 1145.3.5 Review, 123

5.4 Multiple CSTR Steady States and ISOLAS, 1255.4.1 Introduction, 1255.4.2 Step 1: Define the Problem, 1255.4.3 Step 2: AR Construction, 1265.4.4 Steps 3–5: Interpretation and Optimization, 130

5.5 Constructions in Residence Time Space, 1315.5.1 Significance of Residence Time Constructions, 1315.5.2 Mixing in Residence Time Space, 1325.5.3 Visualizing Residence Time Data, 1325.5.4 Unbounded Regions, 1335.5.5 Example: Optimal Reactor Structure for Minimum

Residence Time, 1345.6 Chapter Review, 141

References, 141

SECTION II EXTENDED TOPICS 143

6 Higher Dimensional AR Theory 145

6.1 Introduction, 1456.2 Dimension and Stoichiometry, 146

6.2.1 The Stoichiometric Subspace, 1466.2.2 Concentrations Orthogonal to the Stoichiometric

Subspace, 1526.2.3 Number of Independent Reactor Structures, 158

6.3 The Three Fundamental Reactor Types Used in AR Theory, 1596.3.1 Introduction, 1596.3.2 Extreme Points and Reaction, 1596.3.3 Two Important Theorems, 162

6.4 Critical DSRs and CSTRs, 1666.4.1 Overview, 1666.4.2 Controllability, 1666.4.3 Computing Critical DSR Trajectories, 1696.4.4 Computing Critical CSTR Points, 182

6.5 Chapter Review, 189References, 190

7 Applications of AR Theory 191

7.1 Introduction, 1917.2 Higher Dimensional Constructions, 191

7.2.1 Three-Dimensional Van de Vusse Kinetics, 1917.2.2 BTX Kinetics, 198

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viii CONTENTS

7.3 Nonisothermal Constructions and Reactor TypeConstraints, 2057.3.1 Adiabatic Reaction, 2057.3.2 Constrained AR Construction Using Only PFRs, 2087.3.3 Insights into Interstage and Cold-Shot Cooling

Operation, 2117.4 AR Theory for Batch Reactors, 222

7.4.1 Introduction, 2227.4.2 Similarities Between Batch and Continuous Reactive

Equipment, 2237.4.3 Example: Three-Dimensional Van de Vusse Kinetics

Revisited, 2307.5 Chapter Review, 232

References, 233

8 AR Construction Algorithms 235

8.1 Introduction, 2358.2 Preliminaries, 235

8.2.1 Hyperplanes, 2358.2.2 Computing the Stoichiometric Subspace S, 237

8.3 Overview of AR Construction Methods, 2468.3.1 Introduction, 2468.3.2 Inside-Out versus Outside-In Methods, 247

8.4 Inside-out Construction Methods, 2488.4.1 The Recursive Constant Control Policy Algorithm, 2488.4.2 The Iso-State Method, 2538.4.3 The Complement Method, 258

8.5 Outside-in Construction Methods, 2628.5.1 Overview, 2628.5.2 The Method of Bounding Hyperplanes, 2628.5.3 The Shrink-Wrap Algorithm, 267

8.6 Superstructure Methods, 2708.6.1 LP Formulations, 2708.6.2 IDEAS Approach, 276

8.7 Chapter Review, 279References, 279

9 Attainable Regions for Variable Density Systems 281

9.1 Introduction, 2819.2 Common Conversions to Mass Fraction Space, 281

9.2.1 Preliminary Notation, 2819.2.2 Conversions Involving Molar Quantities, 2839.2.3 Average Density, 2859.2.4 Mixing and Reaction, 2859.2.5 Residence Time in Mass Fraction Space, 2879.2.6 Fundamental Reactor Types, 2889.2.7 Computing the Stoichiometric Subspace, 289

9.3 Examples, 2939.3.1 Three-Dimensional Van de Vusse Kinetics, 2939.3.2 Steam Reforming and Water–gas Shift Reaction, 295

9.4 Chapter Review, 298References, 299

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CONTENTS ix

10 Final Remarks, Further Reading, and Future Directions 301

10.1 Introduction, 30110.2 Chapter Summaries and Final Remarks, 30110.3 Further Reading, 304

10.3.1 AR-related Papers, 30410.3.2 Non-reactor-Related Papers, 305

10.4 Future Directions, 30510.4.1 The Search for a Sufficiency Condition, 30510.4.2 Applying AR Theory to Real Systems, 30610.4.3 Automated AR Construction, 306References, 307

Appendix A Fundamental Reactor Types 309

A.1 The Plug Flow Reactor, 309A.2 The Continuous-Flow Stirred Tank Reactor, 309A.3 The Differential Sidestream Reactor, 310

Appendix B Mathematical Topics 311

B.1 Set Notation, 311B.2 Aspects of Linear Algebra, 311

B.2.1 General Definitions, 311B.2.2 Properties of Determinants, 313

B.3 The Complement Principle, 313B.3.1 Introduction, 313B.3.2 Description: A Generalized Mole Balance over a

Reactor Sub-Network, 313References, 315

Appendix C Companion Software and Website 317

C.1 Introduction, 317C.1.1 The Python Programming Language, 317C.1.2 Jupyter Notebooks, 317C.1.3 Companion Website, 317

C.2 Obtaining Python and Jupyter, 318C.2.1 Anaconda, 318C.2.2 Downloading and Installing the Anaconda

Distribution, 318C.2.3 Downloading Jupyter Notebooks, 318

Index 321

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PREFACE

The problem that this book sets out to solve can be formu-lated very simply. Given a set of chemical reactions withknown kinetics, what is the best reactor that can be used tocarry out the set of reactions. As easy as the question is toask, the answer is not obvious, as on the face of it there arean infinite number of possibilities. What this preface will dois to outline the 50-year journey to solve this problem. Thishistory also embodies a cautionary tale for those doing (andwanting to do) research.

I did my PhD looking at a reaction and its kinetics. As aresult of this, I became interested in chemical reactor theory.I remember, in particular, the excellent books of Denbigh andLevenspiel. Both of them talked about the aforementionedproblem and documented some heuristics to help industrial-ists to come up with some solutions. However, it was recog-nized that these heuristics were sometimes contradictory.

At that stage, I became interested in optimization ingeneral. Also, on my first sabbatical leave, I was able towork with some of the greats of chemical engineering(Stanley Katz, Reuel Shinnar, and Fritz Horn) and becameinvolved solving some more limited optimization problems,such as using contact times for catalytic reactions, andminimizing holding times for a series of continuous-flowstirred tank reactors (CSTRs). Also it was at this time that Ibecame interested in using Pontryagin’s maximum principle(developed for space exploration problems) on chemicalreactor problems. This was the stage that I, at first, thoughtwe could solve the main problem by extending residencetime distribution theory to nonlinear kinetics. There was thewell-known result for segregated systems that one couldwork out conversions for all kinetics using the residencetime distribution. I thought we should be able to extend thistheory for all possible reactors and then use the maximumprinciple to solve the general problem.

On the next two sabbaticals, I worked with Roy Jacksonand Cam Crowe to take this idea further, and in the end, much

to our joy, Roy Jackson and I solved the problem (after 15years!). The joy soon evaporated as we realized that eventhough we had a complete description for all possible reac-tors with nonlinear kinetics, in order to solve the main prob-lem we had to find a function that could have a non-countableinfinity of changes with possibly some of the values going toinfinity. This was clearly an impossible problem to solve.

This meant going back to the drawing board! I startedto fiddle with some simple problems from the literature bydrawing simple two-dimensional graphs. What I soon real-ized is that if there were concavities on the graphs, they couldbe filled in with straight lines to make what are called con-vex hulls. What was really exciting was that these lines werenothing more than mixing between two points on the graph.Out of this came the idea that a reactor was a system that wasmade up of two processes: reaction and mixing. Each of theseprocesses can be represented on a graph as vectors. Suddenly,the problem I was looking at changed to a geometric one,as I was now looking at making the region in the space aslarge as possible using the two vector processes. When visit-ing Martin Feinberg at Rochester, he drew my attention to apaper by Fritz Horn that talked about what he termed attain-able regions (ARs). That is, in this context, the largest regionin some component space that one could obtain using anyprocesses. He showed that if one had this AR, the optimiza-tion problem was relatively simple. Without at first realizingwhat I was doing, I had found a geometric way of findingthe AR for reacting systems. At this stage, I was only doingit graphically on problems that could be represented in twodimensions.

At this stage, it became clear why one could not solvethis problem using standard optimization methods. This wasbecause mixing is not a differential process in the ordinarysense. That is, one could not use methods that only lookedin the immediate neighborhood as one could mix fromany point that was itself attainable, and this could be far

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xii PREFACE

from the neighborhood of the point we were examining.An interesting point is that the AR method can solve thesenon-continuous problems, and so is essentially a newmethod of optimization, but to the best of my knowledgethis idea has not been taken up in other fields of study.

When I arrived back from sabbatical, I looked for a stu-dent to work on the problem to extend the results. At thisstage, Diane Hildebrandt, who I had taught as an undergrad-uate, and had done her MSc with me , was looking for a PhDproject. I warned her that the project was a little open-ended.However, she was sufficiently interested in the topic (was thisbrave or foolhardy?) to want to work on it. In the end, out ofher work, came the foundations of our work on AR theory.At last, we had a method that could solve the problem I hadstarted out to look at more than 20 years earlier.

What is really interesting (and important) is that we wereable to solve a problem that was generally regarded as impos-sible by thinking about it in a very different way. Instead oftrying to answer the problem of finding the optimal reactorin one step, we had first asked what processes are occurringin a chemical reactor, how can we represent them, and howcan we then use these processes to ask what are all possibleoutcomes. Only at this stage do we look at how to performthe optimization. By breaking the problem up into smallerstages, in the end, we were able to solve it.

Publishing the first AR paper turned out to be a bit of anightmare! As reviewers knew solving this problem was notpossible, faith even in scientists was more important than rea-son. The paper kept being rejected with reviewers telling usthere must be something wrong with what we were doing,even though they could not say what it was. In the end, wehad to say to the editor if the reviewers could not find themistake, he should publish it and wait for someone to comeup and show us the fallacy. Luckily, in the end he agreed, andwe are still waiting for someone to show us why what we didwas wrong!

At this stage, my son Benjamin, also a chemical engineer,worked on AR theory for his MSc, for the first industriallyinteresting problem. Diane then went on sabbatical whereshe worked with Martin Feinberg to formalize some of theideas she had started on her PhD. Out of this arose the ideathat there are only three types of reactor needed to find theAR in any dimensions, namely plug flow reactors (PFRs),continuous-flow stirred tank reactors (CSTRs), and differen-tial sidestream reactors (DSRs). Suddenly, from there beingan infinite number of possible reactors needed in an optimalsystem, there are only three types. The significance of this isthat while initially we needed to find the AR before doing theoptimization, it is now, in principle, possible to generate analgorithm to solve large problems without necessarily findingthe full AR, which may potentially be a very large problemin higher dimensions.

The method became a topic of research for us, and someothers, and we published many papers in the area. Also the

method became accepted and now appears as standard inmany textbooks on chemical reactor theory and design. Inparticular, we began to look at ways of calculating ARs inhigher dimensions in order to be able to have a successfulalgorithm that could be used fairly generally. David Ming hasworked on this and also extended the ideas from steady-flowreactors to batch reactors. Matt Metzger who spent time withus from the United States found the topic interesting andhelped to gather all the information in a logical and consis-tent way and helped to see that it could be presented in areadable way.

Thus, even though the story at this stage is complete, Ibelieve there is a lesson we can learn from this. I call this thefour P’s of research. These are as follows:

1. Passion

2. Patience

3. Perseverance

4. Persuasion

I believe the history of this book typifies all of theseaspects.

PASSION

If I (and others) had not believed that this project was worthyof spending time on it, it would not have been done.

PATIENCE

I worked patiently on it for 15 years to get an answer that wasof no real use.

PERSEVERANCE

Perseverance is the hard work you do after you get tired ofdoing the hard work you already did. —Newt Gingrich. Ibelieve this speaks for itself in the context of this work.

PERSUASION

It is not sufficient to do good work on its own; one needs topersuade the rest of the world. We really struggled with this!Thomas Grey said it all.

Full many a gem of purest ray serene,the dark unfathomed caves of ocean bear,full many a flower is born to blush unseen

to waste its sweetness on the desert airDavid Glasser

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ACKNOWLEDGMENTS

The dilemma that we face with attainable region (AR) the-ory is that some aspects of the field are very easy to grasp,whereas other parts are very difficult. Over the years, we havedone our best to accommodate for these different aspects,yet there has not been a single, consistent, treatment of ARtheory that could be used to address multiple audiences andinterest groups. It is our hope that this textbook will not onlyprovide this consistency, but it will also be used by those whowish to understand the field of AR theory for the first time.

The opportunity to write a textbook comes with a mixof excitement and trepidation. On one hand, it is exciting tobe given an opportunity to share our ideas—many of whichhave been developed over many years by many people—andto articulate them in our own way. On the other hand, it isdaunting to think of how best to do this, so that we live up tothe expectations of all those who have contributed to theseideas, and also to those prospective readers who are comingto the field for the first time. Indeed, attempting to writea coherent set of work, suitable for many points of view,has been a challenging task, and this textbook would notexist without the help and support from our family, friends,and colleagues who have helped at various stages alongthe way. In particular, we are very grateful to Cody Bishopand Kwame Owusu-Ansah for helping to proofread (andcorrect) early versions of the work. We would also like tothank Michelle Low, not only for supporting us throughout

the course of this work but also for contributing her timeand creativity to take many of the excellent photographsused in this book. Your diligence and continuous enthusiasmthroughout this time has been invaluable to us. We arealso grateful to Professor Arun Moharir for helping tosupply and organize many of the examples and ideas forthe companion website. We thank Linda Anthony and EricLam, for allowing much of the early stages of this work tobe developed at Rutgers University.

Last, we thank our family and friends who have supportedand encouraged us throughout this journey. Thank you forunderstanding and also for sometimes not understanding,but still supporting us! In the book, we try to emphasize acommon theme of you can’t fix what you don’t know, asa way to explain the importance of understanding perfor-mance bounds. In many ways, this theme is mirrored byyour support. You have helped us to understand our ownlimits, and your encouragement has made this task muchmore achievable. Thank you.

David MingDavid Glasser

Diane HildebrandtBenjamin Glasser

Matthew Metzger

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PRIOR KNOWLEDGE

This book assumes a basic understanding of chemicalreaction stoichiometry and material balances, equivalent tothat administered in an undergraduate course in chemicalengineering or chemistry. Knowledge of introductorychemical reactor theory is also beneficial, although it is nota requirement.

We shall be interested in a broad, macroscopic, view ofreactors—one where elementary mass balances are appliedover entire process units. That is, we are not interested inmodeling the system in detail, and attainable region (AR)theory does not demand the use of microscopic transportequations for instance.

AR theory has a geometric flavor. Hence, readers whopossess a basic understanding of calculus (derivatives andintegration) and linear algebra (vectors and matrices) mayfind the introductory concepts of AR theory easier to follow.Familiarity with plotting graphs in two and three dimensionson a computer is also helpful. Nevertheless, we shall alwaysstrive to provide the necessary theory and tools, when appro-priate. This fulfills the dual purpose of introducing the readerto the particular terminology and nomenclature adopted inthis book.

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HOW THIS BOOK IS STRUCTURED

The material covered in this book is organized into twosections. It may be helpful to refer to Figure P.1 for anoverview of the organization of chapters. Section I (Chapters1–5) focuses on the basics of attainable region (AR) the-ory. Importantly, this section introduces a different wayof viewing chemical reactors and reactor networks. Theexamples discussed in Section I are of a simpler nature, withan emphasis on describing all problems in two dimensionsonly. Section I is best read in a sequential fashion.

Section II (Chapters 6–10) focuses on Extended topicsin AR theory, and also discusses specific fields of researchthat may be of interest to select readers. These chapters aremore mathematical in nature, and may appeal to readerswith a stronger background in computer programming andmathematical modeling. Whereas Section I is intended forsenior undergraduates, Section II is more suited for graduate

Attainable RegionTheory

Basic

principles

Applications

Section I

Basic theory Extended topics

Section II

Final remarks,

further reading and

future directions

Ch 10Higher dimensional

AR theory

Ch 6Two-dimensional

constructions

Ch 5Problem and

introduction

Ch 1

Concentration

and mixing

Ch 2

The

Attainable Region

Ch 3

Reaction

Ch 4

Applications

of AR theory

Ch 7

Attainable Regions

for variable

density systems

Ch 9

AR construction

algorithms

Ch 8

Figure P.1 Organization of chapters.

students and working professionals. We hope that thesechapters will be used to gain a deeper appreciation for whatis an exciting field of study in chemical reactor networksynthesis. Although there is more flexibility in how thissection can be tackled, we suggest reading Chapters 6 and 7together.

Chapter 1: In this chapter, we introduce the idea of thereactor network synthesis problem and performance target-ing. Ultimately, we attempt to articulate two important mes-sages: how do you know you have achieved the best? Youcannot fix what you don’t know.

Chapter 2: In this chapter, we introduce the idea of con-centration and mixing from a geometric perspective. We lookat how data may be interpreted in the phase plane (concen-tration space) and the link between mixing, convex hulls, andattainability. Readers already familiar with vectors, and the

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xvi HOW THIS BOOK IS STRUCTURED

rudiments of ordinary differential equations, may choose toskip Chapter 2 and move on to Chapter 3, where the AR isdescribed.

Chapter 3: In this chapter, we apply the concepts devel-oped in Chapter 2 to a simple thought experiment and showhow system performance may be improved with these ideas.This leads us to introduce, for the first time, the idea of theAR for a reactive system. Basic properties of the AR are alsodiscussed.

Chapter 4: In this chapter, we discuss how reaction mayalso be viewed from a geometric perspective. We introducethe three fundamental reactor types used in AR theory, andwe also discuss additional properties of the AR related toreaction.

Chapter 5: In this chapter, we look to apply our knowl-edge of concentration, reaction, and mixing to simple sys-tems. We generate the AR for a number of two-dimensionalsystems and show how the AR approach can lead to insightsinto the design of chemical reactor networks.

Chapter 6: In this chapter, we describe the nature of theAR boundary in higher dimensional spaces, and we explaina number of important results used in AR theory. We alsolook at ways of calculating conditions to reside on the ARboundary in any dimension.

Chapter 7: In this chapter, we discuss a number ofworked examples for more complex systems, includinghigher dimensional systems and constructions where thereis an equipment constraint. We also discuss how AR theorycan be applied to batch reactors.

Chapter 8: In this chapter, we describe a number ofAR construction algorithms that may be implemented on acomputer.

Chapter 9: In this chapter, we look at how to extend con-cepts of the AR to variable density systems, such as non-idealsystems in the gas phase.

Chapter 10: We conclude by providing summaries andkey take-away points from each chapter. This chapter alsoprovides a small set of literature for further reading as wellas potential research interests.

ICONS

Throughout the book, look out for icons to signal an interest-ing or important aspect of the text. These could be workedexamples, important ideas, or simply extra information.

CONCEPT: An idea that may be useful in laterdiscussions.

EXAMPLE: A question, or set of questions,that are intended to test your understandingof an idea, or to extend on a specific topic.

ILLUSTRATION: A worked example ordiscussion demonstrating an application ofAR theory or an extension of an idea.

IMPORTANT: An important concept thatshould be kept in mind.

OBSERVATION: An idea or a result that maynot be obvious that we wish to highlight.

RECALL: A concept or relation developedpreviously that we wish to highlight again.

SIDE NOTE: Additional information or aninteresting fact that is related to the presentdiscussion.

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SOFTWARE AND COMPANION WEBSITE

Due to the geometric nature of attainable region (AR) theory,as well as the complexity of the systems considered, we oftenneed software tools to help us interpret and visualize ourproblems. Rather than attempting to populate a conventionalCD-ROM with software and additional examples that cannotbe adapted over time, we have decided to release this mate-rial on a companion website (http://attainableregions.com),which has been developed for this book. We hope that thisapproach will allow us to cater to the changing needs of thereader and AR community as a whole, where these softwareresources can be tailored accordingly over time.

Figures and demonstrations supplied in the book havebeen produced primarily in MATLAB. However, the use

of MATLAB is not a requirement for understanding theexamples in this book or using the companion softwareon the website. Instead, Python (specifically the Anacondadistribution) has been chosen for the website examplesand demonstrations. Python may be used without anyrestriction, even for commercial use. It is freely available onthe Internet, and, importantly it is also well supported withinthe scientific and engineering community. Information onhow to install the Anaconda distribution and how to accessthe website examples can be found both in Appendix C, andon the companion website.

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NOMENCLATURE

Commonly used symbols and notation adopted in this bookare listed in the following table. Symbols that are only usedonce for a particular description and that are defined at thepoint of use may not be listed. Also note that some symbolsmay be used in different chapters for different situations, andthus more than one definition may apply.

VECTOR, MATRICES, AND SETS

A Stoichiometriccoefficient matrix fora system of reactions

C Molar concentrationvector

C = [cA, cB, … , cn]T

C0 DSR sidestream mixingconcentration

C0 A point residing on ahyperplane H(n, b)

b = nTC0

C* Concentration vectorfrom a mixture

C* = 𝜆C1 + (1− 𝜆)C2

C# Equilibriumconcentrationachieved in a reactor

E DSR controllabilitymatrix

E = [K, N]T

F Set of feedconcentrations

F = {Cf1, Cf2, … , CfN}

G Mass flow rate vectorfor variable densitysystems

G = [g1, g2, … , gn]T

I The identity matrixJ(C) Jacobian matrix of r(C).

Sometimes also givenas dr(C)

L Set of points belongingto a lineation

K Controllability matrixfor a critical DSRexcluding null spaceN

N Null space of thestoichiometriccoefficient matrix A

N = null(AT)

n Hyperplane normalvector

ni Molar flow rate vector ni = [cAi, cBi, … , cni]T

z Mass fraction vector z = [zA, zB, … , zn]T

r(C) Rate vector of a functionof C

r(C) = [rA(C),rB(C), … , rn(C)]T

S Set of pointsrepresenting thestoichiometricsubspace

S\X The complement regionof X to S.

t(C) CSTR tangent vector t(C) = [I − τJ(C)]−1r(C)v(C) Mixing vector v(C) = C2 −C1

W Molecular weight matrixX Set of points

representing a currentset of achievablestates

z[k] The kth iterated Liebracket

𝜀 Extent of reaction vector 𝜺 = [𝜀1, 𝜀2, … , 𝜀n]T0 The zero vector 0 = [0, 0, … , 0]T

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xx NOMENCLATURE

LETTERS

d Number of independent reactionsci Molar concentration of

component iF(t) Fed-batch feeding rate as a

function of reaction timeG Total mass flow rate through a

reactor or reactor networkG =

∑gi

gi Mass flow rate of stream i in anetwork

H(n, b) Hyperplane defined by normalvector n and scalar b

ki Rate constant i in a rate expressionM Total mass M =

∑mi

mi Mass of component in Number of components

participating in a reactionP Reactor pressurePk Polytope representing the AR at

iteration kQ Volumetric flow rate through a

reactor or reactor networkri(C) Rate expression for component i

as a function of CT Reactor temperaturet Batch reaction timeV Reactor volume or total volume of

a reactor networkV(t) Fed-batch volume as a function of

reaction timeWi Molecular weight of component iWavg Average molecular weight of a

mixtureYi Yield of component iZ The compressibility constant for

real gasesPV̂ = ZRT

zi Mass fraction of component iℝ The set of all real numbersℝn The set of all vectors (n-tuples)

with n real numbers(n-dimensional space)

GREEK SYMBOLS

𝛼 DSR sidestream mixingparameter for constantdensity systems

𝛽 DSR sidestream mixingparameter for variabledensity systems

𝜀i Extent of reaction forreaction i in a system ofreactions.

𝜃 Angle between two vectors xTy = ‖x‖‖y‖ cos(𝜃)Λ(C) Critical CSTR determinant

surface

𝜆 Mixing fraction

𝜈i Stoichiometric coefficientof a component i in areaction

𝜌avg Average density of amixture (by mass)

𝜎 Reactor residence time inmass fraction space

𝜎 = V∕G

Σij Instantaneous selectivityof component i to j

𝜏 Reactor residence time(constant density)

𝜏 = V∕Q

Υi Instantaneous yield ofcomponent i

𝜑(C) vDelR critical DSRcondition for a systemin ℝ3 (d = 3)

𝜑(C) =[J(C)v(C)]T[v(C) ×r(C)]

MATHEMATICAL OPERATIONSAND ABBREVIATIONS

𝛻 Del operator (nabla) 𝛻 =[

𝜕

𝜕cA,𝜕

𝜕cB, … ,

𝜕

𝜕cn

]T

||x|| The norm of a vectorx

‖x‖ =2√

x21 + x2

2 + · · · + x2n

Lfh(C) Lie derivative of ascalar functionh(C) with respectto a vectorfunction f(C).

[f, g](C) Lie bracket of vectorfunctions f(C) andg(C)

[f, g](C) =dg(C)f(C)− df(C)g(C)

Det(A) Determinant of amatrix A

conv(X) Convex hull of a setof points X

cl (X) The closure of a setof points X

rank(A) Rank of matrix A

null(A) Null space of matrixA

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NOMENCLATURE xxi

SUBSCRIPTS

f Refers to feedstreams

Cf = [cAf, cBf, … , cnf]T

i, j, k, n General referenceto elements in alist, or indexcounter

SUPERSCRIPTS

T Matrix transpose# Relating to an equilibrium

stateC# is an equilibrium

concentration* Relating to a mixture C* is a concentration

obtained from amixture

ABBREVIATIONS

AR Attainable regionBTX Benzene–toluene–xylene systemCAS Computer algebra systemCSTR Continuous-flow stirred tank reactorDSR Differential sidestream reactorIDEAS Infinite DimEnsionAl State-spaceLP Linear program or linear programmingMILP Mixed integer linear programMINLP Mixed integer nonlinear programPFR Plug flow reactorRCC Recursive constant control policy

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SECTION I

BASIC THEORY

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1INTRODUCTION

1.1 INTRODUCTION

This book is concerned with a field of study called attainableregions (ARs), which is a set of ideas intended to addressa generalized problem, often encountered in chemical reac-tor and process design. Although the problem can becomequite detailed if we allow it, the basic idea is simple to under-stand. This chapter serves to articulate the type of problemsAR theory could help address. To gain a sense of the scien-tific discipline that we are interested in, many (but not all) ofthe problems we are concerned with can be represented byFigure 1.1.

Assume that we are supplied feed material of a knownstate (i.e., composition, temperature, and flow rate) andasked to design a process that optimizes the production ofa desired product. The system in question is represented bya single block, a “black box.” This is a process that accepts afeed and converts it into something of higher economic orsocial value, chiefly via reactive unit operations. The blockcould be a batch or continuous process. In this book, wewill mainly focus on continuous reactors. Products fromthe block are given by a single effluent stream. In reality,this could come about from a number of intricate recyclesand bypass streams, occurring amongst a vast set of unitoperations within the block—that is, the physical processrepresented by the block in Figure 1.1 is likely to be verycomplex in reality.

For many problems, we might specifically wish to findthe best concentration of a desired component exiting thenetwork, or perhaps the total cost of equipment to meeta minimum level of production—at a minimum productquality—within the network. For now, we shall leavethese descriptions open so that we can describe the centralproblem that is related to them all.

A relevant question to ask in this context is the following:How do we best design the internals of the block given inFigure 1.1? Since we are specifically interested with reactiveprocesses, how do we best design a process involving

Attainable Region Theory: An Introduction to Choosing an Optimal Reactor, First Edition. David Ming, David Glasser, Diane Hildebrandt,Benjamin Glasser, and Matthew Metzger.© 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.Companion website: http://attainableregions.com

reaction? There is no assumption of a predefined designinside the block. We are concerned with the generalizeddesign of the internals, and not simply the optimization of aset design. (Some readers may recognize this as synthesis.)These questions often constitute what is referred to as thereactor network synthesis problem. Moreover, once we havesynthesized a design, how do we measure a “good” designfrom other competing designs? Although it may appear asthe former problem is our chief concern, it is actually thelatter problem that we interested in. Our goal in this bookis to understand how we find and measure “good” designsusing AR theory.

AR theory is hence a set of ideas in chemical reactordesign that aims to understand the reactor network synthesisproblem. But to understand this problem will ultimatelyrequire us to understand a broader problem of what it meansto be “the best,” for various designs within the block mayachieve the same outcome. This is our primary concern inAR theory. It is related to the reactor network synthesisproblem, but it is also distinct.

1.2 MOTIVATION

1.2.1 Toluene Production as a Case Study

To motivate why AR theory is useful, let us begin with astory, involving a team of three young engineers, who areinterested in the production of toluene. We would like to gaina sense of some of the typical design considerations encoun-tered by the team, which may also be addressed with AR the-ory. Sections 1.2.2–1.2.5 describe the team’s story, whereasSection 1.2.6 explains the story in relation to AR theory.Sections 1.3 and 1.4 are intended for readers who are alreadyfamiliar with the rudiments of chemical reactor design.

Sam, Alex, and Donald are promising young engineersworking for a large chemical company that produces toluene(C6H5CH3) as one of its products. One day, their boss asks

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4 ATTAINABLE REGION THEORY

Feed

(specified input)

Cin

Qin, Tin, etc.

Cout

Qout, Tout, etc.

Product

(desired output)

“black box”

reactor network

Figure 1.1 Overview of reactor network.

them to investigate how the company could improve thequality of their toluene product. Specifically, he asks:

How can we maximize the amount of toluene producedfrom a feed of 0.5 moles of ethylene and 1.0 moles ofbenzene?

Although there are potentially many unit operations thatmight improve the amount of toluene in the system, we aremainly interested in unit operations that involve reaction.The objective may be refined slightly to say that we wish tofind a reactive system that maximizes the amount of tolueneproduced.

The three engineers decide to first conduct a number ofexperiments in the company’s lab, as a way to gain a deeperunderstanding of the system of reactions. The discussionsgiven in the following text chronicle their experimental pro-cess, their discoveries, and ultimately their recommendationsin addressing this problem.

1.2.2 Part One: Initial Investigations

Having no prior knowledge or experience with toluene and itsproduction, Sam first consults one of the senior chemists inthe company, and gathers information regarding the reactionchemistry. The following set of reactions explain the system,including all significant products and by-products.

C6H6 +12

C2H4 ↔ C6H5(CH3) (1.1a)

Benzene+ 1∕2 ethylene↔ toluene

C6H5(CH3) +12

C2H4 ↔ C6H4(CH3)2 (1.1b)

Toluene+ 1∕2 ethylene↔ xylene

2C6H6 ↔ C6H5C6H5 + H2 (1.1c)

2 benzene↔ diphenyl+ hydrogenBenzene (B) reacts with ethylene (E) to form toluene (T).

Toluene and ethylene further react to form xylene (X). Atthe same time, benzene reacts to form two side-products ina competing side reaction: diphenyl (D) and hydrogen (H).This system may be classified as a series–parallel reaction,with toluene as an intermediate product, which shall here-after be referred to as the BTX system.

TABLE 1.1 Sam’s First Few Experiments

Experiment ReactionTime (h)

Toluene Concentration(mol/L)

1 5.0 0.00872 2.1 0.03733 1.1 0.05604 0.1 0.0361

For the purposes of the investigation, the BTX reaction isindependent of both temperature and pressure. It is assumedthat the toluene concentration inside the reactor is simple tomeasure, so that concentrations are easily recorded, and thatit is also possible to initiate and terminate the reaction easily.

Sam then begins her investigation in the simplest mannerpossible: by experimenting in a batch reactor (a beaker) in thelab, using an initial feed concentration of 0.5 mol/L ethyleneand 1.0 mol/L benzene.

She starts by carrying out the experiment for a number of(arbitrary) reaction times using the same initial conditionseach time, and then by recording the associated tolueneconcentration achieved after each run. Sam conducts fourexperiments in total, and a summary of her results isdisplayed in Table 1.1.

From the data, there appears to be an ideal reaction time,between 0.1 and 2.1 h, where the toluene concentration ismaximized. Using this insight, Sam carries out a series ofadditional experiments, attempting to isolate the exact reac-tion time that maximizes toluene concentration. In effect, herapproach is to use reaction time as a parameter to find themaximum toluene concentration.

After many experiments, Sam finds that a reaction time of35.4 min (0.59 h) produces the highest toluene concentration:0.0619 mol/L. Confident in her results, Sam reports back toher boss with the good news.

OBSERVATION: Initial investigations

Many of us tend to begin our investigations in a sim-ilar manner to Sam’s approach—with little previousknowledge or known insight, we simply begin withthat is known and then attempt to improve the currentperformance of the system using known parameters. Thisis a reasonable approach, but additional insight into thesystem might bring about further improvements.

1.2.3 Part Two: Iterative Improvement

1.2.3.1 Accidental Improvement Happy with Sam’sinitial results, the team decides to run the reaction on alarger scale. The reaction is predictable and simple tooperate, and they manage to consistently achieve the toluene

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INTRODUCTION 5

concentration expected with little complications. A standardoperating regime is eventually developed, involving foursteps: filling, reacting, emptying, and cleaning.

One day, whilst preparing the reactor for the followingday, Alex forgets to empty the entire contents of the reactorvessel, and a small portion of the reactor product is retainedin the reactor overnight. Alex enters the lab the next day,ready to complete a new set of batches, unaware of his mis-take the previous day. He fills the reactor with fresh feedmaterial, and runs a new batch.

To his surprise, when he expects to obtain the sametoluene concentrations as in the previous experiments, herealizes that the toluene concentration has changed. It hasincreased.

Indeed, his absentmindedness has turned out be some-thing of great interest—not only is it possible to adjust thereaction time of a particular batch, but it is also possible toadjust the starting concentration of the batch, using productfrom a previous run. With this vision, Alex proposes a newthree-step reaction procedure as follows:

1. First, running a fresh batch with feed material for arestricted period;

2. Then, retaining a small fraction of product;

3. Last, refilling the reaction vessel with fresh feed again,thereby altering the starting concentration of the nextbatch. This batch is then run to the desired exit con-centration.

Alex runs the reaction with the new operating procedure(for an arbitrary reaction time and mixture concentration) andfinds that he is able to achieve a toluene concentration of0.0652 mol/L in the following batch, which is an improve-ment over what Sam could achieve.

1.2.3.2 Experimenting with Different CombinationsNoting Alex’s achievement, Donald begins to experimentwith different retained fractions, in an attempt to find furtherimprovements. He begins with an arbitrary combination ofreaction times and mixing fraction, but he finds that some

Figure 1.2 Donald’s investigation, summarized in a matrix for themaximum toluene concentration achieved.

combinations produce worse results, whereas others offermoderate improvements. He writes down his results in amatrix format, which are illustrated in Figure 1.2.

The optimal combination from Donald’s finding suggeststhat 60% of the reactor product volume must be retained anda reaction time of 0.5 h must be used. This gives a tolueneconcentration of 0.0690 mol/L, or approximately 11% overSam’s investigation. Happy with this result, Sam, Alex, andDonald report back to their boss with the updated operatingprocedure.

OBSERVATION: Refining the approach

Although Donald, Sam, and Alex have managed toimprove the toluene concentration, finding improve-ments are becoming more complicated in general. Thisprocedure now involves finding an optimal combinationof parameters, as opposed to just one parameter withSam’s investigation.

1.2.4 Part Three: Coffee

One morning, Donald walks into the office kitchen to makea cup of coffee. He likes his coffee strong and black. He sitsthe cup down on the counter and accidentally brushes past anearby jug of milk. The jug knocks over and some of the milkspills into Donald’s cup, mixing with the coffee and changingthe color of its contents slightly.

Donald notes the change in the color of the coffee. Ratherthan discarding it and pouring a new cup, he realizes that hiscup is like a beaker, and the addition of milk has changedthe beaker’s contents. An interesting idea appears in Don-ald’s mind—could material be added into the toluene reactorduring reaction? He realizes that Sam and Alex have beenconstraining themselves to the operation of standard batchesonly. But the reaction could be carried out in a fed-batch reac-tor, such as in Figure 1.3.

Donald sets out to see if he can improve the tolueneconcentration using fed-batch reaction. He allows materialto flow into the reactor and mix with the reactor contentsfor the entire duration of the run, recording the maximumtoluene concentration achieved after each run. After manyexperiments, Donald believes that he has found the optimumtoluene concentration: 0.0761 mol/L; obtained by feedingat a constant rate of 1.0 L/h during reaction. This result isagain an improvement over previous methods.

SIDE NOTE: New options available

The introduction of a new mode of operation has openedup a wider choice of potential operating approaches topursue.

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6 ATTAINABLE REGION THEORY

Figure 1.3 Coffee and fed-batch reactors.

1.2.5 Part Four: Additional Improvements

The team believes that they are close to reaching a pointwhere no further improvements can be made. One day, Samforgets to switch off the reactor just after a batch has reacted.She turns off the feed to the fed-batch reactor but forgets toturn off the reactor itself, and the reaction proceeds with afeeding rate of zero.

To her surprise, an even higher toluene concentration isobtained. Sam has, in effect, created an operating sequenceinvolving the following two distinct reaction steps:

1. Reaction period involving fed-batch operation;

2. Reaction period of standard batch operation with noadditional feeding.

The team discovers that if the fed-batch portion of thesequence is allowed to run for 1.6 h with a feeding rateof 1.0 L/h, followed by period of no feeding for another15.6 min (0.26 h) as a standard batch, then a maximumtoluene concentration of 0.0793 mol/L may be achieved,which again is an improvement. A summary of all the team’sdiscoveries and recommendations for this investigation isprovided in Table 1.2.

The team has managed to improve the toluene concentra-tion by 28%: starting from 0.0619 mol/L, initially obtainedby a standard batch reactor, to 0.0793 mol/L, obtained by acombined fed-batch and batch sequence. Satisfied with theirprogress—as well as a little tired of having to do so many

experiments—Sam, Alex, and Donald report their findingsas final recommendations.

SIDE NOTE: Additional modifications

Note that concepts from Section 1.2.4 could also be incor-porated into this operating sequence. Depending on howlong we react, and how much product is mixed, the resultmight be improved even further. Each time a new dis-covery is found, many more combinations with existingprocedures can be explored.

1.2.5.1 A Change in Objective Sam, Alex, and Donald’sboss is so impressed with their findings; he proposes anotherproblem for them to tackle (one that the company has not yetbeen able to fully understand): instead of seeking to maxi-mize toluene concentration, what can be done to minimizethe production of hydrogen as a by-product?

Although the team has developed useful insights fromtheir toluene experiments, a new set of experiments mustnow be carried out for hydrogen, and they have little under-standing that could elicit an immediate recommendation forthis new objective. Their boss’ request leaves them feeling abit bewildered, and they walk away hoping that they will beas fortunate with their new investigation as they were withthe previous experiments.

TABLE 1.2 All Discoveries and Recommendations Made by Sam, Alex, and Donald

Toluene Concentration(mol/L)

Associated Method Used

Part One 0.0619 Standard batch reactor. Optimized for reactiontime.

Part Two 0.0690 Repeated batch with partial mixing. Optimizedfor reaction time and starting concentration.

Part Three 0.0761 Fed-batch reactor at 1.0 L/h. Optimized forfeeding rate.

Part Four 0.0793 Fed-batch for 1.6 h at 1.0 L/h, followed by astandard batch optimized for 15.6 min (0.26 h).