Modeling of Process Intensification 9783527311439, 3527311432

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Modeling of Process Intensification Edited by Frerich Johannes Keil

1807–2007 Knowledge for Generations Each generation has its unique needs and aspirations. When Charles Wiley first opened his small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity. And we were there, helping to define a new American literary tradition. Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused on building the future. Once again, we were there, supplying the critical scientific, technical, and engineering knowledge that helped frame the world. Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own borders and a new international community was born. Wiley was there, expanding its operations around the world to enable a global exchange of ideas, opinions, and know-how. For 200 years, Wiley has been an integral part of each generation’s journey, enabling the flow of information and understanding necessary to meet their needs and fulfi ll their aspirations. Today, bold new technologies are changing the way we live and learn. Wiley will be there, providing you the must-have knowledge you need to imagine new worlds, new possibilities, and new opportunities. Generations come and go, but you can always count on Wiley to provide you the knowledge you need, when and where you need it!

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Peter Booth Wiley Chairman of the Board

Modeling of Process Intensification Edited by Frerich Johannes Keil

The Editor Prof. Dr. Dr. h.c. Frerich J. Keil Hamburg University of Technology Institute of Chemical Reaction Engineering Eißendorferstr. 38 21073 Hamburg Germany

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at . © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfi lm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Composion SNP Best-set Typesetter Ltd., Hong Kong Printing

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Wiley Bicentennial logo: Richard J. Pacifico Printed in the Federal Republic of Germany Printed on acid-free paper ISBN 978-3-527-31143-9

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Contents

Preface XI List of Contributors XIII 1

Modeling of Process Intensification – An Introduction and Overview 1 Frerich J. Keil

2

Process Intensification – An Industrial Point of View 9 Robert Franke Introduction 9 Remarks on the Term Process Intensification 9 Management Aspects 10 Microreaction Technology 11 Principal Features 11 Catalytic Wall Reactors 12 Simulation 15 Introduction 15 Molecular Simulations 16 Quantum-chemical Calculations 16 COSMO-RS Calculations 18 Molecular-dynamics Calculations 19 Monte Carlo Simulations in Project Valuation under Risk

2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.2.1 2.3.2.2 2.3.2.3 2.3.3 3 3.1 3.2 3.2.1 3.2.2 3.2.3 3.3

Modeling and Simulation of Microreactors 25 Steffen Hardt Introduction 25 Flow Distributions 30 Straight Microchannels 30 Periodic and Curved Channel Geometries 32 Multichannel Flow Domains 34 Heat Transfer 39

Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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Contents

3.3.1 3.3.2 3.3.3 3.3.4 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.5 3.5.1 3.5.2 3.5.3 3.6

Straight Microchannels 39 Periodic and Curved Channel Geometries 41 Multichannel Flow Domains 43 Micro Heat Exchangers 47 Mass Transfer and Mixing 50 Simple Mixing Channels 51 Chaotic Micromixers 53 Multilamination Micromixers 59 Hydrodynamic Dispersion 62 Chemical Kinetics 66 Numerical Methods for Reacting Flows 66 Reacting Channel Flows 68 Heat-exchanger Reactors 71 Conclusions and Outlook 74

4

Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors 79 Rüdiger Lange Introduction 79 Modeling 82 Reactor Model 85 Simulation 88 Conclusion 93

4.1 4.2 4.3 4.4 4.5 5

5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5

Packed-bed Membrane Reactors 99 Ákos Tóta, Dzmitry Hlushkou, Evangelos Tsotsas, and Andreas Seidel-Morgenstern Introduction 99 The PBMR Principle 101 Case Study 101 Porous Membranes 102 Outline 103 One-dimensional Modeling of Packed-bed Membrane Reactors 103 One-dimensional Pseudohomogeneous Model 103 Cofeed (FBR) vs. Distributed Dosing of Reactants (PBMR) – Nonreactive Conditions 105 Comparison between FBR and PBMR – Reactive Conditions 107 Nonisothermal Operation 111 Two-dimensional Modeling of Packed-bed Membrane Reactors 114 Two-dimensional Model of PBMR – The Momentum-balance Equation 115 Two-dimensional Model of PBMR – The Mass-balance Equation 117 Two-dimensional Model of PBMR – The Energy-balance Equation 118 Boundary Conditions 119 Numerical Solution of the Two-dimensional Model 121

Contents

5.3.6 5.3.7 5.3.8 5.3.9 5.3.10 5.4 5.4.1 5.4.2 5.4.3 5.5 6

6.1 6.2 6.3 6.3.1 6.3.2 6.4 6.5 6.6 6.6.1 6.7 7

7.1 7.2 7.3 7.3.1 7.3.2 7.4 7.4.1 7.4.2 7.4.3 7.5 7.5.1 7.6

Velocity Field in a Packed-bed Membrane Reactor 122 The Influence of Membrane Permeability on the Boundary Conditions 123 Effect of Porosity Profi le 126 Effect of Radial Mass-transport Limitations 127 Comparison of the l(r)- and aw-model Concepts – Temperature Profi les in a PBMR 129 Three-dimensional Modeling of a Packed-bed Membrane Reactor 132 Introduction to the Large-scale Simulation Methods in Fluid Mechanics and Mass Transport 132 Pressure and Velocity Field (Varying the Flow Distribution) – Comparison between FBR and PBMR 133 Advective-diffusive Mass Transport in PBMR 137 Summary and Conclusion 140 The Focused Action of Surface Tension versus the Brute Force of Turbulence – Scaleable Microchannel-based Process Intensification using Monoliths 149 Michiel T. Kreutzer, Annelies van Diepen, Freek Kapteijn, and Jacob A. Moulijn Introduction 149 Monoliths – A Scalable Microchannel Technology 150 Power Required for Gas–Liquid Dispersion 152 Turbulent Contactors 152 Laminar Contactors 154 Physical Adsorption of Oxygen 156 Three-phase Processes 158 Discussion 160 Design Considerations for Intensified Monolith Processes 160 Conclusions 161 Chemical Reaction Modeling in Supercritical Fluids in Special Consideration of Reactions in Supercritical Water 165 Andrea Kruse and Eckhard Dinjus Introduction 165 Properties of Supercritical Fluids 167 The C—C-bond Splitting of Tert.-butylbenzene 169 Experimental 170 Modeling 170 Total Oxidation in Supercritical Fluids 173 Is Oxidation in CO2 the same as Oxidation in Water? 175 Experimental 176 Modeling 177 Glycerol Degradation 179 Modeling 180 Conclusion 186

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VIII

Contents

8 8.1

8.1.1 8.1.2 8.1.2.1 8.1.2.2 8.1.2.3 8.1.2.4 8.1.2.5 8.1.2.6 8.1.2.7 8.1.2.8 8.1.2.9 8.1.3 8.1.4 8.1.4.1 8.1.4.2 8.1.5 8.1.6 8.2 8.2.1 8.2.2 8.2.2.1 8.2.2.2 8.2.2.3 8.2.2.4 8.2.3 8.2.4 8.2.4.1 8.2.4.2 8.2.5 8.2.5.1 8.2.5.2 8.2.5.3 8.2.6 8.2.6.1 8.2.7 8.2.8

Ultrasound Reactors 193 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors 193 Christian Horst Introduction 193 Bubble Behavior in Acoustic Fields 196 Equations for the Motion of the Bubble Wall 196 Cavitation Thresholds 200 Influence of Parameters on Cavitation Behavior 202 Cavitation near Solid Boundaries 204 Finite Amplitude Waves and Shock Waves 205 Streaming 206 Bjerknes Forces 206 Forces on Small Particles 207 Sonochemical Effects 207 Modeling of Sound Fields 209 Examples of Sound Fields in Ultrasound Reactors 214 High-amplitude–High Energy Density Conical Reactor 214 Low-amplitude–High Energy Density Crevice Reactor 217 Modeling of Sonochemical Effects in Ultrasound Reactors 218 Summary 225 Design of Cavitational Reactors 226 Parag R. Gogate and Aniruddha B. Pandit Introduction 226 Theoretical Approach 228 Identification of Stable or Transient Cavitation 231 Effect of Compressibility of Medium 233 Optimization of Operating Parameters 235 Development of Design Equations 235 Distribution of the Cavitational Activity in the Reactors 240 Intensification of Cavitational Activity in the Sonochemical Reactors 247 Use of Process-intensifying Parameters 249 Use of Combination of Cavitation and Advanced Oxidation Processes 250 Design of a Pilot-scale Reactor and its Experimental Evaluation 252 Rational for the Design of a Reactor 252 Actual Design of the Novel Reactor 253 Comparison of the Efficacy of the Hexagonal Flow-cell Reactor with Conventional Designs 254 Hydrodynamic Cavitation Reactors 257 Engineering Design of Hydrodynamic Cavitation Reactors 258 Comparison of Cavitational Yields in Acoustic and Hydrodynamic Cavitation 268 Qualitative Considerations for Reactor Choice, Scaleup and Optimization 268

Contents

271

8.2.9 8.2.10

Efforts Needed in the Future Concluding Remarks 272

9

Modeling of Simulated Moving-bed Chromatography 279 Monika Johannsen Introduction 279 From Elution Chromatography to Countercurrent Chromatography 280 The TMB Process 281 The SMB Process 282 Applications of SMB Chromatography 284 Modeling of Chromatographic Processes 287 Modeling of Single-column Chromatography 288 Ideal Model 290 Dispersive Model 291 Equilibrium Dispersive Model 292 Linear Driving Force (LDF) Model 293 General Rate Model 294 Initial and Boundary Conditions for Single-column Chromatography 295 Modeling of SMB Chromatography 296 Model Classification 296 Modeling of the TMB Process 297 Modeling of the SMB Process 299 Solution of the TMB Model with the Triangle Theory 299 Consideration of Different Adsorption Coefficients in the Zones 303 Modeling of Pressure Gradient in SFC 306 Numerical Algorithms 307 Simulations of SMB-SFC Chromatography 308 Separation of Cis/trans-Isomers of Phytol 308 Separation of Enantiomers of (R,S)-Ibuprofen 311 Separation of Tocopherols 314 Conclusion 317

9.1 9.2 9.2.1 9.2.2 9.3 9.4 9.4.1 9.4.1.1 9.4.1.2 9.4.1.3 9.4.1.4 9.4.1.5 9.4.1.6 9.4.2 9.4.2.1 9.4.2.2 9.4.2.3 9.4.2.4 9.4.2.5 9.4.2.6 9.4.3 9.5 9.5.1 9.5.2 9.5.3 9.6 10 10.1 10.2 10.2.1 10.3 10.3.1 10.3.2 10.3.3 10.3.3.1 10.3.3.2 10.3.3.3

Modeling of Reactive Distillation 323 Eugeny Y. Kenig and Andrzej Górak Introduction 323 Characteristics of Reactive Distillation 324 Column Internals for Reactive Distillation 324 Modeling Principles of Reactive Distillation 326 General Aspects 326 Equilibrium-stage Model 328 Rate-based Approach 329 Balance Equations 330 Mass Transfer and Reaction Coupling in the Fluid Film 332 Nonideal Flow Behavior in Catalytic Column Internals 333

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Contents

10.3.3.4 10.4 10.4.1 10.4.1.1 10.4.1.2 10.4.1.3 10.4.2 10.4.2.1 10.4.2.2 10.4.2.3 10.4.3 10.4.3.1 10.4.3.2 10.4.3.3 10.4.4 10.4.4.1 10.4.4.2 10.4.4.3 10.5

Dynamic Modeling 336 Case Studies 336 Methyl Acetate Synthesis 336 Process Description 336 Process Modeling 338 Results and Discussion 339 Methyl Tertiary Butyl Ether 341 Process Description 341 Process Modeling 341 Results and Discussion 342 Ethyl Acetate Synthesis 343 Process Description 343 Process Modeling 347 Results and Discussion 347 Dymethyl Carbonate Transesterification 350 Process Description 350 Process Modeling 351 Results and Discussion 352 Conclusions and Outlook 355

11

Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields in Chemical Multiphase Processes 365 Faïçal Larachi Background 365 Nonmagnetic Fluids 366 Magnetic Fluids 367 Nonmagnetic Fluids 368 Principle 368 Theory 370 Experimental Results and Discussion 371 Magnetic Fluids 378 Principle 378 Theory 380 Local Description 380 Upscaling 381 Closure Problem 384 Zero-order Axisymmetric Volume-average Model 386 Results and Discussion 386 Concluding Remarks 394

11.1 11.1.1 11.1.2 11.2 11.2.1 11.2.2 11.2.3 11.3 11.3.1 11.3.2 11.3.2.1 11.3.2.2 11.3.2.3 11.3.2.4 11.3.3 11.4

Index

401

XI

Preface Process intensification has been on the upswing since a review written by A. Stankiewicz and J. Moulijn was issued in 2000. Meanwhile, companies and academia are addressing problems in process intensification, organizing workshops and even establishing departments on this subject. Process intensification is a very broad discipline and includes expertise in many diverse fields. It is applied to the development of novel apparatuses and techniques that either dramatically improve chemical or biological processes with respect to reduced equipment size, increased energy efficiency, less waste production, improved inherent safety, or even break new ground in process engineering by introducing newly developed equipment and production procedures. The present book focuses on modeling in process intensification. Experts in various areas of process intensification, from both industry and academia, have contributed to this book, which does not cover all the developments in this field; rather it demonstrates the activities in modeling for some representative problems. New equipment like microreactors, membrane reactors, ultrasound reactors, and those in simulated moving-bed chromatography, magnetic fields in multiphase processes or reactive distillation, requires new modeling approaches. The same applies to nonstationary process operation or the use of supercritical media. Process intensification is an emerging discipline that will result in many surprising developments in the future. The editor is grateful to all the authors who contributed to this volume, and to Dr. Rainer Muenz from Wiley-VCH. F. J. Keil Hamburg, January 2007

Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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List of Contributors Eckhard Dinjus Forschungszentrum Karlsruhe Institut für Technische Chemie P.O. Box 36 40 76021 Karlsruhe Germany Robert Franke Degussa AG Creavis Technologies & Innovation Project House Process Intensification Rodenbacher Chaussee 4 63457 Hanau Germany Parag R. Gogate Institute of Chemical Technology Chemical Engineering Department Matunga, Mumbai-400019 India Andrzej Gorak Universität Dortmund Lehrstuhl für Thermische Verfahrenstechnik FB Chemietechnik 44221 Dortmund Germany

Steffen Hardt Leibniz Universität Hannover Institut für Nano- und Mikroprozesstechnik Callinstr. 36 30167 Hannover Germany Dzmitry Hlushkou Otto von Guericke University Magdeburg Institute of Process Engineering Chair of Thermal Process Engineering P.O. Box 4120 39106 Magdeburg Germany Christian Horst Infraserv Gendorf 84504 Burgkirchen Germany Monika Johannsen Technische Universität Hamburg-Harburg Institut für Thermische Verfahrenstechnik Eißendorferstraße 38 21073 Hamburg Germany

Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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List of Contributors

Freek Kapteijn University of Technology Reactor/Cat. Engineering Group Julianalaan 136 2628 BL Delft The Netherlands Eugeny Y. Kenig Universität Dortmund Lehrstuhl für Thermische Verfahrenstechnik FB Chemietechnik 44221 Dortmund Germany Frerich J. Keil Hamburg University of Technology Institute of Chemical Reaction Engineering Eißendorferstraße 38 21073 Hamburg Germany Michiel T. Kreutzer University of Technology Reactor/Cat. Engineering Group Julianalaan 136 2628 BL Delft Netherlands Andrea Kruse Forschungszentrum Karlsruhe Institut für Technische Chemie P.O. Box 36 40 76021 Karlsruhe Germany Rüdiger Lange Technische Universität Dresden Verfahrentechnik / Zi. A208 Münchener Platz 3 01062 Dresden Germany

Faical Larachi Laval University Department of Chemical Engineering Cité Universitaire Avenue de la Médecine Pouliot Building G1K 7P4 Quebec Canada Jacob A. Moulijn University of Technology Reactor/Cat. Engineering Group Julianalaan 136 2628 BL Delft The Netherlands Andreas Seidel-Morgenstern Otto von Guericke University Magdeburg Institute of Process Engineering Chair of Chemical Process Engineering P.O. Box 4120 39106 Magdeburg Germany Aniruddha B. Pandit Chemical Engineering Department Institute of Chemical Technology Matunga, Mumbai-400019 India Ákos Tóta Otto von Guericke University Magdeburg Institute of Process Engineering Chair of Thermal Process Engineering P.O. Box 4120 39106 Magdeburg Germany

List of Contributors

Evangelos Tsotsas Otto von Guericke University Magdeburg Institute of Process Engineering Chair of Chemical Process Engineering P.O. Box 4120 39106 Magdeburg Germany

Annelies van Diepen University of Technology Reactor/Cat. Engineering Group Julianalaan 136 2628 BL Delft The Netherlands

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1

1 Modeling of Process Intensification – An Introduction and Overview Frerich J. Keil

As noted by Hüther et al. [1], the term “process intensification” (PI) was probably first mentioned in the 1970s by Kleemann et al. [2] and Ramshaw [3]. Ramshaw, among others, pioneered work in the field of process intensification. What does “process intensification” (PI) mean? Over the last two decades, different definitions of this term were published. Cross and Ramshaw defined PI as follows: “Process intensification is a term used to describe the strategy of reducing the size of chemical plant needed to achieve a given production objective” [4]. In a review of PI, Stankiewicz and Moulijn [5] proposed: “Any chemical engineering development that leads to a substantially smaller, cleaner, and more energyefficient technology is process intensification”. The BHR Group describes PI as follows [6]: “Process Intensification is a revolutionary approach to process and plant design, development and implementation. Providing a chemical process with the precise environment it needs to flourish results in better products, and processes which are safer, cleaner, smaller, and cheaper. PI does not just replace old, inefficient plant with new, intensified equipment. It can challenge business models, opening up opportunities for new patentable products and process chemistry and change to just-in-time or distributed manufacture”. To bring forward PI, Degussa established a so-called “project house” whose research activities are focused on PI. Degussa expanded the meaning of the concept “process intensification”: “Process intensification defines a holistic approach starting with an analysis of economic constraints followed by the selection or development of a production process. Process intensification aims at drastic improvements of performance of a process, by rethinking the process as a whole. In particular it can lead to the manufacture of new products which could not be produced by conventional process technology. The process-intensification process itself is “constantly financially evaluated” [1, 7]. As can be recognized from the above definitions, process intensification is a developing field of research and far away from a mature status. The chemical industry and academia are very interested in PI developments. For example, some German chemical engineering associations (DECHEMA, VDI-GVC) established a subject division on process intensification, which has already more than 180 members. In the opening session of this division Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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1 Modeling of Process Intensification – An Introduction and Overview

several sceptical questions arose, like: “Are any new options offered by PI which are not known already from other fields of chemical engineering, e.g. optimization or process integration?” “How large should be the improvement of a process for PI?” “What is the difference between the aims of PI and neighbouring disciplines?” [8]. There is an agreement that PI is an interdisciplinary field of research that needs an integrated approach. In PI, the journey is the reward. PI has inspired already many new developments of equipment, processintensifying methods and design approaches. As thermodynamic equilibrium and reaction kinetic properties are fi xed values for given mixtures under fi xed conditions like temperature, pressure and catalysts, most efforts were directed towards the improvement of transport properties, alternative energy resources, and process fluids. Examples of new equipment are the Sulzer SMR static mixer, which has mixing elements made of heat-transfer tubes, Sulzer’s open-crossflowstructure catalysts, so-called Katapaks, monolithic catalyst supports covered with washcoat layers, microreactors, ICI’s High Gravity Technology (HIGEE), HIGRAVITEC’s rotating packed beds, centrifugal adsorbers made by Bird engineering, BHR’s improved mixing equipment and HEX reactors, high-pressure homogenizers for emulsifications, the spinning-disc reactor (SDR) developed by Ramshaw’s group at Newcastle University, and the supersonic gas/liquid reactor developed by Praxair Inc. (Danbury). Various ultrasonic transducers and reactors are now commercially available. The efforts in PI have been compiled in several books [9–14]. A general introductory paper was presented by Stankiewicz and Moulijn [5]. Process intensification by miniaturization has been reviewed by Charpentier [15]. Jachuck [16] reviewed PI for responsive processing. Other subjects related to process intensification have also been reviewed, for example, trickle-bed reactors [17], multifunctional reactors [18], rotating packed beds [19], multiphase monolith reactors [20], heat-integrated reactors for high-temperature millisecond contact-time catalysis [21], microengineered reactors [22, 23], monoliths as biocatalytic reactors [24], membrane separations [25], two-phase flow under magnetic-field gradients [26], and applications of ultrasound in membrane separation processes [27]. In Fig. 1.1 an overview of equipment and methods employed in PI is presented. PI leads to a higher process flexibility, improved inherent safety and energy efficiency, distributed manufacturing capability, and ability to use reactants at higher concentrations. These goals are achieved by multifunctional reactors, e.g. reactive distillation or membrane reactors, and miniaturization that can be done by employing microreactors and/or improving heat and mass transfer. Microfluidic systems enable very high heat- and mass-transfer rates so that reactions can be executed under more severe conditions with higher yields than conventional reactors. New reaction pathways, for example, direct fluorination of aromatic compounds, are possible, and scaleup of reactors is easier. This feature may enhance instationary reactor operation, like reverse flow, in industrial applications. These are just a few examples. Intensification of heat and mass transfer can be achieved by using supersonic flow, strong gravitational magnetic fields, improved mixing, among other ap-

1 Modeling of Process Intensification – An Introduction and Overview

Figure 1.1 Tools of Process Intensification.

proaches. For example, the spinning-disc reactor technology utilizes the effects of high centrifugal force, which is capable of producing highly sheared films on the surface of rotating discs/cones. Convective heat-transfer coefficients as high as 14 kW m−2 K and mass transfer coefficients, K L values as high as 30 × 10 −5 m s−1 and KG values as high as 12 × 10 −8 m s−1, can be achieved whilst providing micromixing and an appropriate fluid dynamic environment for achieving faster reaction kinetics. A further possibility is offered by external magnetic fields that exert a body force on electrically nonconducting magnetically permeable fluids, and this force can be used to compensate or to amplify the gravitational body force, which can be employed to influence two-phase flow in, for example, trickle-flow reactors. Ultrasound can either be used for enhancing mass transfer or reaction engineering [28]. Cavitation generates conditions of locally very high temperatures (>6000 ºC) and pressure (>10 000 bars) along with the release of active radicals, which results in intensification of many of the physical and chemical transformations. Sonochemistry opened new possibilities for chemical synthesis. Sonoreactors can be thought of as high-energy microreactors. A compilation of chemistry and extreme and nonclassical conditions was edited by van Eldik and Hubbard [29]. The present book reviews recent developments in modeling of process intensification. It is divided into eleven chapters. After an introduction and overview, Robert Franke from Degussa AG describes in Chapter 2 the efforts on PI from an industrial point of view in their “project house”. A special feature is the use

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1 Modeling of Process Intensification – An Introduction and Overview

of molecular simulations on various levels, like quantum chemistry, and classical molecular dynamics or Monte Carlo simulations. For liquid/liquid equilibria COSMO/RS is in use. Cash-flow analysis and project valuation under risk are investigated by Monte Carlo approaches. Chapter 3 has been written by Steffen Hardt from Darmstadt University of Technology on modeling and simulation of microreactors. Flow distributions and heat transfer in various microchannels are described. Fast mass transfer and mixing are key aspects of microreactors. Modeling of micromixers is discussed in detail. The chapter is completed by a review of reacting flows in microchannels. The discussion of modeling and simulation techniques for microreactors shows that the toolbox available at present is quite diverse and goes well beyond the standard capabilities of CFD methods available in commercial solvers. Most of the effects are described by the standard continuum equations, but there are a number of problems that are extremely difficult, and require very fine computational grids. Among these problems is the numerical study of mixing in liquids that often severely suffers from discretization artefacts. Chapter 4 by Rüdiger Lange from the University of Technology Dresden is on modeling and simulation of unsteady-state operated trickle-flow reactors. The behavior of these three-phase reactors is rather complex due to cocurrent flow of gas and liquid downward through a catalyst packing. Periodic change of reactantfeed concentration and/or volumetric flow rate are suitable for a considerable improvement of reaction conversion. A review of unsteady-state operated trickle-flow reactors is presented and a dynamic reactor model developed by Lange’s group, based on an extended axial dispersion model, is described in detail. Andreas Seidel-Morgenstern’s group from the Max-Planck-Institute in Magdeburg presents in Chapter 5 an extensive review of packed-bed membrane reactors (PBMR), and analysis of the properties of this type of reactors by means of models developed in their group. In contrast to conventional tubular fi xed-bed reactors, where all reactants together are fed into the reactor inlet, packed-bed membrane reactors allow one or several reactants to be dosed via membranes over the reactor wall along the axial reactor coordinate. Computational results, based on realistic data originating from the important class of partial oxidation reactions, are presented. The oxidative dehydrogenation of ethane to ethylene using a vanadium oxide catalyst was considered. Different membrane permeabilities were studied in the range of currently available porous materials. Investigations by a twodimensional reactor model revealed that flow maldistribution, caused by increased bed porosity close to the membrane wall, leads to local temperature profi les that result in performance predictions different from an integral reactor. Results of a three-dimensional model using the lattice Boltzmann method are also presented. In Chapter 6 Jacob Moulijn’s group from the Delft University of Technology discusses the advantages and disadvantages of using segmented flow in microchannels to intensify catalytic processes. Once bubbles are formed in microchannels, they can no longer coalesce, and hence no energy is required to break up larger bubbles. As a result, the same gas–liquid mass-transfer behavior can be

1 Modeling of Process Intensification – An Introduction and Overview

obtained at an order of magnitude lower power input. This flow pattern can be used for biochemical conversions using cell cultures, provided the channels are not too small, and the operating conditions are such that biofi lm formation is suppressed. If the segmented flow pattern is used for a reaction catalyzed at the walls of the capillary channels, then the mass transfer is actually improved by reducing the amount of energy that is dissipated in the system. This allows the simultaneous achievements of two goals of process intensification: reduction of energy requirement and reduction of equipment size. Chapter 7 focuses on chemical-reaction modeling in supercritical fluids, in particular in supercritical water. This contribution is from Eckhard Dinjus’ group at the Research Center Karlsruhe. The contribution gives detailed presentations of modeling of systems by elementary reactions and their reaction engineering. Chapter 8 consists of two parts. The first contribution by Christian Horst explains some fundamentals of cavitation and its modeling applied to a so-called “High Energy Density Crevice Reactor”. The Grignard reaction of chlorobutane isomers was used as an example. The sound field inside sonochemical reactors can be modeled by treating the liquid bubble mixture as a pseudofluid. The Kirkwood–Bethe–Gilmore equations were used to calculate the bubble motions of bubbles with different sizes. Knowing the bubble-size distribution at a given sound pressure by calculating cavitation thresholds and using this information in an equation for the local total bubble number, the calculation of the complex bulk modulus of the bubbly mixture is possible. The resulting sound velocities and the damping coefficients can be used for calculating the sound field by finiteelement codes. The simulation results have been employed to optimize reactor geometries and to interpret some surprising effects. Sonochemical effects for Grignard reactions were also modeled. The second contribution, written by Pareg Gogate and Anniruddha Pandit from the Institute of Chemical Technology in Mumbai, stresses important factors for efficient scaleup of cavitational reactors and subsequent industrial applications based on the theoretical and experimental analysis of the net cavitational effects. Guidelines for selection of an optimum set of operating parameters have been presented and hydrodynamic cavitation has also been discussed. Chapter 9 on simulated moving-bed chromatography has been written by Monika Johannsen at Hamburg University of Technology. Simulated moving-bed (SMB) chromatography is a powerful purification process allowing the continuous separation of a feed mixture into two product streams. Most of these separations are performed using liquid chromatography. The complexity of preparative chromatography results in highly complex models for the quantitative description compared to analytical chromatography. The models are based on the adsorption theory and the theory of nonlinear chromatography. Various simulation tools for the SMB technique have been developed, which can be used for optimization of the column length, column configuration, flow rates, feed concentration, and switch times. Applications of SMB chromatography and modeling of this process are reviewed.

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1 Modeling of Process Intensification – An Introduction and Overview

Chapter 10 reviews modeling of reactive distillation. This contribution is written by Kenig and Gorak at the University of Dortmund (Germany). In reactive distillation, reaction and distillation take place within the same zone of a distillation column. Reactants are converted to products with simultaneous separation of the products and recycle of unused reactants. The process basics and peculiarities are discussed in detail. Up-to-date applications, reactive distillation modeling and design issues are presented. The theoretical description is illustrated by several case studies and supported by the results of laboratory-, pilot- and industrialscale experimental investigations. Both, steady-state and dynamic issues are treated, and the design of column internals is addressed. An outlook on future research requirements is given. Faïçal Larachi from the Laval University in Quebec (Canada) presents in Chapter 11 experimental and theoretical investigations on artificial gravity (micro- or macrogravity) generated by strong gradient magnetic fields that could potentially open up attractive applications, especially in multiphase catalytic systems where a number of factors can be optimized in an original manner for improving process efficiency. For example, inhomogeneous and strong fields applied to tricklebed reactors are capable of affecting their hydrodynamics. Liquid holdup can be improved that results in better contacting between a liquid and a catalyst surface. Additionally, a theoretical framework is developed based on the application of the volume-averaging theorems in multiphase porous media to analyze the flow of ferrofluids in a special class of porous media presenting pronounced effects of wall-bypass flows. Limitations of the present models are discussed. To sum up, process intensification is a rapidly developing field that has already inspired many ideas in modeling and design of new equipment and operating modes, and whose potential is by far not fully tapped.

References 1 Hüther, A.; Geißelmann, A.; Hahn, H.: Chem. Ing. Tech., 77 (2005), 1829 2 Kleemann, G.; Hartmann, K.: Wiss. Z. Tech. Hochschule “Carl Schorlemmer”, Leuna Merseburg, 20 (1978), 417 3 Ramshaw, C.: Chem. Eng. (London), 389 (1983), 13 4 Cross, W.T.; Ramshaw, C.: Chem. Eng. Res. Des., 64 (1986), 293 5 Stankiewicz, A.I.; Moulijn, J.A.: Chem. Eng. Prog., 96 (2000), 22 6 BHR Group: www.bhrgroup.com/pi/ aboutpi.htm 7 see Chapter 2, this book 8 Machhammer, O.: Chem. Ing. Tech., 77 (2005), 1635 9 Stankiewicz, A.I.; Moulijn, J.A. (Eds.): Re-engineering the chemical processing

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plant: process intensification, Marcel Dekker, New York (2004) Hessel, V.; Hardt, S.; Löwe, H.: Chemical Micro Process Engineering, Wiley-VCH, Weinheim (2004) Marcano, J.G.S.; Tsotsis, T.T.: Catalytic Membranes and Membrane Reactors, Wiley-VCH, Weinheim (2002) Koch, M.V.; VandenBussche, K.M.; Christman, K.M.: Micro Instrumentation in High Throughput Experimentation and Process Intensification, Wiley-VCH, Weinheim (in preparation, January 2007) Wang, Y.; Holladay, J.D. (Eds.): Microreactor Technology and Process Intensification (ACS Symposium Series No. 914), ACS Publ. (2005)

References 14 Jachuck, R. (Ed.): Process Intensification in the Chemical and Related Industries, Blackwell, Oxford (in preparation) 15 Charpentier, J.-C.: Chem. Eng. Technol., 28 (2005), 255 16 Jachuck, R.: Chem. Eng. Res. Des., 80 (2002), 233 17 Nigam, K.D.P.; Larachi, F.: Chem. Eng. Sci., 60 (2005), 5880 18 Agar, D.: Chem. Eng. Sci., 54 (1999), 1299 19 Burns, J.R.; Jamil, J.N.; Ramshaw, C.: Chem. Eng. Sci., 55 (2000), 2401 20 Kreutzer, M.T.; Kapteijn, F.; Moulijn, J.A.; Heiszwolf, J.: Chem. Eng. Sci., 60 (2005), 5895 21 Liu, T.; Gepert, V.; Veser, G.; Chem. Eng. Res. Des., 83 (2005), 611 22 Gavriilidis, A.; Angeli, P.; Cao, E.; Young, K.K., Wan, Y.S.S.: Eng. Chem. Res. Des., 80 (2002), 3

23 Matlosz, M.: Chem. Ing. Tech., 77 (2005), 1393 24 Kreutzer, M.T.; Kapteijn, F.; Moulijn, J.; Ebrahimi, S.; Kleerebezem, R.: Ind. of Eng. Chem. Res., 44 (2005), 9646 25 Drioli, E.; di Profio, G.; Fontananova, E.: Fluid/Particle Separ. J., 16 (2004), 1 26 Iliuta, J.; Larachi, F.: Can. J. Chem. Eng. 81 (2003), 776 27 Muthukumaram, S.; Kentish, S.E.; Stevens, G.W.; Ashokkumar, M.: Rev. Chem. Eng., 22 (2006), 155 28 Keil, F.J.; Swamy, K.M.: Rev. Chem. Eng., 15 (1999), 85 29 van Eldik, R.; Hubbard, C.D. (Eds.): Chemistry under Extreme and Nonclassical Conditions, Wiley & Spektrum Akademischer Verlag, New York, Heidelberg (1997)

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2 Process Intensification – An Industrial Point of View Robert Franke

2.1 Introduction

Process intensification aims at drastic improvements in performance of a process by rethinking the process as a whole. Modeling of process intensification relies partly on well-established engineering software tools that include process simulators and codes on computational fluid dynamics. However, process intensification goes beyond process optimization making the modeling tools – or the modeling Ansätze generally – much more versatile. This chapter will begin with a brief discussion of process intensification from the conceptual and management points of view. In the second section we will report on process-intensification strategies for gas-phase processes based entirely on relatively simple model calculations. Finally we will discuss molecular and financial simulation approaches that are important in the practical work on process-intensification projects.

2.1.1 Remarks on the Term Process Intensification

The definition of process intensification is still an ongoing process. In 1986 Cross and Ramshaw offered the definition: “Process intensification is a term used to describe the strategy of reducing the size of chemical plant needed to achieve a given production objective” [1]. In 2000 Stankiewicz and Moulijn proposed: “Any chemical engineering development that leads to a substantially smaller, cleaner, and more energy-efficient technology is process intensification” [2]. These definitions restrict process intensification to engineering methods and equipment and explicitly exclude the development of a new chemical route or the change in composition of a catalyst (see, e.g., [2]). We are convinced that these narrow definitions are inadequate for the description of process-intensification projects in industry. Following Hüther et al. [3] our working definition is: process intensification defines a holistic approach starting with an analysis of economic constraints followed by the selection or development of a production process. As Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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stated above, process intensification thus aims at drastic improvements of performance of a process, by rethinking the process as a whole. In particular, it can lead to the manufacture of new products which could not be produced by conventional process technology. The process-intensification process itself is constantly financially evaluated. In view of this definition, modeling in process intensification also includes the employment of methods such as computer-assisted organic synthesis or financial modeling software suites. 2.1.2 Management Aspects

A central precondition of process intensification is creative thinking. Creativity is an active process by which a genuinely new idea takes form. Creativity needs to be distinguished from innovation. In industry, an innovation is the introduction of new products or application processes or services to the market. It characterizes the generation of new ideas in order to produce a new product, process or service. Creativity is required if innovation is to be possible. In industrial R&D creativity is usually a product of prior knowledge and thinking, but truly creative acts in science are mostly unplanned. This does not mean that they happen by chance but that they are quite unpredictable and not guaranteed. Although exceptions may exist, significant inventions are usually the results of many years of constant research in a certain field and the accumulation of vast amounts of data. An enormous number of publications exist that discuss how to create an environment for creativity and innovation (see, e.g., [4] and references therein). Since R&D in industry takes place almost exclusively in teams, team work is the key for running a successful research process. Two types of teams seem to be particularly suitable for creative R&D work: crossfunctional and multidisciplinary teams. Crossfunctional teams consist of members who have different functions and specialisms, for instance in research, marketing, controlling and production. In these teams expertise in developing and launching new products or technologies are linked. Multidisciplinary teams are especially suited to developing new technologies by pooling the expertise of, for instance, chemists, physicists and engineers. The organization of R&D addressed to process intensification can be briefly illustrated by considering Degussa’s approach where it is realized in the form of a so-called “project house”. Project houses have been developed by Degussa for multiunit research. They are a part of Creavis Technologies & Innovation, the Degussa R&D unit that attends to crosscompany and crossportfolio research activities to develop new fields of business and technologies. Project houses are formed for a limited time period of three years with the participation of several Degussa business units and Creavis and are linked into internal and external research networks. They include multidisciplinary project teams whose members are recruited from different business and service units. Since process intensification addresses the drastic improvement of a process using novel technologies by rethinking processes as a whole, knowledge of actually existing and realized

2.2 Microreaction Technology

technologies and products must be brought together with special knowledge in novel equipment and process technology. The concept of project houses is ideally tailored to these requirements. Sustainability is guaranteed because at the end of a project house’s existence, either the business units market the results generated in the form of new products and processes or (in cases where concepts cannot be developed within the business units) internal start-ups for business development are founded.

2.2 Microreaction Technology 2.2.1 Principal Features

Microreaction technology is based on chemical microprocessing that is characterized by a continuous flow of matter through well-defined structures with dimensions that characteristically lie in the interval 0.1–1000 µm. When comparing microreaction technology with (conventional) reaction technology the distinguishing phenomena are usually described as “microeffects” or “scaling effects”. Important microeffects are the strongly intensified heat and mass transport in directions of the small lateral dimensions of the apparatus due to diffusion and heat conduction. These are no longer negligible in comparison to convection and are responsible for drastically enhanced heat and mass transfer. Both result in specific advantages over conventional process technology. For example, the usage of microreactors prevents the occurrence of hot spots due to the greatly improved heat dissipation. This makes possible higher reactant concentrations, higher catalyst loading or the use of highly active catalysts when dealing with exothermic reactions. For a comprehensive introduction we refer to [5]. Microreaction technology is used in each of the three phases of product or process development. In the development laboratory it is especially useful in providing precisely defined conditions. Another advantage is its usage in highthroughput experiments. In process development the characteristics mentioned above turn microreaction systems into powerful tools even if the process itself is to be realized using conventional technology. Microreaction technology is particularly useful here in searching for optimal process parameters and in providing benchmarks. A drastically accelerated process development is possible if microstructured equipment is used in production. It is not necessary to perform timeconsuming studies for the scale up because the production scale can be realized in principle by parallelization; i.e. by multiple repetitions of the microstructured elements (“numbering up” the elements). To date, such chemical microprocesses have only been built as pilot plants, largely due to the formidable challenges that must be overcome in the development of technical concepts, and that go beyond the requirements for conventional processes.

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2.2.2 Catalytic Wall Reactors

This section offers a short overview of research and process development in a field that is the target of one of the key projects in Degussa’s project-house process intensification. Most of this survey has been published recently by Hüther et al. [3]. Partial oxidations are one of the most important classes of gas-phase reactions in the chemical industry. Usually these reactions are highly exothermic and solid catalysts are used. Since the product is generated by an only partial oxidation of the reactant, total oxidation has to be avoided as either a side reaction or an elementary step in a series of reactions. The control of residence time and especially reaction temperature are essential for the enhancement of the selectivity of partial oxidations. A brief look at the reaction enthalpies of ethylene oxide, one of the major intermediate products based on ethylene (in 1999 the world capacity was 14.5 million tonnes per year [6]) illustrates the crucial importance of heat control. The reaction enthalpy of the partial oxidation of ethylene to ethylene oxide is − 105 kJ/mol, while the total oxidation of ethylene resulting in carbon dioxide and water is more than one order of magnitude larger at −1327 kJ/mol. Consequently this side reaction causes the occurrence of regions of extremely high temperatures (hot spots) with negative impact on the selectivity and the lifetime of catalysts. In order to enhance space-time yields of catalytic gas-phase reactions, two strategies are in principle possible: the improvement of catalyst activity or the implementation of more intense process conditions. This usually leads to an increase of heat production that can only be partially released, if at all, using conventional reactor technology. A simple estimation of the temperature profi le inside a tube reactor starts from an energy balance for the system [7]. Since we are particularly interested in the performances of a tube reactor and a catalytic wall reactor the following simplifying assumptions are made: 1. A constant heat conduction and viscosity is achieved. 2. Heat conduction in the longitudinal direction is neglected. 3. Friction in the bed of solid catalyst pellets is neglected. 4. It is assumed that the reactor is an ideal tubular flow reactor. 5. The temperature dependence of the reaction is neglected. The final assumption is a drastic simplification that leads to an underestimation of the temperature increase. The temperature T of an arbitrary point in the catalytic bed may be related to the effective thermal conductivity l , the heat of reaction ∆HR and the rate of reaction rɶ by the differential equation  ∂ 2T  λ  2  − ∆HR ⋅ rɶ (c , T ) = 0  ∂r 

(2.1)

2.2 Microreaction Technology

where r denotes the radius of a small volume element in a fi xed-bed catalytic reactor. With the boundary conditions

∂T =0 ∂r

at r = 0

(2.2)

T = Twall

at r = R

(2.3)

one gets for the temperature increase ∆TR between the center of the tube and the wall of the tube at the entrance of a reactor with tube radius R 1 −∆HR ⋅ c 0 rɶ ( c 0 , T0 ) 2 c p ⋅ ρgas ∆TR = ⋅ ⋅R ⋅ ⋅ c0 2 c p ⋅ ρgas λ

(2.4)

where cp denotes the specific heat capacity, rgas the density of the gas in the stream, c 0 the concentration and T0 the temperature at the entrance of the reactor [8]. The second factor of the rhs of Eq. (2.4) has units of temperature and can be regarded as the adiabatic temperature rise ∆Tad. The third factor can be interpreted as the reciprocal of the time constant t of the heat transport. Under the assumption W kJ kg of typical values for the constants: λ = 1 , cp = 1 , ρgas = 0.7 3 m×K kg × K m s we get for the fourth factor: R 2 ⋅ 700 . The quotient of adiabatic temperm2 ature over residence time varies over a broad range of values. In the case of the K synthesis of acrolein via the oxidation of propylene it amounts to 3800 , while s the acetoxylation of ethylene to vinyl acetate shows only a value of 365 K . s Figure 2.1 depicts the increase of temperature in the catalytic bed as a function ∆Tad and the radius of the tubular reactor. It can be seen that of the quotient τ the reduction of the radius of the tube by a factor of ten by using tubes with a radius of 1 mm instead of 10 mm the maximum temperature inside the catalyst can be reduced by more than one order of magnitude. For the synthesis of acrolein the temperature increase amounts to 133 ºC in a tube with a diameter of 10 mm, while it is reduced to 1.3 ºC when tubes with a diameter of 1 mm are used. A technical realization of reactors with such dimensions is possible by changing the reactor concept and switching to catalytic wall reactors. The walls of these reactors are directly coated by the catalyst. Consequently, the reaction takes place near the walls of the reactor with the result that the heat transfer is considerably improved. Drastic enhancements of heat transfer are possible in microstructured wall reactors. For a detailed discussion we refer to [9]. The use of microstructured catalytic wall reactors offers an interesting option for the revamping of existing plants. The key idea of this so-called booster concept

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Figure 2.1 Temperature increase ∆TR as a function of the quotient ∆Tad and the radius of the tubular reactor. τ

Figure 2.2 Combination of a microstructured wall reactor (WR) with a multitubular reactor (MTR).

is the combination of a microstructured wall reactor with an existing conventional multitubular reactor as shown in Fig. 2.2. To illustrate the potential of the booster concept a highly exothermic reaction is considered. Model calculations for a firstorder reaction are given in Fig. 2.3. The curves show the production rate and hence also the released heat as a function of effective volume of the reactor. The effective volume is defined as the product of the reactor volume with its spacetime yield. Curve A represents the production rate for an isothermic multitubular reactor with amount of heat released ∆H. The production rate shows a steep growth in the first quarter of the reactor with a corresponding heat production. Since in conventional multitubular reactors the heat transport is limited the heat production limits the level of conversion possible. Now the capacity of the multitubular reactor is increased by 20% by inserting the microstructured catalytic wall reactor. If it is assumed that this reactor shows an unlimited heat release, then curve B results. The maximum production rate – and hence the maximum

2.3 Simulation

Figure 2.3 Production rate as function of reactor volume.

heat release – occurs inside the wall reactor. The multitubular reactor no longer works at its limit because the heat released (∆H1) is considerably diminished. To use the multitubular reactor to capacity the production rate could now be increased either by using more active catalysts or by increasing temperature, pressure or gas flow or by rejection of inert gases. The full economic potential of the booster concept can be seen from curve C. An increase of conversion to 250% is realized if the same level of heat release ∆H is reached in the subsequent multitubular reactor by only adding 20% nominal capacity enhancement due to a microstructured wall reactor.

2.3 Simulation 2.3.1 Introduction

Since process intensification represents a genuinely holistic approach, simulation is an integral part during the course of a project. The anticipation of phenomena by virtual means is one of the fundamental sources for intensifying strategies. For process simulation, widely employed software packages such as ASPEN+, SPEEDUP, etc., do the bulk of the work. The success of process simulation depends largely on the accessibility and accuracy of physical and kinetic information for the chemicals involved. The thermophysical properties and kinetic models of these programs are still strongly dependent on experimental measurements. This

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is a severe limitation and often hinders progress and the exploration of unconventional strategies. Molecular-simulation methods based on quantum-mechanical methods can in principle predict the chemical and physical properties of molecules ab initio, i.e. without provision of any experimental information. Simulation methods based on empirical force fields allow atomistic simulations on molecular systems consisting of hundreds of thousands of atoms, and therefore allow predictions for almost any material. Unfortunately, the CPU time scaling of molecular-simulation calculations and process-simulation calculations is entirely different. Therefore a simple direct integration of molecular-simulation methods into process simulation is still not possible. However, the combination of both classes of simulation tools is extremely useful. An obvious and commonly practised way is the replacement of experimental measurement by molecular simulation as the first step and followed by the performance of process simulation runs. In the following subsections we will outline the state-of-the-art of quantumchemical calculations for molecules in the gas phase and briefly discuss two entirely different molecular-simulation methods for the prediction of the thermodynamical properties of complex fluid systems. Since the economic impact of process intensifying work is of overwhelming importance, financial control during the whole project is essential. Therefore we will go on to consider a financial simulation method for project management. 2.3.2 Molecular Simulations 2.3.2.1 Quantum-chemical Calculations Wavefunction-based quantum chemistry has reached the stage where gas-phase thermochemical data can be calculated ab initio to a precision of a few kJ/mol for molecules containing first-row atoms. This is partly due to the development of new computational methods and partly due to the spectacular progress in computer technology. In this subsection we limit our discussion to those methods of quantum chemistry that are capable of very high accuracy. This is only a subset of the methods that are currently available for the quantum-chemical simulation of molecular systems. They show the important characteristic that the exact solution of the Schrödinger equation is approached in a systematic manner. An important advantage of this behavior is that a systematic improvement of approximations in practical calculations is possible. The quality of the approximated molecular wavefunctions depends on the basis of atomic orbitals in terms of which molecular orbitals are expanded (the one-electron space) and on the form of the linear combinations of determinants of these molecular orbitals (the n-electron space). For one- and n-electron spaces, hierarchies exist of increasing accuracy. At the lowest level of the n-electron hierarchy we have the Hartree–Fock (HF) wavefunction that is obtained by variational optimization of a single determinant with respect to the shape of the molecular orbitals that are occupied by electrons. However, the accuracy of the Hartree–Fock description is far too low to provide sufficiently accurate thermochemical data. The most successful approach for

2.3 Simulation

highly accurate calculations of molecular electronic structures is the coupled cluster approach (CC) where the wavefunction |CC〉 is written as an exponential of a cluster operator Tˆ working on the Hartree–Fock wavefunction |HF〉 generating a linear combination of all possible determinants in a given one-electron basis. CC = exp(Tˆ) HF

(2.5)

The cluster operator Tˆ creates excitations and can be written as Tˆ = Tˆ1 + Tˆ2 + Tˆ3 + . . .

(2.6)

where Tˆ1 creates single excitations, Tˆ2 creates double excitations, etc. The coupled cluster operator is usually truncated after double excitations leading to the coupled-cluster singles-and-doubles (CCSD) model. A particularly successful approximation is the CCSD(T) model where the triple excitations are included approximately. To illustrate the performance of CCSD(T) calculations of the atomization energies (AEs) of carbon monoxide are documented on Table 2.1 according to [10]. As can be seen from Table 2.1, in order to achieve extremely high accuracy for thermochemical data, it is necessary to include contributions from molecular vibrations and from the special theory of relativity. Although computationally expensive, calculations of comparable quality can be carried out routinely for a broad range of molecules containing first-row atoms. For a recent review we refer to reference [11]. Problems occur when the wavefunctions are not dominated by one determinant. A prominent example is the ozone molecule where an error of 10 kJ/mol occurs at the CCSD(T) level of theory. As we move down the periodic table relativistic effects become increasingly important and the electronic structure becomes more complicated. It is currently not possible to calculate

Table 2.1 Ab-initio calculation of the AE of the CO molecule

at the one-electron basis set limit. Energy values are given in kJ/mol. For calculational details we refer to [10]. AE is defined as: AE = E(C) + E(O) − E(CO).

HF SD (T) Vibrational Relativistic corr. Total Experimental

E(C)

E(O)

E(CO)

AE

−98 964.9 −388.4 −7.7 0.0 −40.1 −99 401.1

−196 437.1 −639.6 −11.9 0.0 −139.0 −197 227.6

−296 132.2 −1 350.2 −54.2 +12.9 −177.1 −297 700.8

730.2 322.2 34.6 −12.9 −2.0 1072.1 1071.8

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thermochemical data for such more complex electronic systems to an accuracy of a few kJ/mol. However, industrial gas-phase chemistry is dominated by comparatively small molecules containing first row atoms. Since for many species formed in side reactions and intermediates no experimental thermochemical data are available, highly accurate quantum-chemical calculations are of great value. 2.3.2.2 COSMO-RS Calculations While theoretical chemistry has long provided accurate models and efficient computational techniques for molecular systems in the gas phase and in assemblies of weakly interacting molecules, approaches attempting the quantitative description of liquid systems have only been developed in the last twenty years. These have emerged as tools that will be used routinely in industrial research in the near future. A particularly interesting approach is the COSMO-RS model (conductor-like screening model for real solvents) developed by Klamt [12], which uses statistical thermodynamics to derive the macroscopic thermodynamic properties of liquids from the microscopic quantities obtained from quantumchemical calculations. The ingenuity of this approach is that it handles the complex geometric arrangements of molecules in real fluids – i.e. all sterically allowed configurations in the ensemble – simply as an ensemble of surface segments characterized by a distribution function. The calculation of thermodynamic properties of pure and mixed fluid systems such as vapor pressure, activity coefficients, solubilities or excess energies using the COSMO-RS model necessitates beginning with a quantum-chemical calculation for each component. This singlemolecule calculation must be done only once per component or per relevant electrostatically different conformation. The input data for performing COSMORS calculations can then be stored in a database. For practical calculations with the COSMO-RS model it is necessary to introduce around 20 adjustable parameters, which are specific neither to functional groups nor to molecule types. These parameters need only be adjusted once according to a given quantum-chemical level of theory and are thereafter fi xed. Thus, given a suitable parameterization it is possible to use quantum-chemical calculations for single molecules to predict the properties of virtually any imaginable mixture with COSMO-RS. The only input data necessary for these calculations is the molecular skeleton of each component, which consists of the network of chemical bonds, including the set of atoms and connections between them. Having performed quantum-chemical calculations once and stored them in a database, the COSMO-RS calculations to yield thermophysical data are extremely fast. Therefore, COSMO-RS is especially useful in virtual screening, e.g. in the search for entrainers or new solvents. Task-specific ionic liquids are of special interest in process intensification. The great variety of existing ionic liquids leads to an unavoidable theoretical screening at the beginning of a project. To date, the only fast and predictive virtual screening method is COSMO-RS, which provides activity coefficients for the relevant species in a set of ionic liquids. From these, selectivites and capacities are then calculated. Figure 2.4 depicts data from the open literature [13] to illustrate the performance of COSMO-RS.

2.3 Simulation

Figure 2.4 Calculated versus experimental logarithmic activity coefficients at infinite dilution of alkanes (䊏), alkenes (ⵧ), alkylbenzenes (䉭), alcohols (+), polar organics (䊊), and chloromethanes (䊉) in [bmpy][BF4] at 314 K.

COSMO-RS is a surprisingly robust method suitable for predictive calculations in a broad range of application areas. For a comprehensive survey we refer to [14]. COSMO-RS has some limitations: for instance, it is unable to treat dynamic properties such as viscosities and diffusion coefficients. Therefore COSMO-RS does not provide a complete description of liquid phases. In principle, the more rigorous molecular dynamics and Monte Carlo approaches offer a complete picture, but on the other hand they are several orders of magnitude more computationally demanding. 2.3.2.3 Molecular-dynamics Calculations Molecular-dynamics (MD) calculations solve Newton’s equation of motion for molecular systems. The result of these calculations is a trajectory in a 6N-dimensional phase space (3N positions and 3N momenta) where N denotes the number of atoms of which the molecular system consists. The trajectory provides a set of configurations distributed according to a certain statistical ensemble. From statistical physics it follows that in the limit of very long simulation times the phase space should be completely sampled and in that limit this averaging process would yield the thermodynamic properties of the system. Of particular interest for process-simulation purposes is the possibility of calculating phase diagrams using MD. The first paper on calculational MD simulations was already intended to investigate the phase diagram of a hard-sphere system [15]. The accuracy of MD simulations crucially depends on the ability of the chosen force field to model the behavior of the atoms in the material studied under the conditions for which the calculation is run. The problem of modeling a molecular system can therefore

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2 Process Intensification – An Industrial Point of View Table 2.2 Henry’s law constants (in bar) of nitrogen, oxygen

and carbon dioxide in ethanol at two different temperatures according to references [16, 17]. The recommended experimental values (for details see [16, 17]) are given in parentheses.

H N2 H O2 HCO2

323 K

373 K

2904 (2536) 1535 (1748) 210 (211)

2373 (2213) 1459 (1599) 323 (297)

be restated as that of finding a force field. The construction of force fields is a tedious task but its importance can hardly be underestimated. Once reliable forcefield parameters exist, MD simulations can provide accurate data for a variety of thermodynamic properties of complex fluid mixtures. To illustrate the current level of achievement of this method, we document in Table 2.2 recently published calculations of Henry’s law constants in ethanol [16, 17]. It can be seen from Table 2.2 that the predicted Henry’s law constants agree well with the high-quality experimental values showing maximum deviations of 20%. The force field used has been constructed by adjustment to experimental data for the pure components. Since thermophysical data published in the literature often shows a high scattering and is frequently not available for the conditions required in process intensification, MD simulations based on high-precision force fields are an alternative to costly and time-consuming measurements. 2.3.3 Monte Carlo Simulations in Project Valuation under Risk

A prerequisite of project management is the financial evaluation of a (large) number of possible scenarios to compare possible ways of realizing the project. For large and complex projects this presents a significant challenge. Any method of estimating risk must address one question in particular: How does the financial descriptor used in project valuation change in response to changes in the underlying sources of risk? This question asks for a mapping from risk factors to a financial quantity. Most R&D projects involve expenditures and savings over a period of years. To connect the value of cash flows with different time periods, it is essential to employ a cash flow analysis method that takes into account the time value of money. In finance, such methods are called discounted cash flow methods. A particularly useful method is the net present value (NPV) analysis. The NPV measures the difference between the present value of cash inflows and the present value of cash outflows. It is defined as

2.3 Simulation N

NPV ≡ ∑ k =0

Ck (1 + i )k

(2.7)

where N is the amount of time in years that cash has been invested in the project, i is the interest rate and Ck the cash flow at the point in time k. Thus the NPV has an obvious meaning: It quantifies the absolute value of a project (e.g. 2.5 × 10 6 1). Therefore, for the case of an independent accept/reject decision a simple rule can be derived: NPV < 0 ⇒ reject project NPV ≥ 0 ⇒ accept project In the case NPV = 0 the project neither destroys nor creates additional value. It exactly earns the yield required by the investor. In the case NPV > 0 the project adds additional value to the company. The NPV is a multivariable function depending on, e.g., the net price of the product, the production costs, the overhead costs, R&D costs, etc. Once the analytic form of the function has been constructed a sensitivity analysis can be performed to identify those variables that have the greatest effect, the so-called value drivers. The relative importance of the uncertain value drivers is usually visualized using tornado diagrams. A typical information generated by graphical sensitivity analysis would be: An x% increase of the variable “net price” would rise the project value to amount X 1, while a y% decrease would decrease the NPV to Y 1. By varying several input variables simultaneously it is possible to assess and model so-called scenarios. Such modeling allows the comparison of alternative projects or the study of a single project under varying environmental conditions. However, since the input variables are uncertain it is generally more useful to consider probability distributions rather than discrete values: these form the basis of Monte Carlo (MC) simulations. From a mathematical point of view, MC methods derive from measurement theory, being based on the analogy between probability and volume. A universe of possible outcomes is sampled randomly and the fraction of random draws that fall in a given set yields an estimate of the volume of the set. The law of large numbers guarantees that this estimate converges to the correct value as the number of draws tends to infinity. MC methods were introduced into finance for pricing options in 1977 by Boyle [18] more than 25 years after the pioneering work on MC by Metropolis and Ulam [19]. Since then, these methods have been widely used for both valuation and risk analysis [20]. MC simulations for project valuation under risk start with the choice of a financial quantity (e.g. the NPV) which has to be a known function of value drivers. For each of these input variables a probability distribution has to be chosen so as to represent the known (assumed) interval and expected probabilities for certain discrete values as precisely as possible. Frequently used probability distributions include the normal distribution, exponential distribution, and uniform

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2 Process Intensification – An Industrial Point of View

Figure 2.5 MC-based risk profile in reverse cumulative representation.

distribution. In any step of the simulation run random values within the range of the given probability are calculated for each value driver and the corresponding financial quantity is calculated and stored. This step is repeated n times to yield a risk profile for the financial quantity. The higher the number of repetitions the more reliable is the corresponding profile. Figure 2.5 depicts a typical histogram for the NPV in reverse cumulative representation. The following information can easily be derived: With a probability of 50% the NPV will meet or exceed 26.1 × 10 6 1. In the worst case with a probability of 100% the NPV will meet or exceed −3.3 × 10 6 1. Even under best conditions the NPV will not exceed 55.9 × 10 6 1. Integrated management software such as the Degussa-specific solution InnoToolBox, which is based on PipelinePlanner, a software suite developed by the Perlitz Strategy Group, provide MC simulations as one of a variety of tools to control the financial aspects of R&D projects. In our opinion, this is of particular importance in process intensification.

References 1 W. T. Cross, C. Ramshaw, Chem. Eng. Res. Des., 64, (1986), 293 2 A. I. Stankiewicz, J. A. Moulijn, Chem. Eng. Prog., 96, (2000), 22 3 A. Hüther, A. Geißelmann, H. Hahn, Chem. Ing. Tech., 77, (2005), 1829 4 W. L. Miller, L. Morris, 4th Generation R&D – Managing Knowledge, Technology, and Innovation, 1999, John Wiley & Sons, Inc., New York 5 V. Hessel, S. Hardt, H. Löwe, Chemical MicroProcess Engineering – Fundamentals, Modeling and Reactions,

6

7

8 9

2004, Wiley-VCH Verlag GmbH & Co. KgaA, Weinheim K. Weissermel, H.-J. Arpe, Industrial Organic Chemistry, 4th Edition, 2003, Wiley-VCH Verlag GmbH & Co. KgaA, Weinheim J.M. Smith, Chemical Engineering Kinetics, 1970, 2nd Edition, McGraw-Hill Kogahusha Ltd., Tokyo E. Klemm, Chem. Eng. Commun., submitted E. Klemm, M. Rudek, G. Markowz, R. Schütte, Mikroverfahrenstechnik, in

References

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11 12 13 14

Winnacker-Küchler, Chemische Technik, Prozesse und Produkte, Vol. 2, 2004, Wiley-VCH Verlag GmbH & Co. KgaA, Weinheim T. Helgaker, T. A. Ruden, P. Jorgensen, J. Olsen, W. Klopper, J. Phys. Org. Chem., 17, (2004), 913 W. Klopper, J. Noga, Chem. Phys. Chem., 4, (2003), 32 A. Klamt, J. Phys. Chem., 99, (1995), 2224 M. Diedenhofen, F. Eckert, A. Klamt, J. Chem. Eng. Data, 48, (2003), 475 A. Klamt, COSMO-RS – From Quantum Chemistry to Fluid Phase Thermodynamics

15 16 17 18 19 20

and Drug Design, 2005, Elsevier B. V., Amsterdam B. J. Alder, T. E. Wainwright, J. Chem. Phys., 27, (1957), 1208 T. Schnabel, J. Vrabec, H. Hasse, Fluid Phase Equilib., 233, (2005), 134 T. Schnabel, J. Vrabec, H. Hasse, Fluid Phase Equilib., 239, (2006), 125 P. P. Boyle, J. Finan. Econ., 4, (1977), 323 N. Metropolis, S. Ulam, J. Am. Stat. Assoc., 44, (1949), 335 D. L. McLeish, Monte Carlo – Simulation & Finance, 2005, John Wiley & Sons, Inc., Hoboken, New Jersey

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3 Modeling and Simulation of Microreactors Steffen Hardt

3.1 Introduction

In the field of conventional, macroscopic process technology, modeling and simulation approaches are by now used on a routine basis to design and optimize processes and equipment. Many of the models employed have been developed for and carefully adjusted to specific processes and reactors and allow to predict flow as well as heat and mass transfer, often with a high degree of accuracy. Compared to that, modeling and simulation approaches for microreactors are more immature, but bear a great potential for even more reliable computer-based process engineering, as will be discussed below. In general, the purposes of computer simulations are manifold, such as feasibility studies, optimization of process equipment, failure modeling or modeling of process data. For each of these tasks within the field of chemical engineering simulation methods have been applied successfully in the last couple of years. Microreactors are developed for a variety of different purposes, specifically for applications that require high heat- and mass-transfer coefficients and welldefined flow patterns. The spectrum of applications includes gas and liquid flow as well as gas/liquid or liquid/liquid multiphase flow. The variety and complexity of flow phenomena clearly poses major challenges to the modeling approaches, especially when additional effects such as mass transfer and chemical kinetics have to be taken into account. However, there is one aspect that makes the modeling of microreactors in some sense much simpler than that of macroscopic equipment: the laminarity of the flow. Typically, in macroscopic reactors the conditions are such that a turbulent flow pattern develops, thus making the use of turbulence models [1] necessary. With turbulence models the stochastic velocity fluctuations below the scale of grid resolution are accounted for in an effective manner, without the need to explicitly model the time evolution of these fine details of the flow field. Heat- and mass-transfer processes strongly depend on the turbulent velocity fluctuations, for this reason the accuracy of the turbulence model is of paramount importance for a reliable prediction of reactor performance. However, to the Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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3 Modeling and Simulation of Microreactors

current point there is no model available that is capable of describing turbulent flow phenomena in a universal manner and is computationally inexpensive at the same time. For this reason, simulation approaches for microreactors, while usually not necessitating turbulence models, offer some potential to make predictions with a degree of accuracy unparalleled by models of macroscopic reactors. When comparing processes in microreactors with those in conventional systems, a few general differences can be identified: • Flow in microstructures is usually laminar (as mentioned above), in contrast to the turbulent flow patterns on the macroscale. • The diffusion paths for heat and mass transfer are very small, making microreactors ideal candidates for heat- or mass-transfer-limited reactions. • The surface-to-volume ratio of microstructures is very high. Thus, surface effects are likely to dominate over volumetric effects. • The share of solid wall material is typically much higher than in macroscopic equipment. Thus, solid heat transfer plays an important role and has to be accounted for when designing microreactors. While the absence of turbulence simplifies many modeling tasks, the predominance of surface effects introduces additional complications, especially in the case of multiphase flow. Some of the fundamental mechanism of, e.g., dynamic wetting and spreading phenomena are not yet well understood, thus adding some degree of uncertainty to the modeling of these processes. As more and more practical applications of microfluidic systems emerge, research in the field of fluidic surface and interfacial phenomena gets additional impetus. It is thus hoped that in the following years refined models for microfluidic multiphase systems will be formulated and will add an additional degree of predictability to flow phenomena in microreactors. When facing the task to formulate a model of a specific microfluidic system, the question arises if the conventional macroscopic equations describing fluid flow, heat and mass transfer are still valid on the microscale. Systems for chemical processing rarely contain structures with dimensions less than 10 µm, the relevant length scale is often in the range of 100 µm. The standard approach of modeling transport processes in microreactors relies on a continuum description, i.e. continuum field quantities such as pressure and velocity are introduced to represent the mean dynamics of the fluid molecules. The corresponding mathematical formulation is based on the Navier–Stokes equation and convection-diffusion equations for heat and mass transfer. The question to answer is whether or not this mathematical framework is suitable to describe transport processes in microreactors on the relevant length scales. Besides the continuum assumption there is one additional assumption the Navier–Stokes and related equations rest on. This is related to the local statistical distribution of the particles in phase space:

3.1 Introduction

The standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell–Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference comoving with the fluid. When gas flow in microreactors at high temperature or low pressure is considered, this assumption may break down. The principle quantity determining the flow regime of gases and deviations from the standard continuum description is the Knudsen number, defined as Kn =

λ L

(3.1)

Kn is the ratio of two length scales, the mean free path of the gas molecules l and a characteristic length scale of the flow domain L, for example the channel diameter. When Kn is larger than one, a gas molecule is more likely to collide with the channel wall than with another molecule. As the transport of momentum or enthalpy is to a large extent governed by the collisions between molecules, major changes in the flow behavior are expected when the Knudsen number is of the order one. This may happen when gas flow through narrow channels is considered, but also when the temperature is high and/or the pressure is low. Based on the Knudsen number, four different flow regimes can be distinguished [2]: • Continuum flow with no-slip boundary conditions (Kn ≤ 10 −2 ); • Continuum flow with slip boundary conditions (10 −2 < Kn ≤ 10 −1); • Transition flow (10 −1 < Kn ≤ 10); • Free molecular flow (Kn > 10). In the first two cases the Navier–Stokes equation can be applied, in the second case with modified boundary conditions. The computationally most difficult case is the transition flow regime, which, however, might be encountered in microreactor systems. Clearly, the defined ranges of Knudsen numbers are not rigid; they rather vary from case to case. However, the numbers given above are guidelines applicable to many situations encountered in practice. For applications in the field of microreaction engineering, the conclusion may be drawn that the Navier–Stokes equation and other continuum models are valid in many cases, as Knudsen numbers greater than 10 −1 are rarely obtained. This observation is supported by the fact that a typical mean free path of nitrogen at 1 atm is about 70 nm. However, it might be necessary to use slip boundary conditions to describe the flow in microchannels. The first theoretical investigations on slip flow of gases were done already in the 19th century by Maxwell and von Smoluchowski. The basic concept relies on a so-called slip length L s, which relates the local shear strain to the relative flow velocity at the wall ugas − u wall = Ls

∂ ugas ∂y

(3.2) wall

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3 Modeling and Simulation of Microreactors

where ugas, uwall refer to the gas and wall velocity parallel to the wall and y is the coordinate normal to the wall. The slip length L s depends on the wall roughness and can be determined experimentally. Similarly, there is a relationship for the temperature jump across a solid surface in the slip-flow regime [2]. In the transition flow regime, which is rarely encountered in microreactors, the usual convection-diffusion equations cease to be valid. The general formulation to be applied in this regime is given by the Boltzmann equation from which lessinvolved model descriptions such as the Burnett equation can be derived via a low-Knudsen number expansion [3]. However, rather than by a continuum model, transition flows are often computed based on the direct simulation Monte Carlo (DSMC) method [4]. Instead of modeling gas flow by a set of differential or integrodifferential equations, this method is based on tracking the trajectories and interactions of gas molecules directly. DSMC is a time-marching approach based on a time discretization with steps smaller than the collision time of the molecules. It suggests itself that the computational demand for DSMC is much higher than for continuum approaches. Similar to the case of gas flows, the question on the range of applicability of continuum models for liquid flow arises. While the kinetic theory of gases provides clear indications of the limits of continuum models and of the onset of rarefaction effects on the microscale, there is no general framework explaining possible deviations of liquid-flow phenomena from their macroscopic behavior. The concept of mean free path ceases to be useful for liquids, as the molecules interact with their neighbors in a permanent manner (in a sense, the molecular mean free path is zero for liquids). A couple of groups have conducted experiments on liquid flows in microstructures and measured pressure drop and heattransfer coefficients, with largely contradictory results [5–11]. A compilation of the published results does not seem to leave room for any simple conclusions of general importance. Furthermore, the physics of Newtonian liquids does not suggest any universal mechanisms by which the flow behavior in channels of several hundred micrometers width could differ considerably from macroscopic behavior at the same Reynolds number. However, in some specific situations liquid microflows exhibit a behavior that does not occur on the macroscale, but that can be reproduced experimentally and explained by theoretical models. Boundary slip of liquids constitutes a first example of such effects. During the last decade it was confirmed that not only gases, but also liquids can exhibit boundary slip, with corresponding experiments indicating a slip length of a few ten nanometers [12–14]. The phenomenon has also been studied using numerical simulations, especially the molecular dynamics (MD) method that is based on solving Newton’s equations of motion for an ensemble of mutually interacting molecules (see, e.g., [15]). MD simulation results confirm the existence of boundary slip for liquids and indicate how the slip length depends on molecular parameters and shear rate [16]. While slip-flow effects are mostly expected to have an impact on flow behavior on the submicrometer scale, there is another phenomenon that may lead to deviations from the usual model predictions on a larger length scale. On a solid surface exposed to an electrolyte, charges may accumulate, causing the formation of a

3.1 Introduction

Debye layer above the surface to which an additional layer of immobilized charges is attached [17]. This so-called electric double layer (EDL) is related to accumulation of charges inside the liquid that act as sources of an electric field. In turn, this field is sensitive to changes in the EDL structure that may be induced by the flow in such a manner that inside a channel a field component in the streamwise direction builds up. When ionic charges start to migrate in the electric field, liquid molecules are dragged along with them and the apparent viscosity for channel flow changes. The phenomenon is known as the electroviscous effect [18] and may cause a considerable modification of the friction factor in comparatively large channels. The electroviscous effect was studied theoretically by Yang et al. [19] for rectangular microchannels. The channel width and height considered was in the range between 20 µm and 40 µm. Depending on the solute concentration, they found the friction factor to be increased by up to 30%, which shows that the electroviscous effect manifests itself on length scales much larger than the electric double-layer thickness typically being of the order of a few nm. Thus, even in typical microchannel geometries ionic liquids may exhibit some “unexpected” behavior. Many of the flow phenomena discussed in this introductory section can be incorporated into continuum models such as the Navier–Stokes equation, an exception being gas flows in the transition and free-molecular regime. Only a modification of the boundary conditions and not of the model equations themselves is usually necessary, i.e. the nonstandard physics characteristic of the micro- and nanoscale mainly appears as a boundary effect. Thus, when attempting to model transport processes in microreactors, the continuum hypothesis is a far-reaching concept that covers most of the scenarios encountered in practice. For a more detailed discussion of the special aspects of micro- and nanoflows the reader is referred to the book by Karniadakis and Beskok [20]. Along this line of argument it becomes clear that to a large extent the problem of modeling the transport phenomena in microreactors is a scaling problem: The familiar macroscopic model equations may be used, however, they have to be applied in an unfamiliar and possibly unexplored parameter range as far as the dimensionless groups characterizing the problem are concerned [21]. In this context, care has to be taken to pay attention to effects that are usually neglected macroscopically, such as surface tension or viscous heating. Most of the simulation techniques and results presented in the following sections are based on a numerical solution of the macroscopic transport equations. As already mentioned above, owing to the laminar-flow conditions many of these studies can be regarded as first-principles calculations that are free of adjustable model parameters. Consequently, simulation results for microreactors are expected to have a high predictive power. The major limitation for the predictive power of the models is the large number of degrees of freedom that often excludes the simulation of complete reactors and only allows focusing on components such as single channels. Thus, the true art of reactor modeling consists of selecting a suitable subsystem and to represent the portions of the reactor neglected by appropriate boundary conditions. For many of the results reported in the following sections such a strategy has been employed. When it is not clear how to select an

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appropriate subsystem or genuine system-level behavior should be studied, a macromodel approach may be chosen. One strategy of doing so is based on a partitioning of the reactor domain into certain sections each of which is characterized by a specific input-output behavior. As input to the model sections, thermodynamic forces such as pressure or temperature gradients are applied by which thermodynamic fluxes such as mass or heat flux are induced. An alternative strategy of macromodel generation is based on an averaging approach. By averaging over a large number of microscopic structures (e.g., channels) effective equations for volume-averaged field quantities can be derived, thus allowing the number of degrees of freedom for system-level modeling to be reduced considerably. In the following sections, some examples of both of these modeling strategies that allow study of transport phenomena on the level of complete systems or at least larger subsystems will be given. With continuum models being the basis for the simulation of microreactors, computational fluid dynamics (CFD) is the numerical method to be applied. Now, CFD techniques are routinely used in conventional chemical process engineering. In principle, the same methods can be employed for microreactors, with the possible drawback that the numerical algorithms have been optimized mainly for the parameter regime characteristic of macroscopic flows. The fundamentals of CFD will not be discussed here, suitable introductions can be found elsewhere (see, e.g., [22]). Instead, a certain degree of familiarity with CFD is assumed. However, the following sections may also provide valuable insights for those mainly interested in microreactor phenomenology with little background in numerical methods. 3.2 Flow Distributions

Due to the specific microstructuring technology employed to build up microreactors, the geometric shape of the flow domain is often different from that in macroscopic equipment. While the elements for fluid transport such as pipes are usually of circular cross section, the channels in microreactors have a rectangular or trapezoidal cross section. Depending on the specific microstructuring technology, also tub-like grooves or close-to triangular shapes might be obtained. Furthermore, a characteristic feature many microreactors have in common is a comparatively wide flow distributor followed by a large number of parallel microchannels. In the following, the flow distributions in characteristic geometries and methods to obtain approximate flow distributions in highly parallel flow domains will be discussed. 3.2.1 Straight Microchannels

For laminar flow in channels of rectangular cross section, the velocity profi le can be determined analytically. For this purpose, incompressible flow is assumed.

3.2 Flow Distributions

The flow profi le can be expressed in the form of a series expansion (see [23] and references therein) that, however, is not always useful for practical applications where often only a fair approximation of the velocity field over the channel cross section is needed. Purday [24] suggested an approximate solution of the form  x s y r u(x , y ) = u max 1 −    1 −      a     b 

(3.3)

for a rectangular channel oriented along the z-axis with a width 2a in the xdirection and a depth 2b in y-direction, where u is the local flow velocity and umax the maximum velocity. The exponents s and r depend on the aspect ratio b/a of the channel, the most common correlations can be looked up in [23]. Typically, Eq. (3.3) approximates the exact velocity profi le with an accuracy of a few per cent. Often, the detailed velocity profi le is not of interest, but only the friction factor f that determines the pressure drop over a channel of a given length. The fanning friction factor is defined as f =−

dp Dh dz 2ρU 2

(3.4)

with dp/dz being the pressure gradient along the channel, r the density, D h the hydraulic diameter and U the mean flow velocity. The hydraulic diameter is given by 4A/P, with A being the cross-sectional area of the channel and P its perimeter. Shah and London [25] derived a comparatively simple expression for the friction factor in rectangular channels that deviates from the analytical solution by less than 0.05% f =

24 (1 − 1.3553α + 1.9467α 2 − 1.7012α 3 + 0.9564α 4 − 0.2537α 5 ) Re

(3.5)

where a is the aspect ratio of the channel ( a ≤ 1 by definition) and the length scale entering the Reynolds number Re is the hydraulic diameter. Most expressions for the flow profi le in rectangular channels assume that the flow is fully developed, i.e. that the flow velocity is oriented along the z-axis and does not change in the streamwise direction. Close to the entrance of a channel this assumption is not valid, and the flow undergoes a development from an entrance distribution to a fully developed profile. Correspondingly, the pressure distribution deviates from that observed in a fully developed flow and expressions for the friction factor such as Eq. (3.5) are not valid in the entrance region. In order to determine the developing velocity distribution and friction factor, various approaches have been employed, such as analytical calculations based on a linearized inertia term, numerical solutions of the Navier–Stokes equation or experimental velocity and pressure measurements. An overview of the results was given by Hartnett and Kostic [23]. In general, the pressure drop per unit length of an

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3 Modeling and Simulation of Microreactors

entrance flow is higher than that of a fully developed flow. In order to compare entrance flow effects, a hydrodynamic entrance length L hy can be defined that is the length necessary to achieve a centerline velocity equal to 99% of the fully developed value. Usually, a nondimensional quantity is used, defined as L+hy =

Lhy Dh Re

(3.6)

Depending on the aspect ratio of the channel, values between 0.01 and 0.1 are found for the nondimensional entrance length [23]. From Eq. (3.6) it can be deduced – with L +hy being Re independent – that L hy increases linearly with the hydraulic diameter and the Reynolds number. A number of authors have considered channel cross sections other than rectangular [25–27]. In general, an analytical solution of the Navier–Stokes and the enthalpy equations in such channel geometries would be quite involved due to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 3.2.2 Periodic and Curved Channel Geometries

A key advantage of microreactors is their potential for rapid heat and mass transfer. Naturally, the speed up of transport processes is related to the decrease of diffusion paths on the microscale. In addition to this, special channel geometries are explored for which a further speed up of heat and mass transfer is found. Three different channel shapes having been studied are shown in Fig. 3.1, a zigzag (upper left) a sinusoidally curved (upper right) and a converging-diverging channel (bottom). Asako and Faghri [28] computed velocity and temperature fields in zigzag channels based on the finite-volume method for Reynolds numbers up to 1500. They found that above a specific Reynolds number depending on the geometry parameters, flow separation occurs, i.e. in the corners of the channel recirculation zones are being formed. While the friction factor for straight

Figure 3.1 Different channel shapes for which flow distributions have been computed, a zigzag (upper left), a sinusoidally curved (upper right) and a converging-diverging channel (bottom).

3.2 Flow Distributions

channels displays a linear decrease as a function of Reynolds number, as is apparent from Eq. (3.5), the friction factor in zigzag channels becomes nearly independent of the Reynolds number for Re > 1000. Data obtained by Xin and Tao [29] for a zigzag channel with rounded corners show a similar trend. Flow in sinusoidally curved channels for Reynolds numbers up to 500 was studied by Garg and Maji [30]. Again, a finite-volume discretization was used. For most of the parameter space explored, flow separation was not observed. However, for Re = 500 and a large enough sine-wave amplitude as compared to the period, there were some indications of recirculation zones forming in the recesses of the channel. Unfortunately, no attempt was made to compute friction factors. Guzmán and Amon [31] studied flow in converging-diverging channels and paid special attention to the transition between laminar and turbulent flow. They used a spectral-element method where the flow domain is divided into macroelements over which a set of polynomial test functions is defined. Their method allowed to suppress damping of small-scale fluctuations due to numerical viscosity and is thus well suited to study the transition from stationary to oscillatory and chaotic flow. The computed streamline patterns indicate a transition from a nonseparated flow at low Reynolds numbers to a flow with recirculation zones within the recesses of the channel, occurring at a comparatively small Reynolds number between 10 and 20. Streamline patterns for various Reynolds numbers are depicted in Fig. 3.2. The phenomenology of flow distributions in such channel domains was found to be quite diverse. At low Reynolds numbers the flow is stationary, but at a Reynolds number of 150 oscillations begin to develop, with the vortices still being confined in the recesses. At Re = 400 the viscous forces are no longer strong enough to confine the vortices in the recesses and vortex ejection is observed. At even higher Reynolds numbers the flow becomes aperiodic and chaotic. In some cases, for example when large residence times are desired, comparatively long microchannels have to be integrated into a compact microreaction device, a task that can only be achieved with curved channels. In addition to this, straight channels are often not suited for connecting a microflow domain to the external world. One of the first theoretical studies on flow in curved channels was done by Dean [32, 33], who investigated the secondary flow perpendicular to the main flow direction induced by inertial forces using a perturbative analysis. The dimensionless group characterizing such transversal flows is the Dean number, defined as K = Re

Dh R

(3.7)

where D h is the hydraulic diameter and R the mean radius of curvature of the channel. The typical secondary flow pattern in curved channels is given by two counter-rotating vortices separated by the plane of curvature. The strength of these vortices increases with increasing Dean number.

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3 Modeling and Simulation of Microreactors

Figure 3.2 Streamline patterns in a converging-diverging channel for various Reynolds numbers, taken from [31].

Originally, the studies of secondary flow in curved channels were performed for pipes of circular cross section. Of much higher relevance in microreactors are rectangular channels, which are the subject of detailed numerical investigations by Wang and Yang [34]. Specifically, they studied the formation of vortices in channels of square cross section and found a complex branch structure describing the transition between flow patterns with two and four counter-rotating vortices. Up to a Dean number of about 113, a two-vortex solution is found, followed by a transition regime between Dean numbers of 113 and 130 with coexisting two- and four-vortex configurations. For K > 130 the flow exhibits a four-vortex pattern that, however, coexists with a two-vortex pattern of broken symmetry. The friction factor in curved channels was found to be larger than in straight channels for most of the Dean number range considered, and not surprisingly different branches of the flow map are associated with different friction values. 3.2.3 Multichannel Flow Domains

In chemical microprocess technology it is, on the one hand, important to guarantee well-defined and reproducible reaction conditions in microchannels, on the

3.2 Flow Distributions

other hand a high throughput should be achieved. For this purpose, the process fluid is guided through a large number of parallel microchannels, where heat exchange and/or chemical reactions occur. One of the characteristic problems of microprocess technology is to equally distribute the incoming fluid over the microchannels. A fluid maldistribution would induce an unequal residence time in different channels, with undesired consequences for the product distribution of a chemical reaction being conducted inside the reactor. When a process with various competing side reactions and byproducts is considered, the contact time of the process fluid in the reaction region should be as well defined as possible in order to maximize the selectivity of the process. However, it should be pointed out that not only does the maldistribution of the fluid over a multitude of microchannels induce variations of the contact time, but also hydrodynamic dispersion of concentration tracers in the channels themselves. The corresponding effects will be discussed in a subsequent section devoted to mass transfer. Various concepts for the equipartition of fluid over a multitude of microchannels have been developed. One concept relies on guiding the incoming fluid through a flow splitter with subsequent bifurcations [35]. The most widely used design is based on a comparatively wide inlet region leading into a multitude of narrow microchannels. A corresponding geometry, developed for methanol reforming is shown in Fig. 3.3. The figure shows the computational domain of the CFD model of the reactor together with the computed streamlines, where use of reflection symmetry was made and only one quarter of the portion of the reactor to be considered was modeled. An inlet pipe leads to a flow-distribution chamber connected with a multitude of microchannels. The essence of such designs is to be seen in the pressure barrier of the microchannels. The narrower the microchannels, the higher will be the pressure drop in the channels themselves as compared to the pressure drop in the flow-distribution chamber, and the more uniform the flow distribution will be. During the development of the microreformer of Fig. 3.3 one of the goals was to design the flow manifold in such a way that the volume flows in the different reaction channels are approximately the same [36]. In spite of the recirculation zones found, for the chosen design a flow variation of about 2% between different channels was predicted from the CFD simulations. In the application under study, a washcoat catalyst layer is applied to the microchannels. The thickness

Figure 3.3 Streamlines in the flow-distribution chamber of a multichannel reactor.

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3 Modeling and Simulation of Microreactors

variations of the catalyst layer are likely to play the leading role in the nonuniformities in flow distribution to be expected. Hence, a flow manifold with intrinsic volume-flow variations of a few per cent over different reaction channels is usually satisfactory. For the computation of the flow distribution in the methanol-reforming reactor, a reduced-order flow model for the microchannels was used. In such a model, a fully developed flow profi le as given by Eq. (3.3) is assumed, which means that entrance flow effects are neglected. This approximation is justified when the entrance length is small compared to the total length of the channel. A major advantage of a reduced-order flow model is the significant reduction of the degrees of freedom entering the simulation. Each microchannel is then only represented by a single degree of freedom that is the total volume flow, and a resolution of the corresponding flow domain by a computational grid is no longer required. In this way the simulation of microfluidic devices comprising a large number of channels becomes possible at moderate computational costs. Finite-volume grids in combination with reduced-order flow models for microchannels allow flow-distribution problems to be solved in 3D for which the standard approach would be computationally too expensive. However, when the goal is to find designs with an optimum flow equipartition by tuning specific geometric parameters, it is advisable to set up models with significantly fewer degrees of freedom. Commenge et al. [37] studied the flow distribution in multichannel microreactors with the help of macromodels. The geometry they considered was a microstructured plate of a heat-exchanger stack developed at the Institute of Microtechnology, Mainz. The plate, together with arrows indicating the flow direction, is shown on the left side of Fig. 3.4, on the right side the model set up by Commenge et al. together with the geometric parameters is displayed. The model is based on the idea to subdivide the flow domain into a number of virtual channel segments with rectangular cross section over which the flow is distrib-

Figure 3.4 Geometry of the microstructured plates (left) and subdivision of the flow domain into channel segments (right) as considered in [37].

3.2 Flow Distributions

uted. Through the channel segments in the inlet zone the volume flows V1, V2 , . . . , VNc are transported, where a part of the flow branches off into the actual microchannels of width Wc and depth e. For each of the channel segments a relationship between the pressure drop ∆p and the average flow velocity u of the form ∆p = 32λnc

µLu Dh2

(3.8)

was used, where the hydraulic diameter is defined as Dh =

2we w +e

(3.9)

In these expressions, m is the dynamic viscosity, L the length of the channel segment, w and e its width and depth, and l nc a correction factor accounting for the noncircularity of the channels. Clearly, the above formulas rely on the assumption of a hydrodynamically developed flow. By means of the model depicted in Fig. 3.4, Eqs. (3.8) and (3.9) it is possible to compute the flow distribution just by solving a comparatively low-dimensional system of linear algebraic equations. The problem resembles the task of computing the current distribution of an electric circuit. A priori it is not clear if the approximation to subdivide the flow domain in the described way is justified. In order to assess the quality of the chosen approximation, Commenge et al. [37] computed the flow field by means of the finite-volume method. The results obtained suggest that, owing to the orientation of the isobars of the flow and the absence of recirculation loops, the chosen subdivision into channel segments is a reasonable simplification at least for the geometry and the flow regime considered. The model allows the flow distribution to be studied for a variety of different geometries with a minimum computational effort. As a result of simulations, optimized flow-chamber geometries leading to a maximum of flow equipartition were determined. While the strategy described above is a way to obtain effective, low-dimensional models for multichannel reactors in various cases, it clearly has its limitations. In geometries with hundreds of microchannels, another method can be utilized to compute the flow field. In a situation where the flow is distributed over a large number of parallel microchannels, the multitude of channels can be regarded as a porous medium. For porous media effective, volume-averaged transport equations have been known for a long time (for an overview, see [38]). In order to solve a flow-distribution problem for a multichannel geometry, the flow domain can be split into several regions, one of them being the flow-distribution chamber, another being the region comprising the multitude of channels. In the flowdistribution chamber, the ordinary transport equations are solved, whereas in the multichannel domain, the effective, volume-averaged description of the transport processes is used. A corresponding 2D geometry is shown in Fig. 3.5. The

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3 Modeling and Simulation of Microreactors

Figure 3.5 Multichannel geometry with channels and separating walls of uniform width.

incoming flow distributes over a large number of channels with a width of wc separated by walls with a width of w − wc. The most straightforward porous media model that can be used to describe the flow in the multichannel domain is the Darcy equation [38]. The Darcy equation represents a simple model used to relate the pressure drop and the flow velocity inside a porous medium. Applied to the geometry of Fig. 3.5 it is written as d µ p f +ε u f =0 dy K

(3.10)

with the porosity e and the permeability K given as

ε=

ε w c2 wc , K= 12 w

(3.11)

The flow velocity, pressure and dynamic viscosity are denoted as u, p and m and the symbol 〈. . .〉f represents an average over the fluid phase. Kim et al. [39] used an extended Darcy equation to model the flow distribution in a microchannel cooling device. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat-transfer effects in porous media.

3.3 Heat Transfer

Compared to the use of reduced-order flow models, the porous-medium approach allows an even larger multitude of microchannels to be dealt with. Furthermore, for comparatively simple geometries with only a limited number of channels, it represents a simple way to provide qualitative estimates of the flow distribution. However, as a course-grained description it does not reach the same level of accuracy as reduced-order models. Compared to the macromodel approach as propagated by Commenge et al., the porous-medium approach has a broader scope of applicability and can also be applied when recirculation zones appear in the flow-distribution chamber. However, the macromodel approach is computationally less expensive and can ideally be used for optimization studies.

3.3 Heat Transfer

Heat-transfer phenomena belong to the key issues to be studied in microreactors. Due to the small thermal diffusion paths, microreactors bear the potential to enable a fast heat transfer and to control temperature distributions with a very high accuracy. Correspondingly, modeling and simulation of heat-transfer phenomena and their reliable prediction is of paramount importance for process design. The simulation of temperature distributions in microreactors often requires the solution of a conjugate heat-transfer problem, i.e. the temperature fields in the fluid phase as well as in the solid wall material have to be computed. Owing to the microstructuring technologies used to fabricate microreactors, the share of wall material in the total reactor volume is often higher as in conventional equip ment. Hence, solid heat-conduction effects become important and usually have to be taken into account when the temperature field inside a reactor is to be computed. 3.3.1 Straight Microchannels

As stated in the previous section, a standard geometry of channels contained in microfluidic systems is a rectangular or close-to-rectangular cross section. With a given velocity profi le, the temperature field inside rectangular channels can often be determined analytically. However, it has to be pointed out that the problem of determining a temperature profi le in a channel geometry is much more multifaceted than the computation of a flow distribution. While in most cases a zerovelocity boundary condition at the channel walls is prescribed for the flow, the wall-boundary conditions for the temperature field can be diverse. On the one hand, either a heat flux or a temperature can be prescribed. On the other hand, the thermal boundary conditions on the four walls of a rectangular channel might be different. Due to this complexity, the solution for the temperature field itself is usually not reported. Rather, the Nusselt number, defined as

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3 Modeling and Simulation of Microreactors

Nu =

α Dh k

(3.12)

is determined. In this expression, D h is the hydraulic diameter, k the thermal conductivity of the fluid and a the heat-transfer coefficient measuring the transmitted thermal power per unit area divided by a characteristic temperature difference. The Nusselt number is a dimensionless quantity characterizing the efficiency of heat transfer. Similar to the velocity field, the temperature field assumes an invariant profi le far enough downstream from the channel entrance. However, due to the continuous heat transfer from or to the channel walls, only the shape of the temperature distribution stays invariant, but the normalization changes. Close to the entrance of the channel a thermally developing flow may be observed. A thermal entrance flow is a priori not related to a hydrodynamic entrance flow, i.e. a thermally developing flow might be observed even in regions where the flow is hydrodynamically developed and vice versa. Hartnett and Kostic [23] give an overview of the Nusselt numbers obtained for rectangular channels with various aspect ratios and various thermal boundary conditions. Depending on the thermal boundary condition, the Nusselt number for thermally fully developed flow either increases or decreases when the aspect ratio is increased. When the Nusselt number for thermally developing and hydrodynamically developed flow is plotted as a function of position along the channel, a divergence is observed when the channel entrance is approached. This means that with decreasing distance to the channel entrance, an increasing heat-transfer efficiency is found. The same observation is made when a simultaneously (i.e. hydrodynamically and thermally) developing flow is considered. Analogously to hydrodynamically developing flows, a thermal entrance length Lth can be defined. It is given as the duct length required for the Nusselt number to fall within a 5% interval of the fully developed value. Again, a dimensionless quantity L+th =

Lth Dh RePr

(3.13)

is used, where the Prandtl number Pr is the ratio of the momentum diffusivity (i.e. the kinematic viscosity) and the thermal diffusivity. The dimensionless thermal entrance length is a quantity only depending on the aspect ratio of the channel and the thermal boundary conditions. Hence, Lth is a linear function of the hydraulic diameter, the Reynolds and the Prandtl number. Apart from rectangular channels, heat transfer has also been studied in channels with different cross-sectional geometries. Along with their studies of friction factors, Shah [26] and Shah and London [25] also computed heat-transfer properties. Their results include both the Nusselt numbers for fi xed-temperature and fi xed-heat-flux wall boundary conditions and are given as tabulated values for different geometric parameters.

3.3 Heat Transfer

3.3.2 Periodic and Curved Channel Geometries

Similarly as with the computation of friction factors, heat transfer has not only been studied in straight channels, but also in channels with specific periodic shapes. Asako and Faghri [28] computed Nusselt numbers in the zigzag channels shown in Fig. 3.1 using a finite-volume approach in two dimensions. With increasing Reynolds number they found a significant increase of the Nusselt number as compared to a straight channel, i.e. a parallel-plates arrangement. For suitable geometry parameters, a Reynolds number in the range of 1000 and a Prandtl number of 8, a heat-transfer-enhancement factor of about 13 was computed. For a Prandtl number of 0.7 (corresponding to air) they compared the ratio of the heat fluxes achieved with the zigzag and the straight channels at identical values of pumping power and pressure drop. It was found that the zigzag channel outperforms the straight channel by up to a factor of 6, i.e. for the same pumping power or pressure drop a considerably higher heat flux is achieved. This is an interesting result in view of the design of microheat exchangers, where the goal is often to increase the heat flux while limiting the pressure drop. A zigzag geometry with rounded corners was studied in 2D by Xin and Tao [40] using the finite-volume method. The Prandtl number was kept fi xed at 0.7 and the Reynolds number was varied between 100 and 1000. For the range of geometry parameters considered, a heat-transfer-enhancement factor of about 3 was found at Re = 1000, thus confirming the potential of corrugated channels for heat-transfer enhancement. Heat transfer in sinusoidally curved channels was studied by Garg and Maji [30] for a Prandtl number of 1.0. Surprisingly, at a Reynolds number of 100 they obtain a value for the Nusselt number that is lower than the corresponding value in a parallel-plates channel. Such a suppression of heat transfer stands in contradiction to results obtained by Hardt et al. [41] who considered sine-wave channel walls of comparatively large amplitude. They found a heat-transfer enhancement increasing with Reynolds number, with an enhancement factor of about 4 for Re = 800. Both of these studies were restricted to steady-state flows, and steady-state solutions could be determined for the range of Reynolds numbers considered. Experimental work on flow and heat transfer in sinusoidally curved channels was conducted by Rush et al. [42]. Their results indicate heat-transfer enhancement and do not show evidence of a Nusselt number reduction in any range of Reynolds numbers. However, the flow patterns begin to exhibit some unsteady behavior for Reynolds numbers greater than about 200 in the geometry considered. Interestingly, oscillations were predominantly observed close to the exit of the channel and were found to move upstream when the Reynolds number was increased. Rush et al. attributed much of the heat-transfer enhancement found to the unsteady character of the flow. The converging-diverging channel geometry for which Guzmán and Amon [31] computed flow fields was also considered by Wang and Vanka [43]. They computed the flow and temperature field in an elementary cell of the channel based

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3 Modeling and Simulation of Microreactors

on the finite-volume method using periodic boundary conditions. As far as the transition between a steady-state and an unsteady flow is concerned, they find similar results as Guzmán and Amon. At Reynolds numbers beyond the transition to the unsteady flow regime, the temperature field having been evaluated for a Prandtl number of 0.7 becomes quite complex. This is indicated in Fig. 3.6, which shows the evolution of (normalized) temperature contour lines at a Reynolds number of 328. In order to quantify heat-transfer enhancement, the timeaveraged Nusselt number was computed. The increase in Nusselt number compared to a straight channel was found to be significant and reached values as high as 7.54 at a Reynolds number of 520. Sawyers et al. [44] analyzed flow and heat transfer in converging-diverging channels based on a perturbation analysis derived from lubrication theory and on the finite-volume method. Most of their studies were limited to Reynolds numbers up to 100. In this regime the increase in Nusselt number compared to a parallel-plates geometry was found to be moderate, amounting to a factor of about 1.3 at Re = 100 and Pr = 1. The results also showed an approximately linear increase of heat-transfer enhancement as a function of Prandtl number. The studies of Wang and Chen [45] indicated that heat-transfer enhancement in converging-diverging channels is significantly reduced if the separation between the wavy channel walls is bigger than the wavelength of the undulations. They used a cubic spline collocation method to numerically solve the transport equations and, in contrast to the other researchers, did not assume a cyclic structure of the flow and dimensionless temperature fields with the period of the wall structures. The results obtained at Re = 500 and

Figure 3.6 Evolution of the temperature field in the recesses of a converging-diverging channel, as obtained in [43].

3.3 Heat Transfer

Pr = 6.93 show a rather moderate heat-transfer enhancement factor of about 1.8 compared to a parallel-plates geometry that is further reduced for smaller Prandtl numbers. Similarly to the structure of the flow field, heat transfer has also been studied in curved channel geometries. The complicated branch structure with competing patterns of two and four counter-rotating vortices in channels of square cross section is reflected in the Nusselt number [34]. When plotting the Nusselt number as a function of Dean number, different branches are found corresponding to symmetric and asymmetric secondary flow patterns with two and four vortices. However, the relative difference between the different branches is not very pronounced and should be hard to measure experimentally. For a Dean number of 210 and a Prandtl number of 0.7 a heat-transfer-enhancement factor of about 2.8 was determined, thus showing that curved channels as well as other channels with specific periodically varying cross sections may be used for applications where rapid heat transfer is desired. 3.3.3 Multichannel Flow Domains

When a large number of parallel microchannels are considered, the problem of computing the temperature distribution inside the channels and the channel walls becomes quite involved. In such a case the use of the method of reducedorder flow models discussed in Section 3.2.3 is not as straightforward as in the case of a pure flow-distribution problem, due to thermal cross talk between the different channels. In principle, heat transfer in a multichannel domain can be described based on a porous-medium model, where an average is taken over an ensemble of channels and continuous field quantities are introduced. However, in the standard models for porous media, a thermal equilibrium between the fluid and the solid phase is assumed, i.e. only a single temperature field is used. Especially in the regime of large Péclet numbers the thermal-equilibrium assumption may break down and it may become necessary to define separate temperature fields for the fluid and the solid phase. It is thus advisable to take one step back and to derive the mean-field model equations allowing heat transfer in multichannel domains to be described based on a volume-averaging approach. A generic geometry of a multichannel stack the mean-field equations should be applied to is shown in Fig. 3.7. Many microchannels, usually several hundred up to several thousand, are arranged in layers stacked onto each other. As a whole the multichannel stack forms a cuboid of extension L x, Ly, L z in x-, y- and zdirection, respectively. The microchannels of width wf and depth hf are assumed to be parallel to the z-axis and are separated by solid walls of thickness ws and hs, respectively. The need for modeling single channels and their specific geometry is eliminated when regarding the reaction fluid and wall material as interpenetrating continua. Hence, in the following it is assumed that the reactor stack is filled with interpenetrating fluid and solid phases that interact via the exchange of heat.

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Figure 3.7 Multichannel stack comprising a large number of parallel microchannels.

Momentum exchange between the two phases is not accounted for explicitly and it is assumed that the velocity distribution over the cross section of the reactor is known a priori. The phase-volume fractions can be derived from the geometric parameters hf, hs, wf and ws and are given as Φf =

w f hf , Φs = 1 − Φ f (w f + w s )(hf + hs )

(3.14)

The temperature field of the solid phase can change due to heat conduction or due to heat transfer from the fluid phase. For the fluid, heat conduction does not have to be taken into account since the flow domain consists of disconnected channels but convective heat transfer due to the flow velocity as well as heat transfer from the solid is included. In the x-y plane no conduction within the fluid without intermediate transfer to the solid walls can occur. The reason for not including heat conduction in the z-direction is the fact that in most cases of practical relevance it is negligible when compared to convective transport. Following the derivation of Hardt and Baier [46], the fluid and solid temperature fields Tf and Ts are obtained as solutions of the equations

∂  ∂ Tf = −α a V (Tf − Ts ) + Sc ρf c f  + ui  ∂t ∂ x i  ∂ ∂ ∂ Ts − α a V (Ts − Tf ) ρsc s Ts = kij ∂t ∂ xi ∂ x j

(3.15)

where cf, cs are the fluid and solid specific heats, ui is the average velocity of the fluid (averaged over a channel cross section), a the heat-transfer coefficient from

3.3 Heat Transfer

the channel walls to the fluid and aV the specific surface area (wall surface/reactor volume) and the Einstein convention of summation over repeated indices has been used. This convention will be used throughout this chapter when vector or tensor quantities are involved. The fluid and solid densities are volume-averaged quantities and given as

ρf = Φ f ρf( 0 ), ρs = Φs ρs(0)

(3.16)

where the superscript (0) indicates the corresponding material densities. The equation for the fluid temperature contains a source term S c that could, e.g., represent the input of thermal energy from a chemical reaction. Owing to the anisotropicy of the solid matrix the thermal conductivity is direction dependent. For this reason a thermal conductivity tensor kij appears in Eq. (3.15). The heattransfer coefficient a can be obtained from well-known correlations for rectangular channels. The model of Eq. (3.15) can be regarded as a mean-field model expected to describe the average temperatures of the fluid and the solid phase without incorporating the local fluctuations that are due to temperature gradients within single channels or channel walls. In order to compute the thermal conductivity tensor kij, the multitude of walls shown in Fig. 3.7 can be regarded as a network of thermal resistors. On this basis, the components of the thermal conductivity tensor are derived as −1

−1

 h + hs   w + ws  hf + hs  ks w f + w s  ks , kyy ≈ (hf + hs )  f kxx ≈ (w f + w s )  f  hs   ws  kzz ≈ Φsks ,

(3.17)

where ks represents the thermal conductivity of the solid material. All other components of the thermal conductivity tensor vanish. The mean-field model described here has been developed by Hardt and Baier to compute temperature fields in microreactors with low computational effort. Before that, Kim et al. [39] had used a 2D version of such a model to determine the temperature distribution in a microchannel heat sink. In order to assess the quality of the 3D mean-field model, the resulting temperature fields were compared with results from a full conjugate heat-transfer model explicitly accounting for all of the geometrical details of the multichannel stack. The results of such a comparison are displayed in Fig. 3.8. The temperatures were evaluated along the centerline of the reactor pointing in z direction. The heated region with nonzero source term extends up to 1 mm downstream from the inlet plane. The temperature curves show that in the inlet region the fluid and the solid temperature are not equilibrated, thus underlining the necessity of a two-temperature model. The same conclusion was drawn by Kim et al. [39] based on their 2D model. Overall, a good agreement between the mean-field and the full model is found, with local differences in temperature of 2–3 K at maximum and a total temperature range of about 35 K.

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3 Modeling and Simulation of Microreactors

Figure 3.8 Fluid (above) and solid (below) temperatures in a multichannel stack as obtained from the full (dots) and the mean-field model (lines).

The times needed to numerically solve the equations of the two models were found to differ by orders of magnitude. This makes the mean-field description an especially promising method for a fast evaluation of reactor designs. When developing a microreaction system it is often unclear how exactly the properties of the channel walls (geometry, thermal conductivity) influence the reaction performance. Usually, the goal is to achieve a temperature distribution as uniform as possible, in order to suppress unwanted side reactions and to increase

3.3 Heat Transfer

the selectivity of the process. Based on the volume-averaging approach presented here it should be easy to assess and compare the thermal performance of different reactors and to identify favorable designs. 3.3.4 Micro Heat Exchangers

Micro Heat exchangers are used for rapid heat transfer between a hot and a cold fluid, where, owing to the small thermal-diffusion paths, the size of the system can be reduced compared to conventional devices. Among the different flow schemes (cocurrent, countercurrent and crosscurrent) especially the countercurrent scheme has been studied theoretically, because of the practical relevance of countercurrent heat exchangers on the one hand, and the fairly simple model structure compared to crosscurrent heat exchangers on the other hand. A schematic drawing of a countercurrent micro heat exchanger is shown in Fig. 3.9. A characteristic feature of such devices is the fact that the thickness of the wall material separating the channels is of the same order of magnitude as the channel depth itself. This distinguishes micro heat exchangers from their macroscopic counterparts in which the width of the flow passage is usually well above the wall thickness. Owing to the comparatively large share of wall material, longitudinal heat conduction in the channel walls has to be taken into account when attempting to formulate a heat-exchanger model. In contrast, the models for conventional heat exchangers usually account for transversal heat transfer from one fluid to another only. Stief et al. [47] investigated the performance of countercurrent micro heat exchangers, especially with respect to the wall conduction effects and the choice of the wall material. They considered a flow of nitrogen in channels of length 10 mm, depth 50 µm and separating walls of a thickness between 125 µm and

Figure 3.9 Schematic drawing of a countercurrent microheat exchanger.

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3 Modeling and Simulation of Microreactors

500 µm. In order to reduce the computational effort, the heat-exchanger geometry was only discretized in axial direction, i.e. in the direction of the flow. In the direction perpendicular to the flow a fi xed heat-transfer coefficient was used to describe the exchange of heat between the fluid and the channel walls. When the thermal conductivity of the wall material is varied, characteristic temperature profi les for the channels with the hot and cold fluid and the channel wall are obtained. For very small values of the thermal conductivity, no heat is exchanged and the temperatures stay approximately constant. At intermediate wall thermal conductivities, almost linear temperature profi les are obtained, while at very high thermal conductivities the wall assumes a constant temperature and the temperature of the fluids changes rapidly in the entrance regions of the channels. A quantity characterizing the performance of a heat exchanger is the efficiency, which is the ratio of the transmitted heat and the maximum transmittable heat, given as

ε=

hot T inhot − T out cold T inhot − T in

(3.18)

in the case of equal volume flows of two identical incompressible fluids. Tin and Tout denote the temperatures at the channel in- and outlet, respectively. Stief et al. computed the heat-exchanger efficiency for a couple of characteristic geometries as a function of the wall thermal conductivity. A typical result for equal volume flows is displayed in Fig. 3.10. At low conductivities almost no heat is transferred, as expected. At high conductivities the wall assumes a uniform temperature due to axial heat conduction and the device becomes indistinguishable from a cocurrent heat exchanger that is limited to an efficiency of 50% at equal volume flows. At values around 1 W/(m K) the curve assumes a maximum, as axial heat conduction inside the walls is suppressed to a sufficient degree and transversal conduction to the neighboring channels is still efficient. Hence, in

Figure 3.10 Heat-exchanger efficiency as a function of the thermal conductivity of the wall, taken from [47].

3.3 Heat Transfer

typical micro heat exchangers the use of low-conductivity materials such as glass of polymers is preferable, and the common stainless steel materials are expected to reduce the efficiency. Heat losses to the exterior by conduction through the walls belong to the aspects not considered in the analysis of Stief et al. The vertical edges of the rectangular walls shown in Fig. 3.9 are in thermal contact with the housing or the flow-distribution manifold of the heat exchanger, and by these means heat is transferred to the surroundings. This effect was studied by Peterson [48] using a similar pseudo-1D model as Stief et al. discretized only in flow direction. Equal mass flow rates and identical incompressible fluids were assumed. Peterson expressed his results as functions of the two independent dimensionless parameters of the problem, the conduction parameter

λ=

kA ɺ pL mc

(3.19)

and the number of transfer units NTU =

αA ɺ p mc

(3.20)

In these expressions, k denotes the thermal conductivity of the wall material, A ɺ mass flow, cp the specific heat of the fluid, L the cross-sectional area of a wall, m the channel length and a the heat-transfer coefficient between the fluid and the wall. As a result of the computational model, the heat-exchanger efficiency (or effectiveness) and the ratio of the conduction to the flow losses was obtained, as displayed in Fig. 3.11. The conduction losses represent the heat flux dissipated to the surroundings via conduction through the walls, while the flow losses account for the fact that the cold fluid is not heated up to the inlet temperature of the hot fluid (the thermodynamic limit) and vice versa. Figure 3.11 shows that the conduction losses are usually larger than the flow losses. Especially for a highly conductive wall material they exceed the flow losses by one order of magnitude or more. Thus, in combination with the results of Fig. 3.10 the need for a low thermal conductivity construction material for countercurrent micro heat exchangers is underpinned in the most distinct way. The studies of Stief et al. and Peterson are both based on a description that assumes a vanishing transversal heat-transfer resistance within the channel walls, i.e. temperature gradients within the walls pointing in the streamwise direction. A more elaborate model allowing for computation of the complete 2D temperature field in the wall material based on a Fourier expansion was developed by Maranzana et al. [49]. Their results confirm that with the exception of the close vicinity of the channel inlets and within the parameter range studied, transversal gradients are negligible, thus corroborating the usefulness of pseudo-1D models for micro heat exchangers.

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Figure 3.11 Effectiveness and ratio of conduction to flow loss as a function of NTU and conduction parameter for a countercurrent micro heat exchanger, taken from [48].

3.4 Mass Transfer and Mixing

Similar to heat transfer, fast mass transfer is one of the key aspects of microreactors. Again, due to the short diffusion paths, microreactors enable a rapid mass transfer and a uniform solute concentration within the flow domain. A good control of reactant concentration throughout the whole reactor volume is a prerequisite for highly selective chemical reactions and helps to avoid hazardous operation regimes. In addition to this, overcoming mass-transfer limitations by rapid mixing allows the rapid intrinsic kinetics of chemical reactions to be exploited and enables a higher yield and conversion. However, when dealing with liquid-phase reactions, fast mixing remains a challenge even at length scales of 100 µm or less due to the small diffusion constants in liquids. Micromixing has been a very active field of research in the past few years (for overviews, see [50–52]). Not all of the micromixers reported in the literature are suitable candidates for micro process engineering. On the one hand, the Reynolds numbers in microreactors are usually higher than those in lab-on-a-chip devices that are often of the order of one. On the other hand, microreactor processes are typically continuous-flow processes for which batch-type micromixers are of little value. Owing to these differentiations, this section will mainly concentrate on mixing concepts and designs suitable for micro process engineering and will highlight some specific simulation methods for micromixers.

3.4 Mass Transfer and Mixing

3.4.1 Simple Mixing Channels

The simplest micromixer is the so-called mixing tee that is displayed on the left side of Fig. 3.12. Two inlet channels merge into a common mixing channel where mixing of the two coflowing fluid streams occurs. The mixing characteristics of T-type mixers were investigated by Gobby et al. [53] using CFD methods. Mixing of gases in a channel of 500 µm width was considered and certain geometric parameters such as the aspect ratio of the mixing channel or the angle at which the two inlet channels meet were varied. In order to quantify mixing, a mixing length was defined as the length in flow direction after which the gas composition over all positions of a channel cross section deviates by no more than 1% of the equilibrium composition. The CFD results for the mixing length as a function of Péclet number Pe = udh/D are displayed on the right side of Fig. 3.12, where u, dh, D are the average flow velocity, hydraulic channel diameter, and molecular diffusivity, respectively. The maximum Reynolds number considered was about 17. In addition to the CFD results, estimates of the mixing length based on the Fourier number Fo =

Dt l2

(3.21)

are shown in the figure. The Fourier number relates a residence time t in the mixing channel to the molecular diffusion constant D and a characteristic length scale l, which is the width of the channel. For a given value of Fo and a given flow rate, the length along the mixing channel necessary to achieve the corresponding Fourier number was determined. As is apparent from Fig. 3.12, the Fourier

Figure 3.12 Schematic design of a mixing tee (left) and CFD results for mixing of gases in a channel of 500 µm width and 300 µm depth, taken from [53] (right).

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3 Modeling and Simulation of Microreactors

number is a reasonable indicator for mixing that occurs for Fourier numbers between 0.1 and 1.0. Consequently, l2/D is an order-of-magnitude representation of the mixing time scale. The linear increase of the CFD-based mixing length as a function of Péclet number points to a very simple mixing mechanism via diffusion between coflowing fluid lamellae. Obviously, complex convectiondominated mixing mechanisms (for example driven by swirls or recirculating flows) are absent in the simple mixing-tee configuration for the range of Reynolds numbers studied. When the Reynolds number is increased, a regime with a flow pattern of broken symmetry is reached, as found by Bothe et al. [54]. At a Reynolds number of about 140 the symmetric flow pattern becomes unstable and an engulfment flow with an increased area of material interface between the two fluids is found. This is illustrated in Fig. 3.13 that shows maps of the computed concentration field over cross sections of the mixing channel for Reynolds numbers ranging from 119 (a) to 239 (f). The increase in material interface is related to an increase in mixing efficiency, thus making a mixing tee more efficient than predicted by simple

Figure 3.13 Concentration field over the channel cross section of a mixing tee for Reynolds numbers of 119 (a), 139 (b), 146 (c), 159 (d), 186 (e), and 239 (f), reproduced from [54].

3.4 Mass Transfer and Mixing

models accounting for mass transfer through an interface located in the symmetry plane of the mixer. In cases where symmetry breaking and the corresponding engulfment flows are absent (i.e. at sufficiently small Reynolds numbers) the steady-state concentration field in a mixing tee may be computed based on an analytical approach. The underlying assumption is that the flow moves with uniform velocity. Nguyen and Wu [51] give the following Fourier series expansion for the concentration field in a mixing channel of width w with two streams of equal viscosity and a volume flow ratio of r/(1 − r)  c( x , y ) nπ y x 2 ∞ sin nrπ 2n 2π 2 =r + ∑ cos exp  − 2 2 2 w c0 π n =1 n  Pe + Pe + 4n π w 

(3.22)

where the concentrations in the two streams at the channel entrance are c = c 0 and c = 0 and Pe is the Péclet number based on the channel width. The coordinates in the length and width directions of the microchannel are denoted as x and y, respectively. With such an analytical expression the evaluation of mixing performance in a simple microchannel is very time efficient. It should be noted, however, that an even simpler analysis based on the Fourier number already gives the correct order of magnitude for the mixing length, as indicated in Fig. 3.12. 3.4.2 Chaotic Micromixers

The performance of mixing tees is usually not sufficient for liquid-mixing applications, since corresponding diffusion constants are of the order of 10 −9 m2/s or smaller and mixing times far below one second may be required. As described above, mixing in straight channels occurs mainly by diffusion between coflowing streams with a characteristic time constant of l2/D. A reduction of the mixing time scale can be achieved if diffusion is superposed by convective transport. On the macroscale, convective transport is often due to turbulent eddies, and only in the very final stages of the mixing process does diffusion play a role. Unfortunately, in microscopic geometries flow is usually laminar, and a different convection mechanism has to be devised to speed up mixing. In the past few years, much attention has been devoted to creating flow schemes in micromixers that enhance the mixing performance by convective transport. In this context it was found that chaotic flows bear a great potential for speeding up mixing. The corresponding principle of chaotic advection has been studied for quite some time [55], but it has only recently been implemented in microchannel flows. Chaotic advection is synonymous with complex flow patterns in which extreme stretching and backfolding of fluid volumes occurs. It turns out that it is very difficult to numerically study chaotic micromixing of liquids. The reason lies in the discretization artifacts most CFD schemes suffer from, specifically the phenomenon of numerical diffusion [56, 57]. Owing to discretization errors artificial fluxes are induced that, for specific differencing schemes, take the form of

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3 Modeling and Simulation of Microreactors

diffusive fluxes and thus artificially increase the diffusion constant. The magnitude of numerical diffusion depends on the cell-based Péclet number (evaluated with the extension of a grid cell as length scale) and the relative orientation of the grid cells and the flow velocity. Specifically, if the flow is parallel to the grid lines, the diffusivity perpendicular to the flow direction is zero. In the case of a simple, regular flow pattern it is often possible to align the computational grid with the velocity field and thereby suppress numerical diffusion without having to prohibitively increase grid resolution. However, in the case of a chaotic flow this is no longer possible and numerical diffusion often severely falsifies the computational results. In order to be able to evaluate the performance of chaotic micromixers, the standard CFD methods for solving convection-diffusion equations have to be abandoned and replaced by a method that is largely free of discretization artifacts. Such a method is Lagrangian particle tracking that is based on tracking the paths of massless particles in a given flow field. Assuming that the flow velocity ui is given, the particle trajectories are obtained as solutions of the equation dx i = ui dt

(3.23)

where xi denotes the position vector of a particle. The mixing process is then studied by seeding the inlet of the mixer with a multitude of particles and recording how these particles have been redistributed by the flow at a downstream position. The fact that Eq. (3.23) does not contain any contribution of Brownian particle motion is often not a real drawback. If the flow itself displays a large degree of chaoticity, much of the redistribution of the initial concentration field is done solely by convective transport, and diffusion only completes the mixing process. In a coarse-grained description in which concentrations are evaluated by computing the particle numbers in specific grid cells, the inclusion of Brownian motion would not modify the results significantly in many cases. The computational effort for Lagrangian particle tracking can be reduced if the mixing section exhibits some periodicity. The structures inducing the chaotic flow are often arranged in a certain spatial period for which the flow field may be computed using periodic boundary conditions. Particles are then seeded onto the entrance plane of this section of the mixing channel and tracked up to its exit plane. By the fluid streamlines each point of a channel cross section (xk, yk ) is mapped to another point (xk+1, yk+1) of the corresponding cross section after one period of the fluidic structures, as shown in Fig. 3.14. Such a so-called Poincaré map can be written formally as (x k +1 , yk +1 ) = P[(x k , yk )]

(3.24)

The function P can be computed either from an analytical or a numerical representation of the flow field, making use of Eq. (3.23). In such a way, a 3D convection problem is essentially reduced to a mapping between two-dimensional

3.4 Mass Transfer and Mixing

Figure 3.14 Section of a mixing channel with a map P connecting the points of two cross-sectional planes.

Figure 3.15 Micromixer geometry with staggered groove structures on the bottom wall, as considered in [58] (reprinted with permission, © 2002, AAAS). The top of the figure shows a schematic view of the channel

cross section with the vortices induced by the grooves. On the bottom, confocal micrographs showing the distribution of two liquids over the cross section are displayed. Flow is from left to right.

Poincaré sections. The complete mixing process can be analyzed by successive applications of the Poincaré map. Lagrangian particle-tracking techniques have been used to evaluate the performance of several chaotic micromixers. One of the first micromixers for which convincing evidence for chaotic mixing was given is based on a design developed by Stroock et al. [58] and was named the “staggered herringbone mixer” (SHM). The corresponding mixer geometry is shown in Fig. 3.15. The bottom of the channel of height h and width w contains a staggered arrangement of grooves, where a fraction p of the bottom channel wall has grooves inducing a helical flow with a left-handed recirculation and the remaining bottom-wall fraction induces a right-handed recirculation. Schematic views of the channel cross section including projections of the streamlines are shown on top of Fig. 3.15. If p ≠ 1/2, the

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vortex pattern is asymmetric, and a superposition of two patterns exhibiting the larger of the two vortices on the left and on the right side, respectively, could result in chaotic flow. In a generalized sense, this is an implementation of the “blinking-vortex principle” proposed by Aref [59] some time ago. Aref showed that two vortex structures may be superposed in an alternating fashion in order to create a chaotic flow pattern. In the SHM the superposition is achieved by the alternating, staggered groove patterns shown in Fig. 3.15. The lower part of the figure shows channel cross sections with two streams of fluorescent and clear solutions after 0, 0.5 and 1 cycle. The images were recorded using a confocal microscope. The two different vortices are clearly visible and the third frame shows the first indications of a chaotic disturbance of the flow. Mixing in the SHM has been analyzed using Lagrangian particle tracking by Kang and Kwon [60]. By comparison with a mixer containing only simple groove structures (as opposed to a staggered herringbone structures) they could give evidence for the superior performance of chaotic micromixers. The Poincaré sections obtained from particle tracking were compared to the confocal micrographs of Stroock et al. (Fig. 3.16). The computed particle distributions show strong

Figure 3.16 Comparison of the confocal micrographs of Stroock et al. [58] (left) with the Poincaré sections obtained from Lagrangian particle tracking by Kang and Kwon [60] (right).

3.4 Mass Transfer and Mixing

similarities to the distributions of fluorescence intensity, thus proving that mixing in complex flows can realistically be described by Lagrangian particle tracking. With standard CFD techniques resting on the solution of a convectiondiffusion equation for the concentration field this is often not possible, since discretization errors would have blurred the thin lamellae visible in Fig. 3.16. Lagrangian particle tracking has also been applied to analyze chaotic mixing in a number of other channel designs. Stremler et al. [61] studied a twisted microchannel previously proposed by Liu et al. [62] and could show that for Reynolds numbers above 10 mixing is mostly chaotic, apart from a few unmixed, regular islands. Crosschannel micromixers have been studied by a number of authors [63–65]. In such devices there are multiple side channels intersecting with the main mixing channel. An oscillating pressure is imposed onto the side channels such that the time-averaged flow rate entering into the main channel is zero. The flow entering from the side channels distorts the material interface of the two fluids to be mixed, thereby increasing the interfacial area and promoting mixing. The analysis of mass transfer based on Lagrangian particle tracking has shown that for suitable values of the pressure excitation amplitude and frequency chaotic mixing is induced [63]. Many of the studies on micromixing reported in this section are more or less geared towards lab-on-a-chip applications, i.e. the target Reynolds number is about 10 or smaller. In microreactors, throughput is often a more important issue than in lab chips, and correspondingly, Reynolds numbers are higher and may easily reach values of several hundreds. A quite simple micromixer that is well suited for this Reynolds number regime was analyzed theoretically and experimentally by Jiang et al. [66]. The design of the mixer is shown on the bottom of Fig. 3.17. The mixing channel consists of a number of arc-like segments of alternating curvature, two of which are shown in the figure. As discussed in Section 3.2.2, a helical flow develops in such curved channels, with vortices appearing in the projection of the velocity field onto the channel cross section. For Dean numbers larger than 140 the flow exhibits four counter-rotating vortices, as indicated by the CFD results appearing in the middle of Fig. 3.17. In the mixer proposed by Jiang et al. a chaotic flow is created by alternating between the flow patterns in different channel segments, with the smaller pair of vortices either appearing in the right or the left half of the cross-sectional plane. In order to study the mixing performance, a modified version of the usual Lagrangian particle tracking was developed. Usually, two types of tracer particles are distributed over the entire entrance plane and the mixing quality is computed by counting the number of particles in grid cells on a plane located at a downstream position. Based on this method it is difficult to obtain reliable results for states close to complete mixing. Jiang et al. used an alternative method where only the material interface is seeded with particles (indicated as the dotted line in the upper half of one channel cross section in Fig. 3.17) and the length of the intersection of this interface with cross-sectional planes at various downstream positions is computed. This length scale should provide a good measure for mixing, since all the mass-transfer proceeds through the material interface, and can

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Figure 3.17 Two segments of a meandering mixing channel as analyzed by Jiang et al. [66] (bottom) and CFD results showing the projected velocity fields on different crosssectional planes (middle). The upper part of the figure shows the positions of massless particles after 6 channel segments initially seeded along the material interface.

be computed with high accuracy, since the same number of particles usually distributed over a plane can now be distributed along a line. The upper part of Fig. 3.17 shows the distribution of particles along the material interface after 6 channel segments for a Dean number of 200. The particles are distributed over almost the entire cross-sectional plane, thus documenting the pronounced and very complex stretching and folding of the interface. For Dean numbers above 100 an exponential growth of interfacial length scale as a function of residence time in the channel could clearly be proven. From these results it is apparent that mixing proceeds in a chaotic manner. This statement was underpinned by experimental results that showed that the position of close-to-complete mixing along the channel is virtually independent of the flow rate.

3.4 Mass Transfer and Mixing

3.4.3 Multilamination Micromixers

The chaotic micromixers described in the previous section offer a quite elegant solution to the micromixing problem, as complicated manifolds for distributing the fluid streams are no longer required and mixing is achieved by the intrinsic structure of the flow field. However, if the flow is chaotic in one part of the channel domain and regular in the other, unmixed islands that usually deteriorate process performance will remain. Thus, before using a chaotic micromixer in practice its mixing performance should be carefully evaluated over the whole range of Reynolds numbers considered. In comparison, multilamination mixers often require much more complex distribution manifolds and channel structures, but their mixing performance can usually be better controlled and predicted over the operation range envisioned. The principle of mixing by multilamination relies on the creation of a multitude of thin fluid lamellae between which mixing occurs by diffusion. Thus, the mixing times achievable with multlilamination mixers are directly related to the diffusion time scales between neighboring fluid lamellae. Figure 3.18 exemplifies the basic flow structure in a multilamination mixer: A multitude of fluid streams is guided through a flow chamber of varying width. For the mixer in the figure the streams are focused into a narrow channel by which the characteristic diffusion path is drastically reduced and mixing is enhanced. In a similar way as in Section 3.4.1 an analytical formula can be derived that allows for a fast computation of the concentration field in a multilamination mixer. When transforming the spatial coordinate in the streamwise direction (x in Eq. (3.22)) into a time coordinate by dividing the spatial differerential by the average flow velocity and integrating from the inlet to the outlet of the mixer, a 2D spatial problem is reformulated as a problem with one space and one time coordinate. Thus, in a frame-of-reference comoving with the flow and assuming a plug-flow profi le, the convection-diffusion problem is transformed into a pure

Figure 3.18 Multilamination mixer focusing a multitude of fluid streams through an inlet chamber into a narrow mixing channel. Alternating fluid inlets are distributed over a circular-arc-shaped manifold.

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diffusion problem. The resulting diffusion equation can be solved by the usual Fourier-transform techniques. A corresponding analytical model for multilamination mixing has been developed by Hardt et al. [67]. This model takes into account changes of the channel width along the flow path, i.e. it incorporates effects of focusing or defocusing, with the former often being used to speed up mixing. Specifically, a coordinate s orthogonal to the fluid lamellae is introduced that may be a curvilinear coordinate in regions where focusing or defocusing occurs. It is assumed that s lies between −w/2 and w/2, where w is a measure for the channel width. The concentration field is then expanded in harmonics c (s , t ) 1 ∞ ks = + ∑ an (t )sin n , kn = (2n + 1)π w c0 2 n =0

(3.25)

where only the sin terms contribute if the total number of fluid lamellae is even and initial concentrations of c 0 and 0 are considered. The time dependence of the concentration field arises from the fact that a frame-of-reference comoving with the flow is chosen, as described above. Inserting this expansion into the diffusion equation in the comoving frame of reference allows an (t) to be determined as t  dt′  an (t ) = an (0)exp  −Dkn2 ∫ 2  w (t′)   0

(3.26)

where D is the molecular diffusivity. The function w(t) incorporates the changes in channel width. In contrast to a channel of constant width an integral has to be evaluated to determine the Fourier coefficients. This integral represents the history of the mixing process and accounts for the changes in mixing speed due to width variations. Clearly, the model of Eqs. (3.25) and (3.26) should also be applicable to a simple mixing channel with only two fluid streams as considered in Section 3.4.1, i.e. Eq. (3.22) should be recovered when transforming to a formulation with two spatial coordinates instead of a space-time description. However, the difference between the two formulations is that Eq. (3.22) also takes into account diffusion in the streamwise direction, an effect becoming important at small Péclet numbers. Nevertheless the model for multilamination mixing is useful in most practical cases, since diffusion in the streamwise direction is usually negligible. It should also be pointed out that this model is free of numerical diffusion and thus not only allows a faster, but also a more reliable prediction of mixing performance in specific cases. As exemplified by Drese [68], the fast evaluation of mixing performance allows conducting parameter studies that would have been extremely time consuming when using standard CFD techniques. Reducing the solution of the convection-diffusion equation for the concentration field to a 1D problem in the comoving frame-of-reference clearly reaches its limits if the fluid lamellae get deformed and are no longer arranged in parallel. Such a situation may occur if the channel depth is not small compared to its width and the design comprises a sudden expansion or contraction of the flow

3.4 Mass Transfer and Mixing

passage. Hardt and Schönfeld [69] have analyzed multilamination mixers with cornerflow geometries using CFD techniques and identified situations where the multilamination pattern suffers strong deformations. The complex flow patterns found in such mixers may also mislead experimentalists who attempt to evaluate the mixing performance by photometric techniques based on the diffusion of a solute [69]. Creating a multitude of lamellae such as in Fig. 3.18 usually requires a quite complex flow-distribution manifold. There is, however, an alternative multilamination principle that enables a multiplication of the number of lamellae along the mixing channel: split-and-recombine (SAR) mixing. A SAR mixer consists of successive mixing elements each of which causes a doubling of the number of lamellae. An example of such a mixing element is shown in Fig. 3.19. The flow entering the element in the positive z-direction is split into two substreams that are then recombined. Thereby the number of fluid lamellae is increased from two to four. For n successive mixing elements, the number of lamellae is increased to 2n+1 and the diffusion paths are reduced correspondingly. The exponential reduction of diffusion paths resembles mass transfer in chaotic flows where fluid volumes are stretched and folded such that the width of fluid fi laments decreases exponentially. For this reason, SAR mixers are usually very efficient, without the need of incorporating very small channels. Furthermore, in contrast to some chaotic mixers, SAR mixers guarantee a largely uniform mixing quality over the entire channel cross section. For the channel design shown in Fig. 3.19 it could be shown that lamellae multiplication works in a close-to-ideal manner for Reynolds numbers below about 30 [70]. For higher Reynolds numbers inertia forces become important and cause a deformation and finally a merging of lamellae of the same type. Altogether, experiments indicate that at comparatively small Reynolds numbers the design shown enables an efficient liquid mixing at low pressure drops, thus

Figure 3.19 Mixing element of a SAR mixer (left) and optical micrographs showing lamellae multiplication in the first three mixing elements (right).

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qualifying the SAR mixer especially for the processing of highly viscous liquids [70]. As far as the computation of concentration fields in SAR mixers is concerned, the challenges are higher than in ordinary multilamination mixers. The reason for this is that the number of lamellae does not remain constant, rather there are discontinuous transitions between patterns with n and 2n lamellae. Hardt et al. [67] have formulated a strategy to deal with this problem. Based on the formulation of solving a 1D diffusion equation in a comoving frame-of-reference reproduced in Eqs. (3.25) and (3.26) the mixing element shown in Fig. 3.19 is subdivided in two sections. In the first section, the flow passes through two separated branches, in the second section it only passes through one branch. In each of these sections the time evolution of the Fourier coefficients is computed by Eq. (3.26). On the interface, the downstream Fourier coefficients are obtained from a sum involving the upstream Fourier coefficients. In addition, the evolution of the concentration field in a complete SAR mixer comprising several successive mixing elements is obtained in an iterative manner from the analysis of a single mixing element, i.e. the Fourier coefficients on the outlet plane shown in Fig. 3.19 are identified with the coefficients on the inlet plane of the next element. In this way Hardt et al. computed the degree of mixing as a function of position along the channel and found a reasonable agreement with experimental data. Thus, the usefulness of the 1D model for multilamination mixing is corroborated by experimental data and reasonable results can be obtained even for such complex situations involving flow passages of varying width as well as flow splitting and recombination. There are alternative flow topologies for SAR mixers that have been proven to cause a multiplication of fluid lamellae. The design of a mixing element shown in Fig. 3.19 was especially optimized with regard to extending the Reynolds number range in which the SAR principle is implemented, consistent with the throughput requirements typical for microreactors. The disadvantage is that this mixing element poses big challenges for microfabrication. When only very small Reynolds numbers need to be considered the design constraints are less severe, and Chen and Meiners [71] have proposed a design that is easier to fabricate. The flow topology is such that a doubling of the number of fluid lamellae is achieved after two SAR steps, and the functionality with respect to lamella multiplication has been proven using dyed aqueous solutions. In the same spirit as above, the model of Eqs. 25 and 26 can be used to compute the evolution of concentration fields in such a mixer if the upstream and downstream Fourier coefficients at the branching points are matched with each other. 3.4.4 Hydrodynamic Dispersion

Mass transfer of a solute dissolved in a fluid is not only the fundamental mechanism of mixing processes; it also determines the residence-time distribution in microfluidic systems. As mentioned earlier in Section 3.2.3, in many applications

3.4 Mass Transfer and Mixing

it is desirable to have a narrow residence-time distribution of concentration tracers being transported through a microfluidic system. An initially narrow concentration tracer will suffer a broadening (i.e. a dispersion) due to two different effects. First, in some regions of the flow domain of a system the fluid velocity will be smaller than in others, thus leading to a longer residence time of molecules being transported preferably through these parts of the domain. However, due to Brownian motion the molecules will also sample some of the other regions with higher flow velocity. Hence, molecular diffusion might reduce the dispersion of a concentration tracer. On the other hand, by diffusion an initially localized concentration tracer in a fluid at rest will get dispersed. From these arguments it becomes clear that hydrodynamic dispersion depends on a quite subtle interplay of convective and diffusive mass transfer, and the evolution of a concentration tracer as it gets transported through the flow domain depends on various factors such as the flow profi le, the magnitudes of the flow velocity and the diffusion constant. The key analysis of hydrodynamic dispersion of a solute flowing through a tube was done by Taylor [72] and Aris [73]. They assumed a Poiseuille flow profi le in a tube of circular cross section and were able to show that for large enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation

∂c ∂c ∂ 2c +u = De 2 ∂t ∂x ∂x

(3.27)

In this expression, c denotes the concentration averaged over the cross section of the tube, u the average velocity and De an effective diffusivity, also denoted dispersion coefficient, which is given by De = D +

u 2R 2 48D

(3.28)

where D is the molecular diffusivity and R the radius of the tube. The factor 1/48 multiplying the velocity-dependent term is generic for tubes of circular cross section and gets modified when other geometries are considered. In many cases the second term, which can be rewritten as DPe2/48, dominates over the first one, which is a purely diffusive contribution. Hence, due to convection, a concentration tracer is usually dispersed much more strongly than it would have been by diffusion alone. A notable feature of Eqs. (3.27) and (3.28) is their independence of any initial condition. Independent of how the solute is distributed over the channel cross section and along the channel initially, the description given by Taylor and Aris will be valid in the limit of large times (t → ∞). When exactly this limit is reached with a given accuracy depends on the initialization of the concentration field. A rough guideline is provided by the Fourier number of Eq. (3.21) evaluated with the tube radius as length scale. The Fourier number can be regarded as a dimensionless time coordinate that compares the actual time to the

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time a molecule needs to sample the cross-sectional area of the tube. The validity of the Taylor–Aris description should be related to the condition that the Fourier number assumes values of order 1 or larger. The analysis of Taylor and Aris was extended to arbitrary time values by Gill and Sankarasubramanian [74] for the dispersion of an initially plug-like profi le, i.e. c 0 ( x ≤ l / 2) c( x , r , 0) =   0 ( x > l / 2)

(3.29)

where the radial coordinate of the tubular geometry is denoted by r and l is the length of the plug. They derived a generalized evolution equation for the areaaveraged concentration of the form ∞ ∂c ∂nc = ∑ kn (t ) n ∂ t n =1 ∂x

(3.30)

which is valid without any restriction on t. The derivatives in the infinite series appearing on the right side are multiplied by time-dependent dispersion coefficients kn. In the Taylor–Aris limit, all of the dispersion coefficients except k1, which describes the convection of the tracer with the flow, and k2, which determines the spreading of the tracer, are negligible. When moving to smaller times, the time dependence of k2 needs to be taken into account, while all higher dispersion coefficients are still negligible [74]. Only at very small times does the higher dispersion coefficients become important. For the case they considered, Gill and Sankarasubramanian found that k2 can be regarded as time independent for Fourier numbers greater than about 0.5. Even if the results discussed above highlight some of the most important aspects of hydrodynamic dispersion, they were based on cylindrical ducts that are not the generic geometry used in the field of microfluidics. In chemical microprocess technology, tubular sections are used to connect different units, however, the channels contained in microreactors typically have a rectangular or tub-like cross section. Dispersion in rectangular channels was studied in detail by Doshi et al. [75]. The evolution equation Eq. (3.27) is still valid in this case, however, the expression for the dispersion coefficient Eq. (3.28) needs to be modified. While Aris [76] was still able to obtain a simple analytical expression for the dispersion coefficient related to flow between parallel plates, the corresponding expression for rectangular channels is a complicated series expansion. This is not very surprising, since the exact form of the flow profi le in a rectangular channel is given in the form of an infinite series as well. Dutta and Leighton [77] found a simpler relationship that approximates the exact expression for the dispersion coefficient in rectangular channels within an error of 10%. In addition to this, they considered tub-like channel cross sections that are typically obtained by isotropic etching processes. For the latter, they employed a numerical scheme allowing the dispersion coefficient to be computed. On this basis, they compared

3.4 Mass Transfer and Mixing

different channel geometries and identified favorable and less-favorable designs. In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary flow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form Decur = D (1 + Pe2 f (K , Sc))

(3.31)

where Dcur e is the dispersion coefficient in curved ducts, D the molecular diffusivity, Pe the Péclet number of the flow, and the function f depends on the Dean number K defined in Eq. (3.7) and on the Schmidt number Sc, which is the ratio of the kinematic viscosity and the diffusivity. Daskopoulos and Lenhoff found the following asymptotic behavior for the ratio of the dispersion coefficients in curved and straight ducts Decur De

 =1 K →0  −1 ∝ K K → ∞

(3.32)

As mentioned earlier, in curved channels secondary flow patterns of two or four counter-rotating vortices are formed. These vortices redistribute fluid volumes in a plane perpendicular to the main flow direction. Such a transversal mass transfer reduces dispersion, a fact reflected in the K −1 dependence in Eq. (3.32) at large Dean numbers. For small Dean numbers, the secondary flow is negligible, and the dispersion in curved ducts equals the Taylor–Aris dispersion of straight ducts. Similarly as in the case of mixing, the transversal redistribution of matter in a curved channel can be improved in a sequence of channel segments of alternating curvature. By superposition of different vortex patterns a chaotic flow is induced, as described by Jiang et al. [66]. By Lagrangian particle tracking in combination with a CFD computation of the flow field it can be shown that the residence-time distribution of tracers in the channel built from segments as shown in Fig. 3.17 becomes very narrow and approaches a delta-like distribution of zero width. In practice, however, the width of the residence-time distribution is limited by the molecular diffusivity of the tracers that was neglected in the simulations of Jiang et al. Nevertheless, a substantial reduction of hydrodynamic dispersion

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can be achieved in a meandering channel when compared to a straight channel.

3.5 Chemical Kinetics

Most plants or reactors in chemical microprocess technology inevitably contain a unit where chemical conversion takes place. The goal might be to produce fine chemicals with a high yield and selectivity or to screen a large number of reactions in parallel. Hence, a thorough understanding of chemical kinetics is a key requirement for the successful design and optimization of microreaction devices. For this purpose, reliable models of reaction kinetics coupled to the transport equations of momentum, heat and matter are needed. The type of kinetic model to be used depends on the type of reaction considered. For a homogeneous reaction occurring in the bulk of the fluid, a power-law kinetic model is often appropriate (see, e.g., [79]). In such models the rate of a certain reaction depends on a product of powers of the species concentration. On the other hand, heterogeneously catalyzed reactions are often conducted in microreactors. In a strict sense, power-law kinetics does not capture the dynamics of such processes over the full range of pressure, temperature and concentrations. Rather, a more complicated kinetic model of, e.g., Langmuir–Hinshelwood type [80] would have to be used. Nevertheless, power-law kinetics is frequently applied to heterogeneously catalyzed processes in a limited parameter range to simplify the description. Independent of the specific modeling strategy, the kinetic equations often exhibit a nonlinear dependence on the species concentrations and a reaction rate rapidly increasing with temperature. In combination with the transport equations for mass, momentum and heat, the resulting numerical problem is usually challenging due to the nonlinearities and the multitude of time scales involved. For this reason, methods are needed to eliminate some of these difficulties and to simplify the numerical structure. 3.5.1 Numerical Methods for Reacting Flows

The solution of the species concentration equations in combination with the momentum and the enthalpy equation generally requires an iterative procedure. A rough sketch of the numerical structure of a stationary reacting-flow problem is given as  Acc A  uc  ATc

Acu Auu ATu

AcT   c   bc  AuT   u  = bu      ATT  T  bT 

(3.33)

3.5 Chemical Kinetics

where c, u and T denote the vector of concentration, velocity and temperature fields, respectively. Owing to the nonlinear nature of the problem, the coefficients of the different matrices A ab still depend on the unknowns c, u and T. The crosscoupling between different field quantities is provided by those matrices A ab with a ≠ b . The set of nonlinear algebraic equations is solved iteratively, i.e. starting with an initial guess the approximation is successively improved until convergence is reached. Depending on the nature of the chemical-reaction term entering the species-concentration equation, different strategies may be applied to solve Eq. (3.33). For an intrinsic kinetics characterized by a much shorter time scale than transport of momentum, heat and matter, it is often preferable to set up an iteration scheme where a number of iterations of the species-concentration equation are performed during one iteration cycle of the remaining equations. However, for a fast reaction that is heat- and mass-transfer limited (for example in a situation where the reactants are not premixed), comparable iteration cycles of the species-concentration equation and the remaining equations might be sufficient. Apart from the coupling of chemical kinetics to the transport equations, the chemical-reaction dynamics itself may pose numerical challenges when a number of different reactions are superposed. In such a case the rate of disappearance of a chemical species i can be written as −

nɺ i = Rij r j V

(3.34)

where r j is the rate of the jth reaction and Rij is a matrix defining how a specific reaction contributes to a change in concentration of the chemical species involved. For brevity, the Einstein convention of summation over repeated indices was used. Quite frequently it occurs that the time scales characterizing the different reactions vary by orders of magnitude, such that the fast reactions are already completed while the slow reactions have not yet progressed to any appreciable degree. The corresponding stiff differential equations are usually solved using an implicit time-integration scheme that allows comparatively large time steps without suffering from numerical instabilities or predicting unrealistic asymptotic states [81]. However, implicit time integrators involve the solution of a (generally nonlinear) algebraic system of equation for each time step that is done by some iterative scheme such as Newton’s method. For reaction systems with a broad spectrum of time scales these iteration schemes can fail to converge, with the consequence that very small time steps have to be chosen. Such a situation is related to high computational costs, and methods are needed to simulate extremely stiff reaction systems more efficiently. Methods based on the partitioning of a reaction system into fast and slow components were proposed by several authors [82–84]. A key assumption made in this context is the separation of the space of concentration variables into two orthogonal subspaces Q s and Q f spanned by the slow and fast reactions. With this assumption the time variation of the species concentrations is given as

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−

nɺ i = (Q s , j )i ( yɺs ) j + (Q f , j )i ( yɺ f ) j V

(3.35)

The notation is such that (Qs,j ) i, (Q f,j ) i denotes the ith component of the jth basis vector in the subspace of slow and fast reactions, respectively. The corresponding expansion coefficients are ( yɺ s ) j and ( yɺ f ) j, respectively, and are expressed by the reaction rates via ( yɺs )i = (Q sT, j )i R jk rk

(3.36)

( yɺ f )i = (Q Tf , j )i R jk rk

(3.37)

If the time scale of the fast reactions is much shorter than that of the slow reactions, it can be assumed that the former are completed at an initial stage of the latter. Mathematically, this assumption reads

(Q Tf , j )i R jk rk = 0

(3.38)

Equation (3.38) represents a set of algebraic constraints for the vector of species concentrations expressing the fact that the fast reactions are in equilibrium. The introduction of constraints reduces the number of degrees of freedom of the problem, which now exclusively lie in the subspace of slow reactions. In such a way, the fast degrees of freedom have been eliminated, and the problem is now much better suited for numerical solution methods. It has been shown that, depending on the specific problem to be solved, the use of simplified kinetic models allows the computational time to be reduced by two to three orders of magnitude [85]. 3.5.2 Reacting Channel Flows

In chemical micro process technology there exists one class of reactor designs that deserves the term “generic”, since many of the microreactors reported in the literature are based on this design concept. The design comprises at least one rectangular microchannel, often a multitude thereof, with a solid catalyst attached to the channel walls. The reacting fluid flows through the channel, while the reagents diffuse to the channel wall where they undergo chemical reactions. There exist two versions of this design concept, as displayed in Fig. 3.20. Either a smooth surface, often a metal layer, acts as catalyst, or the reaction occurs in a catalytically active porous medium. Clearly, the advantage of the porous catalyst layer is the higher specific surface area offering more reaction sites to the reagents. However, fully coupled simulations of reaction-convection-diffusion processes in catalyst-coated channels are quite rare, so most of the studies reported in this section are based on the concept of wall-catalyzed reactions.

3.5 Chemical Kinetics

Figure 3.20 Reaction channels with a smooth surface (left) and a porous medium (right) as catalyst.

Figure 3.21 Two-dimensional model geometry of a microchannel with a reaction occurring at the lower channel wall.

For the reasons described above, reaction-convection-diffusion problems tend to be difficult to solve numerically. Hence, the simulation of reacting flows in three dimensions or parameter studies of microreaction devices may be very time consuming. In order to enable rapid prototyping of microreactors, efficient modeling strategies with a minimum expenditure of computational resources are needed. The modeling approach developed by Gobby et al. [86] allows rapid assessment of a limited class of reacting microchannel flows. They assumed a microchannel of length L and depth h with a first-order reaction occurring at one of the channel walls, as depicted in Fig. 3.21. In cases where the flow profi le is independent of the axial position in the channel and the problem can be approximated by a twodimensional model, the mass-transport equation for a chemical species a can be written in dimensionless form as Peh ∂c ∂ 2c a u(η) a = ∂ζ ∂η2 L

(3.39)

where the axial and transverse coordinates z and h were nondimensionalized by the channel length L and the channel depth h. The reactant concentration is denoted by ca, the velocity by u, the Péclet number is expressed by the average velocity u and the diffusion constant D as Pe = uh/D. Equation (3.39) is solved subject to the boundary conditions of an impermeable upper channel wall and a firstorder reaction with rate constant k occurring at the lower channel wall. Such a first-order reaction term surely does not adequately capture the mechanism of heterogeneous catalysis, it might, however, be a reasonable approximation to the

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kinetics in a limited parameter or operation range. An important dimensionless group characterizing the reactive flow is the Damköhler number, defined as Da =

kh D

(3.40)

which characterizes the ratio of the diffusive and the reactive time scale. The mass-transport equation has a separable solution of the form c a (ζ , η) = ca (ζ ) f a (η)

(3.41)

where 1

ca (ζ ) = ∫ c a (ζ , η)dη

(3.42)

0

When inserting this ansatz into Eq. (3.39), the solution can be determined in the form of an eigenfunction expansion, as shown by Walker [87]. The parameter controlling the number of terms of this expansion having to be taken into account is Peh/L, which is usually of the order 0.01–1 in microreactors. For this reason, often only the first term contributes. With the entrance condition c a ( z ) = 1 the axial dependence can then be written as ca (ζ ) = exp( − λaζ )

(3.43)

where the eigenvalue l a is given as the solution of a nonlinear algebraic equation. Gobby et al. compared their analytical results to full numerical simulations and found good agreement. In addition to isothermal flows, they also determined analytical solutions for nonisothermal reacting flows and extended their model to second-order kinetics. Hence, they developed a class of models that may provide a simple characterization of reacting flows in microchannels without the need to do a full numerical simulation. Commenge et al. [88] used a similar analytical model for reacting flows in microchannels to assess the quality of simple plug-flow models that may be used to estimate reaction-rate constants. Microreactors lend themselves to measure intrinsic rate constants of chemical reactions, as due to the short diffusion paths, heat- and mass-transfer limitations can be eliminated. The simplest way to deduce the rate constant k of a first-order heterogeneously catalyzed reaction at the walls of a tube is by assuming the reaction to occur in the volume of a plug-flow reactor. In this way the wall reaction is replaced by a pseudohomogeneous reaction and the velocity profi le of the flow is ignored, which means that effectively a onedimensional model is used. By measuring the inlet and outlet concentration of the reacting component, the rate constant is then obtained as k=

uR  c a (ζ = 1)  ln 2L  c a (ζ = 0)

(3.44)

3.5 Chemical Kinetics

where the notation is chosen similar to the previous paragraph and L, R measure the length and radius of the tube, respectively. Two effects are not taken into account by this expression. First, radial concentration gradients are ignored. Secondly, dispersion in the tube as discussed in Section 3.4.4 is neglected. Commenge et al. extended the one-dimensional model of reacting flows to include Taylor–Aris dispersion, i.e. they considered an equation of the form d 2c a Pe* dc a − − βc a = 0 dζ 2 2 dζ

(3.45)

where Pe* is a modified Péclet number containing the Taylor–Aris dispersion constant instead of the diffusivity and b is a dimensionless parameter representing the pseudohomogeneous reaction. In order to study the influence of dispersion on chemical conversion, the solution of Eq. (3.45) was compared to the solution of the corresponding two-dimensional problem, obtained in a similar way as sketched in the previous paragraph. It turned out that for a Damköhler number of 1, no satisfactory agreement between the one- and the two-dimensional model was achieved. The inclusion of Taylor–Aris dispersion improved the concentration profi les to a certain degree with respect to a plug-flow model, however, the main reason for the deviations are the radial concentration gradients that are not accounted for in the one-dimensional models. Hence, when attempting to extract intrinsic reaction-rate constants from comparisons of experimental results with results of one-dimensional reactor models, care should be taken to work in a regime of Damköhler numbers significantly smaller than 1. 3.5.3 Heat-exchanger Reactors

The design of multichannel microreactors for gas-phase reactions is typically based on a stack of microstructured platelets. For strongly endothermic or exothermic reactions, it lends itself to alternate between layers of reaction channels and heating or cooling gas channels that supply energy to or withdraw it from the reaction. Such a setup is similar to the heat-exchanger design depicted in Fig. 3.9. Within this class of microreactor designs a choice can be made between different flow schemes of the gas streams in adjacent layers (co-, counter- or crosscurrent). The countercurrent coupling of an endothermic reaction to a heating gas stream in a multilayer architecture was studied by Hardt et al. [41]. The 2D geometry their model was based on is displayed in Fig. 3.22. The dynamics of a heterogeneously catalyzed gas-phase reaction occurring in a nanoporous medium in combination with heat and mass transfer was simulated using a finite-volume approach. In contrast to other studies of similar type, heat and mass transfer in the nanoporous medium was explicitly accounted for by solving volume-averaged transport equations in the porous medium. Such an approach made it possible to compare the transport resistances in the gas phase and in the porous medium and to study the tradeoff between maximization of

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Figure 3.22 2D model of a countercurrent heat-exchanger reactor with a nanoporous catalyst layer deposited on the channel wall.

Figure 3.23 Normalized concentration profile of a reacting species across a microchannel of 500 µm width with a 100- µm catalyst layer deposited on the wall.

catalyst mass and minimization of mass-transfer resistance due to pore diffusion. A typical concentration profi le of a reacting chemical species that is converted by the catalyst is displayed in Fig. 3.23. Due to the small pore size with an average diameter of 40 nm, the effective diffusivity in the porous medium is quite small and large concentration gradients build up, whereas in the microchannel the gradients are negligible. Typical catalyst effectiveness factors for a 100 µm catalyst layer were found to be of the order of 0.4. One of the outstanding potentials of microreactors is an efficient utilization of the catalyst material. In conventional fi xed-bed technology, catalyst pellets for liquid reactions are usually of a size of 2–5 mm [79]. Due to diffusive limitations in such comparatively large pellets, reactions often occur in a region close to the surface. A main objective of the work of Hardt et al. was to study the influence of heat transfer on the achievable molar flux per unit reactor volume of the product species. They compared unstructured channels to channels containing microfins. Heat-transfer enhancement due to microfins resulted in a different axial tempera-

3.5 Chemical Kinetics

ture profi le with a higher outlet temperature in the reaction gas channel. Due to this effect and by virtue of the temperature dependence of the reaction rate, an improvement of heat-transfer resulted in a significantly higher specific product molar flux. For the system under study, the heat-transfer enhancement achievable with microfins was found to increase the specific molar flux by about a factor of two. Such model studies show that a complex interplay between flow, heat and mass transfer may occur in microreactors and underline the need for fully coupled simulations incorporating conjugate heat transfer and transport in porous media. The optimization of heat transfer in a heat-exchanger reactor was also the objective of the work of TeGrotenhuis et al. [89]. Specifically, the exothermic water-gasshift (WGS) reaction CO + H2O ↔ H2 + CO2

(3.46)

that is utilized in fuel reformers to reduce the level of carbon monoxide, was considered. When the temperature level of an exothermic, reversible reaction such as the WGS reaction increases, the kinetics is accelerated but the equilibrium is shifted more towards the feed components. Due to this fact, neither very low nor very high temperatures are optimal when the goal is to maximize the space-time yield for a given conversion. Rather there is a specific temperature trajectory, i.e. a specific functional dependence of the reaction temperature on time, which allows the space-time yield to be maximized. Due to their short thermal diffusion paths, microreactors allow the temperature profile in a reaction channel to be controlled much better than conventional equipment. TeGrotenhuis et al. analyzed a countercurrent heat-exchanger reactor for the WGS reaction with integrated cooling gas channels for removal of the reaction heat. The computational domain of their 2D model on the basis of the finiteelement method is sketched in Fig. 3.24. The reactor design does not allow for a detailed adjustment of the temperature profile in the reaction gas channel, however, by varying the cooling gas inlet temperature and the ratio of cooling gas and reaction gas flow rate, different temperature profi les can be imposed. The reaction dynamics studied in the simulations is such that by release of reaction heat close to the inlet, the reaction gas temperature rises considerably.

Figure 3.24 Model of a countercurrent heat-exchanger reactor for exothermic reactions. The dashed lines indicate symmetry planes.

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Owing to the fast kinetics at high temperatures, much of the CO conversion is already achieved in the inlet region. The remaining conversion is then mainly due to a reduction of the temperature level and the corresponding shift in chemical equilibrium. This part of the process requires a large channel length. Thus, even minor differences in conversion achievable by an improved temperature control may result in a considerable reduction of reactor size and the required amount of catalyst. When considering more advanced reactor designs allowing for fine tuning of the temperature trajectory, CFD simulations are indispensable for performance optimization.

3.6 Conclusions and Outlook

The discussion of modeling and simulation techniques for microreactors shows that the toolbox available at present is quite diverse and goes well beyond the standard capabilities of CFD methods available in commercial solvers. In microreactors, special methods needed for the modeling of noncontinuum physics play only a minor role and most of the effects are described by the standard continuum equations. However, even if the laminar nature of the flow somehow reduces the difficulty of simulation problems compared to macroscopic flows, there are a number of problems that are extremely difficult and require very fine computational grids. Among these problems is the numerical study of mixing in liquids that often suffers severely from discretization artefacts. An ample subject of growing importance that has not been discussed in this overview due to space limitations is modeling and simulation of multiphase microflows. Macroscopic multiphase flows are often modeled using an Euler–Euler formulation of interpenetrating continua that is a volume-averaged description where phase boundaries are no longer explicitly resolved. When the spatial extension of bubbles or droplets becomes comparable to the channel dimensions, the gas/liquid or liquid/liquid interfaces need to be modeled explicitly. There are methods available for such kind of problems (e.g. the volume-of-fluid method) that are, however, computationally much more expensive than the Euler–Euler formulations employed for macroscopic flows. It is not only for the computational requirements that the predictive power of multiphase microflow computations is much lower than that of their counterparts for single-phase flow. As an example, moving contact lines are relavant for gas-liquid flows and in some cases flow is accompanied by phase-change phenomena. Both the motion of a gas-liquid-solid contact line and the evaporation of liquid heavily depend on complex transport phenomena in the close vicinity of the contact-line region that are only partially understood until now. Consequently, the simulation of such phenomena as flow boiling in microchannels is in its infancy and its predictive power is very limited. Such examples underline the necessity of further research on multiphase microflows. Currently, corresponding numerical schemes are available, but the models do not adequately account for the boundary conditions at the contact line.

References

A further topic that has not been discussed in this overview and that is only beginning to evolve refers to system-level simulations of micro process devices. In conventional process technology there is often an interplay between detailed CFD and system-level simulations of physicochemical processes. Owing to their large computational requirements, CFD methods can usually not be employed for the simulation of chemical reactors or plants consisting of a number of different subunits. Rather, CFD results enter into system models that treat the subunits in a simplified manner by just referring to characteristic curves specifying and input-output behavior. In such a way the flow in a pipe, as an example, is characterized by a flow rate as a function of applied pressure difference. In conventional process technology a library of component models already exists that may be further extended and refined by incorporating the results of CFD simulations. However, in micro process technology such system-level approaches do not yet seem to have been developed to an appreciable degree. A natural step for the further development of modeling and simulation techniques for microreactors would be to formulate system-level models that allow a coupling to CFD simulations. In such a way the commercial application of microreactors could also gain additional impetus, as system-level models would allow a fast evaluation of processes unparalleled by the comparatively time-consuming approaches presently available.

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors Rüdiger Lange

4.1 Introduction

Gas-liquid-solid reactors with a trickle-flow regime are the most widely used type of three-phase reactors and are usually operated under steady-state conditions. The behavior of this kind of reactor under the other three-phase fi xed-bed reactors is rather complex due to gas and liquid flow concurrently downward through a catalyst packing. For process intensification it is required to improve some of the specific process steps in such chemical reactors. Figure 4.1 shows an overview of different factors that influenced the trickle-bed reactor performance. In the last decade trickle-flow reactors under unsteady-state operation, i.e. periodic change of reactant feed concentration and/or volumetric flow rate were studied for improving the trickle-bed reactor performance. Unsteady-stateoperated trickle-flow reactors aimed at enhancing their productivities have been proposed and studied above all by Silveston et al. [1, 2], Haure and coworkers [3– 5], Hanika et al. [6, 7] and Lange and coworkers [7–12] under so-called slow-mode conditions. Khadilkar et al. [13–15] also studied alpha-methylstyrene hydrogenation in a trickle-bed reactor with a 26-cm long catalyst bed (0.5% Pd on aluminum, spheres). In the on/off mode, it was possible to achieve an efficiency increase for the gas-limited reaction (up to 60% at a lowest liquid load of L = 0.068 kg/m2 s), while mean conversion increases with decreasing mass transfer inhibition of the gas component (low split and long gas-residence time). In addition to the on/off mode, investigations were carried out to study periodic process operation in trickle-bed reactors while changing between two liquid throughputs (base/peak mode). The authors (Haure et al. [3], Lange et al. [8] and Khadilkar et al. [13]) have found consistently that potential conversion increases tend to be lower. By variation of the liquid throughput in the base/peak mode, Khadilkar et al. [13] were able to increase the reactor efficiency by 12% despite liquid limitation in the reaction. The higher liquid velocity in the peak phase improves the liquid distribution and, as described by Lange et al. [8], reopens obstructed channels Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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Figure 4.1 Influence factors on the trickle-bed reactor performance.

thereby activating the stagnating liquid portions (from the base phase) for the reaction. The experimental and simulation results of Liu et al. [16] showed also for a complex reaction system (2-ethylanthrahydroquione-hydrogenation) an improved trickle-bed reactor performance under unsteady-state-operation conditions. The selectivity of the consecutive reaction was increased up to 12% in comparison to steady-state conditions. In the following papers are presented further theoretical studies including used mathematical models of unsteady-state-operated TBRs [17–20]. Schubert [21] analyzed dynamic reactor models of recent years and presented a comparison of their useful applications for the simulation of periodically operated reactors. Generally, an improvement is attained if the time-average performance of the periodic operation is larger than that of the corresponding steady-state process. The following will explain one way of essentially process intensification of trickle-bed reactors by periodic operation. The unsteady-state operation was considered as square waves cycling liquid flow rate at the reactor inlet. The hydrogenation of alpha-methylstyrene to cumene: C6H5(CH3 ) = CH2(L) + H2(G) → C6H5CH(CH3 )2(L) over a palladium catalyst (0.7% Pd/γ -Al2O3 ) was selected as a model reaction. Own experimental trickle-bed reactor investigations showed for this reaction system that the periodic variations of liquid flow rates by on/off

4.1 Introduction Table 4.1 Experimental conversions of alpha-methylstyrene

under periodic operation conditions depending on periodical time, (on/off mode: ReL = 0.23/0, S = 0.5, T = 303 K, P = 3 bar, UA,steady-state = 0.33). tperiod, min

UA,us experimental, –

1.0 2.0 4.0 10.0

0.36 0.38 0.41 0.44

Figure 4.2 Flow regimes in fixed-bed reactors under gas/liquid downflow conditions.

modus can lead to a substantial improvement of the TBR performance (see Table 4.1). Further experimental data and explanation of the experimental set up may be found in earlier papers of Lange et al. [8–10]. The subject of this work is the theoretical investigations of periodical operation of trickle-bed reactors in comparison to steady-state operation based on these experimental results. Generally, gas-liquid-solid fi xed-bed reactors can operate in hydrodynamically different regimes whose boundaries depend on gas and liquid superficial velocities, catalyst bed and fluid properties. After Charpentier and Favier [22] and Gianetto and Specchia [23] in gas-liquidsolid reactors under different ratio of gas and liquid flow conditions, various flow regimes (e.g. trickling, pulsing, bubble and spray) might exist (see Fig. 4.2). Most common in practice is a trickling flow that occurs at relatively low gas and liquid flow rates. Under these conditions in trickle-bed reactors (TBRs) complete liquid films around the particles would break up into partial fi lms, rivulets and droplets and a so-called partial wetting trickling regime exist. It is easier for hydrogen to enter the no wetted or very thin wetted part of partially wetted catalyst pellets because the mass-transfer resistance between a gas and the particle sur-

81

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors

face would be less than between a liquid and the particle surface or very thin wetted particle surface. The performance of trickle-bed reactors may be affected by many factors, such as interphase mass transfer, intraparticle diffusion, axial dispersion and incomplete catalyst wetting. Therefore, knowledge about these influenced factors is important for their mathematical description by an unsteady-state reactor model. Until now, the literature analysis shows the experimental and theoretical understanding of trickle-bed reactors under unsteady-state-operation conditions has improved, but not considerably. The following studies are focused on the trickling regime under unsteady-state-operation conditions.

4.2 Modeling

An essential task in chemical reaction engineering focuses on the modeling, simulation and optimization of the different reactor types. This approach is based on the mathematical description (modeling) of the part processes in the reactors using the laws of mass, energy and momentum balances [24–26]. For a mathematical model of the trickle-bed reactor, the said balance equations are generally made for a spatially limited section, a differential volume element of the reactor. Account is to be taken of transport phenomena like convection, dispersion, phase changes and reaction, as well as their mutual interactions within the gas and liquid phases flowing through the catalyst packing. To solve the resultant differential balance equations, additional information on the conditions prevailing at the interfaces, especially at the beginning and at the end of the catalyst packing (reaction zone) as well as at the wall of the reactor is necessary. In the end, however, the level of detail of the mathematical models is crucially determined by the data available and especially by the required model parameters actually obtained, because the determination of parameters, if possible at all, is very difficult metrologically. Figure 4.3 shows typical models largely used for the simulation of fi xed-bed reactors with gas/liquid flows.

Figure 4.3 Basic types of fixed-bed reactor models.

4.2 Modeling

The large number of models presented in prior literature for the simulation of two- and three-phase reactors (e.g. trickle-bed reactors) can generally be divided into the following major classes or types: • continuum models; • cell models; • statistical models. The latter model type describes the experimentally determined relations between dependent and independent variables with the help of statistical methods and neglects the known physicochemical relations. Such models are primarily used on reactor types difficult to describe deterministically. The cell models are composed of specific networks of mixing cells (e.g. stirred reactor cascades) or of combinations of mixing cells and transport cells (ideal tube reactors). The socalled continuum models, however, handle each phase as a continuum. The continuum models are further distinguished as homogeneous and heterogeneous reactor models. In the heterogeneous reactor model, the fluid phases and the solid phase (catalyst) are considered and mathematically described as individual items. The homogeneous reactor model neglects said individual approach and assumes in such case for modeling purposes that the three phases existing in the tricklebed reactor constitute one “homogeneous” phase. In addition, the continuum models are distinguished in whether the concentration and temperature profi les are considered one-dimensional (e.g. axial direction) or two-dimensional (both axial and radial space directions). The complex interactions, as found in trickle-bed reactors in particular, between the chemical reaction in the catalyst particle and the simultaneous part processes for material, energy and momentum transport can be described mathematically by means of the balances of mass, energy and momentum. The mathematical models show different complexity depending on whether certain transport steps are considered or neglected. The balance of the mass or material change of a component j over time (known as material balance) is generally calculated for a defined reactor volume. For reactor-modeling purposes, an infinitesimal volume element (see Fig. 4.4) is assumed for said balance space. For reaction considerations, it is sufficient to limit the enthalpy balance to a heat balance, i.e. to consider the prevailing temperatures or temperature profi les. The general mathematical structures of the balances are detailed in Jakubith [26]. To formulate the heat balances for each phase, it is necessary to make various assumptions or generalizations for the particular application case. To be com-

Figure 4.4 Infinitesimal volume element in cylinder coordinates.

83

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors

plete, the mathematical model of a fi xed-bed reactor must include the momentum balance to be able to take into account the time-related and local velocity changes of the fluids especially in irregular-packed catalyst beds. The balance calculation of the flow-inducing forces finally leads to the complete set of Navier–Stokes equations for all three local directions [26]. It has not been possible so far to come to a closed solution of the Navier–Stokes equations taking into account the mass and heat balances. Widely different assumptions or simplifications are therefore used in the literature. The momentum balance will not be discussed in more detail here, because the first simulation tests for periodic process control of trickle-bed reactors do not consider the momentum balance. A complete mathematical model for a threephase reactor would thus be made up of the respective material, heat and momentum balances for the “gas phase”, for the “liquid phase”, and for the “solid phase (catalyst)”, but their complete solution currently encounters major difficulties. In the monograph by Ramachandran and Chaudhari [27] regarding catalytic three-phase reactors, it is generally pointed out that the balance calculations for the creation of mathematical models in many international papers regard the liquid outside the catalyst grain as the “liquid phase” and the grain fi lled with liquid and/or gas pseudohomogeneously as the “catalyst (solid phase)”. Also, Valerius et al. [28, 29] show by exact formulation that the balance of the “solid-phase (catalyst)” actually is a balance of the liquid phase and/or gas phase in the catalyst pellet. Nevertheless, this text will follow the international trend and designate the trickle-bed reactor balance equations for the gas/liquid phase in the catalyst grain as “balance for the solid phase” or “catalyst”. The complex hydrodynamic relations in trickle-bed reactors together with the various assumptions or neglects result in widely different structured mathematical models. Two basic types of models, the differential models and the cascade or cell models, are most frequently used in the literature. For the creation or selection of mathematical models, it is important to solve key problems such as: • deviation from ideal tube reactor flow behavior; • possibilities for pseudohomogeneous considerations; • consideration of dispersions effects in the gas and liquid phases; • consideration of independent gas and liquid flows; • consideration of the heterogeneity of the liquid phase (dynamic and static liquid holdup); • necessity of multidimensional considerations; • possibilities for exact determination of parameters; • existence of suitable computer software, etc. Depending on which of the key problems are emphasized, there will be simple dispersion models, extended dispersion models (e.g. dispersion exchange models), belonging to the class of differentially formulated models. The division

4.3 Reactor Model

of the reactor into individual sections or cells results in so-called cell models, which combine a series of stirred-tank reactors or constitute extensions of such stirred-tank cascades (e.g. cell-exchange models). Examples of several models and their numerical solutions were given by Attou et al. [30], whose model is limited to the descriptions of hydrodynamics, and by Khadilkar et al. [31] as well as Jiang et al. [32], who present general trickle-bed reactor models. The two latter papers originate from the well-known Dudukovic research group and show the latest state-of-the-art modeling of trickle-bed reactors in stationary operation.

4.3 Reactor Model

The developed dynamic reactor model for the simulation studies of the unsteadystate-operated trickle-flow reactor is based on an extended axial dispersion model to predict the overall reactor performance incorporating partial wetting. This heterogeneous model consists of unsteady-state mass and enthalpy balances of the reaction components within the gas, liquid and catalyst phase. The individual mass-transfer steps at a partially wetted catalyst particle are shown in Fig. 4.5. The dynamic trickle-bed reactor model consists of the following set of coupled second order partial differential equations (PDEs).

Figure 4.5 Important mass-transfer steps in trickle-flow reactors under partial-wetting conditions.

85

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors

Mass Balances: Total mass balance liquid phase (L): ∂u ∂εL =− L ∂x ∂t

(4.1)

Liquid phase (component H2 ):

∂ (ε Lc H2,L ) ∂ 2c H2,L ∂ (uLc H2,L )  c H ,G  = Dax,L − + (kH2,GLaGL ) 2 − c H2,L  h ∂t ∂ x2 ∂x  H2  − kH2,LSε SaSηCE(c H2,L − c HL 2,S )

(4.2)

Liquid phase (component A):

∂ (ε Lc A,L ) ∂ 2c A,L ∂ (uLc A,L )  c = Dax,L − + (kA,GLaGL ) A,G − c A,L  ∂t ∂ x2 ∂x   hA − kA,LSε SaSηCE(c A,L − c A,S )

(4.3)

Gas phase (G):

εG

∂ c H2,G ∂ 2c H2,G ∂ (uGc H2,G )  c H ,G  = Dax ,G − − (kH2,GLaGL ) 2 − c H2,L  ∂ x2 ∂t ∂x h H  2   c H2,G G  − kH2,GSε SaS(1 − ηCE ) − c H2,S   hH2 

(4.4)

Solid phase (S): Component H2 (from gas phase):

εS

∂ c HG2,S  c H ,G  = kH2,GSε SaS(1 − ηCE ) 2 − c HG2,S  − rG h ∂t H  2 

(4.5)

Component H2 (from liquid phase):

εS

∂ c HL 2,S  c H ,L  = kH2,LSε SaSηCE 2 − c HG2,S  − rL ∂t h  H2 

(4.6)

Component A:

εS

∂ c A,S = kA,LSεSaSηCE (c A,L − c A,S ) − rL − rG ∂t

(4.7)

4.3 Reactor Model

The enthalpy balances were transformed into heat balances. The mathematical equations for the unsteady state description of the temperature variations in the liquid and in the solid phase were formulated as follows: Heat Balances: Liquid phase (L):

∂ TL ∂ 2TL ∂ (uLTL ) = (1 − ε S )λax ,L − ρ Lc P,L 2 ∂t ∂x ∂x − (α LSηCE + α GS(1 − ηCE ))ε SaS (TL − TS ) − (α LWηCE + α GW(1 − ηCE ))a W (TL − TW )

(4.8)

∂ TS ∂ 2TS = ε Sλax ,S + ε S(α LSηCE + α GS(1 − ηCE ))aS(TL − TS ) ∂t ∂ x2 + ε S (rL + rG )(−∆HR )

(4.9)

(1 − ε S ) ρ L c P,L

Solid phase (S):

ε S ρ S c P,S

The necessary boundary and initial conditions are summarized in Table 4.2. The set of coupled second-order partial differential equations (PDEs) was solved with the ACM program package. For these simulation studies correlations of model parameters under steady-state conditions were used (see Table 4.3), Table 4.2 Boundary and initial conditions for the used differential equation system.

Boundary and initial conditions Boundary conditions: Dax,G Dax,L

∂c H2 ,G ∂x ∂c H2 ,L ∂x

Dax,L

∂c A,L ∂x

λax,L

∂TR ∂x

∂c H2 ,L ∂x

x =0

x =0

x =0

x =0

= uG [c H2 ,G x =0 − c H2 ,G x =0− ]

(4.10)

= uL [c H2 ,L x =0 − c H2 ,L x =0− ]

(4.11)

= uL [c A,L x =0 − c A,L x =0− ]

= uL ρL c P,L [TR

= 0; x =L

∂c H2 ,G ∂x

x =0 +

= 0; x =L

− TR

x =0 −

∂c A,L ∂x

(4.12) (4.13)

] = 0;

x =L

∂TR ∂x

=0

(4.14)

x =L

Initial conditions: x < 0, x = 0 − : c j = c 0j ; TL = T R0 +

x > 0, x = (0 ) = 0: c j = 0; TR = T

(4.15) 0 R

(4.16)

87

88

4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors Table 4.3 Correlations of different model parameters,

determined under steady-state-operation conditions for the used extended axial dispersion reactor model. Correlations (authors) Gas-liquid mass transfer (Lange [33]): d kH2 ,GL aGL = 0.33 DH2 ,L  R   dP 

0.46

 ρLuL     µL 

0.14

ScH0.52 ,L

(4.17)

Liquid-solid mass transfer: (Lange [33]): DH2 ,L  dR 0.82  ρLuL 0.9 0.5     ScH2 ,L aSε S  dP   µL 

kH2 ,LS = 0.75

(4.18)

(Goto et al. [34]): kA,LS = 0.45

DA,L  ρLuL    aSε S  µL 

0.56

Sc1A,L3

(4.19)

Gas-solid mass transfer (Nelson and Galloway [35]): kH2 ,GS = 0.18

DH2 ,G 1 3 −1 3 ε S (ε S − 1)ReG ScH2 23,G dP

(4.20)

Liquid holdup (Lange [33]): d ε L = 0.16(1 − ε S ) R   dP 

0.33

ReL0.14

(4.21)

Wetting efficiency (Mills and Dudukovic [36]):   ad  ηCE = 1.0 − exp  −1.35 ⋅ ReL0.333 FrL0.235WeL−0.17  S P 2   (1 − ε S )  

−0.0425

  

(4.22)

Gas and liquid axial dispersion (Lange [33]): 0.61 Dax,G = 13Re0.78 G ; D ax,L = 0.55ReL

(4.23)

because until now no correlations, determined under periodic conditions, were available. The first steps in the following simulations have been performed with the aim of showing a process intensification by an unsteady-state-operated trickle-flow reactor in comparison to steady-state-operation conditions.

4.4 Simulation

The simulations have been performed with the aim to analyze the influence of some main parameters (e.g. liquid flow rate, cycle time, liquid holdup) on the re-

4.4 Simulation

actor performance (conversion) and to compare with the experimental results of Table 4.1. One simulation result of the comparison of steady- and unsteady-state operation is demonstrated in Fig. 4.6. Typical simulation curves for steady-state and unsteady-state (periodical) operation conditions under the same time-average conditions can be seen. Further detailed simulation results for different periodical times are presented in Fig. 4.7. For an on/off liquid flow variation at constant cycle split of 0.5 an increasing of the time-averaged alpha-methylstyrene conversion depending on the specific cycle times (tperiod) will be shown. The conversions predicted by the dynamic heterogeneous model were significantly lower than the experimental data (see Table 4.1). The reason for this could be that the correlation equations used for the model parameter are not accurate enough for the prediction of the required model under nonstationary conditions. But the simulation results confirmed the same trend that was found experimentally. The changing hydrodynamics is one important influence factor under periodical liquid flow rate in trickle-bed reactors. Because until now an exact mathematical description of the hydrodynamics in periodically operated trickle-bed reactors especially the liquid flow field and the liquid distribution within the catalyst bed, is impossible and detailed further experimental and theoretical studies of the hydrodynamics under unsteady-state conditions are required. Despite a broad basis of experimental data published in the literature no correlation with general applicability is yet available, if an accuracy of the liquid holdup level better than 30% is desired [37]. It is well known that a variation of the liquid flow rate at the reactor inlet results in pulses of liquid traveling through the catalyst bed and a corresponding variation in the liquid holdup. For the liquid holdup oscillation a mathematical

Figure 4.6 Simulation result for the conversion under steadystate- and unsteady-state-operation conditions in a laboratory trickle-flow reactor.

89

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors

Figure 4.7 Simulation results of the timeaveraged conversion (curve 3: UA,us) and of the reduced concentrations (c′j) for a periodic liquid flow rate (on-off modus, ReL = 0.23/0 at a constant cycle split of

S = 0.5) and different periodic times [a) tperiod = 2 min, b) tperiod = 4 min, c) tperiod = 10 min] (Curve 1 represents the reduced concentration of A and curves 2 and 4 of H2).

4.4 Simulation

equation is necessary that is determined under experimental dynamic conditions. For the following holdup simulation a combination of an empirical holdup correlation (Table 4.3, Lange [33]) with the total mass balance (Eq. (4.1)) is used. This leads to a partial differential equation that describes the liquid holdup oscillation under unsteady-state-operation conditions. The simulation based on this equation combination shows a strongly asymmetrical decay of the liquid pulses (see Fig. 4.8). Figure 4.8 shows typical liquid holdup profi les along the reactor axis and the liquid holdup oscillations at different bed lengths. A decreasing holdup amplitude can be seen depending on longer catalyst beds. Figure 4.9 presents the influence of the specific periodic times on the attenuation of the liquid holdup pulses. It can be seen that the pulse decay strongly depends on cycle time (tperiod) and reactor bed length. If the periodic time is very long the system reaches a so-called quasisteady state in periodic operation and very short periods (see curve (a) in Fig. 4.9) lead to a merging of pulses. The choice of pulsing frequency can be influenced, e.g. by pulse attenuation characteristic, mass storage in catalyst particles and reaction kinetics. However, it has to be noted that, although the above-described approach allows a good representation of the characteristic asymmetrical decay of liquid pulses, for a good quantitative agreement of the holdup level and the velocity of the pulses traveling through the bed a relation between external liquid holdup and liquid

Figure 4.8 Periodic liquid holdup at different axial position of the trickle-bed reactor.

91

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors

Figure 4.9 Attenuation of the liquid holdup pulses.

Figure 4.10 Simulated conversions and catalyst wetting depending on liquid flow rate.

superficial velocity suitable for the particular reaction system has to be found. In Fig. 4.10 the influence of the liquid flow rate and catalyst wetting on the alphamethylstyrene conversion under steady state and periodic (unsteady state) operation conditions will be explained. The detailed data for these simulation results are summarized in Table 4.4. The condition (position 1) represents a steady-state simulation result for a constant liquid flow rate with a corresponded constant conversion of UA,ss = 0.284. Under this operation condition the calculated wetted catalyst surface is 82%. If

4.5 Conclusion Table 4.4 Comparison of alphy-methylstyrene conversion

under steady- and unsteady-state-operation conditions in a trickle-bed reactor. Operation conditions

uL , cm/s

S, –

eL , –

hCE, –

UA, –

Steady state

0.172 0.82 0.024/0.320 0.5 0.15 0.5 0.403

(1.0)

0.27 0.284 0.15/0.30

0.82

0.284

0.5/0.8

0.403

Unsteady state (uL,min / uL,max)

0.5

/ /

0.30 0.9

the liquid flow rate switches between two liquid flow levels (positions 2 and 3 in Fig. 4.10) with a cycle split of 0.5 a time-averaged unsteady-state conversion of UA,us = 0.403 will be reached (see Table 4.4). Under this unsteady-state condition it is very important that the same time-averaged liquid loading of the catalyst bed as under steady-state conditions is realized. Only on this basis is a performance comparison between steady state and unsteady state possible. The periodic liquid flow changes the hydrodynamic regime and influences the external and internal catalyst wetting. Therefore the catalyst particle wetted area will be increased and decreased depending on the liquid duration. The degree of wetting will be crucially influenced. This has a strong effect on the intensity of the process, as it alters the conditions of transport of the reactants to the catalyst surface where adsorption, chemical reactions and desorption of the products take place. The simulation studies demonstrate that the liquid-flow variation has a strong influence on the liquid holdup oscillation and on the catalyst-wetting efficiency. Consequently, the time-averaged alpha-methylstyrene conversion will be increased, because the mass-transfer resistance between the phases (gas-liquid, gas-solid and liquid-solid) affect the overall reaction rate and consequently the conversion will be improved. Generally, the tendency of time-averaged conversion improvement could be simulated.

4.5 Conclusion

The conversions predicted by the dynamic heterogeneous model were significantly lower than the experimental data. The reason for this could be that the correlation equations used for the model parameters are not accurate enough for the prediction of the required model under nonstationary conditions. But the

93

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors

simulation results confirmed the same trend that was found experimentally. These simulation results show the possibility of the improvement of the reactor performance under unsteady-state (periodic) operation in comparison with the steady-state condition. The experimental and theoretical results demonstrated that under liquid-flow interruption the time-averaged conversion is higher than under steady-state operating conditions. The previously described mathematical models for simulating the periodic operation of trickle-bed reactors are limited to liquid-throughput variations both in the trickle and in the pulse regimes, because these strategies have so far given the most efficient improvement of reactor performance. However, the mathematical description of the reactor behavior under unsteadystate operation in currently existing models is not detailed and accurate enough to cover all effects sufficiently. The modeling of the periodic process management is chiefly based on the extension of steady-state models for trickle-bed reactors. Their applications to nonsteady-state systems have several drawbacks, as summarized in the following points: • Dynamic processes in trickle-bed reactors are often described by just one-dimensional macroscopic models. • In most cases, the models start out from coarse assumptions (e.g. plug flow or straight-line flow) and do not consider the spatial and time-line changes of the gas and liquid phase velocities and the different holdups. • Many models were simplified by assuming equilibrium between the phases and by a pseudohomogeneous approach. • Models describe pseudotransient behavior and thus neglect the real accumulation terms of the nonlimited components. • Material transport terms for each of the components contain coefficients determined for ideal fluid conditions. • The spatial terms are neglected on the catalyst level to simplify the simulations. • Frequently, reaction and material transport are considered for the limiting components only, while the remaining concentrations are regarded as invariable. • Often, the models consider just a single reaction while neglecting consecutive and parallel reactions. • Most models do not consider multicomponent mass transfer between the phases and thus neglect the mass and energy transport through the interfaces. • Effects like evaporation and condensation are neglected. • Model parameters are determined with correlations not determined under the conditions actually applied (material system, operating conditions, etc . . .).

Notation

In general, it can be concluded that substantial progresses have been made in the experimental and theoretical analysis of trickle-bed reactors under unsteady-state conditions. But until now these results are not sufficient for a priori design and scale-up of a periodically operated trickle-bed reactor. The mathematical reactor models, which are now available are not detailed enough to simulate all of the main transient behavior observed. For solving this problem specific correlations for specific model parameters (e.g. liquid holdup, mass transfer gas-solid and liquid-solid, intrinsic chemical kinetic, etc.) determined under dynamic conditions are required. The available correlations for important hydrodynamic, massand heat-transfer parameters for periodically operated trickle-bed reactors leave a lot to be desired. Indeed, work for unsteady-state conditions on a larger scale may also be necessary. Furthermore, it is concluded that under determined conditions (e.g., an optimal combination of periodic variations of selected parameters, such as forced concentration-oscillations and periodic variation of the liquid flow rate at the same time) could be used in order to achieve further significant performance and especially selectivity improvements for complex reactions systems in trickle-bed reactors. The knowledge until now is only “the end of the beginning” of the experimental and theoretical investigations of the periodic operation procedure for trickle-bed reactors. Additional research in this area is needed. In the near future, further progress may be expected.

Notation

a aS aW c cP Dax Dj d FrL g G h Kj k kr L L P

interfacial area, m2/m3 specific area of particle per unit reactor volume, m2/m3 reactor wall area, m2 concentration, mol/m3 molar heat capacity, J/(mol K) axial dispersion coefficient, m2/s molecular diffusivity, m2/s diameter, m dimensionless liquid Froude number, u2L/g dp, – acceleration due to gravity, m2/s superficial mass velocity of gas, G = u GrG, kg/(m2 s) Henry’s coefficient, (mol/m3 ) liquid/(mol/m3 ) gas adsorption equation constant for Langmuir–Hinshelwood kinetics, m3/mol mass-transfer coefficient, s−1 reaction-rate constant, mol/(m3 s) length, m superficial mass velocity of liquid, L = uLrL , kg/(m2 s) pressure, bar

95

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors

R ReG ReL rG rL ∆HR S ScH ,G ScH ,L Sc A,L t t′ T u U WeL x 2

2

radius, m dimensionless gas Reynolds number, u GdPrG/ eGmG, – dimensionless liquid Reynolds number, uL dPrL/ eLmL , – reaction rate (H2 adsorbed from gas phase), mol/(m3 s) reaction rate (H2 adsorbed from liquid phase), mol/(m3 s) heat of reaction, J/mol cycle split, – dimensionless gas Schmidt number for H2, mG/ rGD H ,G dimensionless liquid Schmidt number for H2, mL/ rLD H ,L dimensionless liquid Schmidt number for A, mL/ rLDA,L time, s reduced time, – temperature, K superficial velocity, m/s conversion, – dimensionless liquid Weber number, u2L dPrL/ sL axial coordinate, m 2

2

Greek Letters a e eB hCE l

L m r s y

heat-transfer coefficient, W/(m2 K) phase holdup, m3/m3 reactor, – bed porosity, m3/m3 reactor, – external liquid-solid contacting efficiency, – thermal conductivity, W/(m K) specific factor, L = ( rG/ rH O ) ( r L/ rAir), – dynamic viscosity, kg/(m s) mass density, kg/m3 surface tension, N/m specific factor, y = ( sH O/ sL) [( hL/ hH O ) ( rH O/ rL)2 ] 0.33 2

2

Subscripts and Superscripts

A ax eff G H2 i j L max min p R

alpha-methylstyrene axial effective gas hydrogen reaction component j liquid maximum minimum particle reactor

2

2

References

S ss us W

solid (catalyst) steady state unsteady state wall

References 1 Silveston, P. L., Periodic operation of chemical reactors – a review of the experimental literature. Sadhana, 1987, 10(1–2), 217–246 2 Silveston, P. L., Hanika, J., Challenges for the periodic operation of trickle-bed catalytic reactors. Chemical Engineering Science, 2002, 57, 3373 3 Haure, P. M., Hudgins, R. R., Silveston, P. L., Periodic operation of a trickle-bed reactor. AIChE Journal, 1989, 35(9), 1437–1444 4 Haure, P. M., Bogdashev, S. M., Bunimovich, M., Stegasov, A. N., Hudgins, R. R., Silveston, P. L., Thermal waves in the periodic operation of a trickle-bed reactor. Chemical Engineering Science, 1990, 45, 2255 5 Castellari, A. T., Haure, P. M., Experimental study of the periodic operation of a trickle-bed reactor. AIChE Journal, 1995, 41, 1593 6 Hanika, J., Lange, R., Turek, F., Computer-aided control of a laboratory trickle-bed reactor. Chemical Engineering and Processing, 1990, 28, 23 7 Hanika, J., Lange, R., Dynamic aspects of adiabatic trickle-bed reactor control near the boiling point of reaction mixture. Chemical Engineering Science, 1996, 51(11), 3145–3150 8 Lange, R., Hanika, J., Problems encountered in the periodic operation of catalytic three-phase reactors. Presented at the 9th Congress CHISA, 1987, Prague 9 Lange, R., Hanika, J., Hudgins, R. R., Silveston, P. L., Experimental study on the periodical process control of a tricklebed reactor. Chemie Ingenieur Technik, 1994, 66(3), 365 10 Lange, R., Hanika, J., Stradiotto, D. A., Hudgins, R. R., Silveston, P. L.,

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Investigations of periodic operated trickle-bed reactors. Chemical Engineering Science, 1994, 49(24B), 5615 Lange, R., Gutsche, R., Hanika, J., Forced periodic operation of a trickle-bed reactor. Chemical Engineering Science, 1999, 54(13–14), 2569–2573 Schubert, S., Grünewald, M., Lange, R., Dynamic modeling and simulation of trickle-bed reactors under cycling operation mode. Presented at the 4th European Congress on Chemical Engineering, Granada, 2003, Spain Khadilkar, M. R., Wu, Y. X., Al-Dahhan, M. H., Dudukovic, M. P., Colakyan, M., Comparison of trickle-bed and upflow reactor performance at high pressure: model predictions and experimental observations. Chemical Engineering Science, 1996, 51, 2139 Khadilkar, M. R., Al-Dahhan, M. H., Dudukovic, M. P., Parametric study of unsteady-state flow modulation in tricklebed reactors. Chemical Engineering Science, 1999, 54, 2585 Khadilkar, M. R., Performance studies of trickle-bed reactors. PhD Thesis, 1998, Washington University of St. Louis, USA Liu, G., Duan, Y., Wang, Y., Wang, L., Mi, Z., Periodically operated trickle-bed reactor for EAQs hydrogenation: experiments and modeling. Chemical Engineering Science, 2005, 60, 6270 Stegasov, A. N., Kirillov, V. A., Silveston, P. L., Modeling of catalytic SO2 oxidation for continuous and periodic liquid flow through a trickle bed. Chemical Engineering Science, 1994, 49, 3699 Warna, J., Salmi, T., Dynamic modeling of catalytic three-phase reactors. Computers & Chemical Engineering, 1996, 20, 39

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4 Modeling and Simulation of Unsteady-state-operated Trickle-flow Reactors 19 Gabarain, L., Castellari, A. T., Cechini, J., Tobolski, A., Haure, P. M., Analysis of rate enhancement in a periodically operated trickle-bed reactor. AIChE Journal, 1997, 43, 166 20 Lange, R., Schubert, M., Dietrich, W., Grünewald, M., Unsteady-state operation of trickle-bed reactors. Chemical Engineering Science, 2004, 59, 5355 21 Schubert, M., Process intensification in trickle-bed reactors by unsteady state mode of operation. PhD Thesis, 2006, Dresden University of Technology, Germany 22 Charpentier, J. C., Favier, M., Some liquid holdup experimental data in trickle-bed reactors for foaming and nonfoaming hydrocarbons, AIChE Journal, 1975, 21, 1213–1218 23 Gianetto, A., Specchia, V., Trickle-bed reactors: State-of-the-art and perspectives, Chemical Engineering Science, 1992, 47, 13/14, 3197 24 Matros, Y. S., Bunimovich, G. A., Unsteady-state reactor operation, in: Handbook of Heterogeneous Catalysis, Vol. 3, 1464–1479 (Editors: G. Ertl, H. Knözinger, J. Weitkamp), Wiley-VCH, 1998 25 Budde, K., Reaktionstechnik I – Lehrbuch, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig, 1988 26 Jakubith, M., Chemische Verfahrenstechnik, Einführung in die Reaktionstechnik und Grundoperationen, VCH Verlagsgesellschaft, Weinheim, 1991 27 Ramachandran, P. A., Chaudhari, R. V., Three-Phase Catalytic Reactors, Gordon and Breach Science Publishers, New York, 1983 28 Valerius, G., Zhu, X., Hofmann, H., Modeling of a trickle-bed reactor. I: Extended definitions and new approximations. Chemical Engineering Processing, 1996, 35, 1–9 29 Valerius, G., Zhu, X., Hofmann, H., Arntz, D., Haas, T., Modeling of a

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trickle-bed reactor. II: The hydrogenation of 3-hydroxypropanal to 1,3 propanediol, Chemical Engineering Processing, 1996, 35, 11–19 Attou, A., Boyer, C., Ferschneider, G., Modeling of the hydrodynamics of the cocurrent gas-liquid trickle flow through a trickle-bed reactor. Chemical Engineering Science, 1999, 54, 785–802 Khadilkar, M. R., Mills, P. L., Dudukovic, M. P., Trickle-bed reactor models for systems with a volatile liquid phase. Chemical Engineering Science, 1999, 54, 2421–2431 Jiang, Y., Khadilkar, M. R., Al-Dahhan, M. H., Dudukovic, M. P., Single phase flow modeling in packed beds: Discrete cell approach revisited. Chemical Engineering Science, 2000, 55, 10, 1829–1844 Lange, R., Beitrag zur experimentellen Untersuchung und Modellierung von Teilprozessen für katalytische Dreiphasenreaktionen im Rieselbettreaktor. PhD Thesis, 1978, Technical University of LeunaMerseburg, Germany Goto, S., Levec, J., Smith, J. M., Mass transfer in packed beds with two-phase flow. Industrial Engineering Chemistry, Process Design and Development, 1975, 14(4), 473–478 Nelson, P. A., Galloway, T. R., Particle to fluid heat and mass transfer in dense systems of fine particles. Chemical Engineering Science, 1975, 30(1), 1 Mills, P. L., Dudukovic, M. P., Evaluation of liquid-solid contacting in trickle-bed reactors by tracer methods. AIChE Journal, 1981, 27(6), 893–904 Dudukovic, M. P., Larachi, F., Mills, P. L., Multiphase catalytic reactors: a perspective on current knowledge and future trends. Catalysis Reviews – Science and Engineering, 2002, 44(1), 123–246

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5 Packed-bed Membrane Reactors Ákos Tóta, Dzmitry Hlushkou, Evangelos Tsotsas, and Andreas Seidel-Morgenstern

5.1 Introduction

Various possibilities of combining reaction and separation processes in one apparatus have been reviewed recently (Krishna, 2002). One of the options suggested is concerned with the application of membranes in chemical reactors. The main idea of membrane reactors consists in dividing a reactor into sections of different functions using membranes that are permeable only for certain reactants. The concept is capable of tackling several of the classical problems encountered in chemical reaction engineering. Membranes have been most successfully applied in membrane reactors designed to trap homogeneous catalysts in the reactor during continuous operation (Sirkar et al., 1999; Greiner et al., 2003). Alternatively there are intense activities devoted to applying membranes for the selective removal of products from the reaction zone formed in reversible reactions in order to suppress backward reactions (van de Graaf et al., 1999; Assabumrungrat et al., 2001; Itoh et al., 2003; Wang et al., 2003). Another interesting area concerns the improvement of selectivity and yield with respect to a certain target component that is formed in a network of consecutive and parallel reactions. In this context again, the selective removal of the desired intermediate via a membrane is attractive. On the other hand, improved operation can be achieved if reactants converted in reaction networks are contacted spatially distributed in optimized amounts (Harold and Lee, 1997; Lu et al., 1997a; Tellez et al., 1999; Diakov and Varma, 2004; Thomas et al., 2004). This goal can be achieved, e.g., by controlled reactant dosing via suitable membranes. Further, there are interesting activities attempting to use a catalytically active membrane as a contactor where reactants entering from both sides meet and form the products inside the membrane (Capannelli et al., 1996; Saracco et al., 1999; Dittmeyer et al., 2004; Ozdemir et al., 2006). The state-of-the-art regarding the various membrane-reactor configurations has been reviewed in several comprehensive summaries (e.g. Sanchez Marcano and Tsotsis, 2002; Dixon, 2003; Seidel-Morgenstern, 2005). Modern Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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developments were reported on a regular basis during the “International Congresses on Catalysis in Membrane Reactors – ICCMR” (Dittmeyer, 2005). Due to the availability of the mentioned overviews it is not the goal of this chapter to consider the whole field of membrane reactors. Rather, the discussion below will be focused on presenting simplified and more detailed mathematical models capable of describing the performance of membrane reactors. Although there are several studies available for analyzing the combination of reaction and membrane separation (e.g. Salomon et al., 2000; Struis and Stucki, 2001; Wieland et al., 2002; Patil et al., 2005; Rohde et al., 2005) there is a need to analyze in more detail specific features of membrane reactors. The focus of this chapter will be the development and application of simplified and also more detailed mathematical models for packed-bed membrane reactors in which certain reactants are dosed over the reactor wall using nonselective membranes. This type of membrane reactor is sometimes also-called a “distributor” (Dalmon, 1997; Julbe et al., 2001). Despite this restricted focus of the work, most of the concepts considered should be applicable also in the analysis of other types of membrane reactors. There are several theoretical and laboratory-scale studies available focusing on the application of various configurations of membrane reactors in order to improve selectivity–conversion relations in complex reaction systems. In particular, several industrially relevant partial oxidation reactions were investigated. Relevant studies were performed, e.g., for the oxidative coupling of methane (Tonkovich et al., 1996a; Diakov et al., 2001), for the partial oxidation of ethane (Coronas et al., 1995; Al-Juaied et al., 2001), propane (Ziaka et al., 1993; Ramos et al., 2000; Schäfer et al., 2003) and for the oxidative dehydrogenation (ODH) of butane (Tellez et al., 1997; Mallada et al., 2000). If there is the possibility of undesired consecutive reactions, typically tubular reactors are applied to minimize backmixing (Levenspiel, 1999). In order to increase reaction rates and to improve the selectivity towards the desired product, solid (heterogeneous) catalysts are positioned in reactor tubes (tubular fi xed-bed reactors, FBR). Theoretical studies reveal that in networks of consecutive and parallel reactions the local concentrations have a severe influence on the differential selectivity with respect to the target component and thus on integral selectivity and the yield achievable at the reactor outlet (Reyes et al., 1993; Tellez et al., 1999; Alfonso et al., 2001; Tota et al., 2004). A possibility of altering the internal concentration profi les is offered by using membranes as porous reactor walls. To answer the question whether a certain dosing strategy, conventional cofeed mode in the FBR or distributed feed mode in a packed-bed membrane reactor (PBMR), is useful or not, there is detailed knowledge regarding the dependencies of the reaction rates on concentrations required. Systematic theoretical studies are available regarding the significance of the reaction orders for a proper selection of the components that should be dosed (Lu et al., 1997c; Lafarga and Varma, 2000; Hamel et al., 2003). The probably most extensive theoretical work on reactant dosing in packed-bed membrane reactors up to now was carried out by Kürten, 2003. All of these studies revealed that a proper knowledge of the concrete reac-

5.1 Introduction

tion kinetics is essential to analyze the potential of process intensification in “distributor”-type packed-bed membrane reactors. 5.1.1 The PBMR Principle

Figure 5.1 illustrates schematically the principle of the packed-bed membrane reactor applied for controlled dosing of reactants through the reactor wall. The reactants are fed into the PBMR spatially separated on the inner (tube side, TS) and outer side (shell side, SS) of the membrane. The membrane itself is considered to be catalytically inert. The reactants are converted on the alternatively also catalyst particles, positioned in the tube side of the membrane (e.g., catalyst can be placed in the shell side). The “dead-end” reactor configuration shown in the figure (i.e. closed shell-side outlet) allows dosing reactants through the membrane in a controlled manner, with predefined flow rates. All reactants dosed have to permeate through the membrane. The products and the unconverted reactants leave the reactor at the tube side outlet. The PBMR shown in Fig. 5.1 differs from the conventional FBR in the residence-time behavior and with respect to the local concentration and temperature profiles. 5.1.2 Case Study

To be close to realistic situations the theoretical results presented below will be often in close relation to an actual case study. For this, the oxidative dehydrogenation (ODH) of ethane to ethylene on a VOx /γ -Al2O3 catalyst was chosen as a model reaction. A detailed analysis of the network of the five main reactions taking place was recently given by Klose et al., 2004a. According to the scheme given in Fig. 5.2 ethane is converted in a parallel reaction to CO2 and to the desired product ethylene. This parallel reaction limits the maximal achievable ethylene selectivity in the reaction to approximately 85%. Additionally, the consecutive reaction to CO and the total oxidation path can further decrease the olefin selectivity. The authors suggested a kinetic model based on a Mars–van-Krevelen-type redox mechanism for the ethylene production and Langmuir–Hinshelwood–Hougen– Watson (LHHW) kinetics for the deep oxidation reactions. The kinetic equations suggested and the parameters estimated are presented in the Appendix. The

Figure 5.1 Distributed dosing in inert packed-bed membrane reactors (PBMR) with “dead-end” configuration for a reaction A + B → P.

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5 Packed-bed Membrane Reactors Figure 5.2 Reaction network for the oxidative dehydrogenation of ethane to ethylene (Klose et al., 2004a).

derived model was found to describe a large set of experimental data from laboratory-scale FBR and PBMR apparatus with good accuracy. 5.1.3 Porous Membranes

Many partial oxidation reactions and selective hydrogenations require higher temperatures to occur at a sufficient rate (e.g. 500–650 ºC for the ODH of ethane). Thus, in reactors using membranes to dose oxygen or hydrogen, typically inorganic membranes are considered. Currently available membranes allow operation temperatures up to 1200 K (Bouwmeester, 2003). Regarding the preferential permeation of species through inert membranes one can distinguish between permselective (mostly dense) and nonpermselective (usually porous) membrane materials. In this work, based on the case study investigated, the latter are considered. The transport through porous membranes is governed by diffusion and viscous flow (convection). According to Darcy’s law, the pressure-driven viscous flow through a porous medium depends on the permeability (B 0 ) of the membrane. Using Hagen–Poiseuille’s law and assuming cylindrical pores, the permeability is related to the mean diameter of the membrane pores (B 0 ∼ d2por). Depending on the pore size, pore structure, characterized by the porosity to tortuosity ratio ( e / τ and fluid properties, the diffusive transport mechanism is ɺɺ classified into molecular, Knudsen, configurational and surface diffusion (Bird et al., 2002). For the mathematical description of the component transport through a porous membrane there are two modeling approaches common. The first is the so-called extended Fick model (EFM), which can be applied to describe the transport of diluted, nonadsorbable gases in mesoporous membrane materials at low pressure (Veldsink et al., 1995; Papavassiliou et al., 1997; Al-Juaied et al., 2001). The second, the more general dusty gas model, is based on the Stefan–Maxwell approach for multicomponent diffusion (Mason et al., 1967; Krishna and Wesselingh, 2000). Both models require knowledge of the above-mentioned membrane properties (e.g., B 0, e / t ). Because for a specific membrane material these parameters are a priori not predictable, they have to be determined experimentally. Typical membrane materials and characteristic transport parameters used in this work are listed in Table 5.1.

5.2 One-dimensional Modeling of Packed-bed Membrane Reactors Table 5.1 Structural properties of typical membrane

materials for inert membrane reactor applications (Thomas et al., 2001). Membrane material

B0 [m2]

Mean dpor [ m m]

Sinter metal (SS316, Inconel, etc.) Ceramic ( α -Al2O3, support) Ceramic ( γ -Al2O3, coating layer) Silica (SiO2, coating layer)

10 −10 –10 −13 10 −13 –10 −15 10 −18 –10 −19 10 −19 –10 −20

1–100 0.5–3 0.005–0.02 0.001–0.002

In order to study general differences between FBR and PBMR, in the present work a simplified modeling approach has been followed, neglecting the diffusive transport in the membrane. Predefined component fluxes through the membrane were considered, making the application of a more sophisticated transport model unnecessary. 5.1.4 Outline

The structure of this chapter is as follows. In the next section a comparison of the FBR and PBMR will be presented using a simplified 1D model. Advantages and drawbacks of the PBMR will be illustrated. Subsequently, a more detailed 2D model will be developed to study important aspects of radial mass and heat transfer, as well as scale-up problems that might occur in a PBMR. In the last section a short outlook to more sophisticated 3D membrane reactor models is given. Such models are still not suitable for extensive parametric studies. However, they enable a deeper investigation of local velocity and concentration profiles that develop in such reactors.

5.2 One-dimensional Modeling of Packed-bed Membrane Reactors 5.2.1 One-dimensional Pseudohomogeneous Model

A 1D model can be formulated to describe both the FBR and the PBMR using the following simplifying assumptions: • steady-state conditions; • pseudohomogeneous description; • plug-flow (i.e. convection dominated flow, no axial dispersion); • constant reactor pressure; • predefined fluxes through the membrane into the PBMR.

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Based on these assumptions, the mass-balance equation for species i in the tube side can be written as: NR dωi ρg ⋅FT = VR ⋅ ρcat (1 − ε )⋅Mi ∑ vi, j ⋅r j + A w ⋅ ji (ξ ) i = 1, . . . ,N C dξ j =1

(5.1)

The component balance equation is expressed with mass fractions of the individual species, w i and the density of the reaction mixture, rg. In Eq. (5.1) FT is the total volumetric flow rate, x = z/L the dimensionless reactor length. The first term on the right-hand side corresponds to sources by chemical reactions; the second term quantifies the local mass fluxes, ji ( x ), entering through the membrane. In the FBR model the flux term is omitted. The catalyst density is denoted by rcat and ε is the averaged porosity of the catalyst bed. VR is the reactor volume, Mi the molecular mass of species i, Aw the surface area of the membrane wall. The reaction rate r j represents the jth rate of the overall NR reactions taking place. To solve the mass-balance equation, the feed composition and the flow need to be defined at the reactor entrance as the initial condition of the system of ordinary differential equations given with Eq. (5.1):

ω i |ξ = 0 = ω i0 ; ρgFT |ξ = 0 = ρg0FT0

(5.2)

In packed-bed reactors typically temperature profiles develop that influence the reactor performance considerably. Axial temperature profi les can be calculated using the following simplified 1D energy balance: NC NR dρg ⋅FT ⋅ c pT = VR ⋅ ρcat (1 − ε )⋅ ∑ (−∆Hr , j )⋅r j + c p ⋅ A w ⋅ ∑ ji (ξ )⋅Tw dξ i j =1

+ κ ⋅ A w ⋅(T − Tw )

(5.3)

where cp is the specific heat capacity of the reaction mixture, Tw is the wall temperature and κ is the overall heat-transfer coefficient. The latter can be calculated ɺ by applying the concept of thermal resistances. The wall heat-transfer coefficient can be calculated according to Martin and Nilles, 1993. The feed temperature at the reactor inlet was set as: T |ξ = 0 = T 0

(5.4)

With the exception of a few cases (simple reaction orders, isothermal condition), the solution of the above system of ordinary differential equations can be obtained only numerically. For this, there are several numerical software tools available, (e.g., Matlab, Mathematica, GNU Octave, etc.) that provide advanced solvers for such initial value problems. A good overview can be found, e.g., in Press et al., 2002.

5.2 One-dimensional Modeling of Packed-bed Membrane Reactors

In the following sections selected results are presented that illustrate the main differences between the conventional FBR and the PBMR, based on numerical solution of Eqs. (5.1)–(5.4). 5.2.2 Cofeed (FBR) vs. Distributed Dosing of Reactants (PBMR) – Nonreactive Conditions

Before discussing the reactor performance in the presence of chemical reactions, it is instructive to consider the main differences between FBR and PBMR in the residence-time behavior and in the local component concentration profi les caused by the differences in dosing strategy. In accordance with the case study illustrated in Fig. 5.2, ethane and oxygen will be considered as two feed components. Both can be dosed together at the reactor inlet (FBR) or in a distributed manner (PBMR). In both cases it is assumed at first that no reactions take place between them. In the case of the PBMR it is further assumed that there is a uniform dosing profi le, i.e. ji = const. The set of simulation parameters used in the present study is summarized in Table 5.2. In the following it is referred as the “standard parameter set”. The reactant mass fractions in the PBMR, w i, are related below to the overall feed according to Eq. (5.5) and can be translated easely into concentration units used later.

ωi =

SS ω 0i ρ g0 ⋅FTS + ω SS 1 Nc i ρ g ⋅ FSS , with FSS = SS ⋅ ∑ A w ⋅ ji FT ρg i

and FTS = FT − FSS

(5.5)

Figure 5.3a shows typical total flow rates, FT, for the FBR and for the PBMR for identical overall feed flow rates. The total flow rate in the FBR remains constant along the reactor length. All molecules are fed exclusively at the reactor entrance and have, therefore, the same mean residence time in the reactor. For comparison, Table 5.2 Standard simulation parameters for the 1D PBMR model.

Reactor length (length of the porous zone) – L Inner/outer diameter of the membrane – di/d o Particle diameter – dp Catalyst density – r cat Average bulk porosity of the catalyst bed – ε Contac time – wcat/F T Oxygen concentration range – c O2 Ethane concentration – c ethane Tube- to shell-side flow ratio – F TS/FSS Inlet and wall temperatures – T 0, Tw

60 mm 7/10 mm 1.8 mm 1020 kg m−3 0.47 50–500 kg s/m3 0.5–15 vol.% 1 vol.% 1/9–3/1 600 ºC

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Figure 5.3 (a) Local flow rates, (b) Local oxygen concentration in the FBR (solid) and PBMR. Identical overall feed flow for both reactors. FTS/FSS = 1/9 (dashed); 2/1 (dotted); 1/1 (dashed-dotted); 9/1 (dotted solid) in the PBMR.

PBMR profi les are shown for four different ratios of tube side to shell side flow rate (FTS/FSS = 1 : 9, 2 : 1, 1 : 1 and 9 : 1). For a uniformly distributed flux through the membrane there is a linear increase in the total flow rate along the reactor length. The slope decreases as the FTS/FSS flow ratio increases. This causes a decreasing residence time of the molecules fed at the entrance of the reactor. Molecules fed via the shell side, have a different residence-time distribution. This is due to the fact that molecules entering the reaction zone next to the inlet pass a longer reactor distance than molecules entering close to the outlet. The average residence time of the reactant dosed in a PBMR through the membrane is lower than that in a FBR (Tonkovich et al., 1996b). Therefore, an important issue in PBMR design is the specification of the reactant that should be distributed along the reactor length. This specification can be made if the kinetics of the particular reaction system are known. More detailed studies on this topic can be found elsewhere (e.g. Alfonso et al., 1999., Lu et al., 1997b). Just like the residence-time behavior, the local concentration of the reactants influences the reactor performance significantly. They affect both conversion and selectivity. Illustrating the differences between the dosing concepts, Fig. 5.3b shows the internal oxygen concentration profi les. Hereby the total amount of oxygen dosed over the membrane in the PBMR was the same as for the FBR. In the absence of a chemical reaction in the FBR the molar fraction of oxygen is constant along the reactor length. Considering the actual reaction system, to reach a high ethylene selectivity a high oxygen level is undesirable, because it favors the consecutive reactions (Klose et al., 2003). Via distributed feeding, the local oxygen concentration can be reduced in the PBMR as shown in the figure. At low FTS/FSS ratios (i.e. for high transmembrane fluxes) the concentration of oxygen increases rapidly at the entrance. The local, and hence also the averaged oxygen concentration is always lower than in the FBR, which should be favorable regarding ethyl-

5.2 One-dimensional Modeling of Packed-bed Membrane Reactors

ene selectivity. Simultaneously, rapid oxygen supply (see profi le for FTS/FSS = 1/9) can avoid losses in ethane conversion. It is worthwhile to note again that under these flow conditions the local residence time of the hydrocarbon molecules is relatively high near the PBMR inlet. This behavior can, together with the rapidly increasing oxygen concentration, significantly influence the thermal behavior of the reactor (e.g., hot-spot formation). To suppress this undesirable effect, the FTS/FSS ratio can be increased, which results in decreased contact time, and lower oxygen concentration at the entrance in the PBMR. This aspect will be discussed in Section 5.2.4 in more detail. Note that there are also several experimental studies concerning the effect of residence-time distribution and local concentration profi les on the integral performance of a PBMR available (Tonkovich et al., 1996a; Pena et al., 1998; Klose et al., 2004b). 5.2.3 Comparison between FBR and PBMR – Reactive Conditions

In this section the reactor concepts are compared under reactive conditions, using the oxidative dehydrogenation of ethane for illustration of the reactor behavior. In Fig. 5.4 the effect of oxygen concentration on the reactor performance is depicted for both reactor types. The simulation results were obtained with the isothermal model, thus neglecting the effect of temperature on the reactor performance. The space-time (wcat/FT ) was 200 kg s/m3 in the calculations; otherwise the parameters given in Table 5.2 were used. In Fig. 5.4a the ethane conversion is plotted as a function of the oxygen concentration. Below 8 vol.% the conversion increases rapidly with increasing oxygen concentration in both reactors. Above this value the oxygen level hardly affects the ethane conversion. For lower oxygen concentrations the ethane conversion is higher in the FBR than in the PBMR, which can be explained by the higher local oxygen concentration in the FBR. The ethylene selectivity is depicted in Fig. 5.4b. As expected from the kinetics, the maximum values are obtained for small amounts of oxygen in the feed. In agreement with the reaction rate characteristics, an increase of the oxygen level leads in both reactors to ethylene loss, due to the enhancement of the consecutive reactions. The simulations clearly show that due to shorter residence times the deep oxidation reaction was suppressed in FBR. For the parameter range considered the application of PBMR would not be favorable, if high ethylene selectivity is the objective. Regarding ethylene yield the PBMR is superior compared to the FBR (Fig. 5.4c). In the space-velocity region considered the maximum ethylene yields are achieved at high oxygen concentrations. Compared to the FBR higher conversion is achievable in the PBMR, while the ethylene selectivity is decreased only moderately, which leads to higher ethylene yields. Comparing the reactors at the same conversion in the classical selectivity vs. conversion plot (Fig. 5.4d) one can see that at a low conversion level the applica-

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Figure 5.4 Comparison between FBR and PBMR at different oxygen concentrations cO2 = 0.5–15 vol.%, Temperature 600 ºC, wcat/FT = 200 kg s/m3, c ethane = 1 vol.%, FTS/FSS = 1/9; (a) ethane conversion; (b) ethylene selectivity; (c) ethylene yield, (d) ethylene selectivity vs. conversion.

tion of a FBR is favorable. In this region the ethylene selectivity is very sensitive to changes in the oxygen concentration. Therefore, operation at higher conversion using a PBMR appears to be advantageous. In Fig. 5.4 the reactor concepts were compared at only one selected space-time. Thus, it is desirable to inspect a broader range of this important parameter. In Fig. 5.5 simulation results are shown. In the conversion plot shown in Fig. 5.5a again benefits of the PBMR can be recognized. Conversion differences of up to 20% can be obtained in a broad residence time range. In contrast, below 2 vol.% oxygen concentration, the FBR shows slightly higher conversions. This region is extended towards higher oxygen concentrations with increasing residence time. Differences in the mean residence time of the reactants between FBR and PBMR are still present, but not so significant, as below 200 kg s/m3. With increasing contact time the oxygen level is the only conversion determining factor. The PBMR shows higher ethylene selectivity than the FBR only in the oxygencontrolled region (Fig. 5.5b). Increasing the oxygen level, or decreasing the spacetime (i.e. moving towards contact-time-controlled regions), leads to a selectivity

5.2 One-dimensional Modeling of Packed-bed Membrane Reactors

Figure 5.5 Comparison between FBR (dark grey) and PBMR (light grey). Temperature 600 ºC, wcat/FT = 50–500 kg s/m3, c ethane = 1 vol.%, cO2 = 0.5–15 vol.%, FTS/FSS = 1/9. (a) ethane conversion; (b) ethylene selectivity; (c) ethylene yield.

decrease. Because of unavoidable uncertainties of the kinetic model (Klose et al., 2004a) the simulation results below an oxygen level of 0.7 vol.% should be evaluated with care. The resulting ethylene yields (Fig. 5.5c) show that there are two preferable regions for the application of a PBMR. The first one is in the residence-timecontrolled region (in this case below ∼200 kg s/m3 ). In this region the oxygen concentration can be relatively high. The higher ethylene yields achievable here result from the high educt conversion. The second region, where the 1D model postulates the highest PBMR yields is at longer contact times for oxygen concentrations below 1.5 vol.%. Finally, a comparison between the dosing concepts (FBR vs. PBMR) will be given at the same conversion. To get an insight in the selectivity vs. conversion behavior of the PBMR for the whole parameter range studied, one can extract all the simulation results for ethylene selectivity and ethane conversion and plot the corresponding values together in one diagram (Fig. 5.6a). From these points, those at the upper boundary are of interest. These points are presented in Fig. 5.6b. The resulting envelope is the attainable region for the reactor (here at 600 ºC), for the flow rates and reactant concentrations considered. In this special case the borders of the envelope are the edges indicated in Fig. 5.5b as lines A, B, C and D.

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Figure 5.6 Ethylene selectivity vs. ethane conversion under isothermal conditions in FBR and PBMR. (a), all data points according to Fig. 5.5; (b), envelop of border points shown in (a) Temperature 600 ºC, wcat/FT = 50–500 kg s/m3, c ethane = 1 vol.%, cO2 = 0.5–15 vol.% FBR (black), FTS/FSS = 1/9 (gray).

The result shows that for low residence times (line C) and at high oxygen concentration (line B) the PBMR outperforms the FBR. However, in this region the yields are quite low in both reactors. Furthermore, the curves of the PBMR lie inside the operation window of the FBR, thus one can easily find operating parameters where the FBR is superior compared to the PBMR. For ethane per pass conversions below ∼65% the FBR allows for higher ethylene selectivity, and therefore higher yields. If higher per pass conversions are needed, the reaction should be carried out in a PBMR, because the ethylene selectivity does not decrease as rapidly as in the FBR. The maximal ethylene yields, marked in Fig. 5.6b by white dots, are hardly different in both reactors. According to the simulation result, 25.2% of the fed ethane is converted to ethylene. This was achieved in the membrane reactor for approximately 10% higher conversion, while the ethylene selectivity was approximately 7% below that of the FBR. It is important to note that under these operating conditions the reaction kinetics and also the applied simple pseudohomogeneous reactor model can lead to uncertainties in the predicted reactor performance. The use of more detailed models, which account for transfer limitations between the gas phase and the catalyst, could be necessary. As has already been indicated, the PBMR performance can be significantly influenced by varying only the FTS/FSS ratio, while keeping the total flow rates and the fed oxygen amounts constant. In Fig. 5.7 is shown the influence of FTS/FSS ratio that was varied as follows: 3/1, 1/1 and 1/9. Also in these calculations, the oxidant was dosed exclusively on the shell side. Hereby, the maximal oxygen concentration in the feed was 20.5 vol.% (air) according to the experimental conditions reported by Klose et al., 2004b. Therefore, limited by the FTS/FSS ratio, the overall oxygen concentration was varied between 0.5 vol.% and 5 vol.%. Otherwise the same parameters were used as in the previous calculations. Increasing the tube-side flow fraction in the PBMR the ethylene selectivity is increasing, while the conversion is decreasing. The obtained conversions for the

5.2 One-dimensional Modeling of Packed-bed Membrane Reactors

Figure 5.7 Effect of FTS/FSS ratio in the PBMR. Temperature 600 ºC, wcat/FT = 50–500 kg s/m3, c ethane = 1 vol.%, cO2 = 0.5–5 vol.% FTS/FSS = 1/9, FTS/FSS = 3/1, FTS/FSS = 1/1.

highest FTS/FSS ratio (3/1) are, especially in the high residence time region, distinctly lower than for the 1/9 case. This is not surprising, as the local oxygen concentrations are considerably lower for FTS/FSS = 3/1 (see Fig. 5.3b). This leads to higher ethylene selectivity, but does not influence the maximal achievable ethylene yield. The maximal yield is obtained for the largest shell-side flow fraction. However, below 55% ethane per pass conversion the higher FTS/FSS ratio is apparently more advantageous. It is worth noting that there is obviously an important limitation on the application of a high FTS/FSS ratio in PBMR. Below a certain flow through the membrane, back diffusion of educts and products from the tube to the shell side can take place. It can influence negatively both the conversion and the intermediate selectivity. Therefore, while looking for optimal reactor parameters, one should be aware of minimal flow rates or pressure differences over the membrane and use this information as an additional constraint during optimization. 5.2.4 Nonisothermal Operation

In this section the effect of internal temperature profi les on the reactor performance will be considered based on the predictions of the 1D model. On the laboratory scale these effects are often not so relevant. However, in possible large-scale applications temperature gradients are highly probable and can be critical. One of the main advantages of the PBMR, or more generally of the distributed dosing concept is the fact that due to the sidestream runaway behavior and hot-spot formation can be suppressed considerably (Hwang and Smith, 2004). However, it is important to note that feed segregation alone is not necessarily sufficient to suppress undesirable temperature effects. Indeed, the following simulation results are intended to illustrate the possibility of an opposite behavior. The simple 1D reactor model (Eqs. (5.1)–(5.4)) was used to calculate the temperature profiles in both reactors. The gas properties were calculated following

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5 Packed-bed Membrane Reactors

Figure 5.8 Temperature profiles in FBR and PBMR. Inlet and wall temperature 600 ºC, wcat/FT = 225 kg s/m3, c ethane = 1 vol.%, cO2 = 5 vol.%, FTS/FSS = 1/9 in PBMR.

Reid et al., 1989, using pure component data from Himmelblau, 1974. As in the previous isothermal calculations the feed and wall temperature was fi xed at 600 ºC. In Fig. 5.8 are shown temperature profi les in both reactors. Because of the small reactor scale considered the maximum temperatures differ not much from the inlet values, but there are clear tendencies. It can be seen that in this particular case the different contact-time behavior results in a temperature maximum near the reactor inlet for the PBMR, while in the FBR the maximal temperature arises approximately in the middle of the catalyst bed. Predominantly, the residence time, rather than the transport through the membrane and therefore the oxygen supply, is responsible for these temperature profi les. It is instructive to inspect a wider range of operation conditions and to compare the two reactor concepts. Figure 5.9 shows the maximal temperature increase, ∆T = Tmax − T 0, in both reactors. The PBMR results correspond to FTS/FSS ratios 1 : 9 and 3 : 1. As before, oxygen is fed only in the shell side. The heat generation in the PBMR with high dosing rates through the membrane (FTS/FSS = 1/9) is more pronounced than in the FBR. This can be explained by the distinctly higher contact times of ethane near the reactor entrance, which favors the total oxidation and leads to undesired hot-spot formation. If the reaction heat could be removed more efficiently from the reaction zone, using e.g., a specific PBMR construction, this flow rate ratio could be attractive, because both ethylene selectivity and yield is higher in the PBMR than in the FBR (compare with Fig. 5.5). For larger tube side flows (FTS/FSS = 3 : 1) the PBMR shows the expected smoother temperature profi les. In the whole operation range studied, the maximal reactor temperature is lower than in the FBR. As shown earlier, this FTS/FSS flow ratio delivers higher ethylene yields in operation regions where the conversion is below 55%.

5.2 One-dimensional Modeling of Packed-bed Membrane Reactors

Figure 5.9 Maximum temperature increase in FBR and PBMR. Inlet and wall temperature 600 ºC, wcat/FT = 50–500 kg s/m3, c ethane = 1 vol.%, cO2 = 0.5–5 vol.%. FBR (black), PBMR FTS/FSS – 1/9 (gray, transparent), FTS/FSS – 3/1 (light gray).

Figure 5.10 (a) Oxygen concentration and (b) Temperature profile in PBMR with cofeeding of oxygen on the tube side. Inlet and wall temperature 600 ºC, FTS/FSS = 3/1, wcat/FT = 500 kg s/m3, c ethane = 1 vol.%, cO2 = 5 vol.%, Parameter: O2 distribution between TS and SS (W = 0–0.2–0.5–0.8–1).

In the cases discussed above oxygen was fed exclusively on the shell side. However, it might be useful to dose one of the reactants simultaneously on both sides of the membrane. With this combination of cofeed and distributed dosing it can be possible to suppress negative effects at high FTS/FSS ratios in the PBMR (e.g. low conversion, catalyst deactivation, etc.). In Fig. 5.10 is illustrated the effect of simultaneous cofeeding of oxygen on the internal temperature profile. In these calculations an oxygen-distribution factor, W = w O0 2 / w O2 (inlet oxygen mass fraction/overall oxygen mass fraction, compare Eq. (5.5)), was varied between zero and one. The FTS/FSS ratio was fi xed at 3 : 1, the oxygen concentration was 5 vol.% and the space-time was 500 kg s/m3.

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On decreasing the factor W, i.e. dosing less oxygen on the tube side, the maximal temperature in the reactor can be considerably reduced and removed from the reactor entrance. This leads to the “temperature-smoothing” effect of distributed dosing of reactants, which makes the application of PBMR attractive, especially for highly exothermic reactions. However, with decreasing oxygen amount entering via the shell-side feed, the ethane conversion can decrease too. Therefore, while optimizing the oxygen distribution to obtain maximal reactor performance, considering, e.g., constraints set by the explosion limits of the reactants, the hot-spot temperature should additionally be considered (Dixon, 2001). An important advantage of the 1D model described (Eqs. (5.1)–(5.4)) is its simplicity. It enables extensive parametric studies to be carried out in a very short time. The results might provide already a good accuracy, especially for laboratoryscale reactors. However, often deeper insight and more detailed models are required to understand better the interactions between hydrodynamics, transport processes and catalytic reaction in packed-bed membrane reactors. With a more realistic 2D reactor model, presented in the following section, one can answer more specific questions related to these complex processes occurring in a PBMR.

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors

In the previous section a one-dimensional model was developed and used to demonstrate the possible benefits of packed-bed membrane reactors as compared to the established fi xed-bed reactors. Basic phenomena can be described with sufficient accuracy even with this simple model. However, when the goal is to predict reactor behavior in more detail the one-dimensional model may reach its limits due to radial mass- and heat-transfer limitations. Additionally, flowmaldistribution effects can also not be captured. Taking advantage of improvements in computation speed, it is nowadays possible to predict the influence of these phenomena with two- or even three-dimensional models and use this knowledge to optimize reactor performance. However, detailed modeling of membrane reactors is not as straightforward as in the case of fi xed-bed reactors. There are still a couple of open questions; e.g. whether semiempirical correlations obtained under nonreactive conditions in fi xed beds are applicable also to membrane reactors or not. Until these questions have been completely clarified one has to rely on the available database and correlations as the best possible estimate. In this section, a two-dimensional, pseudohomogeneous reactor model will be developed neglecting heat- and mass-transfer limitations between the bulk phase and catalyst particles, as well as inside the catalyst pellets. The two-dimensional formulation presented takes advantage of the cylindrical reactor geometry shown in Fig. 5.11. Neglecting the third (angular) coordinate can distinctly speed up the calculations compared to the solution of a 3D model (see Section 5.4). Further reduction

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors

Figure 5.11 Model domain and boundary notation considered by the 2D description of a PBMR.

of the computational time can be achieved by expressing mass and heat transport through the membrane and the reactor walls favorably in the boundary conditions. The choice of appropriate boundary conditions and control volumes is essential for the reliability and accuracy of the results. The issue of boundary conditions and respective simplifications will be considered below. 5.3.1 Two-dimensional Model of PBMR – The Momentum-balance Equation

To calculate the flow field for the reactor geometry shown in Fig. 5.11 the extended Navier–Stokes equation ∂ρgε v + ∇⋅ ( ερg vv ) = −ε∇p −ψρgε v − ∇⋅ ε ⋅τ ∂t

( )

(5.6)

and the mass-continuity equation

∂ερg + ∇⋅ (ερg v ) = 0 ∂t

(5.7)

were solved. Since for this type of membrane reactors mainly the steady-state solution is of interest, the accumulation terms can be omitted from the equations above. However, solving the time-dependent problem can speed up the convergence of the numerical solution and is therefore frequently used in simulation tools. The fluid is treated as Newtonian and follows the ideal gas law so that the gas density is

ρg =

pM . RT

The shear stress tensor is defined as

(5.8)

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5 Packed-bed Membrane Reactors

2 T τ = − µg ∇v + ( ∇v )  +  µg − ζ B  ( ∇v ) I  3

(5.9)

where the bulk or dilatational viscosity zB, which contributes to the normal stress, is zero for low-density monoatomic gases, and neglected for low Mach number flows. The friction force arising due to the flow through the packed bed can be calculated according to Ergun, 1952, by defining the friction coefficient as

ψ = 150

(1 − ε )2 µ g ε

3

ρg d

2 p

+ 1.75

1−ε ε v ε 3 dp

(5.10)

It is worth noting that there were numerous Ergun-type correlations derived in the past for the calculation of drag forces based on the representative elementary volume approach (Bear, 1988). To account for nonuniform voidage profi le in packed beds, local porosity values may be substituted in Eqs. (5.6), (5.7) and (5.10) as suggested by among others Stanek and Szekely, 1974, Delmas and Froment, 1988. In spite of some discussion about the applicability of the Ergun equation with local quantities (Molerus, 1982), the mentioned substitution allows for a good qualitative description of numerous experimental data (Martin and Nilles, 1993; Bey and Eigenberger, 1997) and is therefore used in the following calculations. If the tube-to-particle diameter ratio is low (di/dp < 20) a significant part of the flow shifts from the core of the bed in the direction of the reactor wall. This socalled flow maldistribution is caused by the nonuniform porosity profi le of the packed bed. The effect on the local velocity is pronounced, especially at low particle Reynolds number (Re0 < 25, Winterberg and Tsotsas, 2000b). To describe the experimentally found damped oscillatory behavior of the radial porosity profi les, various voidage models have been presented (Goodling et al., 1983; Küfner and Hofmann, 1990). The oscillations were found to vanish for not perfectly spherical particles of a certain size distribution. In these cases a simple, exponentially declining function is sufficient to predict the radial porosity profi le (Vortmeyer and Schuster, 1983; Hunt and Tien, 1990). In terms of local velocity some overestimation has been observed near the reactor wall, which can be avoided by introducing an effective dynamic viscosity matching the calculations to the experimental data (Giese et al., 1998; Bey and Eigenberger, 2001). This second-order effect has been neglected in the present analysis. Consequently, the dynamic viscosity of the fluid, mg, was used in the calculations. The porosity profi le by Hunt and Tien, 1990:  R −r  ε (r ) = ε ∞ + (1 − ε ∞ )exp  −6 ⋅  dp  

(5.11)

was preferred over the profi le suggested by Giese et al., 1998, because it has been developed under comparable assumptions and always meets the theoretically

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors

expected porosity value at the wall ( e = 1). For the sake of comparison, calculations were also conducted with averaged porosity values, ε . 5.3.2 Two-dimensional Model of PBMR – The Mass-balance Equation

Concentration profi les along the radial and axial reactor coordinates can be calculated by solving the species-conservation equation: NR ∂ (ερgω i ) + ∇⋅(−Die ⋅ ρg∇ω i + ερgω i ⋅ v) = (1 − ε )ρcatMi ⋅ ∑ ν ij ⋅r j , ∂t j

i = 1, . . . , N C

(5.12)

where w i is the mass fraction of component i and rcat is the density of the catalyst pellet. The right-hand side of the equation accounts for the chemical reaction. Since the description of the reaction volume is pseudohomogeneous, effective dispersion coefficients (Die ) appear on the left-hand side. In this work, radial effective dispersion coefficients used in combination with an averaged porosity were calculated according to the correlation Die,r = Dim + D t = (1 − 1 − ε )Dim +

ud p K ∞ ⋅ f (d p di )

(5.13)

with the correction factor: f (dp di ) = 2 − [1 − 2 (dp di )]

2

(5.14)

(Tsotsas and Schlünder, 1988a). The above correlation accounts for both molecular and crossmixing (quasiturbulent) dispersion. The mixture averaged diffusion coefficients D im in the molecular term can be calculated after Kee et al., 2003 using binary diffusion coefficients according to Fuller et al., 1966. The highly diluted reaction mixture justifies this procedure. The crossmixing term is calculated with the averaged superficial velocity, u, and with the constant K∞ . The latter has a theoretically deducible value of 8 (Schlünder, 1966). In calculations with a radial porosity profi le the local superficial velocity was used. In this case the correction factor in the denominator of Eq. (5.13) is omitted. The resulting relationship: Die,r = Dim + D t = (1 − 1 − ε (r ) ) Dim +

ud p 8

(5.15)

has also been applied by Kürten et al., 2004. The influence of the voidage profi le is taken into account by the local values of the bed porosity.

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5 Packed-bed Membrane Reactors e The axial dispersion coefficient, Di,ax , can be calculated by essentially the same formula, with the only difference that the constant in the mixing term has a value of 2 (Aris and Amundson, 1957; Tsotsas and Schlünder, 1988b).

5.3.3 Two-dimensional Model of PBMR – The Energy-balance Equation

In analogy to the component mass-balance equation, the thermal energy equation is

∂ (ερg ⋅ c pT + (1 − ε )ρsc p,sT ) + ∇⋅(−λ e ⋅∇T + ερg ⋅ c pT ⋅ v ) ∂t NR

= (1 − ε )ρcat ⋅ ∑ (−∆Hr , j )⋅r j

(5.16)

j =1

where rs, and cp,s are the density and specific heat capacity of the solid, respectively. Equation (5.16) can be used to calculate two-dimensional temperature profi les in the PBMR. The term on the right-hand side represents the heat source related to chemical reactions. Important model parameters for a PBMR are the effective radial and axial heat dispersion coefficients, which were calculated according to two different approaches described in Tsotsas, 2002. The first approach presumes heat dispersion with different but radially constant coefficients in both directions. For the radial heat dispersion one can write:

λ er = λ bed + λg

Pe0 K ∞ f (d p di )

(5.17)

where lbed is the thermal conductivity of the bed without gas flow that can be calculated by the Zehner–Bauer–Schlünder model (Bauer and Schlünder, 1978). The second term is completely analogous to the second term of Eq. (5.13) and accounts for dispersion due to crossmixing. The molecular Péclet number for heat transfer, Pe0, is calculated with the average superficial velocity and the factor K∞, which has the same value as for mass dispersion (K∞ = 8). For the axial direction the same correlation can be applied, however with a 4-times larger convective term. The second approach, called the l (r)-model, was first suggested by Kwong and Smith, 1957, was readdressed by Cheng and Vortmeyer, 1988, and refined in an extensive comparison with experimental data by Winterberg and Tsotsas, 2000a, Winterberg et al., 2000. The authors stress that especially in reactive flow problems at low Reynolds numbers the assumption of an inhibiting laminar sublayer at the wall is not appropriate. Therefore, the use of a wall heat-transfer coefficient, aw, can only be artificial. Instead, they suggest correlating the radial heat dispersion coefficient as:

λre (r ) = λbed (r ) + λg

Pe0uc f (R − r ) 8u0

(5.18)

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors

with

 R − r 2   f (R − r ) =  K 2d p   1

if 0 < R − r < K 2d p

(5.19)

if K 2d p < R − r < R

Re and K 2 = 0.44 + 4 ⋅ exp  − 0   70 

(5.20)

In Eq. (5.18) u c is the superficial velocity of the fluid in the core of the packed bed (i.e. at r = 0). The dimensionless numbers Re0 and Pe0 are calculated with the average superficial velocity. The axial dispersion coefficient is again calculated in analogy to Eq. (5.17):

λaxe (r ) = λbed (r ) + λg

Pe0 2

(5.21)

Haidegger et al., 1989, have studied the total oxidation of ethane in a fi xed-bed reactor and found a better agreement between experiment and simulation with the l (r)-model, compared to the model using a wall resistance (1/ aw). Simulation results for models with and without the radial porosity profi le will be compared below. In the model neglecting the influence of the voidage profi le, constant mass and heat dispersion and a wall resistance (1/ aw ≠ 0) will be applied. The effects of the radial voidage profi le will be studied using the same correlation as used in Kürten, 2003, for the mass and the l (r)-model for the heat dispersion. 5.3.4 Boundary Conditions

To obtain a unique solution of the resulting system of coupled partial differential equations (Eqs. (5.6), (5.7), (5.12) and (5.16)), boundary conditions are needed at the borders of the model domain (Fig. 5.11). The boundary conditions applied to the momentum, thermal energy and component mass-balance equations are given in Table 5.3. On boundary IV, the mass flux through the membrane, denoted by ji (z/L), can be formulated in different ways, depending on the reactor configuration and/or membrane characteristics. In the special case of a closed shell-side outlet considered here (the so-called “dead-end” configuration) the entire amount fed on the shell side has to permeate through the membrane. Together with the assumptions that the membrane is impermeable for all other components than those present in the shell side feed, and that the shell-side flow is uniformly distributed along the reactor length, the fluxes are constant and can be expressed as follows:  ρ gSS ⋅ω SS i ⋅ u M , for components fed on the shell side ji =  0 , for other components 

(5.22)

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5 Packed-bed Membrane Reactors

Table 5.3 Boundary conditions for the 2D PBMR model.

Boundary (Fig. 5.11) Component mass balance

Momentum balance

w i = w 0i

I. z = 0 0 < r < R II. z = L 0 < r < R III. r = 0 0 < z < L

␯ = ␯0 − n · (−Dei · rg∇w i ) = 0 − t · ␯ = 0; p = 0 − n · (−Dei · rg∇w i + erg ␯ ) = 0 − n · ␯ = 0

IV. r = R 0 < z < L

− n · (−Dei · rg∇w i + erg ␯ ) = j

−n · ␯ = u M

Energy balance T = T0 − n · (−l e · rgcp∇T) = 0 − n · (−l e · rgcp∇T + rgcpT · ␯ ) = 0 − n · (−l e · rgcp∇T) = a w (T − Tw ) or T = Tw

Figure 5.12 Extended model domain of the PBMR illustrated in Fig. 5.11.

With the superficial velocity of the fluid in the membrane, uM, the following simple Dirichlet boundary conditions can be defined on boundary IV for the momentum balance: v z |r =R = 0; vr |r =R = uM

(5.23)

However, this simplification can not be applied without restrictions. Depending on operation parameters, mixture composition and membrane properties it may be necessary to treat the boundary condition at the membrane in a different manner. In the most general case the balance equations have to be solved simultaneously not only for the tube side but also for the membrane and for the shell side, corresponding to an extended model domain illustrated in Fig. 5.12. Typical examples where this extended analysis is necessary are conditions of possible back diffusion of components from the tube to the shell side. Additionally, more concentrated mixtures, Knudsen and/or surface diffusion may require the application of the Stefan–Maxwell approach in order to describe the mass transport in the porous medium (Krishna and Wesselingh, 1997). To illustrate some of the above-mentioned aspects, results for the pressure and velocity field based on the solution of the extended model are presented below. If not mentioned otherwise, the two-dimensional simulations were conducted with the set of reference parameters presented in Table 5.4.

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors Table 5.4 Standard simulation parameters for the 2D PBMR model.

Reactor length (length of the porous zone) – L Inner/outer diameter of the membrane – di/d o Inner/outer diameter of the reactor wall – Di/Do Particle diameter – dp Envelop catalyst density – r cat Bulk porosity of the catalyst bed – ε Porosity in the core of the bed – e ∞ Contact time – wcat/F T Oxygen concentration – c O2 Ethane concentration – c ethane Tube- to shell-side flow ratio – F TS/FSS Inlet and wall temperatures – T 0, Tw

105 mm 21/32 mm 39/45 mm 1.8 mm 1020 kg m−3 0.43 0.385 250 kg s/m3 1 vol.% 1 vol.% 1/9 600 ºC

5.3.5 Numerical Solution of the Two-dimensional Model

To solve 2D reactive flow problems various numerical strategies can be applied. Dependent on the complexity of model geometry and the structure of the resulting set of equations, which in turn is determined by, e.g., the nonlinearity of the reaction source term or the degree of coupling of different transport phenomena, different numerical methods and differentiation techniques can be beneficial (Ferziger and Peric, 2001). Therefore, the spectrum of available commercial or in-house numerical tools and software codes is also quite wide. Most widespread tools are based on discretization by the finite volume (FV) or the finite element (FE) method. These methods are in many respects similar to each other. In both, the model domain is divided into a finite number of discrete volumes or elements, respectively, where the conservation equations are applied. In the FVM the solution is to be calculated in the centers of the control volumes (node points). Fluxes through the control volume surface are obtained by interpolation. In the FEM an approximate solution is developed for each of the elements using a linear combination of appropriate functions (shape functions). This approximation is substituted in the original conservation equations. According to the method of weighted residuals the resulting algebraic equations are multiplied with a weight or test function (which is constant in the FVM) and afterwards integrated over the entire domain. The problem is formulated in a way that the weighted integral vanishes for the exact solution. The usually nonlinear equation system is linearized and solved by direct or iterative methods. In this work, the commercial software package Comsol Multiphysics 3.2 (Comsol AB, Stockholm, Sweden) was used to solve the system of nonlinear partial differential equations. Apart from predefined built-in applications, this tool enables the user to define and solve PDE systems by finite-element discretization. Taking advantage of the axial symmetry of the problem, the model domain can be simplified to two dimensions. In this domain, a structured quadrilateral mesh

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was generated. The mesh was refined at the reactor entrance (boundary I) and at the reactor wall (boundary IV) to achieve a smooth solution. Typically, 30 nodes in the radial and 60 in the axial direction were necessary to achieve a solution independent of the mesh size. Second-order Lagrange elements were used to calculate the temperature and concentration profi les in the reactor. For the solution of the pressure field linear Lagrange elements were applied. The resulting set of nonlinear algebraic equations was solved by a damped Newton method using a direct solver for the linearized problem. The relative tolerance values for the solution were set to 10 −5 –10 −6. Typical solution times for the stationary coupled problem with 10 dependent variables (200 000 DOF) were about 45 min on a FSC Celsius 810 workstation (Fedora Core 3, Linux operating system, kernel 2.6.9) with an AMD Opteron® 248 processor and 4 GB of physical memory. 5.3.6 Velocity Field in a Packed-bed Membrane Reactor

In this section, steady-state two-dimensional velocity profi les in the PBMR will be analyzed, considering the relevance of boundary conditions (BC) and the effect of flow maldistribution. The BC for the PBMR model (Table 5.3) at the entrance, the symmetry axis and at the reactor outlet are identical to those for a fi xed-bed reactor. Setting up the BC at the reactor entrance, one should take care to avoid singularities at the reactor walls. This can be achieved by use of a smoothened velocity profi le, e.g. by a damped Heaviside function, which helps to avoid oscillation in the solution and makes convergence faster. The definition of a proper BC at the porous membrane wall is not as evident as for the FBR, where no-slip conditions ( v = 0 ) can be applied. Flow at the interface of a fluid and of a porous medium was subject of numerous experimental and theoretical investigations in the past. The pioneers of the topic, Beavers and Joseph, 1967, have examined pressure-driven flow past a homogeneous porous material. According to the two-domain approach, they considered Stokes flow in the fluid and Darcy flow in the porous domain. Because of the different order of the set of corresponding PDE a semiempirical boundary condition was introduced to describe the slip velocity at the interface. Note that one can also use the equation of Brinkman, 1947, describing momentum transport in the porous medium. In this case, both PDE are second order, so that continuity of stress and velocity can be fulfi lled. It can be shown that the axial velocity declines with the distance, s, from the surface of the porous half-space according to u z (s ) = us ⋅ exp( −s

B0 ) + u z

s →∞

(5.24)

where B 0 is the permeability of the porous medium. Consequently, the penetration depth of the external fluid flow is of the scale B0 , comparable with the pore diameter dpor. The slip velocity uS at the interface and the drift velocity uz|s→∞

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors

deeply within the porous medium are only of interest for highly porous structures, e.g., foams (B 0 ∼ 10 −8 m2, e ∼ 0.8) structures (James and Davis, 2001). Comparing with permeability values typical for the less porous materials applied in membrane reactors (Table 5.1) it can be expected that the no-slip boundary condition will be usually valid. To demonstrate the above conclusion, selected results of a numerical study using the so-called one-domain approach to describe the momentum transfer at the fluid/porous interface in a membrane reactor will be presented. 5.3.7 The Influence of Membrane Permeability on the Boundary Conditions

In the present study the membrane is treated as a homogeneous porous medium. Furthermore, for the sake of simplicity only single-gas permeation is considered. The extended Navier–Stokes equation is solved on the geometrical domain illustrated in Fig. 5.12. Since the packed bed, the membrane, and the shell side are included in the calculation, boundary conditions do not have to be formulated explicitly at the interface (Goyeau et al., 2003). To account for the friction forces the following coefficients were substituted in Eq. (5.6):

ψ = 150

(1 − ε )2 µ g ε

3

ρg d

2 p

+ 1.75

1−ε ε v ε 3 dp

(5.10)

in the catalyst bed,

ψ=

µg ερgB0

(5.25)

in the membrane according to Darcy’s law, assuming constant porosity and

ψ =0

(5.26)

in the annulus. Note that the inertia and the surface force terms are present in Eq. (5.6) but the friction term is predominant in the membrane (Eq. (5.25)). Since the shell side of the reactor is considered as an empty annulus the friction term is zero (Eq. (5.26)). In Fig. 5.13 profi les of axial and radial velocity are shown at three axial positions for the annulus, the membrane and the packed bed. The profi les correspond to an α -alumina membrane with a relatively low permeability (B 0 = 9.5 × 10 −14 m2 ) and a mean pore diameter of dpor = 3 µm (compare with Table 5.1). The Knudsen number is approximately 10 for this membrane at the given conditions, so that only viscous flow occurs. Because the annulus is empty and the flow within it is laminar, the resulting profi le for the axial velocity component is parabolic. Compared to the annulus flow, the axial velocity component in the membrane is

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5 Packed-bed Membrane Reactors

Figure 5.13 (a) Axial and (b) radial superficial velocity component at different axial positions – Low membrane permeability B 0 = 9.5 × 10 −14 m2.

Figure 5.14 (a) Axial and (b) radial superficial velocity component at different axial positions – high membrane permeability B 0 = 8.6 × 10 −11 m2.

negligible. Axial pressure gradients in the annulus, the membrane and the packed bed are small in comparison with the pressure difference between the shell side and the tube side. Consequently, identical profi les of radial velocity are obtained at different axial positions (Fig. 5.13b). In other words, a uniform, self-similar flow distribution is predicted along the reactor. The slip velocity at the porous solid interface is also negligible. Therefore, simplified boundary conditions of vz = 0 and vr ≠ 0 = const. can be applied at the reactor wall. The effect of the porosity profi le in the tube side will be addressed later. The effect of increased membrane permeability is depicted in Fig. 5.14. The material properties considered now correspond to that of a sintered metallic macrofi ltration membrane (B 0 = 8.6 × 10 −11 m2, mean dpor = 100 µm). In this calculation, the same total flow rate and the same FTS/FSS ratio was applied as in Fig. 5.13. As can be seen, both profi les of the axial and radial velocity differ significantly at different axial positions (Fig. 5.14). In fact, flow from the tube to the shell side (ur positive) is predicted at the entrance of the permeable part of the membrane. The flow direction is inverted (ur is negative) at z/L = 0.2. Further downstream, the absolute value of ur is increasing in a nonlinear way. The pressure drop through the catalyst bed is now

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors

comparable to the pressure drop through the highly permeable membrane (Fig. 5.15) and is responsible for the mentioned penetration of fluid from the tube into the membrane at low z/L. The slip velocity caused by viscous forces at the interface is higher here (compare Fig. 5.14a and Fig. 5.13a), however, it is still quite small with respect to the pressure-induced flow and, therefore, of low interest. The sealed regions of the membrane are also included in the plot in Fig. 5.15. There, because of the fully developed flow (the radial velocity is zero, the axial velocity is constant), the pressure decreases linearly with the axial coordinate. In Fig. 5.16 the resulting profi les of average axial velocity in the catalyst bed are shown. It can be seen that the axial tube-side velocity increases linearly with the reactor length when a large pressure drop is imposed over the membrane (uniform dosing profile, constant pressure difference across the membrane). In contrast, a nonlinear profi le is obtained with a highly permeable membrane including a minimum caused by the discussed inversion of flow. Due to this inversion, a considerable amount of hydrocarbon could permeate to the shell side under reactive conditions, resulting in an undesirable bypass of the catalyst bed. A nonuniform dosing profi le can also significantly influence the contact time and, therefore, the reactor performance. Whether these effects can improve the reactor performance, or not, obviously depends on the reaction system and on the operating conditions. Klose et al., 2004b, have shown how one can utilize the shape of imposed profi les to significantly influence selectivity and yield of intermediates. In general, membrane reactors with highly porous membranes can not be modeled properly by only applying a boundary condition at the inner membrane wall.

Figure 5.15 Pressure field in the PBMR – high membrane permeability B 0 = 8.6 × 10 −11 m2.

Figure 5.16 Average axial velocity profile in the catalyst bed along the PBMR for two different membrane permeabilities.

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Flow and transport in the membrane and on the shell side have to be also considered in more detail. This increases the computational costs considerably. 5.3.8 Effect of Porosity Profile

In the following membranes with a low permeability will be focused on, i.e. reactors where simplified boundary conditions at the membrane wall can be applied. Flow maldistribution is a problem well known from conventional fi xed-bed reactors. At low aspect ratios (tube to particle diameter) a considerable amount of fluid streams preferentially near the reactor wall. The resulting effect on the integral performance in a PBMR is shown in the next section. At this point only fluid dynamic aspects are considered. In Fig. 5.17 the developed superficial velocity profi les at z/L = 0.5 are depicted. The black curve corresponds to calculations with the bed-porosity profi le by Hunt and Tien, 1990 (Eq. (5.7)), the light gray curve was simulated using the constant volume-averaged porosity value. Comparing both models it can be seen that a high maximal velocity is predicted at the vicinity of the wall when using the radial voidage profi le. Maximal velocity was found to decrease with increasing Reynolds number. This behavior was also observed in numerous former studies on fi xed-bed reactors, e.g. by Bey and Eigenberger, 1996. The analogy of these results to that in fi xed-bed simulations without sidestream is pronounced and can be explained with the predominance of the bed frictionforce term in the extended Navier–Stokes equation (Eq. (5.6)). Hydrodynamically developed flow is achieved after a distance of just about one particle diameter in the axial direction. However, the developed profi le in a PBMR is characterized by a radial velocity different from zero. One can prove analytically that for reactive flow problems with negligible change in the physical properties (density, viscosity) the superficial radial velocity decreases linearly towards the core (Kürten, 2003). In Fig. 5.17b the superficial radial velocities are compared. Using the radial porosity profi le, smaller absolute values of the local superficial velocity are calcu-

Figure 5.17 (a) Axial and (b) radial velocity profile at z/L = 0.5 in the PBMR – effect of porosity profile.

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors

lated than in the case of constant porosity. The radial component of the interstitial velocity goes through a maximum and decreases linearly as the local porosity is decreased to the value of the core (not shown here). Because radial velocity directly influences the radial dispersion, an effect on the interparticle reactant transport can be expected. Whether the transport from the membrane to the core is distinctly influenced by convection, or not, is examined in the next section. 5.3.9 Effect of Radial Mass-transport Limitations

Mass transport of the dosed component oxygen from the membrane to the center of the catalyst bed is studied in this section. The boundary conditions for the component mass balances were given in Table 5.3. In setting up these BCs it was assumed that the membrane walls are impermeable for every component except oxygen and an inert gas (nitrogen). This constraint is valid for membrane materials with low permeability and dominating convective transport through the membrane. Reactor models accounting for radial porosity profile were compared with models using the averaged bed-porosity value. Isothermal conditions were applied in order to rule out thermal effects on the concentration profi les. To check the need for two-dimensional models the results were compared with that obtained by using the pseudohomogeneous one-dimensional reactor model (Eqs. (5.1)–(5.4)). In Fig. 5.18 the oxygen concentration profi les in the PBMR are depicted. The model accounting for the radial porosity profi le shows considerably higher oxygen concentrations near the membrane. This result was expected because of two reasons: First, the reduced amount of catalyst at the wall, corresponding to the higher local porosity, leads to lower local oxygen conversion; second, radial dispersion is enhanced due to the higher local fluid velocity. On the other hand, following the same logic, the oxygen supply to the core is worse, because of lower dispersion and longer contact time of the components. In this context, it is instructive to

Figure 5.18 Effect of porosity profile on the local oxygen concentration. (a) 2D oxygen profiles (b) radially averaged axial O2 profile. 1D simulation with: Temperature 600 ºC, wcat/ FT = 250 kg s/m3, c ethane = 1 vol.%, c O2 = 1 vol.%, FTS/FSS = 1/9.

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calculate a radially averaged oxygen profile normalized with respect to the catalyst mass: R

ω O2 =

2π ⋅ r (1 − ε(r ))ρcat ⋅ω O2 dr 2 R π (1 − ε )ρcat ∫0

(5.27)

With this averaging formula, the high oxygen concentration near the membrane, which contributes less to the reaction, has a lower weight. A comparison of the respective oxygen profi les along the reactor shows that the model with a radial porosity profi le predicts lower oxygen concentrations (Fig. 5.18b). This could be a reason for the slightly higher ethylene selectivity and lower ethane conversions visible in Fig. 5.19. However, one should be careful when assessing the integral reactor performance on the basis of axial oxygen profiles. Then, even though the oxygen concentrations calculated with the 1D model are considerably lower than those of the 2D model, the predicted conversion values (Fig. 5.19) are in this case 7–8% higher. Hereby, the lower oxygen concentration results from higher oxygen consumption. Picking up again the important effect of the FTS/FSS ratio studied already with the 1D model, the role of radial convective mass transfer was examined. The standard parameter set was used for these calculations, varying only the FTS/FSS ratio. The two-dimensional oxygen concentration profiles in Fig. 5.20a show the expected tendencies, which have already been discussed in Section 5.2. In the case of higher shell side flow (FTS/FSS = 1/9) the oxygen concentration rises faster, despite the lower radial mass dispersion, resulting from the lower fluid velocity at the entrance of the reactor (see also Fig. 5.20b). Under tube-side-dominated flow conditions a significant part of the catalyst bed is unreachable for the oxygen molecules. This might lead to undesired coke formation, catalyst deactivation and secondary reactions. In such cases not only the radial mass transfer in the bed, but also the oxygen supply through the membrane may be limiting. With

Figure 5.19 Effect of porosity profile on the ethylene selectivity – Temperature 600 ºC, wcat/FT = 250 kg s/m3, c ethane = 1 vol.%, c O2 = 1–5 vol.%, FTS/FSS = 1/9.

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors

Figure 5.20 Influence of the flow distribution on the oxygen concentration profile (a) 2D oxygen profiles (b) radially averaged axial O2 profiles. Results obtained for standard parameter set (Table 5.4).

additional cofeed of oxygen at the tube-side entrance one could utilize these, otherwise unused, regions of the bed. In this case one should be, however, prepared for higher heat generation. Summarizing, the effect of the porosity profi le on the integral reactor performance is rather small for the conditions studied. This can, however, change for systems with kinetics more sensitive to the educt concentration (higher reaction orders). In comparison with the 1D model results it was found that the simple model overpredicts the achievable intermediate yields in the PBMR. Consequently, radial mass-transfer limitations can not be neglected if more precise predictions are required. 5.3.10 Comparison of the l (r)- and a w-model Concepts – Temperature Profiles in a PBMR

In this section, the 2D analysis is extended to nonisothermal PBMR operation. In particular, differences between the two heat-transfer models, the l (r)- and the aw-model, are examined. In the first instance, only the computational domain of Fig. 5.11 is considered, with boundary conditions at the membrane wall according to Table 5.3. The standard parameter set of Table 5.4 is used, though with 1–5 vol.% overall oxygen concentration. Calculated temperature profi les are shown in Fig. 5.21. By comparison, the 2D model with a heat-transfer resistance at the wall gives slightly higher hot-spot temperatures than the l (r)-model. This implies a somewhat better cooling performance according to the l (r)-model, in spite of the pronounced bypass flow near the membrane and the resulting lower fluid velocity in the core of the bed. In addition to the radially averaged temperature profi les, the results from the one-dimensional model are also depicted in Fig. 5.21a. While the maximal difference between predictions of the two-dimensional models is less than 4 K, the one-dimensional model overestimates those results by about

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Figure 5.21 Comparison between the heat-transfer models in the PBMR (a) averaged temperature profiles, c O2 = 5 vol.% (b) ethylene selectivity vs. conversion, c O2 = 1–5 vol.% – otherwise standard parameters used (Table 5.4).

15 K. Comparing the reactor performance, it can be seen that – just as in the isothermal case – 7 to 8% higher conversion values are obtained with the simple 1D model (Fig. 5.21b). This higher conversion is responsible for the increased hotspot temperature, and shows once more the limitations of the 1D model. The ethylene selectivity and also the conversion obtained by the 2D models are identical within 1%. Up to now, the heat-transfer resistances of the membrane and of the shell side have been neglected. This is reasonable only in exceptional cases, because the highest heat-transfer resistance of the system is expected – due to the reactor construction – at the shell side. Assuming quasistationary heat transfer, the calculation of the overall heat-transfer coefficient between reactor wall and catalyst bed is easy to implement using a series connection of thermal resistances for both the aw- and l (r)-models. Because of the small differences in the predicted temperature profi les and the lower computational cost, the aw-model with the overall heat-transfer coefficients has been used in the following calculations. In the previous discussion of the one-dimensional nonisothermal simulation results it has been shown that for certain operating conditions the ethane conversion can be increased considerably in a PBMR compared to conventional fi xed-bed reactors. The price, which had to be paid, was the higher local heat generation and insufficient heat removal. The problem is more pronounced in the large-scale apparatus. For illustration, the temperature profi le in the PBMR calculated with the extended version of the aw-model is depicted in Fig. 5.22. Accounting for the thermal resistance of the shell side and of the membrane, a temperature maximum of more than 20 K above the inlet and outer reactor wall temperature is predicted. For the sake of completeness it has to be noted that the thermal resistance of the shell side was calculated for an annulus fi lled with inert particles. This constructional modification is, compared to a reactor with an empty annulus, necessary, otherwise the reaction is becoming uncontrollable. Because of

5.3 Two-dimensional Modeling of Packed-bed Membrane Reactors Figure 5.22 Hot-spot formation in PBMR – 5 vol.% oxygen concentration, otherwise standard parameters (Table 5.4).

Figure 5.23 Averaged temperature profile – comparison between PBMR and FBR – 5 vol.% oxygen, standard parameter set (Table 5.4).

laminar flow, the gas behaves like an insulation layer in the empty shell side, leading to a runaway of the reaction under the simulated conditions. Another option, rather than fi lling the shell side with inert particles, is to place the catalyst in the annulus, while dosing the oxygen from the tube side. Considering the controllability of the reaction, it can be beneficial to increase the FTS/FSS ratio (Fig. 5.23). As mentioned above, the negative effect of the radial mass-transport limitations can be suppressed by cofeeding a small amount of oxygen on the tube side. Decreasing the shell-side flow can be impractical in a laboratory-scale reactor because of back diffusion of hydrocarbon molecules to the shell side. This problem is not so relevant on the large scale or in the case of larger amounts of catalyst, because of higher flow rates through the membrane at the same space velocities, wcat/FT. However, on increasing the FTS/FSS ratio one has to accept a decrease of per pass conversion in the PBMR in order to outperform the fi xed-bed reactor. Nevertheless, superior ethylene selectivity can make this operation region attractive. Finally, one can state that for the studied model reaction and the operating conditions considered, the thermal effects are very important in terms of their

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influence on reactor behavior and on product distribution. It was found that the radial voidage profi le has a considerable effect on the flow field and on radial mass transfer. However, its influence on the integral reactor performance was relatively small. As long as simulations indicate similar results with and without consideration of the porosity profi le, the relatively simple model with constant bed porosity is preferred. Two-dimensional simulations enable analysis of the complex and coupled physicochemical processes that occur in packed-bed membrane reactors more deeply and comprehensively than is possible with a simple one-dimensional model. Special attention was given above to the analysis of hydrodynamic effects caused by local variation in bed porosity and to the definition of appropriate boundary conditions depending on the particular membrane properties. It was demonstrated that larger-scale applications require a precise treatment of, especially, radial heat transfer, which possesses a large effect on the integral reactor performance. Uncertainties have been identified regarding the applicability of existing correlations for effective properties like the radial and axial dispersion coefficients under conditions of radial inflow. Hence, there is a need for targeted experiments in reactors with permeable or semipermeable walls as described, e.g., in Hussain et al., 2006. To this end, the application of more sophisticated 3D models may also contribute to an improved understanding of the reactor behavior. Even though the complete design of large-scale reactors with such models seems to be still utopistic, 3D simulations can help to understand local phenomena more profoundly. Utilizing the flexibility and accuracy of such in silico experiments more reliable correlations could be developed for, e.g., the mass- and heat-transfer parameters in pseudohomogeneous models (Freund et al., 2005). As an illustration of the capabilities of 3D numerical approaches, in the next section a deeper insight in the fluid flow and mass transport in packed-bed membrane reactors is given with the help of a 3D reactor model, based on the lattice Boltzmann equation method.

5.4 Three-dimensional Modeling of a Packed-bed Membrane Reactor 5.4.1 Introduction to the Large-scale Simulation Methods in Fluid Mechanics and Mass Transport

In the last decade, we have found a rapid development of the method known as the lattice Boltzmann equation (LBE) method. Because of its physical soundness and outstanding amenability to parallel processing, the LBE method has been successfully applied for the simulation of a variety of flow- and mass-transport situations, including flows through porous media, turbulence, advective diffusion, multiphase and reactive flows, to name but a few. In particular, an effective numerical approach to simulate advective-diffusion transport (the moment-

5.4 Three-dimensional Modeling of a Packed-bed Membrane Reactor

propagation method), adopted to parallel implementation and based on the LBE methodology, was developed a few years ago (Lowe and Frenkel, 1995, Merks et al., 2002). Together with modern parallel computers, the LBE method has become a powerful computational method for studying various complex systems. Unlike conventional Navier–Stokes solvers, the LBE method considers flows to be composed of a collection of pseudoparticles that are represented by a velocity distribution function. The macroscopic quantities of the fluid, such as the density r , velocity v, and temperature T (average internal energy), can all be determined from the given distribution function f by calculating its statistical velocity moments. In fact, the lattice Boltzmann equation is a minimal form of the Boltzmann kinetic equation in which all details of molecular motion are removed except those that are strictly needed to recover hydrodynamics at the macroscopic scale (mass, momentum and energy conservation). This results in a very simple equation for a discrete distribution function f i (x,t) = f(x,ci,t) describing the probability to find a pseudoparticle at lattice site x at time t with speed ci. It has been shown (He and Luo, 1997, Abe, 1997) that this discrete-velocity model can be considered as a special finite-difference approximation of the Boltzmann equation, obtained by standard projection of the continuum Boltzmann equation on Hermite polynomials and subsequent numerical evaluation of the kinetic moments by Gaussian quadrature. This route highlights the role of the discrete speed set ci, as a very effective sampling device of velocity space. This property is indeed the key to capture the complexity of fluid flows by means of only a couple of discrete speeds. For the simulations presented in this chapter, a D 3Q 19 lattice (where D and Q denote the dimension of the lattice and the number of links per lattice node, respectively) was employed that is obtained as 3D projection of a 4D face-centered hypercubic lattice. As a model system a cylindrical reactor of the length L = 60 mm and inner diameter di = 7 mm packed with 400 uniform, nonporous spherical particles of the diameter dp = 1.8 mm was studied. The geometrical dimensions, as well as the average porosity, ε = 0.47, of the packed bed were adjusted to those used in Section 5.2. Spatial discretization with the resolution of 30 lattice constant per sphere diameter was performed resulting in the computational domain of dimension 1300 × 117 × 117 points. The selected results given below intend to illustrate substantial differences and characteristics in the fluid dynamics in a FBR (FTS/FSS = ∞) and PBMR. All the simulations presented were carried out on a Hewlett Packard Superdome parallel computer (64 processors, 120 GB RAM). Typical simulation times for the complete model were about 24 h on this architecture. 5.4.2 Pressure and Velocity Field (Varying the Flow Distribution) – Comparison between FBR and PBMR

The interior of a reactor is represented by a random-close arrangement of hard impermeable spherical catalyst particles packed in a confining cylindrical container (Fig. 5.24). The radial-distribution function of interparticle porosity shows

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Figure 5.24 Computer-generated catalyst bed and its radial porosity distribution function averaged over axial coordinate (solid line). Mean bed porosity e = 0.47 (dashed line), aspect ratios di/dp = 3.9, L/dp = 18.5, r/R = 1 corresponds to the inner wall of the membrane.

a damped oscillatory behavior in agreement with the experimental findings by, e.g., Benenati and Brosilow, 1962. The same behavior was observed in the axial direction at the bottom and top of the packing as by Zou and Yu, 1995. However, these oscillations in the porosity profi le damp down to the mean porosity value within a distance of three particle layers in the axial direction (see also Freund et al., 2005). The side wall of the confining container is assumed to be a macroscopically homogeneous membrane with permeability B 0 (in particular, B 0 = 0 corresponds to the impermeable membrane i.e. the FBR). The flow rate through the membrane depends on both, permeability and the local pressure drop across the membrane (by Darcy’s law). Local gas density (and pressure) is assumed to be constant at the membrane shell-side. In addition, fluid flow through the packed bed is assumed to be isothermal, nonreactive and nonturbulent, while gas kinematic viscosity is assumed to be independent on gas density. The following boundary conditions have been imposed to simulate the pressure and velocity fields in the fi xed bed: • no-slip velocity boundary condition at the packing particle surface; • zero axial flow velocity component at the membrane surface;

5.4 Three-dimensional Modeling of a Packed-bed Membrane Reactor

a)

b)

Figure 5.25 Effect of the membrane permeability on the pressure distribution. (a) Pressure distribution, averaged over interparticle space along the angular coordinate; (b) distribution of average pressure along the reactor length.

• normal flow velocity component at the membrane surface defined by the local flow rate through the membrane in dependence on the local pressure drop and gas density; • fi xed pressure values at the membrane shell-side (p SS = 3.45 [a.u.1)]) and for two cross sections located at five particle diameters up- and downstream of the fi xed bed (pin = 1.15 [a.u.] and pout = 1.00 [a.u.], respectively). Simulations of fluid flow in the PBMR with fi xed pin and pout were carried out for three values of membrane permeability, which provided the three different FTS/FSS ratios (0.56, 10 and ∞. It must be noted that for the simple boundary conditions applied the overall flow rates are in this case not identical, unlike in the previous sections. In spite of this, the presented results show local effects, which are only troublesome to obtain by other techniques and demonstrate the high spatial resolution achievable with the LBE method. In Fig. 5.25a are shown the computed (circumferentially averaged in the interstitial space of the packed bed) pressure distributions corresponding to the three different values of FTS/FSS. As expected, with an increasing flow rate through the membrane (for small FTS/FSS ) the average pressure decreases nonlinearly along the packed bed (Fig. 5.25b, compare also with Fig. 5.15). At the same time, the three-dimensional pressure distributions are characterized by a very weak dependence of the local values over the radial coordinate. By contrast, the flow velocity field through the packed bed has a much more nonuniform distribution along the radial direction. This heterogeneity does not disappear even after 1) arbitrary units.

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Figure 5.26 Effect of the membrane permeability on the velocity distribution (only the axial velocity component is presented).

circumferential averaging (Fig. 5.26). It results from the geometrical structure of the packed bed. Its layered structure leads to the occurrence of the extended concentric void regions characterized by, inter alia, a higher local axial flow velocity. Figure 5.27 illustrates the effect of dosing through the membrane on the axial velocity component. In this figure the profi les represent the radial distribution of the axial velocity component (obtained by circumferential averaging over the interstitial pore space) at different axial positions (z/L = 0, 0.5, and 1). It is remarkable that the presented axial velocity profi les show oscillations even in the vicinity of both ends of the fi xed bed. In Fig. 5.27 it can also be seen that there is a difference in the magnitude of the local velocity components but the profi les are qualitatively similar. The flow introduced through the membrane does not appear to influence considerably the axial velocity distribution. However, a more systematic study should be performed to confirm this finding. The differences in the flow distribution in the axial direction are an obvious consequence of the heterogeneities in the local porosity distribution. In Fig. 5.28 one can compare the radial porosity distribution obtained by averaging along the whole packing (black line), with those obtained by averaging only along the crosssectional layers with a thickness of one particle diameter at z/L = 0.25 and 0.75 (light grey and dark grey lines, respectively). Though all the porosity profi les show three maxima, they are characterized by different amplitude at a given radial position. This can be explained by the different geometrical structure of the packing obtained even after circumferential averaging. In particular, the radial porosity profi le corresponding to z/L = 0.75 indicates the presence of a relatively narrow but extended (with the longitudinal dimension of the order of one particle diameter) void in the proximity of the center of the packed bed. The effect of this void

5.4 Three-dimensional Modeling of a Packed-bed Membrane Reactor

Figure 5.27 Effect of FTS/FSS ratio on the axial-velocity distribution averaged over interparticle space along the angular coordinate at three different axial positions.

on the local flow velocity distribution is illustrated in Fig. 5.29 representing the axial flow velocity distribution at the packing cross section z/L = 0.75 for FTS/FSS = 0.56. The dark-gray area in the center corresponds to this high flow velocity region of the packing, which declines in the up- and downstream (axial) directions (see Fig. 5.26). 5.4.3 Advective-diffusive Mass Transport in PBMR

In contrast to FBR, local concentrations of the components in a PBMR are affected additionally by mass transport through the membrane. In order to illustrate how the dosing of a component influences the local concentration distribution in

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Figure 5.28 Radial porosity profile averaged over interparticle space along angular coordinate at two different axial positions and over the entire packing.

Figure 5.29 Axial velocity component at z/L = 0.75 simulated for FTS/FSS = 0.56.

a PBMR, a 3D-simulation study of advective-diffusive mass transport of a twocomponent gas mixture was performed. It was assumed that i) fi xed-bed particles are nonporous; ii) the mixture components are nonreactive both mutually and regarding interaction with a solid phase; iii) the mixture components have the same kinematic and dynamic viscosities. Advective-diffusive transport was simulated by the moment-propagation method (Lowe and Frenkel, 1995; Merks et al., 2002). In this method, a scalar quantity P(x,t), e.g. species concentration, is released in the lattice. At each itera-

5.4 Three-dimensional Modeling of a Packed-bed Membrane Reactor

tion, after the LBE step, a fraction ∆/ r of P(x,t) stays on the lattice node and the remaining fraction is distributed over the neighboring nodes according to the probability f(x − ci,t) that a carrier fluid particle moves with velocity ci after collision, giving P ( x , t + ∆t ) = ∑ i

( f i ( x − ci ) − ∆ b ) P ( x − ci , t ) P ( x, t ) , + ρ ( x − ci ) ρ (x )

(5.28)

where b is the number of discrete velocities in the lattice. The parameter ∆ is used to set the molecular diffusion coefficient D. For the employed D 3Q 19 lattice D=

1 5 ∆ − 6 19 ρ

and b = 19

(5.29)

A halfway bounce-back boundary condition was imposed at the spheres/fluid interface. Gas particles (molecules) which are propagated into a solid point bounce back immediately and stay where they were. The flux through the membrane depended on the local pressure (advective contribution) and local concentration gradient (diffusive contribution) across the membrane. Figure 5.30 illustrates the concentration distributions (circumferentially averaged) of the component dosed through a uniformly permeable membrane for FTS/FSS ratios 10 and 0.56. As expected, the concentration of the dosed component increases along the reactor length. However, the increase is not strictly monotonic, there are small “tunnels”, or, e.g., next to the outlet, even sharp concentration peaks visible. These fluctuations in the concentration profi le are present for both simulated conditions. They result from the bed structure and from the simplifying assumption of impermeable particles.

Figure 5.30 Effect of FTS/FSS ratio on concentration distribution of nonreactive component fed through the membrane, averaged over the interparticle space along the angular coordinate.

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Considering the above results it can be expected that under reacting conditions the structural inhomogeneities in the catalyst bed can have a significant impact on the integral performance of a PBMR. It should be noted that although here only a few results obtained for a nonreactive system with a macroscopically homogeneous membrane were presented, the employed numerical LBE approach is inherently capable of dealing with a wide range of transport-reaction systems. In addition, the above approach can be applied to simulate heat transfer in chemical reactors considering also the dependence of the fluid properties (density, viscosity, mass and heat dispersion coefficients) on the local temperature (He and Li, 2000; Lallemand and Luo, 2003; Yoshino and Inamura, 2003; D’Orazio et al., 2004; Sullivan et al., 2005). In this regard, computer simulations based on the LBE method appear to be well suited for performing more detailed 3D simulations of PBMR in the future.

5.5 Summary and Conclusion

In this chapter a specific type of membrane reactor, the so-called “distributor” was analyzed theoretically. In contrast to conventional tubular fi xed-bed reactors (FBR), where all reactants are introduced together at the reactor inlet (cofeed mode), packed-bed membrane reactors (PBMR) allow dosing of one or several reactants via membranes over the reactor wall along the axial coordinate (distributed-feed mode). The “distributor”-type membrane reactor possesses different residence-time characteristics and different local concentration profi les compared to the conventional FBR. The additional degrees of freedom allow in complex networks of consecutive and parallel reactions, the selectivity and the yield to be enhanced with respect to a certain target product. The concept can be considered as an interesting option in the current attempts to improve and intensify reaction processes. The study presented was based on realistic data originating from the important class of partial oxidation reactions that might be favorably performed in such membrane reactors. The oxidative dehydrogenation of ethane to ethylene using a vanadium oxide catalyst was considered. Concerning the properties of the membranes different permeabilities were studied, being in the range of currently available porous materials. Reactor models of different complexity were developed and applied in order to study the performance and to evaluate the potential of the “distributor”-type PBMR. A simplified 1D model illustrated the general features of the PBMR and demonstrated significant differences between FBR and PBMR, due to the different dosing concepts applied. This simple model revealed the need for a careful design of PBMR. The several variants of the computationally more expensive 2D models presented in this work provide much deeper insight. In particular, the following re-

Notation

sults obtained with this type of model are noteworthy. The radial voidage profi le has a considerable effect on the flow field and on radial mass transfer. However, its influence on the integral reactor performance was found to be relatively small. The thermal effects are very important for the reactor behaviour and the product distribution. Comparing predictions of 1D and 2D models, it was found that the simple model overestimates the reactor performance. Radial mass and heat transfer limitations can not be neglected if more precise predictions are required. In a final section, results from 3D calculations using the lattice Boltzmann equation method were presented. For a low aspect ratio results were obtained using a randomly generated catalyst packing. This still computationally expensive method possesses the potential to evaluate in future in more detail the complicated but most important situation close to the wall of a PBMR. Altogether, the results presented indicate the overall potential of using optimized dosing concepts, (e.g. in a PBMR), in order to improve the performance of complex chemical reactions. The theoretical framework presented allows as for rapid first estimations in early development stages as for more detailed studies required for process design and optimization. Notation Latin Letters

A B0 c c0 cp D d dpor f F I j L M NC NR p r R r T t u

surface area [m2 ] membrane permeability [m2 ] concentration [vol.%]; lattice velocity total concentration [mol/m3 ] specific heat capacity [J/(kg K)] mass diffusion, dispersion coefficient [m2/s]; diameter [m] pore diameter [m] particle probability distribution function volumetric flow rate [m3/s] identity matrix mass flux [kg/(m2s)] reactor length [m] molecular mass [kg/mol] number of components number of reactions pressure [Pa] reaction rate [mol/(kg s)] reactor radius; universal gas constant − 8.3145 J/(mol K) radial coordinate [m] temperature [K] time [s] superficial velocity [m/s]

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u0 uM uc x v VR wcat z

radially averaged superficial velocity [m/s] averaged superficial fluid velocity in the membrance [m/s] superficial velocity in the core of the catalyst bed [m/s] lattice coordinate interstitial velocity (u/ ε ) [m/s] reactor volume [m3 ] catalyst mass (wcat = VR · rcat (1 − ε )) [kg] axial coordinate

Greek Letters e e∞ zB k l m n x r r cat

τ Y w

bed porosity [-] porosity of catalyst bed at infinite distance from the wall [-] dilatational viscosity [Pa s] overall heat-transfer coefficient [W/(m2 K)] thermal conductivity [W/(m K)] dynamic viscosity [Pa s] stoichiometric coefficients dimensionless length [-] density [kg/m3 ] envelope density of the catalyst particle [kg/m3 ] stress tensor [kg/(m s2 )] friction coefficient [1/s] mass fraction [-]

Subscripts and Superscripts

0, in e g i j o p s SS t TS w out

inlet effective properties gas phase component index; inner reaction index outer particle solid phase; slip shell side turbulent dispersion index tube side wall outlet

Dimensionless Numbers

Pe0

molecular Peclét number, Pe0 =

Re0

particle Reynolds number, Re0 =

ud Dim udρg

µg

References

Appendix

Kinetic model for the oxidative dehydrogenation (ODH) of ethane on a VOx /γ– Al2O3 catalyst suggested by Klose et al., 2004a. Reaction

Rate equations*

1

r1 =

kredCC2H6 koxCO0.52 [kredCC2H6 + koxC O0.52 ] K C2H6 CC2H6 K O0.52 CO0.52 r2 = k2 × K O2 )0.5CO0.52 ] [1 + (K C2H6 )CC2H6 + (K CO2 )CCO2 ] [1 + (K 0.5 0.5 K C2H4 CC2H4 K O2 CO2 r3 = k3 × [1 + (K C2H4 )CC2H4 + (K CO )CCO ] [1 + (K O2 )0.5CO0.25 ] K C2H4 CC2H4 K O0.25CO0.25 r4 = k4 × [1 + (K C2H4 )CC2H4 + (K CO2 )CCO2 ] [1 + (K O2 )0.5CO0.25 ] K COCCOK O0.25CO0.25 r5 = k5 2 [1 + (K CO )CCO + (K O2 )0.5CO0.25 + (K CO2 )CCO2 ]

2 3 4 5

* The rate constants follow the Arrhenius law, ki = k 0i · exp(E Ai /RT), with i = red, ox, 1, . . . , 5, where Ci is the molar concentration of component i at 298K and 1 atm in mol/l.

Kinetic coefficients for the ODH of ethane on a VOx /γ–Al2O3 catalyst suggested by Klose et al., 2004a. k 0red = 4.3 × 10 9 l/(kg · h) k 0ox = 1.1 × 10 8 mol0.5 l0.5/(kg · h) k 02 = 1.6 × 107 mol/(kg · h) k 03 = 2.4 × 10 4 mol/(kg · h) k 04 = 1.00 × 103 mol/(kg · h) k 05 = 1.10 × 107 mol/(kg · h)

KC2H6 = 4769.8 l/mol KC2H4 = 3025.6 l/mol KO2 = 1002.6 l/mol KCO = 3233.9 l/mol KCO2 = 3455.8 l/mol

E Aox = E Ared = 94 kJ/mol E A2 = 114 kJ/mol E A3 = 51 kJ/mol E A4 = 51 kJ/mol E A5 = 118 kJ/mol

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6 The Focused Action of Surface Tension versus the Brute Force of Turbulence – Scaleable Microchannel-based Process Intensification using Monoliths Michiel T. Kreutzer, Annelies van Diepen, Freek Kapteijn, and Jacob A. Moulijn

6.1 Introduction

The quest for energy-efficient, compact, safe, environment friendly and sustainable processes has become known in recent years as process intensification. Ramshaw [3], one of the pioneers in the field, defined process intensification as a strategy for making dramatic reductions in the size of a chemical plant so as to reach a given production objective. This reduction can be achieved by reducing the size of the equipment of individual unit operations and by reducing the number of unit operations. Stankiewicz and Moulijn [4] took the definition of process intensification one step further, proposing that process intensification is a revolutionary concept rather than an evolutionary one: instead of squeezing more out of existing plants and hoping to eventually achieve the desired goals, radically new concepts must be adopted. This contribution is concerned with gas–liquid and gas–liquid–solid reactors. Using the definitions briefly summarized above, processes that need such a gas–liquid(–solid) reactor should be designed in such a way that drastically higher rates based on reactor volume can be achieved while maintaining or improving the selectivity towards the desired product. The high volumetric rates then allow a reduction in size of the reactor itself, and high selectivity eliminates the need of separation steps downstream of the reactor. The desired reduction in size not only calls for better catalysts, but also requires much higher mass-transfer rates than can be achieved in conventional reactors. In this work, the quality of the catalyst is not considered, and we focus on the hydrodynamics and the transport phenomena. This by no means indicates that the quality of the catalyst is irrelevant. Quite the contrary, developing good catalysts is the paramount enabling step for catalytic processes. However, the discussion of the catalyst preparation varies from process to process, and here we want Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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to focus on what can be done to achieve order-of-magnitude reductions in reactor size, provided that a fast catalyst is found. Chemical engineers intuitively work under the assumption that to improve the mass-transfer characteristics of a gas–liquid reactor, more energy must be dissipated in the fluids to effect a more vigorous contacting of the fluids. For singlephase flow in turbulent systems, this concept has become known as the Chilton–Colburn analogy. Another example of the coupling of hydrodynamics and mass transfer in multiphase systems is the common correlation of the mass transfer and the power input for stirred-tank reactors. As a result, most of the equipment heralded in process intensification not only intensifies chemical conversion per unit volume, but also intensifies the mechanical-energy dissipation per unit volume. This is the brute force of turbulence that the title alludes to: most intensified technologies rely on turbulence to beat the fluids – in some way or other – into high contact area and thin boundary layers. This coupling hinders the achievement of the goals of process intensification: while high mass transfer may be achieved to reduce the reactor size, the auxiliary equipment needed to achieve the vigorous contacting – be it a stirrer, pump, venturi ejector or a compressor – will grow in size and perhaps more importantly grow in energy requirement. A good example is the venturi–loop reactor [5], in which high mass-transfer rates can be achieved, but a very powerful pump is needed to force the liquid through a small nozzle to create the liquid jet that releases its energy upon entrainment and dispersion of gas.

6.2 Monoliths – A Scalable Microchannel Technology

Honeycomb monoliths (Fig. 6.1) are structured catalyst supports consisting of parallel straight capillary channels. Nowadays, they are widely used for the catalytic exhaust converter in the automobile industry and end-of-pipe gas cleaning. The gas-only application of monoliths stems from the fact that the pressure drop is low: using the surface area of the catalyst as a criterion, the pressure drop in a monolith is an order of magnitude lower than in randomly packed beds. The channels are about a millimeter in diameter, and on the wall (∼100 µm) a washcoat of catalytic material (∼50 µm) is applied. In the last decade, monoliths have been increasingly considered for liquid-phase reactions [7–14], mainly in various forms of a loop configuration. Figure 6.2 shows two possible configurations of such a loop process. In our department, the monolith packings have been applied for a host of gas–liquid–solid applications. Briefly, we have obtained hydrogenation rates for α -methylstyrene that were orders of magnitude higher than in typical packed beds [15], and we have obtained selectivities in the A → B → C hydrogenation of benzaldehyde that were similar to values obtained in fine powder slurry reactors [16]. Monoliths have been

6.2 Monoliths – A Scalable Microchannel Technology

Figure 6.1 A slice of a monolith catalyst packing. The channels have a diameter of the order of 1 mm. The structure is mass produced for automotive applications by extrusion. The structure of parallel straight channels without tortuous bends leads to a low pressure drop.

Figure 6.2 Monolith loop-reactor configurations. From Heiszwolf et al. [6].

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6 The Focused Action of Surface Tension versus the Brute Force of Turbulence

used in biotechnological applications using immobilized enzymes [17] and in air-lift biofi lm reactors [18]. Recently, we have used several carbon-based techniques to prepare monoliths with immobilized enzymes [19] and used these structures to catalyze several model reactions. The high mass-transfer characteristics were obtained in plug-flow [20] conditions and at minimal or even zero pressure drop [6, 21]. The last observation is unlike normal multiphase reactor behavior: quite generally, mechanical energy is used to overcome mass-transfer limitations, be it by using smaller particles in fi xed beds at the expense of pressure drop, or by using higher stirrer speeds or gas throughput in slurry systems. Understanding this story, improved mass transfer by lowering the power input, is the main aim of this chapter. Most of the gas–liquid applications of monoliths have used a heterogeneous catalyst (be it supported noble metals or immobilized enzymes) on the channel walls. Here, we also consider the use of monoliths without a catalyst on the walls in gas–liquid applications, i.e. homogeneously catalyzed liquid-phase reactions. The fluid mechanics of the system do not change appreciably by letting the reaction take place in the liquid bulk instead of in a washcoat layer, and it is interesting to consider such reactions in a discussion of mass transfer and power-input requirement. Of course, the mass-transfer behavior does change by changing the location where the reaction takes place, and we will discuss gas–liquid reactors and gas–liquid–solid reactors separately. One of the gas–liquid applications that would benefit from intensification at low mechanical energy input is industrial fermentation, which frequently is mass-transfer limited in air-lifts, bubble columns and gently stirred tanks. Highshear mechanical energy tends to kill the cells, and the mechanical intensifiers that work well for catalyst powders are much less attractive for fermentations. Fermentations are generally performed at low pressure in very large reactors, even by chemical engineering standards, and reduction of energy costs is often desired. Besides industrial fermentation processes, environmental processes involving gas–liquid mass transfer and microbial conversion exist for which the same arguments apply. Examples include aerobic wastewater treatment and biotechnological gas treatment processes like those developed for NOx and SOx removal from flue gases [22, 23].

6.3 Power Required for Gas–Liquid Dispersion 6.3.1 Turbulent Contactors

A comparison of different gas–liquid contactors best starts with exploring the amount of energy that is needed to generate interfacial area. In conventional turbulent contactors, the bubble size is determined by bubble break-up and coalescence. In essence, one can set up a balance between (1) surface tension, which

6.3 Power Required for Gas–Liquid Dispersion

resists the deformation of the convex bubble shape that leads to bubble break-up, and (2) turbulent eddies, which batter away at the interface. The ratio between these stresses gives a Weber-number criterion for bubble break-up [24],

ρLu(2L ) ~

γ dB

(6.1)

2 can be estimated from the power dissipated per unit volume in in which u (L) eddies of similar length scale as the bubble scale [25]. Such an analysis leads to

dB ~

γ 2/5

(6.2)

2/5

P    V 

ρ L2/5

which indicates that the interfacial area is proportional to (P/V) 0.4. Generally, for the group kL a a slightly higher exponent with respect to power input is observed, which indicates that the mass-transfer coefficient is a mild function of the turbulence intensity surrounding the bubble [26]. Naturally, a higher bubble holdup increases the bubble surface area, and correlations of the following type are widely used: P kLa ~   V 

0.65

 uGs    V 

a

(6.3)

Here, (u Gs/V) accounts for the gas holdup with the constant a typically in the range 0.2–0.3. It should be stressed here that correlations such as Eq. (6.3) are more firmly rooted in turbulence theory than their first empirical appearance suggests. The power of the eddies mainly breaks up bubbles into smaller ones (∼(P/V) 0.4 ), but also reduces the boundary layer surrounding the bubbles (∼(P/ V) 0.25), and the combination of these effects leads to Eq. (6.3). The proportionality constant for Eq. (6.3) varies from equipment to equipment and, of course, is fluidproperty dependent. For oxygen transfer to a clean aqueous medium, a value of 1 × 10 −3 gives values of the right order of magnitude, kL a ≈ 0.05 s−1 at (P/V) = 103 W/m3 and kL a ≈ 0.2 s−1 at (P/V) = 104 W/m3, see, e.g., Linek et al. [27] or Schlüter and Deckwer [28]. Again, the coupling of power dissipation and mass transfer hinders the achievement of the goals of process intensification: while high mass-transfer rates may be achieved to reduce the reactor size, the auxiliary equipment needed to achieve the vigorous contacting grows in size and energy requirement. For intensifying hazardous processes where minimization of toxic or explosive inventory is most important, the energy costs are not driving process intensification. However, for large-scale processes where operating costs are important, reducing the plant size while increasing the energy demand is only a partial solution.

153

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6 The Focused Action of Surface Tension versus the Brute Force of Turbulence

6.3.2 Laminar Contactors

In order to overcome the coupling of power dissipation and mass transfer, we need to consider a different mechanism for gas–liquid contacting. If we turn to laminar flow, an external structure should be used to create or maintain the surface area. For example, in a falling-film reactor the gas/liquid interfacial area is roughly equal to the wall area. In capillaries at moderate velocities, the predominant flow pattern is called Taylor [29] flow, see Fig. 6.3. In Taylor flow, the gas bubbles are too large to retain their spherical shape and are stretched to fit inside the channel. Surface tension pushes the bubble towards the channel wall, and only a thin fi lm remains between the bubble and the wall. The central argument of this chapter is that these bubbles cause recirculation in the slugs, resulting in a very efficient surface renewal near the gas/liquid interface. This is the focused action of surface tension: the energy that is needed to cause flow in the channel leads to caterpillar-like motion that enhances the mass transfer. The thickness d of the fi lm between the bubble and the wall is mainly due to viscous stresses near the bubble caps. From a lubrication analysis [30] we find for a bubble train moving at a velocity U in a capillary of diameter d

δ ~ Ca2/3 d

(6.4)

µU . γ Instead of resisting the break-up into smaller bubbles and thus opposing high mass-transfer rates, the surface-tension forces minimize the thickness of the fi lm separating the bubble from the catalyst. The liquid is effectively sealed between the bubbles and cannot escape the slug that it forms. This prevents bubble coalescence, and surface tension now eliminates the need for turbulent energy to break up the bubbles. Of course, the small bubbles that enter the channel still have to be created, which still costs energy and that might well be done using a turbulent contactor. The important distinction is that once inside the channel, no energy is required to maintain the small bubble size. The energy requirement may actually be rather high for a small reactor, with short channels and a relatively large disperser at the entrance of the channels. On the other hand, in a reactor with long channels with Ca =

Figure 6.3 Computed flow of a bubble in a capillary. A lubricating thin liquid layer (dark gray) persists on the wall. Liquid between the slugs (light gray) circulates. Kreutzer et al. [1] have reported the numerical details.

6.3 Power Required for Gas–Liquid Dispersion

the energy dissipated in the feed section may be ignored, and the energy requirement is very low. As can be seen in Fig. 6.2, the energy requirement for gas circulation can be ignored. In fact, most monolith loop layouts do not require a compressor. Then, apart from the power requirement of a gas–liquid disperser, we need energy (1) to force the gas through the column, (2) to force the liquid through the column and (3) to lift the liquid in the liquid recycle line from the bottom of the column to the top of the column, which yields  P  =  ∆p  (u + u ) + ρ gu     Ls Gs Ls  V   L col

(6.5)

We begin with the simpler analysis of long bubbles and slugs, for which we ignore mass transfer and Laplace pressure terms at the bubble caps. Then, we can simply estimate the pressure drop over the column by  ∆p  = ε  2Fµ U − ρ g     L   d2  L col

(6.6)

Using eL = uLs/U and U = uLs + u Gs, the power input associated with the static head inside the column, r g eL , cancels the power input in the recycle line for lifting the liquid to the top of the column, leaving only a laminar, viscous term to yield 2  P  ~ ε LFµU   2 V d

(6.7)

in which F is the Hagen–Pouseuille proportionality constant for developed flow (16 for round channels, 14.2 for square channels, etc.). Still for long bubbles and slugs, we ignore the contribution from the caps to the mass transfer, and we estimate the mass transfer from the bubble to fi lm by penetration theory [2] kLa ~ ε G

DU d Lbubble

(6.8)

2

and by introducing the unit cell length LUC = Lbubble/ eG and by substituting Eq. (6.7), we find a form that allows a comparison with Eq. (6.3): kLa ~

Dε G dLUC(Fµε L )1/2

1/ 4

P    V 

(6.9)

So, for laminar contactors we predict that the mass transfer is a much weaker function of the amount of energy dissipated than for turbulent contactors. The fact that energy is not required for interfacial area generation is in perfect agreement with this scaling analysis. Of course, there still is an impact of velocity on

155

156

6 The Focused Action of Surface Tension versus the Brute Force of Turbulence

the mass transfer, as a higher velocity reduces the contact times in penetration theory. The group under the root in Eq. (6.9) may be estimated for oxygen transfer to an aqueous medium. Using the following estimates for an approximate order of magnitude analysis, D = O (10 −9 ) m/s2, d = O (10 −3 ) m, LUC = O (10 −2 ) m, m = O (10 −3 ) Pa s and eG = O (10 −1), we find that the entire group is O (10 −1), and we estimate P kLa ≈ 0.1   V 

1/ 4

(6.10)

For low power input, e.g. (P/V) = 1 kW/m3, this gives kL a ≈ 0.5, which is an order of magnitude higher than can be obtained in turbulent contactors at the same power input. Again, this result can be understood by realizing that no power is required for bubble break-up. If the bubbles and slugs are short, then the Laplace pressures associated with each bubble become important. Fortunately, this contribution to the pressure drop is of the same order as the frictional losses [1], e.g., F = 45 for air/water and L slug ≈ 4d, so we can keep Eq. (6.7) with a higher value for F. For short bubbles we no longer ignore the mass transfer through the caps, which we again estimate using penetration theory [2] (kLa )caps ~

DU dL2UC

(6.11)

or, by substituting Eq. (6.7) (kLa )caps ~

D dL (Fµε L )1/2 2 UC

1/ 4

P    V 

(6.12)

This shows that if the unit cell is short, the channel diameter becomes less important, but the bubble length and slug length become more important. The group under the root in Eq. (6.12) is similar in value to the group under the root 2 of Eq. (6.9) if LUC ∼ dLUC or LUC ∼ d, which is self-consistent with the criterion of short bubbles and slugs for which Eq. (6.12) was derived. Because Eq. (6.12) only adds to Eq. (6.9) in the same order of magnitude, we do not need to change the order-of-magnitude estimate of Eq. (6.10).

6.4 Physical Adsorption of Oxygen

In Fig. 6.4, experimental mass-transfer data for the adsorption of oxygen are compared for turbulent contactors and monoliths. The data for stirred tanks and bubble column follow the trend predicted by Eq. (6.3), and the different lines correspond to different (uLs/V).

6.4 Physical Adsorption of Oxygen

Figure 6.4 Gas–liquid mass transfer versus power input for monoliths and turbulent contactors. All data are for the O2/ water system. The dotted line corresponds to Eq. (6.10), the shaded area gives the range of kL a vs. (P/V) for monoliths. The five larger dots in the monolith data are for a set of experiments where the gas holdup eG was kept at 0.6.

The monolith data were taken from Kreutzer et al. [15]. In exactly the same setup, we have also measured the pressure drop, and data in Fig. 6.4 were obtained by calculating the power input from the experimental pressure drop data. The monolith data are consistently higher than the data for the turbulent systems at equal power input. There is significant scatter in the monolith data. The mass transfer was measured at steady state for two lengths of monolith columns. The measured outlet concentration of the short column was used as the inlet concentration for the rest of the longer column, and with the measured outlet concentration the mass-transfer group was determined from kLa =

 C − C out,short  U ln sat Llong − Lshort  Csat − C out,long 

(6.13)

Although the oxygen concentration could be determined with an accuracy of 1.5% of the saturation value, the difference Csat–Cout,long was typically less than 0.1 Csat, and the subtraction of very similar numbers contributed significantly to the experimental error. Apart from the experimental error, the monolith data in Fig. 6.4 were obtained for 0.3 < eG < 0.7. As indicated by Eq. (6.9), kL a is a function of eG. There is a trend in the data that the higher kL a values in Fig. 6.4 correspond to higher gas holdup. The five larger symbols in Fig. 4 are for eG = 0.6. Also, the parameter F was found to be a strongly nonlinear, nonmonotonic func-

157

158

6 The Focused Action of Surface Tension versus the Brute Force of Turbulence

tion of the holdup (see Heiszwolf et al. [31]), which makes it impossible to correct for eG in a straightforward manner. In Fig. 6.4, we also plotted Eq. (10), and the order of magnitude, predicted in the section above, is confirmed by the experimental data. Heiszwolf et al. [6] used a correlation by Bercˇicˇ and Pintar [32] and experimental data in a loop reactor where the pressure drop over the column was kept at zero to obtain P kLa ≈ 7.5 × 10 −2  V 

0.2

(6.14)

which is fairly similar to Eq. (10). Recently, Van Baten and Krishna [2] performed a CFD study that demonstrated that the correlation of Bercˇicˇ and Pintar [32] did predict values of the right order of magnitude, but they improved the modeling by introducing the penetration-theory model that forms the basis of the analysis in this chapter. So, although the correlations used for Eq. (6.14) are now superceded, the absolute values were based on experiment and still hold. Note that the values of Heiszwolf et al. [31] were slightly lower than the data presented here, which is consistent with the fact that their data were obtained at somewhat lower gas holdups: forced gas circulation, which does not require too much power, does help to achieve higher mass transfer. Of course, the choice between the slightly higher mass transfer and the reduced cost of eliminating a fan or compressor in the gas recirculation is an economical one.

6.5 Three-phase Processes

The more established gas–liquid application of monoliths involves a heterogeneous catalyst on the wall. Figure 6.3 shows that the liquid is divided into a recirculation region in the slugs and a lubricating layer separating the bubble-slug train from the wall. For the transport of gas to a catalyst on the wall, the film turns out to be the dominant resistance [33, 34]. It is obvious that the film resistance is dominant for transfer from the bubble to the wall, as it is the only resistance. For transfer from the caps through the slug to the wall, it can be shown, in a first approximation, that the slugs are efficient in transporting gas that dissolves at the bubble caps to the lubricating film, so the fi lm is the dominant resistance here also. As a result, the concentration gradient in the slug is fairly constant and does not change much when the passing bubble is replaced by a saturated liquid slug. Then, we obtain an estimate for the mass-transfer coefficient using fi lm theory kG→S ~

D D ~ δ dCa2/3

(6.15)

6.5 Three-phase Processes

Now, the surface tension actively assists in reducing the film resistance to mass transfer, while viscous stresses thicken the fi lms. Substituting Eq. (6.7) and using the definition of the capillary number, we obtain kG→Sa ~

3

D 3γ 2Fε L µd 8

P    V 

−1/3

(6.16)

An increase in energy input into the system – or, put plainly, an increase in velocity of the bubble train – will increase the viscous stresses and decrease the mass transfer. This effect, improved mass transfer at lower energy input, seems counterintuitive at first but follows readily from the lubrication analysis. Kreutzer et al. [15] demonstrated experimentally using the hydrogenation of α -methylstyrene that at low velocity the mass transfer to the wall indeed increased. So, we again observed that surface tension assists in obtaining high mass transfer. The trick is to use the surface tension in a focused manner in laminar flow, as opposed to trying to fight it in a turbulent environment. For a fair comparison to data obtained in conventional reactors, it should be noted that the experiments by Kreutzer et al. [15] were performed using hydrogen in toluene. For this system, the diffusion coefficient of the dissolved hydrogen is higher than the diffusion coefficient of oxygen in water. We have calculated the mass transfer of oxygen to the wall through an aqueous medium, using 0.66 as the proportionality constant in Eq. (6.4) (see, e.g., Bretherton [30]). Note that the calculated values are for round channels. In coated square channels the masstransfer rate is somewhat lower, but the trend, better mass transfer at lower power input, is the same. The experimental bubble-column data from Linek et al. [27] and stirred-tank data from Schlüter and Deckwer [28] are also plotted in Fig. 6.5. If we ignore shuttling of catalyst particles to the fi lm surrounding bubbles in turbulent contactors, then the highest gas-to-solid mass-transfer rate is obtained at high catalyst loading, and in that case to a first approximation the gas–liquid mass-transfer rate, as defined by Eq. (6.3), can be used to estimate the overall mass transfer. Figure 6.5 highlights the counterintuitive behavior of monoliths: the less energy is introduced, the better the mass transfer. Since commercial-scale stirred-tank reactors can usually not be operated at very high power input, the monolith reactor is an excellent alternative for all processes that benefit from good mass-transfer characteristics. This includes processes for which the catalyzed intrinsic kinetics are very fast, processes where mass-transfer limitations lead to a drop in selectivity and processes where the stability of the catalyst deteriorates at low (hydrogen) concentrations inside the catalyst. It is not very realistic to consider the use of monoliths at high power input. At high velocities, the mass-transfer characteristics deteriorate and the residence time is low. The only reason for not operating a monolith at low velocities is flow stability. If the pressure-drop results of this work are applied to the stability criteria of Kreutzer et al. [20], the stability criterion is roughly uLs > 0.05 m/s.

159

160

6 The Focused Action of Surface Tension versus the Brute Force of Turbulence

Figure 6.5 Gas–liquid–solid mass transfer in various contactors. The data of Linek et al. [27] and Schlüter and Deckwer [28] are the G/L mass-transfer data also found in Fig. 6.4. The data for the monolith are calculated

using d /d = 0.66 Ca0.66, properties of water and oxygen, D = 2.8 × 10 −9 m2/s, eG = 0.5 and F = 60. The data from Kreutzer et al. [15] were obtained using hydrogen/toluene instead of oxygen/water.

6.6 Discussion

The analysis of power input and mass transfer in this chapter was limited to scaling, and the results serve to demonstrate the opportunities that arise when alternatives to the contacting workhorse, turbulence, are considered. The works of Bercˇicˇ and Pintar [32] and Van Baten and Krishna [2] have elucidated the most important aspects of G/L mass transfer, and two-phase pressure drop in capillary channels is discussed in detail by Kreutzer et al. [1], but more experimental masstransfer data is needed, in particular with using different fluids at controlled, independently varied bubble and slug lengths. 6.6.1 Design Considerations for Intensified Monolith Processes

The details of designing a multiphase monolith columns is a topic in itself, and the interested reader is referred to Kreutzer et al. [13]. Here, several aspects that follow immediately from the scaling analysis presented above are discussed. The proper distribution of gas and liquid over the column cross section is crucial for monoliths, because inside a monolith block no redistribution can occur. Fortunately, static mixers can be used that combine a good distribution with

Notation

a low pressure drop [20, 35]. For a G/L contactor, the distributor should create bubbles and slugs that are short inside the channels, because this improves mass transfer (Eq. (6.12)) and increases flow stability. For G/L/S reactors with catalyst on the channel walls, the first argument hardly holds, while the stability argument is equally valid as in G/L reactors. Equations (6.9) and (6.12) show that the mass transfer is only a mild function of the channel diameter. If dust tolerance is important, it seems worthwhile to sacrifice some interfacial area. On the other hand, for G/L/S reactors the channel size is crucial, and channel size should be increased only as a last resort. The G/L/S mass transfer improves with lower throughput, so it makes sense to operate at the lowest possible practical flowrates. The lower limit is set (1) by the requirement of Taylor flow, for which 2–3 cm/s is enough for the smaller channels (>200 cpsi) and (2) the requirement of stable gas/liquid flow, which may require somewhat higher flow rates [20]. The ratio of gas to liquid throughput hardly affects the mass-transfer rate, but a sizeable gas holdup may be required if entire bubbles may otherwise be completely dissolved, resulting in the loss of the Taylor-flow characteristics. The G/L mass transfer scales only with (P/V)1/4, and increasing or decreasing the flow rate has a limited effect on the mass transfer. For most applications, it seems best to be just above the lower limit for stable operation. 6.7 Conclusions

In this chapter, we have proposed the use of capillary channels in general, and monoliths in particular, to intensify processes involving gas–liquid mass transfer. Using a simple scaling analysis, involving (1) viscous pressure drop, (2) hydrostatic pressure drop, (3) interfacial pressure drop and (4) penetration theory for mass transfer, it has been demonstrated that two-phase laminar bubble-train flow in small channels can exhibit better mass transfer for a given power input than turbulent contactors. This flow pattern can be used for biochemical conversions using cells cultures, provided the channels are not too small and the operating conditions are such that biofi lm formation is suppressed. If the segmented flow pattern is used for a reaction catalyzed at the walls of the capillary channels, then the mass transfer is actually improved by reducing the amount of energy that is dissipated in the system. This allows the simultaneous achievement of two goals of process intensification: reduction of energy requirement and reduction of equipment size. Notation

a d

interfacial area, m2/m3 diameter, m

161

162

6 The Focused Action of Surface Tension versus the Brute Force of Turbulence

D g k L p P u uL U V

diffusion coefficient, m2/s gravitational constant, m/s2 mass-transfer coefficient, m/s length, m pressure, Pa power, W velocity, m/s fluctuation of velocity in turbulence sum of gas and liquid superficial velocity, m/s volume, m3

Greek symbols g d e m r

surface tension, N/m fi lm thickness, m holdup, – viscosity, Pa s density, kg/m3

Dimensionless Groups

Ca f F Re We

capillary number (= mU/ g ), – friction factor, – laminar friction constant (= f Re), – Reynolds number (= rUd/ m ), – Weber number (= rU2 d/ g ), –

Subscripts

bubble caps col G long L out s sat short slug S UC

referring to bubble referring to bubble end caps column gas referring to long column liquid referring to outlet condition superficial saturation referring to short column referring to slug solid unit cell, i.e. a bubble and a slug

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29 Taylor, G. I., Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 1961, 10, 161–165. 30 Bretherton, F. P., The motion of long bubbles in tubes. J. Fluid Mech. 1961, 10, 166–188. 31 Heiszwolf, J. J.; Kreutzer, M. T.; van der Eijnden, M. G.; Kapteijn, F.; Moulijn, J. A., Gas-liquid mass transfer of aqueous Taylor flow in monoliths. Cat. Today 2001, 69 (1–4), 51–55. 32 Bercˇicˇ, G.; Pintar, A., The role of gas bubbles and liquid slug lengths on mass transport in the Taylor flow through capillaries. Chem. Eng. Sci. 1997, 52 (21/22), 3709–3719. 33 Kreutzer, M. T., Hydrodynamics of Taylor Flow in Capillaries and Monoliths Channels, Doctoral dissertation, Delft University of Technology, Delft, the Netherlands 2003. 34 Bercˇicˇ, G., Influence of operating conditions on the observed reaction rate in the single channel monolith reactor. Cat. Today 2001, 69 (1–4), 147–152. 35 Welp, K. A.; Cartolano, A. R.; Parillo, D. J.; Boehme, R. P.; Machado, R. M.; Caram, S., Monolith catalytic reactor coupled to static mixer. European Patent 2002, EP 1 287 884 A2.

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7 Chemical Reaction Modeling in Supercritical Fluids in Special Consideration of Reactions in Supercritical Water Andrea Kruse and Eckhard Dinjus

7.1 Introduction

Supercritical solvents have generated an increased interest in the last few decades. One reason is that their solvent properties vary considerable with temperature and density. They are “tunable solvents” [1] and for each purpose – separations or reactions – the optimal properties can be adjusted (see, for example, [1–8]). Usually, supercritical fluids are used as a tool to get homogeneous mixtures. In a homogeneous phase, for example, oxidations are extraordinarily fast and complete. The usually improved heat and mass transfer is a further advantage. Supercritical fluids show their good solvent properties only in the supercritical state. Therefore separation after reaction or extraction is very simply achieved by reducing temperature and pressure. This enables very sustainable processes (for example [1, 9]). Here supercritical carbon dioxide and water are of special interest, because they are cheap, nontoxic or of very low toxicity, in the case of carbon dioxide and nonexplosive. Up to now some studies of modeling the extraction process have been published (for example [10–14]). The kinetic modeling of chemical reactions is a special challenge, because the properties of supercritical fluids, changing with temperature and density, may influence the reaction rate of a selected reaction step. The modeling of chemical reactions by means of systems of elementary reaction is widely used in gas-phase kinetics, especially in oxidation reactions. This technique was tried also for the description/calculation of reactions in supercritical fluids. This was successful to a certain extent, but also problems were observed due to the special properties of a supercritical fluid in comparison to the gas state. The goal of these modeling studies was to get a better understanding of these reactions and how they are influenced by the properties of the supercritical fluid. In view of the technical application it is an advantage to have such a model, Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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because interpolation and extrapolation from the experimental range are more reliable than in the case of a simple global kinetics equation. In addition these models give not only information about the main reaction but also for side reactions, which is useful for the transfer of chemical reactions from the lab-scale into the technical scale. Reactions in supercritical fluids (SCF) are usually reactions at high pressure. In principle the pressure effects on elementary reactions can be considered by implementation of the activation volume into the model. The activation volume is the difference between the partial molar volumes of the activated complex and the reactants. It is often regarded as a sum of two activation volumes: The intrinsic activation volume and the solvent-dependent part. The intrinsic activation volume is usually small, for example for the scission of a C—C bond it is roughly −20 mL/mole. The solvent-activation volumes can become up to orders of magnitude higher, especially if an attractive interaction between, for example, a polar reactant and water as the solvent occurs. For reactions in water near the critical point activation volumes in the range from 100–1000 mL/mole were reported [15–20].1) Intrinsic activation volumes of a single reaction may be estimated; solvent-activation volumes have to be determined experimentally. Usually very few data for solvent-activation volumes at supercritical reactions conditions are available (exceptions: [21–23] and references cited therein). This lack of data is one of the major problems for kinetic modeling of reactions in supercritical fluids. From a microscopic point of view, collisions between reactant molecules with other reactant or solvent molecules have a significant influence on chemical kinetics. The increase in reaction rates up to the so-called “high-pressure plateau” at increased pressure by high collision frequencies is for example expressed in the RRKM theory.2) This is usually only important for the reaction of small species. The RRKM or similar theories are used to transform kinetic data to higher pressure for calculation of oxidations in supercritical data (see below). These nonspecific interactions (independent of the chemical nature of the solvents) are much easier to consider during modeling than specific interactions like, e.g., hydrogen bonds. Therefore for example reactions of polar species in supercritical water are a special challenge. A recent discussion of different solvent effects in supercritical fluids is given by Kajimoto [25]. In the following the properties of SCF are discussed as relevant for chemical reactions. Based on this, three examples of chemical reactions modeling are presented. These three different chemical-reaction modeling studies show from the first to the third reaction an increasing specific impact of the solvent on chemical kinetics. The first example is the C—C-bond splitting of tert.-butylbenzene in supercritical water and for comparison in pure nitrogen as well as in a mixture of water and nitrogen at high pressure. Here, the main effect of the supercritical 1) In some of these cases, the high activation volume might be not due to solvent effects but to a change in the reaction mechanism. 2) Theory by Rice, Ramsperger, Kassel and Marcus; see for example [24].

7.2 Properties of Supercritical Fluids

fluid found is a consequence of the high pressure or dilution by the solvent; no hints for a specific solvent effect between reactant and solvent were observed. The second reaction is the oxidation of methanol in supercritical water and supercritical carbon dioxide. Other working groups had developed very similar kinetic models for oxidation in supercritical water based on gas-phase models. In these models the pressure effect is considered by transformation of the kinetic parameter used in gas-phase models to high-pressure values by the RRKM theory or a similar approach. No specific interaction of water with the reactant was implemented in the model except the reaction with water. Therefore these models should also describe the oxidation in another fluid at the same temperature and pressure. One of these models was used to calculate the oxidation of methanol in water and carbon dioxide. The results were compared with experimental data showing that these models are not complete due to their origin as gas-phase models. The third example is the degradation of glycerol in near- and supercritical water. Here the change of water properties leads to a change in the product spectrum. This change can be described by a kinetic model, consisting of a free radical and an ionic mechanism competing with each other. This model is not sufficient to explain all the details found experimentally and includes no pressure dependence. All these examples show the challenges occurring in view of kinetic modeling of reactions in supercritical fluids, especially in supercritical water. On the other hand the studies presented here give an impression of the opportunities of kinetic modeling after overcoming the lack of fundamental studies concerning solvent effects on elementary reactions steps.

7.2 Properties of Supercritical Fluids

Supercritical fluids are defined as a fluid at a pressure above the critical pressure and a temperature above the critical temperature. Below the critical point, the vapor-the pressure curve separates the liquid and gaseous phase. The vapor pressure ends up at the critical point. Beyond the critical point, the density of the fluids can be varied continuously from liquid-like to gas-like values without phase transition. This variability of density corresponds to diversity of properties. Supercritical fluids are “tunable” solvents [26] for which the properties can be adjusted as a function of temperature and pressure. This chapter focuses on the utilization of supercritical CO2 and water. The properties of these two supercritical fluids will now be introduced. The critical point of CO2 is at 31.1 ºC and 7.38 MPa. Supercritical CO2 possess properties that are between those of the liquid and the gaseous state. It is used as a nonpolar solvent and exhibits a static dielectric constant generally between 1.1 and 1.5, depending upon density. In the case of water, the changes in properties from ambient via the critical point (Tc = 373 ºC, pc = 22.1 MPa, rc = 320 kg/m3 ) up to supercritical conditions

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are particularly drastic (see, e.g., [27–30]). Liquid water at standard conditions (T = 25 ºC, p = 0.1 MPa) is poorly miscible with hydrocarbons and gases. In contrast, it is a good solvent for salts because of its high relative dielectric constant of 78.5 at a high density of 997 kg/m3 at ambient conditions [31]. At nearly critical temperature and pressure the relative dielectric constant is in the region of 10 [31, 32], this is nearly the relative dielectric constant of common polar organic solvents at standard conditions, and it further decreases with rising temperature. Therefore, supercritical water (SCW) at low densities becomes a poor solvent for ionic and highly polar species like inorganic salts. Accordingly, SCW becomes completely miscible with many organic compounds and most gases (see, for example, [32–34]). This complete miscibility makes SCW an excellent solvent for homogeneous reactions of organic compounds with gases, like the oxidation of organic compounds with oxygen and air. The absence of phase boundaries leads to a fast and complete reaction (see, for example, [35–39]). In water below the critical temperature usually no complete miscibility but an increased solubility of organic compounds is found [40]. The ionic product increases slightly with temperature up to around 1010–11 in the range between 200–300 ºC [30, 31]. Above the critical temperature the ionic product decreases drastically with temperature but increases with pressure. In subcritical water and in supercritical water at high pressures it can be some orders of magnitude higher than in ambient water. In these regions water can play the role of an acid or base catalyst because of the rather high concentration of H3O + and OH− ions. Recent studies [41] show that the chemical kinetics of some acid- or base-catalyzed reactions are less sensitive to changes of the ionic product by pressure and temperature than expected. Here the water itself may play the role of a proton donator or acceptor. This opens up opportunities for many applications [42, 43] because reactions that usually need the addition of acids and bases can be conducted without it. Supercritical water as a high-pressure solvent causes certain differences on reaction kinetics compared to gas-phase reactions. Examples are reactions of free radicals: The reaction rates of small free radicals should be increased by the enhanced energy equilibration rate due to the high collision frequency ([44–46] see also [25]). This is very important during, for example, the oxidation in supercritical water because all reactions involving OH and HO2 free radicals are accelerated. They are the most important free radicals of the oxidation chain reaction (see, for example, [47]), therefore this enhancement leads to higher reaction rates. In addition to these pressure effects free-radical reactions, in particular the reactions of OH and HO2, should be influenced by interactions with the surrounding water, for example, by hydrogen bonds. From a microscopic point of view, water is a polar molecule, even if its macroscopic properties are those of a nonpolar solvent as a result of the low dielectric constant at low densities (see Fig. 7.1). This means that water can take part in reactions – as reactant or as part of the activated complex – and it tends to form local-density inhomogeneities. Although the number and lifetime of hydrogen bonds decreases with temperature and decreasing density, they still exist also in

7.3 The C—C-bond Splitting of Tert.-butylbenzene

Figure 7.1 Density r , ion product IP and dielectric constant e of water at high temperature and high pressure (data taken from [31]).

supercritical water (see [48–51] and reference cited therein). Supercritical water – on a short time and length scale – is not a homogeneous fluid: A varying number of higher-density areas occur in which water molecules are connected by hydrogen bonds with varying size, depending on temperature and pressure (see, for example, [49, 52–56]). This tendency to form local-density enhancements connected with high compressibility leads to a high structuring of water around polar solutes. The relative high density of solvations shells influences solvation energies and therefore chemical kinetics (see also [57]). This effect can be considered by using a “local relative dielectric constant” higher than the bulk value (see also, for example, [58–60]. A recent analysis about the role of water in different chemical reactions is given by Akiya and Savage [57].

7.3 The C—C-bond Splitting of Tert.-butylbenzene

During homolytic bond-splitting reactions, that is pyrolysis reactions, in water less tar and coke is formed than in a neat pyrolysis [57]. Sometimes this is explained by the hydrogen-donation ability of water [57]. Alternative or additional explanations may be dilution by solvation or a cage effect [57]. To study this in detail, we have conducted the degradation of tert.-butylbenzene not only in supercritical water but also in nitrogen and nitrogen/water mixtures. Tert. butylbenzene (TBB) was used as a model compound for the degradation of hydrocarbons. It is a molecule with three different type of C—C bonds of differ-

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ent reactivity and a liquid that can be easily handled. C—C bond splitting in supercritical water is interesting, e.g., in view of recycling of polymers and degradation of lignin during biomass gasification. The reaction was conducted at very low conversions of only some per cent. The reason is that the focus of this work was on the early intermediates of TBB degradation, to get information about the reactivity of the different types of bonds. At high conversions the reaction mixture becomes too complex to identify the reaction pathways. 7.3.1 Experimental

The experiments were carried out in a flow reactor at pressures between 5 and 25 MPa, temperatures between 505 and 540 ºC, and reaction times between 15 and 55 s. Water, nitrogen, and water–nitrogen mixtures were used as reaction media [61]. The products found are typical for a free-radical reaction similar but not identical to gas-phase reactions. No significant differences in the product distributions were found among the reaction in the different reaction media. This also hints at a free-radical reaction, because differences should be observed, if for example ionic reactions occur. 7.3.2 Modeling

The degradation of tert.-butylbenzene in HCW is described by a model consisting of 171 elementary reactions, based on the available knowledge of gas-phase pyrolysis (for details see [62]). For the calculation of chemical-kinetic parameters low-pressure data have been used as starting values. These data were directly taken from the literature (e.g., [63, 64]), if available, or estimated from suitable literature data of similar reactions. In an iterative process, the calculation results were compared with the experimental results and the kinetic parameters were varied within certain limits to get a better agreement using the computer package LARKIN [65] (for details see [62]). The main reaction pathways of the resulting reaction mechanism are shown in Fig. 7.2. Compared to gas-phase pyrolysis the unimolecular reactions are preferred over bimolecular reactions and they are relative slow. This difference is more pronounced when the pressure is increased and may simply be a consequence of dilution. Some additional experiments with ethyl benzene in SCW could directly be compared with gas-phase experiments available from the literature [66]. In SCW the reaction was roughly three orders of magnitude slower than in argon at ambient pressure which hints at a certain diffusion limitation or cage effect in SCW (for diffusion limitation and so forth, see also [25]). As mentioned above, no significant difference of the overall reaction rate between the reaction of tert.-butylbenzene in HCW, nitrogen and nitrogen–water mixtures was found in the experimental error range (Fig. 7.3). Does this mean that water is an inert reaction medium without any interaction with the reaction?

7.3 The C—C-bond Splitting of Tert.-butylbenzene

Figure 7.2 Main reaction pathways of the pyrolysis of tert.-butylbenzene (TBB) at high pressure ([61, 62]).

However, a separate experiment with D2O showed that water reacts with intermediates to products containing substantial amounts of deuterium and deuteriumcontaining compounds. As already mentioned the variety and kind of the product species of tert.-butylbenzene degradation indicate that the reaction proceeds via a free-radical mechanism. These free radicals are able to react with water forming hydroxyl free radicals. These hydroxyl free radicals can abstract hydrogen from

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Figure 7.3 Conversion of tert.-butylbenzene in four different reaction media (SCW, two N2—H2O mixtures and N2) at 25 MPa [61]. In mixture I the mole fraction of N2 is 0.03 in water. In mixture II the mole fraction of N2 is 0.35.

Table 7.1 Contents of deuterium in the products and in the

remaining tert.-butylbenzene after reaction in supercritical D2O [61]. Product

Per cent of monodeuterated molecules (±2)

benzene toluene iso-butylbenzene tert-butylbenzene cumene α -methylstyrene

23 19 17 0 14 11

organic compounds or may be added to double bonds. Only in the second case would some oxygen-containing products be formed. Careful analyses of the product mixtures gave no evidence for oxygen-containing substances in measurable concentrations [61]. Hydrogen exchange is another possibility for an impact of the water environment on the reaction and for the formation of deuterium-containing products during the reaction in supercritical D2O. It was confirmed that a catalytic H—D exchange (e.g. at the reactor surface) did not occur under the conditions applied because the unreacted tert.-butylbenzene did not even show traces of deuterium. However, all of the products investigated show substantial contents of deuterium in varying quantities, as detected by mass spectrometry (Table 7.1). This shows that water takes part within the radical-chain mechanism, because free radicals abstract deuterium from water. The relative content of deuterium corresponds with the reactivity of this free radical (Table 7.1). This abstraction of H or D atoms by free radicals has no influence on the overall reaction rate, because the OH free radicals formed are much more reactive than the different hydrocarbon free radicals of the chain reactions. Every OH radical reacts faster with a hydrocarbon than a hydrocarbon free radical would.

7.4 Total Oxidation in Supercritical Fluids

During the degradation of tert.-butlybenzene only nonpolar free radicals are formed with low interaction with the solvent. Oxygen-containing radicals can interact much more with water, which is important, for example, for oxidation reactions. In these cases specific solvent effects are of more importance and significant difference in the chemical kinetics of the reaction in water compared with nitrogen as reaction medium could be assumed. This study concerns only very small conversions. Longer reaction times would lead to more intermediates, which may react with water or add OH free radicals to form a large variety of products. It is likely that after longer reaction times significant amounts of oxygen-containing compounds, e.g. alcohols or ketones, are found. Another C—C-bond splitting reaction or pyrolysis study [67] also shows no significant change when the reaction occurs in argon instead of in SCW. The pyrolysis rate of hexadecane in SCW was almost the same as that in 0.1 MPa argon. The product distribution was also nearly the same for both cases. No carbon acids, alcohols and carbon oxides were detected. Results of polyethylene pyrolysis in SCW were clearly different from pyrolysis results in Argon. In SCW, higher yields of shorter chain hydrocarbons, a higher 1-alkene/n-alkane ratio, and higher conversion were obtained. This is explained by the high solubility of hydrocarbons in SCW. In argon the reaction of PE occurs in a molten PE phase. In SCW the degradation products are dissolved by SCW and then further degraded in the SCW phase. Here, the unimolecular beta-scission is preferred over bimolecular H abstraction than in the molten PE phase [67]. In pyrolysis reactions with nonpolar intermediates the main role of water is solvation and dilution. Dilution leads to a preference of unimolecular reactions by which, e.g., tar formation is avoided. The important free radicals of the pyrolysis chain reactions are large free radicals, which react with a reaction rate in the “high-pressure plateau” even at ambient pressure. At higher pressure the reaction rate could decrease by diffusion limitation. Water can participate in the chain reaction, but without significant influence on the overall kinetics. This does not mean that no solvent effects during pyrolysis reactions exist. Troe and coworkers [68] report a unique increase of the reaction rate of the recombination of benzyl radicals in a high-pressure medium (Ar, N2, CO2 ) due to solvent effects. This effect was more strongly accentuated in CO2. It has to be considered that any solvent effect in single reaction steps of pyrolysis reactions does not necessarily mean that a significant influence on the overall reaction rate is to be found.

7.4 Total Oxidation in Supercritical Fluids

The complete miscibility of organic compounds and gases with supercritical water opens up the possibility to carry out an oxidation in an inert solvent. No

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phase boundaries occur, which avoids incomplete conversion due to incomplete mixing and mass-transfer limitations of the fluid. The oxidation of harmful organic compounds contained in aqueous waste effluents known as supercritical water oxidation (SCWO) has been worked out since the 1980s [reviews: [36, 37, 39, 69–72]. From an engineering point of view, two major difficulties emerged under supercritical operating conditions [69]: • increased corrosion of reactor and heat exchanger construction materials; • drastically decreased solubility of salts resulting in precipitation inside of the reactor and feed preheater causing fouling and even plugging. During the SCWO process, the organic compounds react completely with the oxidant – mostly airborne oxygen-forming CO2 and H2O. The heteroatoms chlorine, sulfur, or phosphorus present in the organic wastes are transformed into the mineral acids HCl, H2SO4, or H3PO3, respectively. Organic-bonded nitrogen predominately forms N2 and small amounts of N2O. Undesired byproducts known from incineration like dioxines, and NOx are usually not formed (see review articles cited above). To achieve the desired conversion efficiency of about 98% (for some hazardous wastes even higher), usually temperatures of 500–600 ºC at pressures between 25 and 35 MPa and a reactor residence time of up to one minute are applied. Essentially, three reactor concepts were developed and studied [36, 37, 69]: The tubular reactor [73, 74], a tank reactor with a reaction zone in the upper part and a cooling zone to dissolve the salts in the lower part of the thank [35], and the “transpiring wall reactor” with an inner porous pipe that is penetrated with water to prevent salt deposition at the wall [69, 75–77]. A fourth concept is the hydrothermal burner, which cools the wall by coaxial injection of large amounts of water [78]. As oxidants, mainly air, oxygen, and hydrogen peroxide were tested. Mostly, Ni-based alloys were used as reactor construction materials. For waste streams with low amounts of salts a simple tubular reactor is useful, because plugging is not likely. In order to find a suitable material for such SCWO reactors, corrosion studies with different metals, alloys, and ceramics have been carried out [39, 79–92]. The conclusion of all these studies is that there is no reactor material suitable for all waste feedstocks but for each single feedstock one suitable material can be found. In the “transpiring wall reactor” not only salt deposition but also corrosion is avoided by the construction [for example [69, 75–77]. Besides the investigation of numerous model compounds, real wastes from chemical, pharmaceutical and food industry, from municipal sewage treatment plants, and from military and nuclear power facilities were tested in bench- and pilot-scale plants [37, 70]. Only one commercial plant is known to be operated due to the relative high costs of this process [93].

7.4 Total Oxidation in Supercritical Fluids

To describe the oxidation of simple compounds in supercritical water different working groups created different kinetic models. These models consist of elementary reactions [94–101] or lumped chemical reactions [95, 102] in the mathematical form of ordinary differential equations. From a chemical point of view the different models with elementary reactions are very similar with small differences in single values of the activation energy and the pre-exponential factor. These modeling studies are supported by spectroscopic determination of intermediates [103–105]. The basis of the models consisting of elementary reactions [94–99] are well-investigated gas-phase models [106, 107]. The gas-phase models are transformed to high-pressure conditions by increasing the reaction rate of the elementary reactions to the value of the “high-pressure plateau” as a consequence of the increased energy transfer at higher pressure (see, for example, [46]). In later studies the reaction rates were changed until they agreed with the experimental high-pressure data for elementary reactions [46]. These high-pressure data were obtained for helium as the pressure medium and also did not consider specific interactions. These models give a relative good description of the reaction but underpredict the increase of the overall reaction rate with increasing pressure [57]. Additionally, changes in the product composition by increasing pressure could not be described [108]. The models include water as reactant, as product and as collision partner. These models do not include specific effects of water such as interaction of the oxygen-containing radicals with a water solvent molecule or the formation of ionic intermediates [109]. This deficit of solvent-effect implementation may be the reason why these models do not describe the pressure dependence well. Experiments [103, 110] and calculations [111, 112] of the decomposition of H2O2 show a significant density impact due to solvation effects. Calculation of the fugacity coefficient of HO2 [113] also shows that water is a solvent with a specific solvent effect. Consideration of these specific influences of water during SCWO may lead to improved elementary reaction models [57]. There are not many measurements of elementary reactions with free radicals in supercritical water. Two examples are the addition of OH radical to nitrobenzene [114] and the reaction of the OH radicals with methanol [115]. The experimental rate constants for hydrogen abstraction from methanol by OH radicals are significantly higher than the values used by a variety of researchers for modeling oxidation of methanol in supercritical water using detailed chemical kinetics models [115].

7.4.1 Is Oxidation in CO2 the Same as Oxidation in Water?

During oxidation in CO2, there is no water (at least in the beginning of the reaction) and CO2 fulfils the role of a collision partner. In the SCWO models no specific effects of water are included, only the presence of a collision partner and

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water as reactant. Therefore, these models should be able to describe oxidations in CO2 also. 7.4.2 Experimental

Experiments of the oxidation of methanol in CO2 [116] show that there are distinct differences between the oxidation in SCW and CO2 : The ignition temperatures of oxidations in CO2 are lower and below ignition the resulting gas compositions are different in SCWO and oxidation in CO2. Very low amounts of water added to CO2 lead to a remarkable decrease of the CO amount formed and a lower O2 consumption (Fig. 7.4). The amount of formic acid found is lower in supercritical CO2 (Fig. 7.4).

Figure 7.4 Conversion to CO a), of O2 b), and formation of formic acid c), after conversion of methanol in supercritical CO2, SCW, and two different CO2—H2O mixtures at 420 ºC and 25 MPa as function of reaction time. Printed with kind permission of Springer Science and Business Media [116]).

7.4 Total Oxidation in Supercritical Fluids

7.4.3 Modeling

The kinetic modeling calculations, using the model of Brock et al. [94] and the computer package CHEMKIN II [117] were performed to investigate oxidation. As illustrated by Fig. 7.5, the main chain free-radical reaction – after a certain starting period of the reaction – are the HO and the HO2 free radicals. This is found for the oxidation in SCW as well as for the oxidation in supercritical CO2. In fact, the calculations lead to very similar results in both cases (Fig. 7.6). As expected, the SCWO model used was also useful for the description of oxidation in CO2 (Fig. 7.6). This is clear, because the differences between oxidation in SCW and supercritical CO2 are relatively small. This also means that the sensitivity of water as a reactant is less pronounced as assumed by some working groups (see also [57]). On the other hand, some details are not satisfactorily described, like the influence of small amounts of water on the CO formation (see Figs. 7.4 and 7.6). In principle the model predicts the decrease of CO with water additions, because it includes water as reactant but the model underpredicts this effect significantly (see Figs. 7.4 and 7.6).

Figure 7.5 Most important reaction pathways of the oxidation of methanol in supercritical water at 377 ºC with an excess of oxygen after 100 s reaction time [118]. Calculated using the model of Brock et al. [94]. Printed with kind permission of Kluwer Academic Publishers [116]).

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Figure 7.6 Conversion to CO (a), conversion of O2 (b), and of methanol (c), at 420 ºC and 25 MPa as function of reaction time using the model of Brock et al. [94]. Printed with kind permission of Springer Science and Business Media [116].

The reason might be, as mentioned before, that nonidealities are not included in the model and/or that the model is not complete. An intermediate found after the methanol oxidation is formic acid. This compound is not included in SCWO models, because the models originate from gas-phase models without formic acid. Formic acid is assumed to be the intermediate of the water-gas-shift reaction. (A summary of studies concerning the water-gas-shift reaction is given in [47].) The participation of water in the activated complex with formic acid leads to a preference of the formation of CO2 and H2 over CO and H2O in supercritical water. This acceleration or preference of decarboxylation was first calculated by Melius et al. [119] by applying ab-initio calculations, based on studies of formicacid decomposition by Ruelle et al. [120, 121]. Influenced by this calculations Rice et al. [20] described the density effect on homogeneous water-gas-shift reaction kinetics. Here, the measured reaction increases much faster than linearly when the water density was raised above 0.35 g/cm3, corresponding to a volume of acti-

7.5 Glycerol Degradation

vation of −1135 cm3/mol. This was taken as a hint for the participation of water molecules leading to a decreased activation energy. Later this was confirmed by detailed ab-initio calculations [122] and experimental studies [123] of the formicacid decomposition in SCW. Studies on the “uncatalyzed” water-gas-shift reaction are most likely influenced by the surface-catalyzed reactions [47]. The experiments reported here were carried out in a reactor “seasoned” (or “aged”) under reaction conditions [116]. The relatively low reaction rate of the methanol oxidation in SCW compared to other studies and the relatively high ignition temperature leads to the assumption that the catalytic activity of the surface was relative low [124], probably because the surface is covered by an oxide layer. This also means that the catalytic activity concerning the surface-catalyzed water-gas-shift reaction is low. And this leads to the assumption that the homogeneous water-gas-shift reaction – as mentioned above – is of major importance and should be considered in models for SCWO. The SCWO model used is – like all these models – based on gas-phase models. Therefore no formic acid is considered. Additionally only a free-radical pathway of the water-gas-shift reaction is part of the model. It is possible that an implementation of the water-gas-shift reaction via formic acid would lead to an improved model, concerning the description of SCWO and/or the description of oxidation in CO2. A really perfect description can only be expected, if specific solvent effects in the form of activation volumes are incorporated. These would be different for SCW and supercritical CO2.

7.5 Glycerol Degradation

Different working groups conducted studies to derive useful products from glycerol [125–127]. Glycerol is formed as a byproduct of biodiesel production; the increasing production of biodiesel leads to an increase of glycerol production predominating requirements of, for example, the cosmetic industry. Glycerol is also considered as a model compound for biomass in basic investigations of biomass conversion [128]. The main products of the glycerol degradation in near- and supercritical water are methanol, acetaldehyde, propionaldehyde, acrolein, allyl alcohol, ethanol, formaldehyde, carbon monoxide, carbon dioxide and hydrogen. Studies show that the measured composition of the product mixture at constant temperature depended on the water density (Fig. 7.7). This was taken as an indication that these products could be formed by competing ionic and free-radical reaction pathways. Usually in gas-phase kinetics the product composition changes with temperature because of the different activation energies and, to a minor extent with pressure, mainly because of the concentration effect on bimolecular elementary reaction steps. In water, the drastic dependence on pressure is likely a consequence of the competition between reactions with different polarity. Free radical reaction rates (involving large free radicals beyond the RRKM highpressure limit, see, for example, [25]) should decrease with pressure as a result

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Figure 7.7 Relative yield of selected products versus pressure. The relative yields, which mean the fraction of a product relative to all liquid products, of several experiments with different conversions and compositions, but at a temperature of 395 ºC, are averaged.

of the cage effect or otherwise remain constant. Ionic reaction rates should increase with pressure either because of the higher dielectric constant stabilizing ionic compounds (see, for example, [25]), or because of the higher ionic product of water at higher densities leading to the formation of more ionic intermediates. Therefore, the density dependence of free-radical reaction rates and ionic reaction rates is opposed and causes a drastic change in the product composition with pressure. The temperature dependence of ionic reactions at constant pressure in nearand supercritical water shows a typical non-Arrhenius3) behavior. If free-radical reactions are important for the global rate, the Arrhenius plot may even become more complicated by the superposition and interaction of the two mechanisms. Figure 7.8 shows the Arrhenius plot of the glycerol degradation in near- and supercritical water at 45 MPa. Here, the overlay of ionic and free-radical reactions is responsible for such an unusual shape. (For details see [128].) 7.5.1 Modeling

For the modeling of the decomposition of glycerol in near- and supercritical water on the basis of elementary reactions, we assumed a combination of a free-radical thermal decomposition model and an acid-catalyzed ionic decomposition model [128]. This is supported by some studies in the literature, which also include the 3) This non-Arrhenius behavior of acid- or base-catalyzed reactions is mainly a consequence of the nonconsideration of the H + concentration in the rate law, see also [57].

7.5 Glycerol Degradation

Figure 7.8 Arrhenius plot of the global rate constant of glycerol degradation (first-order kinetics, 45 MPa, water/ glycerol ratio: 199) from experimental results and from model calculations. In addition, the ionic product for water at 45 MPa is given [31].

competition of ionic and free-radical reactions [129] or polar and nonpolar reactions [130] and the complexity of the experimental results, which cannot be explained by either a pure free-radical or a pure ionic mechanism alone. The radical part of the reaction mechanism was developed similar to other radical mechanisms in supercritical water [61, 62], which are only modifications of free-radical mechanisms at low pressures. The reaction classes considered are: initiation reactions, β -scissions, hydrogen-transfer reactions, radical isomerizations, radical additions, radical dehydratizations, radical substitutions, and radical-termination reactions. For most of the elementary reactions involving free radicals formed in the decomposition of glycerol, no data are reported in the literature. On the other hand, one can find similar or analogous elementary reactions. There are, for example, studies of the thermal decomposition [131, 132] and the combustion of smaller molecules (for example [101]), and also investigations of specific elementary reactions [45]. Some of the reactions used may be considered as a combination of elementary reactions or as elementary reactions lumped into a single reaction step. The primary assignment of kinetic parameters to an elementary reaction was done in analogy to known reactions of the same type or, if known, from the literature. In addition, some rules of thumb for adjusting the reaction rates from normal pressures to supercritical water conditions were used like: reaction rates for bond-breaking reactions are slower; radical isomerization and radical substitution are favored. (For details see [128].) The ionic part of the reaction mechanism is totally based on assumptions. There is no literature known to us, in which similar reaction systems are

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described on the basis of elementary reactions or detailed single steps. The selfdissociation equilibrium of water is known in dependence of temperature and pressure [30, 31]. Therefore – if the self-dissociation forward and backward reaction is assumed to be very fast – the concentrations of protons and OH− ions are known. These ions may catalyze different reactions. This assumption appears to be too simple: Hunter et al. [42] report that there is experimental evidence that water molecules are able to react as proton acceptor and proton donor and enhance acid- or base-catalyzed reactions in some cases as well. Anyway, there were no data available in order to consider this during building of the model. The following ionic reaction classes were considered (including their back reactions): protonation, deprotonation by OH− ions, hydration, keto-enoltautomerization, acetalization and aldol condensation. The autoprotolysis of water is the basis of many suggested mechanisms (overview: [57]) for hydrolysis (e.g., [15, 42, 130]) and elimination (e.g., [19, 127, 133– 137]) reactions, especially in near-critical water. The primary step for these reactions is the protonation (see also Fig. 7.9) of an oxygen-containing group like a carbonyl group [15], an ether linkage [129, 130] or a hydroxyl group [19, 127, 134–137]. In the last case, the protonated alcohol is suggested by Antal et al. [19, 127, 135–137] to undergo dehydration to a carbenium ion. Keto-enol-tautomerization is an important step during the pinacol rearrangement to pinacolone [138]. This reaction occurs in strong acid solutions but also

Figure 7.9 Main reaction pathways of the ionic reaction mechanism, calculated for 45 MPa, 350 ºC and 118 s.

7.5 Glycerol Degradation

in near-critical and supercritical water because of the high autoprotolysis in this medium [138]. During autoprotolysis, both the H3O + and the OH− ions are formed. Therefore, typical reactions usually occurring in basic solutions are found in neutral near- or supercritical water as well. In the mechanism presented here the deprotonation by OH− ions and the aldol condensation are taken into consideration. The aldol condensation of butylaldehyde [139] and crossed aldol condensations of benzaldehyde [139, 140] in subcritical water were found to take place without a basic (or acetic) 4) catalyst. The aim of the modeling was to get a chemical model and the respective kinetic parameter in order to describe the overall reaction rate and the product distribution for all temperatures and initial concentrations for which experimental results have been obtained. The model calculations were performed with CHEMKIN II [117]. A single run calculated the concentrations of all species in dependence of the reaction time for a radial homogeneous plug-flow reactor and for overall isothermal conditions. In addition, sensitivity calculations and flow analysis were performed in order to gain information about important reaction steps. (For details see [128].) Model optimization was done by comparing the experimentally measured concentrations of the main products with the calculated ones, by adjusting the kinetic and thermodynamic parameters and consideration of additional reaction steps [128]. The comparison was done for the experiments at 45 MPa and all temperatures, but the highest and lowest temperature results were weighted most. Because most of the kinetic parameters are usually insensitive to fitting, we used sensitivity calculations to gain information about the most sensitive reactions. The final reaction mechanism and its adjusted kinetic and thermodynamic parameters describe the experimental outcome for the experiments at 45 MPa reasonably well. Details of our calculations, like the kinetic and thermodynamic parameters are given in [128] and in the references cited therein. The experimental and calculated conversions are in good accordance except in the temperature range in which neither the ionic nor the free-radical reaction pathways dominate. This effect can also be seen in Fig. 7.8. At around 420 ºC the experimental reaction-rate constant is higher than the calculated one. At around 392 ºC the experimental reaction-rate constant is lower than the calculated one. The reason for this unsatisfactory discrepancy may be that the two types of reactions are influencing each other, which is not considered in the model and that the values of the kinetic parameters in our model are not dependent on the density of water. Usually, the major products show a better accordance between calculated and experimental yields since the model is optimized for the major products [128]. At higher temperatures, in the regime of the free-radical reactions, the model calculations describe the composition better than at low temperature, where the ionic reactions dominate [128]. The reason might be that the ionic pathways occurring are not completely considered in the model, whereas the 4) The crossed aldol condensation was found to be acid and base catalyzed in high-temperature water [140].

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free-radical paths are considered more completely. As mentioned above, the ionic part of the model is more speculative than the free-radical part, because more information is available from the literature for the free-radical reactions at high temperatures than for ionic reactions. From the final model calculations – which basically describe the temperature behavior at a constant pressure correctly – a flow analysis at a medium reaction time can be used to analyze the most important reaction steps in order to get a more compact reaction mechanism. The simplified ionic mechanism at around 350 ºC is shown in Fig. 7.9. The thickness of arrows symbolize the relative reaction flow of a reaction pathway from the educt point of view. They are calculated from the reaction rates of the elementary reactions. These relative amounts change (slightly) with reaction time. Here, the most important step is the protonation of glycerol. This means that the reaction rates of the ionic reactions strongly depend on the self-dissociation of water. In Fig. 7.10, the simplified reaction schema of the free-radical mechanism at 470 ºC is presented as a result of the flow analysis. Here, the first step is the abstraction of a OH group from glycerol.

Figure 7.10 Main reaction pathways of the free-radical reaction mechanism, calculated for 45 MPa, 470 ºC, 110 s.

7.5 Glycerol Degradation

Calculated molar fraction / mole %

Figure 7.8 shows the ion products of water [31] as well as the measured and calculated global reaction rates. The values of the ion product known from the literature are incorporated in the model. Only at the two highest temperatures, which are dominated by the radical-reaction mechanism, can the experimental results be considered to behave Arrhenius-like and yield an activation energy of about 150 kJ/mole with a pre-exponential factor of about 1018 s−1. At lower temperatures the ionic mechanism gains more and more importance, which depends mostly on the ion product of water, which itself is closely correlated to the density of water. Figure 7.11 shows a comparison between experimental and calculated conversions as well as the experimental and calculated yields of the products. The accordance is satisfying. More such comparisons can be found elsewhere [62]. In conclusion, the model consisting of elementary reaction steps successfully describes the non-Arrhenius behavior of the degradation of glycerol in water and the product composition. The model consists of two parts: a free-radical and an ionic system of reaction pathways. The calculations show that the model is able to reflect the preference of the free-radical reactions at higher temperatures and lower densities and the preference of ionic reactions at lower temperature and higher densities. The reaction model is in reasonable accordance with the global reaction rate of the glycerol degradation found experimentally. Also, the model is able to describe the experimentally detected dependence of the product distribution on temperature and pressure. Improvements and alternatives inside the ionic part of the mechanism and, furthermore, the additions concerning the interference of ionic and free-radical

Conversion Acetaldehyde Acrolein Propionaldehyde Allyl Alcohol Methanol Formaldehyde CO2 CO

5 4 3 2 1 0 0

1

2

3

4

Measured molar fraction / mole % Figure 7.11 Comparison of calculations for and measurements at 45 MPa, 450 ºC and water : glycerol ratios between 131–199. The overall conversion and selected products are considered.

5

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reaction steps may lead to a better and a more general reaction model for the decomposition of glycerol in high-pressurized water. In this modeling approach, the change of the reaction mechanism with temperature and density is induced by the change of the ionic product. The influence of the change of solvent properties on single reaction steps in not considered. It is likely that the solvent effect on ionic reactions is different from that on freeradical reactions, which may result in a change of reaction mechanism. Additionally, inside one mechanism, namely the ionic one, different reaction steps may show varied dependencies on solvent properties (see, for example, [130]). In a theoretical study an ionic reaction and a free-radical reaction were compared: A real-space-grid quantum-mechanical method based on the densityfunctional theory combined with a molecular-mechanical approach (QM/MM) have been applied to investigate the solvation effect of the supercritical water on the association reaction of an OH anion or OH radical to a formaldehyde molecule [141]. Concerning the addition of the OH ion a local dielectric constant of 78.5 was assumed, because the local density in supercritical water is similar to that in ambient water. It was shown that the activation energy of the ionic association in SCW is higher than that in ambient water (10.4 kcal changes 13.6 kcal). The QM/MM simulations on the radical association process have shown that the solvation effect of ambient and supercritical water on the reaction is modest and that the energy profi les are almost in accordance with that in the gas phase. The radical association will be preferred energetically over the ionic one in the high-temperature and low-density region of water.

7.6 Conclusion

Kinetic modeling is a helpful tool to understand and describe chemistry. On applying this kind of modeling to reactions in supercritical fluids usually models based on gas-phase studies were used. This is a good approximation for reactions at low densities; at higher densities significant discrepancies may occur. Here, solvent effects become increasingly important. Water is – like always – a very special reaction medium. Here, not only “normal solvent effects” but even stronger interactions may occur: Water can act as a catalyst, being a proton acceptor and donor or as part of the activation complex. This impact increases with the polarity of the reactants and intermediates. This is shown by the examples given here. In the case of tert.-butylbenzene pyrolysis the reactant and – at low conversions – the intermediates are nonpolar. The interactions with the solvent are low. The differences relative to the gas-phase reaction are due to the role of the solvent as dilution medium and collision partner. Cage effects are assumed to occur. The nature of the solvent, here water and nitrogen as well as mixtures of those, have no significant influence on the global reaction rate. In the case of the oxidation of methanol in SCW and supercritical CO2, the educt and the intermediates are relatively polar and may strongly interact with

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W. A. Peters, K. A. Smith, J. W. Tester, The Journal of Supercritical Fluids 2005, 34, 249–286. M. J. Antal Jr., W. S. L. Mok, J. C. Roy, A. T. Raissi, D. G. M. Anderson, Journal of Analytical and Applied Pyrolysis 1985, 8, 291–303. D. Broll, C. Kaul, A. Kramer, P. Krammer, T. Richter, M. Jung, H. Vogel, P. Zehner, Angewandte Chemie – International Edition 1999, 38, 2998–3014. S. Ramayya, A. Brittain, C. DeAlmeida, W. Mok, J. Antal, Fuel 1987, 66, 1364–1371. W. Bühler, E. Dinjus, H. J. Ederer, A. Kruse, C. Mas, The Journal of Supercritical Fluids 2002, 22, 37–53. J. M. L. Penninger, R. J. A. Kersten, H. C. L. Baur, The Journal of Supercritical Fluids 1999, 16, 119–132. J. D. Taylor, F. A. Pacheco, J. I. Steinfeld, J. W. Tester, Industrial and Engineering Chemistry Research 2002, 41, 1–8. M. Watanabe, H. Hirakoso, S. Sawamoto, T. Adschiri, K. Arai, Pyrolysis of Hydrocarbons in Supercritical Water in K. Arai (Eds.), Tohoku University Press, Sendai, 1997.

132 M. A. B. West, M. R. Gay, Canadian Journal of Chemical Engineering 1987, 65, 645–651. 133 N. Akiya, P. E. Savage, Industrial and Engineering Chemistry Research 2001, 40, 1822–1831. 134 T. Richter, H. Vogel, Chemical Engineering and Technology 2001, 24, 340–343. 135 M. J. J. Antal, M. Carlsson, X. Xu, D. G. M. Anderson, Industrial & Engineering Chemistry Research 1998, 37, 3820–3829. 136 X. Xu, M. J. J. Antal, D. G. M. Anderson, Industrial & Engineering Chemistry Research 1997, 36, 23–41. 137 X. Xu, C. P. De Almeida, M. J. J. Antal, Industrial & Engineering Chemistry Research 1991, 30, 1478–1485. 138 Y. Ikushima, K. Hatakeda, O. Sato, T. Yokoyama, M. Arai, J. Am. Chem. Soc. 2000, 122, 1908–1918. 139 S. A. Nolen, C. L. Liotta, C. A. Eckert, Gläser Roger, Green Chem. 2003, 5, 663–669. 140 C. M. Comisar, P. E. Savage, Green Chem. 2004, 6, 227–231. 141 H. Takahashi, S. Takei, T. Hori, T. Nitta, Journal of Molecular Structure: THEOCHEM 2003, 632, 185–195.

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8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors Christian Horst 8.1.1 Introduction

Applications of ultrasound in processing and synthesis are widespread and offer unusual and beneficial operating conditions [1–6]. Two major principles are used in sonochemistry [8] or sonoprocessing: a) mechanical effects for mixing and disintegration and b) high-energy processes for radical reactions. Most effects occur in liquids, where sound pressures are able to disrupt the continuum. This generates oscillating bubbles, a process called cavitation. Cavitation is observed in shock tubes, pumps and other hydrodynamic devices. Due to the high frequencies in sonochemistry of between 16 kHz and several megahertz, the dynamics of oscillating bubbles create drastic conditions. Temperatures of several thousand Kelvin, extreme heating or cooling rates of 105 K/s, and pressures of up to several hundred MPa are observed in a transient cavity, while the bulk conditions in the liquid remain at ambient temperature and pressure. Radiation forces create intense micro- and macromixing with high shear forces, which are used in emulsification, homogenization, and fragmentation processes. Asymmetrical bubble oscillations in the vicinity of solid particles lead to liquid microjets and shock waves, which are used in cleaning, dispersion, activation, and fragmentation of solid materials. Besides the main industrial applications in cleaning, emulsification, and welding, other uses of ultrasound, such as solids processing, atomization, crystallization, environmental protection and separation are emerging [9, 10]. The term ultrasound describes sound waves with a frequency range from 16 kHz up to several megahertz. Vibrational motions are transmitted by oscillating devices into a fluid and cause pressure waves. This varying sound pressure is superimposed on the static pressure of the liquid. Fluids are generally capable Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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Figure 8.1.1 Classification and range of applications for ultrasound processes.

of transmitting longitudinal waves. Above a critical pressure, liquids are disrupted by the applied sound pressure and so-called cavities are created. Most sonochemical effects are secondary effects generated by oscillating bubbles. Sound waves in liquids are longitudinal density and pressure waves in which particle oscillations occur in the direction of the wave. The displacement around the rest position causes compression and rarefaction. The fundamental quantities of a sound wave are the time-dependent particle displacement x , the particle velocity v and the sound pressure pa:

ξ = ξˆsin(2π ft ) v=

dξ =ˆ v cos(2π ft ) dt

pa = ˆ pa cos(2π ft )

(8.1.1) (8.1.2) (8.1.3)

with the peak values of particle displacement, particle velocity and sound pressure indicated by circumflexes. In plane waves, the particle velocity v and the sound pressure pa are related by the specific impedance of the liquid

ρLc L =

pa v

which is the product of the liquid density rL and velocity of sound c L .

(8.1.4)

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

Figure 8.1.2 Propagation of a traveling wave with density, normalized particle displacement, particle velocity and sound pressure as a function of normalized time or coordinate.

The sound wave transports energy by the kinetic energy of the oscillating particles. If we assume an energy density E given by E = 0.5ρLv 2

(8.1.5)

that passes through a cross-sectional area S with the velocity of sound c we derive the so-called ultrasound intensity I as the energy flux per unit area: I = 0.5ρLc L ˆ v 2 = 0.5

ˆ pa2 ρLc L

(8.1.6)

The determination of local intensities is quite difficult, and ultrasound devices are mostly characterized by calorimetric measurements. Sound-pressure measurements and methods for determining the local intensity with coated thermocouples or by chemical means are still under development. Sufficiently high sound pressures in liquids create voids or gas- and vapor-filled bubbles. Any liquid has a theoretical tensile strength that characterizes the minimum pressure for disruption. Due to the presence of nuclei such as dissolved gases, solid impurities, and rough walls, cavitation occurs at far lower sound pressures than are theoretically necessary. In nearly any liquid, initial nuclei are present that show a distinct size distribution and grow under a certain sound pressure. Bubble growth, multiplication, and disappearance in a sound field is still a very complex phenomenon.

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Figure 8.1.3 Spatial distribution of energy, cavitation, and effects in ultrasound fields.

8.1.2 Bubble Behavior in Acoustic Fields

Several types of bubbles are present in cavitating liquids. Empty cavities (true cavitation), gas-fi lled cavities, vapor-fi lled cavities or mixtures of gas and vapor can be observed. Bubbles may disappear because of dissolution of their contents under the applied sound pressure, some of them oscillate stably over several periods and others collapse violently, followed by the creation of smaller bubbles. To constitute the proper sound field for desired bubble behavior is the key task to achieve maximum efficiency. 8.1.2.1 Equations for the Motion of the Bubble Wall Bubble motions in a sinusoidal sound pressure field can be described, for example, by the Rayleigh–Plesset equation [1, 7] 3γ ɺ   ɺɺ + 3 Rɺ 2 = 1  p∞ + 2σ L − pV   R0  + pV − 2σ L − 4ηLR − p∞ − pa  (8.1.7) RR    ρL  R0 R R 2  R  

that includes the influence of surface tension sL , liquid viscosity hL , polytropic exponent g of the gas inside the bubble, vapor pressure p V and initial bubble radius R0 at rest. The solutions of this equation lead to nonlinear bubble-wall motions. A special case are resonant bubbles, which show damped oscillations. The resonant radius R r of such a bubble can be described by Minnaert’s formula 2σ  2σ  ρLω r2Rr2 = 3γ  P∞ + L  − L  Rr  Rr with the circular frequency w r = 2πf.

(8.1.8)

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

197

More sophisticated equations of motion can be summarized by the formula ɺɺ(1 − M ) + 3 Rɺ 2 1 − M  = (1 + M )H − (1 − M ) R dH RR c dt 2  3

(8.1.9)

with the Mach number M and the enthalpy H, describing the difference between the liquid enthalpy at the bubble/liquid interface and the enthalpy in the bulk liquid. Different models have been proposed by • Rayleigh–Plesset (RP) • Rayleigh–Plesset–Noltingk–Neppiras–Poritsky (RPNNP) • Noltking–Neppiras (NN) • Herring–Flynn (HF) • Kirkwood–Bethe–Gilmore (KBG) The pressure at the bubble/liquid interface can be calculated by 3κ 2σ 2σ   R − pV   0  + pV − L pL =  p0 +   R  R0 R

(8.1.10)

whereas the pressure in the bulk liquid is equal to p∞ = p0 + pa (t )

(8.1.11)

The Kirkwood–Bethe–Gilmore (KBG) model is the most elegant model so far. It incorporates an equation of state for the liquid, the so-called Tait equation that describes the pressure at the bubble wall p in terms of the density of the liquid at the bubble wall compared to the equilibrium density r0.

Table 8.1.1 Terms for enthalpy H and Mach number M regarding different theories.

Theory

Formula for H

Formula for M

RP

H=

1 { pL (t ) − p∞ (t )} ρL

M=0

RPNNP

H=

1 { pL (t ) − p∞ (t )} ρL

M=0

HF

H=

1 { pL (t ) − p∞ (t )} ρL

M=

Rɺ c0

KBG

 2σ R0 n A 1 n   p0 + − pv H= R R  n − 1 ρL  ( n −1) n −(pp0 + pa (t ) + B)

M=

Rɺ c( ρ )

(

)( )

3κ

+ pv −

2σ 2ηRɺ  − + B R R 

( n −1) n

    

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8 Ultrasound Reactors n

ρ p = A   − B.  ρ0 

(8.1.12)

The coefficients A, B and n are specific for the liquid. For water, these values are A = 3001 atm, B = A − 1, and n = 7. The speed of sound in the liquid can be expressed by c( ρ ) = c 02 + (n − 1)H .

(8.1.13)

Results for the time-dependent bubble radius are shown in Figs. 8.1.4–8.1.6. An analysis of the equation of motion for a single transient bubble under a constant driving sound pressure can be performed by solving the Kirkwood– Bethe–Gilmore model. The best combination of sound pressure and initial bubble radius can be found by demanding a maximum bubble radius prior to collapse, a very small final radius and a collapse that is timed to be finished at the maximum positive pressure. A transient bubble with these attributes should give the best results because of its maximum energy content and the high speeds of the bubble wall on disintegration. For a 20 kHz sound field, a pressure of 3.5 bar is sufficient to create this

Figure 8.1.4 Stable cavitation. Time-dependent radius of a bubble with 1 µm rest radius R 0.

Figure 8.1.5 Onset of transient cavitation after passing the transient cavitation threshold.

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

Figure 8.1.6 Overdriven transient cavitation with only one cavitational event during two ultrasound oscillations.

kind of bubble. Lower or higher sound pressures would either lead to smaller maximum radii with less energy or to a hindered collapse because of the reversal of the sound pressure. Sound fields are always characterized in that there are regions of different sound pressure amplitude. The task for a suitable modeling would be therefore to discriminate between regions with proper and improper characteristics and – in a second step – to develop a reactor with a high portion of suitable sound-pressure regions. The theoretical analysis of the single-bubble behavior shows some remarkable results: 1. Bubbles with a great initial radius should be avoided. A degassing cycle or a pulsed operation are suitable methods to remove bubbles beyond the optimum bubble radius of approximately 10 µm. 2. Sound-pressure amplitudes above a frequency-dependent value should be avoided because of the delayed bubble collapse with reduced impact energy. 3. Bubbles with maximum energy content can be created with low-frequency ultrasound. Higher maximum bubble radii and a nearly isothermal collapse are adequate conditions for the mechanical action of asymmetrical bubble collapses. 4. High temperatures should be avoided because of the cushioning effect of vaporous bubble content. The optimum bubble conditions can be confirmed by a spectral analysis of the sound pressure. A secondary sound field is emitted by the oscillating bubbles that can be analyzed by the model and measured inside the reactor. Because of the nonlinear bubble motion under a sinusoidal driving pressure, a FFT-analysis of a hydrophone signal can be used to identify the dominant bubble behavior inside a reactor. Having a short depth of penetration, these signals can be used to map active zones. As a measure for developed and active cavitation the measured ratio

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of the transmission power of the fundamental frequency f 0 and the subharmonic 3/2 f 0 ΨE =

PE( 23 f 0 ) PE( f 0 )

(8.1.14)

can be used. The amount of white noise is a rough indicator of chaotic bubble behavior with lowered cavitation intensity. 8.1.2.2 Cavitation Thresholds A more illustrative description of the bubble motion in an acoustic field can be introduced by the concept of cavitation thresholds for different types of bubble behavior. Depending on the nature of the motion four basic types of cavitating voids are distinguished: stable cavitation, rectified diffusion, dissolving bubbles, and transient cavitation. In the absence of an acoustic field, a gas bubble in a liquid will slowly dissolve owing to the excess internal pressure required to withstand the surface tension pressure 2sL/R0. By applying a changing pressure, the dissolution time can be increased. Very small bubbles will still dissolve, but some bubbles larger than a certain threshold radius can be stabilized in the sound field. These stable oscillating bubbles survive several acoustic cycles. In a process, known as rectified diffusion, such bubbles collect gas from the liquid and grow in size. In the rarefaction cycle, the bubble radius increases and volatile gases or vapor can enter the bubble across the interface. In the following compression cycle, the bubble shrinks, and the diffusion of gases into the liquid is hindered by the smaller exchange surface. This effect is aided by the changing thickness of the liquid boundary layer, which generates high concentration gradients at large radii and small gradients in the compression phase. Rectified diffusion is responsible for the degassing of liquids under ultrasound, even at a low sound pressure of 0.01 MPa. Such a growing bubble may reach the resonance radius, at which very strong oscillations cause surface instabilities and the generation of smaller bubbles. Surface oscillations are responsible for very high microstreaming in the vicinity of stable bubbles, which accelerates heat and mass transport. The behavior of some bubbles can change dramatically at a certain radius. The radius grows rapidly to more than twice the size in half an acoustic period. This is followed by a sudden collapse after which the bubble disappears. These bubbles are known as transient bubbles and cause most of the sonochemical effects. Transient bubbles have a very small collapse time that was first deduced by Rayleigh. An approximation for the collapse time in acoustic sound fields is 1

p   ρ 2  τ B = 0.915⋅Rmax  L   1 + V   pm   pm 

(8.1.15)

where R max is the maximum bubble radius before collapse, pm the mean driving pressure (the sum of hydrostatic and sound pressure), and p V the vapor or gas pressure inside the bubble at maximum expansion.

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

Transient bubble generation is subject to two restrictions: First, a transient bubble must undergo extensive growth, for which its radius must exceed a certain threshold at which the forces acting on it are higher than the surface-tension forces. Secondly, having reached this state, sufficient energy must be concentrated in a very short time, and this requires critical values of the radius before (R max) and after collapse (R min). Only in this case is the energy dissipation (heat conduction, viscosity losses) smaller than the work done by the spherical convergence of the surrounding liquid. These two criteria are described by the so-called Blake threshold radius R B,0 and the Flynn dynamic radius R F,0. Bubbles having initial radii R0 smaller than the Blake threshold and greater than the Flynn dynamic radius will undergo transient collapse. Apfel used these criteria and derived an expression for the maximum bubble radius before collapse. 1

Rmax =

4 2 ( ˆpa − p∞ ) pa ρLˆ 3ω

2  3 ( ˆpa − p∞ )  1 + p 3   ∞

(8.1.16)

Assuming isothermal expansion to maximum bubble radius R max and subsequent adiabatic collapse, the pressures and temperatures generated in a gas-fi lled transient cavity can be derived as R  Tmax = T0(γ − 1)  max   R0 

3

(8.1.17) γ

 p (γ − 1) γ −1 pmax = pg,m ∞  pg,m 

(8.1.18)

where pg,m = p ∞ (R0/R max)3 is the pressure in the bubble at maximum radius.

Figure 8.1.7 Maximum bubble radius for a transient cavity with an initial radius of 1 µm in a sound field with a sound pressure of 4 bar.

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8.1.2.3 Influence of Parameters on Cavitation Behavior Dissolved gases will be present in most applications of ultrasound. The type and amount of these gases in the treated liquid are very important factors in the sonochemical effects [11]. Highly gas saturated liquids have large numbers of cavitation nuclei and low cavitation thresholds. Degassed liquids need higher sound pressures for cavitation, but the sonochemical effects are in most cases more intense. The bubbles can collect gas from the surrounding liquid, and this rectified diffusion creates large bubbles with high gas contents. On collapsing, these bubbles create only moderate temperatures and pressures. As a general rule, degassing a liquid is always favorable in sonochemical applications. Beside the gas content of a cavitating bubble, the used gas is important (Table 3). Gases with high polytropic exponents and small thermal conductivity show the best effects under sonication. Monatomic gases like argon and xenon exhibit good cavitation intensities, whereas diatomic gases such as nitrogen and oxygen tend to decrease the observed effects. Gases with high thermal conductivity such as helium decrease the intensity of cavitational collapse to nearly zero. The vapor pressure of liquids can cushion the bubble collapse like a high gas content. Vapor in a transient bubble can be condensed in the compression cycle and lead to higher cavitation intensities than gas-fi lled bubbles. Experiments with different solvents show that small vapor pressures are necessary for a sufficiently high cavitation intensity. Higher vapor pressures, especially near the boiling point of the liquid, can dampen the cavitation efficiency to nearly zero. If a substrate is subject to treatment within the collapsing bubbles, then a certain number of its molecules must be present in the bubbles and exert an at least measurable vapor pressure. The existence of molecules inside the bubble can easily be proved by means of molecules that exist as ionic or molecular species at different pH values. Ionic species do not enter the bubbles, and high-temperature pyrolysis products can therefore not be created. High viscosity of a sonicated liquid lowers the cavitation threshold markedly. Viscous liquids generate bubbles only at high sound pressures. Bubble motion is damped by the dissipative effect of the viscosity and the smaller maximum bubble radii, and the lower inward wall velocities terminate most sonochemical effects. The effect of temperature on sonochemical processes is explained by its influence on viscosity, gas solubility, vapor pressure and surface tension. Any chemical reaction will be influenced by the temperature-dependent reaction-rate coefficient. In most cases, cavitation is more pronounced at lower temperatures due to the lower vapor pressure of the liquid. This lowers the total gas content of the collapsing bubbles and causes higher cavitation intensities. This paradoxical temperature dependence is highly pronounced for sonochemical reactions involving radical generation in transient bubbles. Lesser effects are generally observed in heterogeneous systems, where intense micromixing by oscillating bubbles favors mass and heat transfer. The effects of temperature and vapor pressure are illustrated in Figs. 8.1.8 and 8.1.9. They show the influence of temperature in different liquids on the number

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

Figure 8.1.8 Influence of temperature on the number of indentations as a function of temperature for different solvents.

Figure 8.1.9 Interpretation of Fig. 8.1.8 in terms of temperature-dependent vapor pressures.

of indentations on a metallic target that are caused by single transient cavities impinging on the target. Different solvents show, beyond a specific threshold temperature, a drastically reduced cavitation effect. This is due to the increased vapor pressure inside the collapsing bubbles cushioning the inward motion of the bubble wall. It can be seen that above a solvent vapor pressure of 0.01 MPa nearly no indentations can be found on the targets.

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The static pressure in a sound field alters the thresholds for rectified diffusion, transient bubbles and other characteristics. A high static pressure can prevent the generation of bubbles by ultrasound or create a different size distribution. For a given amount of energy, smaller bubbles with a higher energy contents are created, which show very strong erosion activities in heterogeneous systems. The effect of frequency on sonochemical reactions is still an active field of research. Some measurements indicate that different frequency ranges are needed for special reaction types. Low frequencies in the range from 16 to 100 kHz, known as power ultrasound, are mainly active in heterogeneous systems with micromixing, cleaning, mechanical action on suspended solids and intense bubble motion. High frequencies favor high temperatures and pressures in the cavitation bubbles, thus creating a large number of radicals. The sonolysis of water seems to be most pronounced at frequencies between 300 and 500 kHz. Frequency is a very important parameter in defining whether a bubble of a given size is transient or stable. The smaller bubbles present in a high-frequency sound field have a lower energy content than the larger bubbles generated at low frequencies. The higher number of cavitational events in high-frequency applications is beneficial for sonochemical effects. In nearly all sonochemical processes, an optimum frequency can be found experimentally. The effect of intensity or power input to the ultrasonic device is complicated. Higher intensities provoke larger amplitudes at the vibrating surface in contact with the liquid. At low intensities, a linear dependency between generated sound pressure and amplitude is observed. Raising this intensity above the cavitation threshold of the liquid causes oscillating bubbles, and under certain circumstances the contact between radiating surface and liquid is lost. The motion of the transducer and the liquid are out of phase, an effect known as decoupling. A second reason for the nonlinear relationship between intensity and sonochemical effects is the creation of cavitation zones. Higher intensities create more and larger bubbles, which may coalesce and lead to fewer transient events. Energy efficiency and sonochemical effects are therefore sometimes smaller at higher intensities, and often the optimum intensity for a specific sonochemical effect must be determined experimentally. 8.1.2.4 Cavitation near Solid Boundaries Undisturbed bubbles in liquids show spherical oscillations in a sound field. Any extended phase interface such as boundaries, suspended solids, or reactor walls in the vicinity of the transient bubble prevent spherical collapse [12]. Asymmetric bubble-wall motion leads to the formation of an involution, which is directed towards the disturbance [13]. In the late stages of the collapse of a transient bubble, a liquid microjet breaks through the remote bubble wall and impinges on the boundary. Liquid microjets can reach velocities in the range of several hundred meters per second and are responsible for cavitation erosion on solids. The pressures generated on solid surfaces can melt soft metals, create pitting, or disrupt inorganic layers on metals. A liquid jet with a diameter of roughly one tenth of the collapsing bubbles diameter creates a water hammer p WH pressure of

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

pWH =

( ρLc sρ TcT )vJ ρLc s + ρTc T

(8.1.19)

where cS is the shock speed (cS ≈ c L + 2v J), v J the jet speed, c T the speed of sound and rT the density of the target. This pressure is maintained in the center of the jet for a time t col = R max/10cS followed by a Bernoulli pressure of rLv J2/2. Most heterogeneous reactions make use of these mechanical effects in cavitating sound fields. Even more intense effects are observed when clouds of bubbles collapse near solid boundaries. The concerted breakdown of such a hemispherical transient bubble cloud generates shock-wave pressures and jetting with intensities some orders of magnitude higher than those of single bubbles. Figure 8.1.10 depicts the action of high-speed water jets on metallic specimens with different fracture behavior. 8.1.2.5 Finite Amplitude Waves and Shock Waves Most applications in ultrasound deal with rather small particle velocities v. This can be described by the acoustic Mach number M = v/c L . A 20-kHz high-intensity step horn with peak amplitude of 100 µm creates a Mach number of about 0.01. For low Mach numbers, the acoustic approximation of incompressible liquids is applicable. In certain cases, for example focused sound fields, much higher Mach numbers are present and lead to distortion of the sinusoidal wave form. Density and sound pressure no longer satisfy a simple linear equation with a fi xed speed of sound c. A pressure-dependent sound speed is observed (material nonlinearity). Convective terms in the equation of motion cannot be neglected at higher Mach numbers and lead to a more complicated situation (convective nonlinearity) where the real speed of the wave front consists of two parts, the speed

Figure 8.1.10 Erosion test on magnesium, aluminum, and zinc targets in toluene at 20 ºC with a 13-mm stepped horn operating at 20 kHz and 200 W for 30 min.

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of sound c and the particle velocity v. The material nonlinearity and the convective nonlinearity tend to increase the propagation velocities of zones with high pressures. This leads to saw-tooth waveforms that develop a shock wave. Consisting of higher harmonics of the initial sound frequency, these saw-tooth waves are strongly attenuated and quickly decay into low-amplitude sinusoidal waves with the initial frequency. Shock waves are also formed in transient bubble collapse. If the bubble wall reaches high velocities, a shock wave is emitted into the surrounding liquid. The Mach number in this case is M = (dR/dt)/c L and reaches values higher than unity. The shock wave is generated some distance from the bubble. Hickling and Plesset calculated, that above a pressure of 100 MPa inside the bubble, a shock wave starts at a distance of r/R min = 5–6 where R min is the minimum bubble radius at collapse. 8.1.2.6 Streaming As an acoustic wave travels through a medium, it can be absorbed. Because of the absorption of momentum in the direction of the sound field, flow is initiated in this direction. Local variations in intensity and energy absorption lead to local streaming velocities of the order of several centimeters per second. Finite amplitude waves create acoustic streaming due high absorption of higher harmonic components [14, 15]. Small obstacles in a sound field create circulation by friction between their boundaries and the vibrating liquid particles. This acoustic microstreaming boundary layer has a thickness d MS of

δ MS =

2ηL,S ρLω

(8.1.20)

where hL,S is the shear viscosity of the liquid. Microstreaming enhances mass and heat transfer and leads to shear forces on the obstacles. 8.1.2.7 Bjerknes Forces Stable oscillating bubbles experience translational radiation forces in a travelingwave field. In standing-wave fields, these so-called primary Bjerknes forces [16] act on oscillating bubbles and are due to a time-average force resulting from sound-field pressure gradients and bubble oscillation. Bubbles below resonance size are attracted to pressure antinodes, whereas bubbles above resonance size travel down pressure gradients towards pressure nodes. Secondary Bjerknes forces arise when two oscillating bubbles are present in a pressure field. Attractive forces between bubbles with inphase pulsation cause coalescence. Bubbles oscillating out of phase are repelled. The bubble oscillation is in phase when both bubbles are smaller or larger than the resonance size and attractive forces dominate. If one bubble is smaller and one larger than the resonance size, they oscillate in and out of phase and repel one another. Primary and secondary Bjerknes forces lead to structures known as cavitation streamers. A large bubble above the resonance size oscillating with surface insta-

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

bilities and located in a pressure node generates microbubbles below the resonance size, which are repelled and forced towards the pressure antinodes of a standing-wave field, where attractive forces cause clouds of oscillating microbubbles. The lines of microbubbles are often called microstreamers. 8.1.2.8 Forces on Small Particles Small particles with radii smaller than the cavitation bubbles are accelerated in the velocity and pressure gradients around oscillating bubbles. These forces are even greater when particles are subject to shock waves. Particle velocities of up to 500 km/h can be reached. On collision of two such particles, oxide layers are removed and metals are melted together. Such treated metals are used in catalysis, where activity enhancements by a factor of up to 10 6 are observed. Like the Bjerknes forces, solid particles whose acoustic properties differ from those of the liquid are subject to primary and secondary forces. Primary forces drive particles into pressure nodes, where secondary forces between particles are responsible for further aggregation. 8.1.2.9 Sonochemical Effects The effects of ultrasound on chemical reactions are in most cases secondary effects caused by cavitation and can be divided into five main groups: 1. Reactions inside the cavitation bubble at high temperatures and pressures 2. Reactions at the gas/liquid interface of the bubbles (secondary reactions of products formed within the bubbles, high-temperature reactions of nonvolatile liquids) 3. Reactions due to high pressures in the surrounding liquid 4. Effects caused by nonlinear bubble collapse near boundaries 5. Enhanced mass and heat transfer by macro- and micromixing

The so-called “hot-spot theory” is frequently used to explain sonochemical effects. Very high temperatures and pressures are generated in a cavity on collapse. Noltking and Neppiras calculated temperatures of up to 10 4 K in a transient cavity. This would lead to black-body thermal radiation that can be observed spectroscopically. Temperatures of 5000 K in the bubble, 2000 K in the liquid boundary layer and pressures of 50 MPa were found in experiments with metal carbonyls. Thus, by creating extreme temperatures in transient bubbles and the surrounding liquid boundaries, high-temperature reactions can be initiated in a liquid that remains at ambient temperature. These calculations are confirmed by the presence of pyrolysis products of irradiated liquids. These include the homolytic cleavage of water, the cleavage of carbon–halogen bonds, and the production of radical products from organic liquids like alkynes and alkenes. The extreme heating and cooling rates of about 105 K/s resemble those in the freezing of molten metals when poured onto a plate with a temperature of −270 ºC. Under the

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extreme conditions of a transient bubble, even plasma reactions or fusion were proposed. Theoretically, conditions similar to those on the surface of the sun are possible. Another approach is the so-called “shock-wave theory”. The compression of transient bubbles leads to an increasing bubble-wall velocity that may eventually reach the speed of sound of the liquid. In true transient cavitation, the bubble vanishes after collapse, creating a shock wave in the liquid. Particles and macromolecules are accelerated in the steep pressure gradient and are shock fragmented. High-speed particles collide and undergo mechanical damage. The “supercritical-water theory” describes the effects of high temperatures and pressures in aqueous systems when conditions are reached under which supercritical water is likely. Supercritical water is known to have a strong solvent action towards organic compounds and extreme chemical activity. Sonochemical effects are possible inside the supercritical water layer surrounding a transient bubble. At the present time, no direct evidence for the generation of supercritical water in ultrasound fields has been found experimentally. Russian workers questioned the existence of true cavitation and postulated a somewhat different sonochemical effect called “charge theory”. Oscillating bubbles undergo very rapid size changes. Friction forces at the gas/liquid bubble boundary can create charged species that can lead to secondary chemical reactions in the bulk liquid. Classical ionic bimolecular nucleophilic substitution reactions (SN2) have been interpreted as a radical mechanism with radical formation by transfer of a single electron according to RX + Y −  → RX • − + Y

(8.1.21)

Ultrasound has an accelerating effect on such single-electron transfer reactions (SETs) and can alter the reaction pathways when ionic SN2 reactions and SET reactions occur simultaneously and lead to different products. Microstreaming, shock waves, and liquid microjets in the vicinity of solid surfaces lead to very efficient cleaning. This effect has been used in industry for more than forty years. Insoluble layers of inorganic salts, polymers, or liquids can be removed by the ultrasonic cleaning effect. In heterogeneous systems such a clean reactive surface leads to improved dissolution rates of metals in acids and enhanced reaction rates. Chemical reactions giving insoluble products are freed from these mass-transport-limiting layers and react rapidly. Several mechanisms for the activation of solids in chemical reactions are induced by the mechanical effects of ultrasound and transient cavities. Highintensity ultrasound is able not only to remove passivating layers on solids, but to break solids like salts and metals as well. Impacting liquid microjets or shock waves create the pitting on solid surfaces. These indentations are free of unreactive layers and react readily with dissolved reactants. Mechanical energy is introduced to the lattice of the solid and leads to slightly excited active centers. In general, such activated solids exhibit higher reaction rates than the untreated solids.

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

Brittle solids can be cleaved by transient cavities. As in milling operations, this leads to higher specific surfaces, high energy content at the disrupted site, and reactive fresh surfaces. Because of the tiny bubble size, very small particles in the micrometer range can be produced. Acceleration of small particles in the shock wave field of a transient bubble can lead to particle agglomeration. Highly accelerated particles collide and melt together on contact. Contact temperatures of nearly 3000 K were measured for very small particles. Particle agglomerates in the vicinity of transient bubbles are fragmented and yield higher specific surfaces [16]. Primary and secondary forces on suspended solids can increase the amount of aggregation in pressure antinodes of a standing-wave field. These effects are used, for example, in separation processes. Many heterogeneous reactions are accelerated by the enhanced micromixing properties of cavitating sound fields. Oscillating and transient bubbles create intense microstreaming in the vicinity of suspended solids. Macromixing is induced by acoustic streaming and the oscillation of bubbles in the sound field. In most cases, a locally different mass-transport coefficient is observed. A tenfold increase in mass-transfer coefficients compared with silent reactions was measured [18]. 8.1.3 Modeling of Sound Fields

The spatial distribution of energy in a sound field and the therefore distributed effective areas can be found by two different approaches. Measurement of the local parameters in a given reactor setup or modeling are suitable methods. Known measuring techniques comprise local physical methods (hydrophones, thermoacoustic sensors, aluminum foil erosion, optical sound pressure and velocity sensors, radiation pressure scale, laser-Doppler anemometry), local chemical methods (Weissler reaction, chemiluminescence, electrochemical sensors), global chemical methods (model reactions), and sonoluminiscence (single-bubble sonoluminescence SBSL, multibubble sonoluminescence MBSL). The modeling of sound fields requires a more fundamental effort than the craftsmanship of pure measurement. The calculation of the single-bubble motion under a given sound pressure is a more or less simple but time-consuming task. Multibubble motion with the involvement of primary and secondary Bjerknes forces is a matter of current investigations [17]. The simplified calculation of sound fields with oscillating and transient bubbles is the latest method to describe the spatial distribution of sound energy and cavitation in different devices. Newer methods include the solution of the Navier–Stokes equation combined with simultaneous calculation of the equation of motion of a representative bubble in the center of a FEM-mesh [18] or the two-phase approach with a liquid and several bubble classes simulated simultaneously [19]. Our own approach uses the effective speed of sound and density of a bubbly liquid calculated by incorporation of a bubble number balance and the solution of the equation of motion for a broad range of bubble sizes.

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Van Wijngaarden-Papanicolaou [20] for example describe the cavitating liquid by a set of equations with mass balance  ∂β 1 ∂p + divv = , 2 ρLc L ∂ t ∂t

(8.1.22)

momentum balance  ∂v 1 + ⋅ gradp = 0, ∂ t ρL

(8.1.23)

and an equation describing the time-dependant bubble volume b (t) ∞

β(t ) = ∫ VB(Re , t )⋅ nB(Re )dRe

(8.1.24)

0

A different approach to decribe the bubbly liquid uses mean fluid properties of the liquid–bubble mixture and starts with the ordinary mass balance  ∂ρ + div(ρ v ) = 0 ∂t

(8.1.25)

momentum balance   1 ∂v  + (v ⋅ grad)v + ⋅ grad p = 0 ∂t ρ

(8.1.26)

All nonlinear effects of the bubble motion are included in the description of the complex modulus B of the bubbly liquid B m = −Vm

dp

(8.1.27)

dVm

This modulus is composed of the modulus BL of the pure liquid and the modulus BB of the bubbles B m = BL + BB

BB = −

(8.1.28) dp(t )

∞

  d  ln ∫ VB(Re , t )⋅ nB(Re )dRe   0 

(8.1.29)

As is easily seen, we need several pieces of information to solve this equation. First, we combine the mass and momentum balances. If we disregard convective terms in these equations we arrive at the so-called acoustical approximation

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

div grad p + k 2 ⋅ p = 0

(8.1.30)

which relates the complex sound pressure p and the wave number k. The wave number k being a complex term allows for the definition of effective speed of sound cm of the bubbly liquid and a damping factor a m. k=

ω ω ω = k ′ + ik ′′ = + cm cm αm

(8.1.31)

The speed of sound is directly given by c 2m =

Bm ρ

(8.1.32)

The effective speed of sound in a bubbly liquid may differ greatly from that of the liquid or the gas, respectively. In water, for example, the speed of sound equals 1500 m/s, in air the speed of sound is 332 m/s but in water containing only a very small fraction of bubbles values for the effective speeds of sound may fall to 40 m/s. This is due to the nonlinear motion of the bubble in the liquid that alters the compressibility drastically. Damping of the sound field originates from this nonlinearity, too. For the calculation of the complex modulus the total number of bubbles in a control volume and their respective bubble-size distribution have to be known. A simple model for the creation and coalescence of bubbles in the sound field can be formulated dnB = kcoll( j − 1) nB,cav − kcoalnB3 . dt

(8.1.33)

The number of bubbles is reduced by the approach and coalescence of bubble under the action of secondary Bjerknes forces. New bubbles are created by the violent collapse of cavitating bubbles. Their number nB,cav and the number j of daughter bubbles created on collapse have to be calculated along with the total number of bubbles. The coefficients kcoll and kcoal have to be adjusted to experimental findings and are generally a function of the applied local sound pressure. A cavitating field will reach steady-state conditions for the total number of bubbles in only a very few ultrasound cycles. Therefore, the total number of bubbles reaches after only some milliseconds the steady-state value 1

 k ( j − 1)  2 nB =  koll   kcoal 

(8.1.34)

The number of bubbles increases sharply beyond a threshold pressure. Beyond this threshold pressure the number of stable and transient bubbles decreases due

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Figure 8.1.11 Number of transient and stable bubbles as a function of normalized sound pressure.

Figure 8.1.12 Bubble volume fractions for stable and transient bubbles as a function of normalized sound pressure.

to the increased coalescence rates. Volume fractions of transient and stable bubbles change differently with higher sound pressures. The number of stable oscillating bubbles decreases with higher sound pressures but the size of these bubbles is altered only marginally. Transient bubbles show a huge increase in bubble

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

radius under higher sound pressure. The volume fraction increases despite the reduced total number. The last piece for the model is the bubble-size distribution function and the limits for the rest radii of bubbles in the sound field. The cavitation thresholds as a function of applied sound pressure indicate the upper and lower size limits for bubbles in a cavitating sound field. A simplifying point of view would differentiate between a) transient bubbles, b) stable bubbles and c) dissolving bubbles. Only bubbles with the attributes stable or transient are likely to exist in a steadystate cavitating sound field. The lower size of a bubble at a given sound pressure is defined by the lower limit for stable cavitation. Any bubble smaller than this size would dissolve rather quickly under the action of its own surface tension. The upper size limit is made up by the transient cavitation threshold. Smaller bubbles may exist as stable oscillating bubbles that eventually grow in size by rectified diffusion or their path into regions with higher sound pressures. Bigger bubbles than this will collapse and create smaller daughter bubbles. The bubble size R0 is rather small and is of the order of several micrometers. If we combine the above-mentioned information then we are able to calculate the effective speed of sound in a bubbly liquid and the damping coefficient as a function of the local sound pressure. These pieces of information can directly be coupled to the sound-field equation and may be calculated by any threedimensional code capable of solving the wave equation for the sound pressure with variable coefficients. div grad p + k(| p|)2 ⋅ p = 0

(8.1.35)

The shape of the curve for the effective speed of sound shows a decrease for higher sound pressures. The damping in the liquid is most pronounced for the

Figure 8.1.13 Regions of stable cavitation, transient cavitation and dissolving bubbles for a frequency of 20 kHz in water at ambient temperature and a static pressure of 1 bar. Bubble radius R is normalized by the resonant bubble radius Rr.

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Figure 8.1.14 Effective speed of sound cm = w /Re{k} in a bubbly liquid as a function of local sound pressure for water with air bubbles at ambient temperature and ambient static pressure.

Figure 8.1.15 Damping coefficient Im{k} in a bubbly liquid as a function of local sound pressure for water with air bubbles at ambient temperature and ambient static pressure.

case that most of the bubbles have reached their optimum collapse conditions. This is obviously the case for sound pressures around 3 to 4 bar. An uptake of energy from the liquid is accompanied by transient bubble collapse. Zones of intense cavitational activity are therefore restricted to a narrow pressure range. Ultrasound reactors can be optimized to contain zones of proper sound pressure predominantly but some of the reactor volume is always inoperative because of the very nature of sound fields. 8.1.4 Examples of Sound Fields in Ultrasound Reactors 8.1.4.1 High-amplitude–High Energy Density Conical Reactor One of the tested ultrasound reactors is a high-amplitude–high energy density conical reactor shown in Fig. 8.1.16 [21]. The sound field is generated by a stepped

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors Table 8.1.2 Typical parameter values used in the modeling

procedure. Parameter

Value

Unit

bubble number bubble number per cycle speed of sound attenuation initial bubble size maximum bubble size maximum bubble pressure maximum bubble temperature microjet velocity bubble collapse time frequency cycle period

105 –1010 10 0 –10 6 20–1500 0–100 2.0 0.2 7.8 2064 100 3.5 20 50.0

dm−3 s−1 dm−3 m s −1 dB/cm−1 µm mm MPa K m s−1 µs Hz µs

Figure 8.1.16 Sketch of the high-amplitude–high energy density conical reactor.

horn (1) indicated in the lower part of the drawing. The sonicated liquid (2) is located in the conical part surrounded by a cooling jacket. The sound field inside this reactor has been investigated by numerous methods. These include calorimetric measurements for the total power input and the energy-conversion efficiency, hydrophone mapping to analyze the local sound pressure and the secondary sound fields of cavitating bubbles, aluminum-foil measurements to determine the local mechanical cavitation effects as well as chemical experiments to bridge the information about the sound-field characteristics and the resulting sonochemical effects.

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The simulation of the sound field reveals that only a very small area is effective concerning the existence and the action of transient bubbles. Figure 8.1.17 depicts the sound pressure in the conical reactor. The face of the stepped horn is located at the left side of the picture. The right side is the interface between liquid and gas. Erosion of aluminum specimen and measurements with hydrophones show that there is a strong correlation between the zones with high absorption coefficients and zones of high cavitational energy. Indentations on aluminum foils can be seen primarily in these distinct areas with proper sound pressure, high sound

Figure 8.1.17 Sound-pressure distribution in a mediumintensity sound field. Radial and axial coordinates are scaled with the speed of sound l0 of pure water at 20 kHz.

Figure 8.1.18 Absorption coefficient in a medium-intensity sound field. Radial and axial coordinates are scaled with the speed of sound l0 of pure water at 20 kHz.

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

absorption and small speeds of sound indicating a high number of transient bubbles. The optimization of the sound field includes the reduction of cavitational erosion at the sound-transmitting surfaces and the utilization of the given reactor volume. Reducing zones of low and high sound pressure can be achieved by a design change of the reactor and the transmitting sound sources. 8.1.4.2 Low-amplitude–High Energy Density Crevice Reactor Calculations and experiments have been used to develop a crevice reactor with a reduced amplitude of the sound-transmitting face, thus reducing cavitational erosion, but a similar energy density regarding the sonicated liquid volume. The sketch of the reactor can be seen in Fig. 8.1.19. The reactor is made up of an inner sound source. This inner wall is operated at 44 kHz and oscillates like a breathing wall. The outer transducers are driven at 25 kHz. The total power uptake of both sound sources is 3 kW for a liquid volume of 1.86 L. A simulation of the sound field revealed an interesting finding during the experiments with a chemical reaction: The performance of the reactor was best as long as only the inner sound source was active. Driven at 44 kHz with a power of 1 kW the direct influence on the chemical reaction outperformed all other operation modes. Using the outer transmitters alone with 2 kW power or a dual mode with inner and outer transmitters with a total of 3 kW power only decreased the influence on the chemical reaction. A simulation of the sound field and a check with aluminum foils showed that the outer transducers tended to deliver too much energy in a limited area directly in front of the sound-transmitting surfaces. The thus disturbed sound field had less volume with medium sound pressures and therefore transient cavities of the proper size. Figure 8.1.20 gives an impression of the sound field pattern inside the crevice reactor under different operation modes.

Figure 8.1.19 Sketch of the low-amplitude–high energy density crevice reactor.

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Figure 8.1.20 Simulation of the soundpressure distribution in the crevice reactor with internal and external transducers. Dark areas represent zones with sound pressures

far below 3 to 4 bar (the optimum sound pressure for transient cavitation at 20 kHz in water) or zones with much higher sound pressures.

8.1.5 Modeling of Sonochemical Effects in Ultrasound Reactors

Solid–fluid reactions like the reaction of organic halides in solvents with solid magnesium occur at dislocations in the solid reactant. In most cases, alloys with some percentage of other metals and stressed material, like magnesium turnings, are used to enhance the reaction rates [22] R − X (l ) + Mg (s ) Solvent  → R -Mg − X (l )

(8.1.36)

Such a reaction can be described by a surface-based rate equation of the form −E r ′ ≅ k0 ⋅ exp  A  ⋅ c R − X (l )  RT 

 mol   m2s 

(8.1.37)

in the nonactivated kinetics. Under ultrasound, higher reaction rates are observed and are compared with the nonactivated ones via an ultrasound-enhancement factor EUS =

kUS k

(8.1.38)

that describes the product of the measured reaction-rate constants with and without ultrasound. Modeling and kinetic data for the Grignard reaction of chlorobutane isomers have been published in [21]. Dynamic experiments revealed the influence of ultrasound on the apparent reaction-rate constant. Very fast reactionrate accelerations were measured after the onset of sonication. The reaction rate reaches a high constant level that falls slowly when the sound is switched off.

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

Figure 8.1.21 Course of product concentration in a dynamic experiment with three time zones: I) reaction without ultrasound, II) reaction with ultrasound switched on and III) reaction with ultrasound switched off. The reactor works in a continuous operation mode and reaction rates in time zone I) are too low to achieve

an increase in product concentration. Shortly after starting ultrasound, a ten-times higher reaction rate leads to an increase in product concentration. A slow decrease in reaction rate is observed in time zone III) where the reaction rate falls slowly to the values before sonication.

Figure 8.1.21 depicts the course of the butyl magnesium chloride concentration in the Grignard reaction with different ultrasound conditions. This behavior is observed in mechanically activated reactions where dislocations in the solid lattice are subject to lower activation energies and higher reaction rates [23]. This comes from the higher energy content of lattice elements that are mechanically stressed. In reactive surroundings, easier reaction and a lower apparent activation energy are observed. The mechanical activation in ultrasound fields comes from transient bubbles, creating liquid microjets, which cause damage and higher dislocation densities on metallic surfaces. Undisturbed bubbles in liquids show spherical oscillations in a sound field. Any extended disturbance like boundaries, suspended solids or reactor walls in the vicinity of the transient bubble prevent the spherical collapse. Asymmetric bubble-wall motion leads to the formation of an involution, which is directed towards the disturbance. In the late stages of the collapse of a transient bubble, a liquid microjet, which breaks through the remote bubble wall, impinges on the boundary. Liquid microjets can reach velocities in the range of several hundred meters per second and are responsible for cavitation erosion on solids. The generated pressures on solid surfaces can melt soft metals, create pitting or disrupt inorganic layers on metals. A liquid jet, with a diameter of roughly one tenth of the collapsing bubbles diameter creates a high water hammer pressure on the solid. This pressure remains in the center of the jet for a time

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Figure 8.1.22 Generation of lattice imperfections as the root cause of mechanochemical activation of solid material.

t col = R max/10(c L + 2vjet) followed by a Bernoulli pressure of rLvjet2/2. Most heterogeneous reactions make use of these mechanical effects in cavitating sound fields. The intensity of this impingement in terms of the local power input per unit area solid depends on the local sound pressure in an ultrasonic device and can be expressed by IE =

ηcav 4π 3 ( p∞ − pV ). f ⋅ nB,cav ⋅ Rmax ε SaS 3

(8.1.39)

with the solid volume fraction eS, the specific surface area of the solid aS, the impact frequency f and hcav as the impact probability. The generated lattice imperfections have a direct influence on the excess enthalpy of a solid material [24, 25]. If we compare the enthalpy of a reference solid (raw material, single crystal) and a mechanically treated one then we get for the enthalpy difference the expression N  ∆H E (T ) = (H E* (T ) − H E(T )) = ∑  i  ∆HiE(T ) i  NL 

(8.1.40)

with the number of lattice imperfections Ni of type i and the Avogadro number NL = 6.0220 × 1026 kmol−1. The enthalpy of a single lattice imperfection HEi is a number that can be evaluated easily by theoretical calculations. For temperatures in the normally present reaction systems we may allow for the simplification ∆GE (T ) ≅ ∆H E (T )

(8.1.41)

Typical lattice imperfections and contributions to the excess enthalpy are • ∆HA: surface enthalpy (NA atoms located at interfaces) • ∆HK: grain-boundary enthalpy (NK atoms at grain boundaries)

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

• ∆HV: point defects and dislocations (NV atoms at defects and dislocations) • ∆HP: phase change enthalpy (NP atoms in metastable phases or amorphous regions) An increase in the excess enthalpy of a solid material has a pronounced effect on the reactivity. Part of the normally necessary activation energy supplied by thermal energy is delivered by the increased initial energy level of the solid. Apparent activation energies can be observed that are generally smaller than the normally measured activation energy. If we follow the formalism of the kinetic theory then we arrive at an equation describing the influence of a reduced activation energy of a mechanically treated solid on the reaction rate r′/(mol/m2/s) between the solid and a liquid component c X,S at the outer surface of the solid.  −(E A − η ⋅ n A* )   −E *  r ′ = k0 ⋅ exp  A  ⋅ c X,S = k0 ⋅ exp   ⋅ c X,S RT    RT 

(8.1.42)

E A* is the apparent activation energy, E A the normally observed activation energy without mechanical activation and h · nA* is the amount of mechanical energy stored in the lattice of the deformed solid. A simple model for the description of reactions between mechanically treated solids and a liquid component has to take into account that not all of the metal’s surface is deformed and therefore mechanically activated. The surface fraction of nonactivated metal surface Q and the activated surface fraction 1 − Q can be used to model the global reaction rate −E   −(E A − η ⋅ n A* )   r ′ = k0 (1 − Θ)⋅ exp  A  + Θ ⋅ exp    ⋅ c X,S RT  RT    

(8.1.43)

This two-center kinetics also needs a description for the surface fraction of activated metal. This can be accomplished by a dynamic model dΘ = −k * ⋅ Θ ⋅ c X,S + kact ⋅(1 − Θ) − kdec ⋅ Θ dt

(8.1.44)

that includes the generation of active surface area by mechanical activation (kact), a thermal annealing of disturbed surface fractions (kdec) and annealing by chemical conversion of a disturbed lattice component by reaction and release into the bulk fluid (k*). With these model assumptions and the so-called ultrasound enhancement factor EUS which describes the quotient of the apparent frequency factor with ultrasound and the frequency factor without ultrasound at the same bulk temperature T, we arrive at the equation EUS (T ) =

k())), T ) T = (1 − Θ) + Θ ⋅ exp Θ ⋅ Ω ⋅γ 0  k(T ) T 

(8.1.45)

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The parameter set Ω=

η ⋅ n A,0 E ≤ 1, γ = A EA RT0

and Θ =

τT ≤1 τ act

(8.1.46)

describes the fraction Ω of the activation energy delivered by the mechanical treatment, g is the Arrhenius number, and Θ is the activation degree that can be found using the time constants for activation, thermal annealing, chemical annealing and the total time constant t T. 1 1 1 1 = + + = kact + kdec + k *c X,S τ T τ act τ dec τ reac

(8.1.47)

Figure 8.1.23 depicts the ultrasound enhancement factor EUS as a function of the surface activation degree Φ with parameter Ω. The circles represent measured values for EUS and Φ. In the case of magnesium as the treated solid with tetrahydrofurane as solvent and different chlorobutanes as reactants a mechanical activation degree of approximately 10% has been observed. This means that 10% of the necessary activation energy has been delivered by the mechanical treatment by impinging liquid microjets. This is a typical value that can be found in other devices like reaction mills, too. This value of 4 to 6 kJ/mol for mechanically treated

Figure 8.1.23 Ultrasound enhancement factor EUS as a function of the surface activation degree Φ with parameter Ω describing the fraction of mechanical stored energy compared to the thermal activation energy.

8.1 Some Fundamentals of Ultrasonics and the Design of High Energy Density Cravice Reactors

magnesium is comparable to values found for nickel (11 kJ/mol) or copper (6–7 kJ/mol) in [25]. The simulation is able to interpret several characteristics of the investigated Grignard reaction in the ultrasound reactors. Most notably, there is a very pronounced effect of the type of chlorine-containing reactant and the activation energy without mechanical activation. Very slow reactions are accelerated to a very high degree under ultrasound, whereas fast reactions do not gain a significant enhancement. The limited influence of ultrasound on fast Grignard reaction can be explained by the two-center model. The reaction rate at the activated and the nonactivated sites is of comparable order for initially fast reactions. The surplus of mechanically activated sites is only of minor importance for the global reaction rate. Initially slow reactions gather a tremendous reaction-rate enhancement in an ultrasound field. Figure 8.1.24 depicts the enhancement factor for three different reactions of magnesium with chlorines. The influence of the reaction rates without ultrasound and the temperature dependency are obvious. At 65 ºC nearly all mechanical action of the ultrasound field ceases. This is due to the increase in vapor pressure cushioning the collapse of transient bubbles. No liquid jets nor shock waves can be observed at such high vapor pressures in sonicated liquids. Typically, a chemical synthesis would be performed at the highest possible temperature to achieve the highest reaction rate. Grignard reactions are therefore often performed under reflux of the solvent. Tetrahydrofurane boils at 65 ºC and the optimum reaction rate for a fast reaction can be found at this temperature without ultrasound. The application of ultrasound for such a fast reaction would only lead to the same reaction rate and space–time yield. Only slow reactions or reactions that have to be operated at lower bulk temperatures due to selectivity issues are prone to be enhanced by mechanical energy. Figure 8.1.25 shows the course of the apparent reaction-rate constants with and without ultrasound for a slow reaction. Under ultrasound, reaction rates reach

Figure 8.1.24 Ultrasound enhancement factor EUS as a function of temperature for three different reactions components. Lines represent the model equation; dots represent the measured enhancement factors. The reaction-rate constants without mechanical

activation are k0 = 11 m/s for chlorobenzene, k0 = 314 m/s for 2-chlorobutane, and k0 = 8800 m/s for 2-chloro-2-methylpropane. The activation energy for all three reactions is 43 kJ/mol.

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Figure 8.1.25 Arrhenius diagram for the slow reaction of chlorobenzene with magnesium in THF (TB = 65 ºC).

Figure 8.1.26 Arrhenius diagram for the fast reaction of 2chloro2-methylpropane with magnesium in THF (TB = 65 ºC).

values for the reaction at the boiling temperature at approximately 32 ºC. Figure 8.1.26 depicts that for a fast reaction there is almost no difference in the reaction rates for silent and sonicated reaction. It is therefore unnecessary to use ultrasound unless there are temperature-dependent side reactions that demand lower temperatures.

References

8.1.6 Summary

The sound field inside a sonochemical reactor can be modeled by treating the liquid bubble mixture as a pseudofluid with mean values. The calculation of the “bubbly mixture” properties is complicated because of the interaction of nonlinear bubble motion depending on the initial radius of the bubbles, the interaction between bubbles among each other and the interaction of the sound fields and the bubbles. The proposed solution is the calculation of the bubble motion of bubbles with different sizes using the Kirkwood–Bethe–Gilmore equations. Knowing the bubble-size distribution at a given sound pressure by calculating cavitation thresholds and using this information in an equation for the local total bubble number, the calculation of the complex bulk modulus of the bubbly mixture is possible. The resulting speeds of sound and the damping coefficients can be integrated in the equation for the acoustical approximation neglecting convective terms in the mass and momentum balance. The solution of the sound field equation with nonlinear coefficients can be accomplished by standard FEM codes. The simulation results have been used to optimize reactor geometries and to interpret surprising effects. The simulation of sonochemical effects on liquid–solid reactions is accomplished by a two-center model that allows for mechanically activated areas on a solid with increased reactivity. The parameters have been fitted to experiments and can be used to find suitable reaction parameters as well as a prediction whether or not a given Grignard reaction will perform better under ultrasound activation or not.

References 1 Leighton T.G. (1997) The Acoustic 8 Einhorn C., Einhorn J., Luche, J.-L. Bubble, Academic Press (1989) Sonochemistry – The Use of 2 Young F.R. (1989) Cavitation. McGrawUltrasonic Waves in Synthetic Hill Book Company, Maidenhead Organic Chemistry. Synthesis: 787– 3 Mason, T.J., Lorimer, J.P. (1989) 813 Sonochemistry: Theory, Applications and 9 Berlan J., Mason T.J. (1992) Uses of Ultrasound in Chemistry, Ellis Sonochemistry: from research Horwood Limited laboratories to industrial plants. 4 Mason, T.J. (1986) Chemistry with Ultrasonics 30: 203–212 Ultrasound, Elsevier Applied Science 10 Hoffmann U., Horst C., Wietelmann U., 5 Suslick, K.S. (1988) Ultrasound: Its Bandelin S., Jung R. (1999) Chemical, Physical, and Biological Sonochemistry. Ullmann’s Encyclopedia Effects, VCH Publishers of Industrial Chemistry, Sixth Edition, 6 Margulis, M.A. (1995) Sonochemistry 1999 Electronic Release; Wiley-VCH, and Cavitation, Gordon and Breach Weinheim Publisher 11 Noltking B.E., Neppiras E.A. (1951) 7 Neppiras E.A. (1980) Acoustic Cavitation, Cavitation produced by ultrasonics. Proc. Phys. Rep. 61: 159–251 Roy. Soc. 64B: 1032–1038

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8 Ultrasound Reactors 12 Benjamin T.B., Ellis A.T. (1966) The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Phil. Trans. R. Soc. Lond. A260: 221–240 13 Chahine G.L., Bovis A.G. (1983) Pressure field generated by nonhemispherical bubble collapse. Trans. ASME, J. Fluids Eng. 105: 356–363 14 Kubanskii P.N. (1962) Acceleration of convective heat exchange by acoustic streaming. Sov. Phys. Acoust. 8: 62–65 15 Okada K., Fuseya S., Nishimura Y., Matsubara M. (1972) Effect of ultrasound on micromixing. Chem. Eng. Sci. 27: 529–535 16 Bjerknes V.F.J. (1906) Fields of Force. Columbia University Press, New York 17 Mettin R., Krefting D., Appel J., Koch P., Lauterborn (2001) 3rd Conference “Applications of Power Ultrasound in Physical and Chemical Processing”, Paris, France, December 2001 18 Dähnke S., Keil F. (1998) Modeling of sound fields in liquids with a nonhomogeneous distribution of cavitation bubbles as the basis for the design of sonochemical reactors. Chem. Eng. Technol. 21: 873–877

19 Servant G., Laborde J.-L., Hita A., Caltagirone J.-P., Gerard A. (2001) 3rd Conference “Applications of Power Ultrasound in Physical and Chemical Processing”, Paris, France, December 2001 20 Commander K.W., Prosperetti A. (1989) Linear pressure waves in bubbly liquids: Comparison between theory and experiments. J. Acoust. Soc. Am. 85: 732–746 21 Horst C., Chen Y.-S., Kunz U., Hoffmann U. (1996) Design, Modeling and Performance of a novel sonochemical reactor for heterogeneous reactions. Chem. Eng. Sci. 51:1837–1846 22 Horst C., Kunz U., Rosenplänter A., Hoffmann U. (1999) Activated SolidFluid Reactions in Ultrasound Reactors. Chem. Eng. Sci. 54: 2849–2858 23 McCormick P.G., Froes F.H. (1998) The fundamentals of Mechanochemical Processing. JOM 1998: 61–65 24 Heegn H. (1990) Mechanische Aktivierung von Festkörpern. Chem. Ing. Tech. 62: 458–464 25 Heinicke G. (1984) Tribochemistry. Carl Hanser Verlag, München

8.2 Design of Cavitational Reactors Parag R. Gogate and Aniruddha B. Pandit 8.2.1 Introduction

Chemical effects associated with the cavitation induced in a liquid, by the passage of ultrasound, are quite distinct from the conventional chemical processes. Very high energy densities (energy released per unit volume) are obtained locally resulting in high pressures (of the order of 100–50 000 bars) and temperatures (in the range of 1000–5000 K) and these effects are observed at millions of locations in the reactor. There are a large number of illustrations where these spectacular effects have been successfully harnessed for a variety of applications worldwide [Suslick, 1988, Luche, 1999, Shah et al., 1999, Mason, 1999]. A few of the important applications can be given as chemical synthesis (in both homogenous and heterogeneous systems by way of increase in the rate and selectivity of many chemical reactions [Petrier et al., 1982, 1984, Pandit and Joshi, 1983, Mason, 1986,

8.2 Design of Cavitational Reactors

Petrier and Luche, 1987, Lindley and Mason, 1987, Thompson and Doraiswamy, 1999, Suslick, 1998, Suslick et al., 1999, Sivakumar and Pandit, 2001a, Sivakumar et al., 2002a]), wastewater treatment (degradation of many of the biorefractory/ complex chemicals [Kotronarou et al., 1991, 1992, Petrier et al., 1992, Bhatnagar and Cheung, 1994, Serpone et al., 1994, Shirgaonkar and Pandit, 1997, Hua and Hoffmann, 1997, Seymore and Gupta, 1997, Gogate, 2001]), textile processing [Thakore, 1990, McCall et al., 1998, Rathi et al., 1997, Yechmenev et al., 1998, 1999], biotechnology (cell disruption [Save et al., 1994, Shirgaonkar et al., 1998 and Agrawal and Pandit, 2003] foam control in bioreactors [Dedhia et al., 2004]) and crystallization [Srinivasan et al., 1995], etc. However, it should be noted that, in spite of extensive research, there are very few illustrations of chemical processing being carried out on an industrial scale owing to the kind of expertise required in diverse fields such as material science, acoustics, chemical engineering, etc. for scaling up successful lab-scale processes. The potential major problems in the design and successful operation of cavitational reactors are: 1. Exact quantification of the cavitation, i.e. collapse intensity as a function of different operating and geometric properties is lacking. 2. Lack of suitable design strategies linking the theoretically available information on cavity dynamics with the experimentally observed results. 3. The existing information is available mainly on the laboratory scale, which would give very large scaleup ratios and hence lack confidence and certainty. 4. Local existence of the cavitation phenomena just near the irradiating surface in the case of ultrasonic transducers and wide distribution of the energy dissipation patterns in the reactor due to uneven energy dissipation. Design information is required from diverse fields such as chemical engineering (gas–liquid hydrodynamics and other reactor operations), material science (for construction of transducers efficiently operating at conditions of high frequency and high power dissipation) and acoustics (for better understanding of the sound and pressure field existing in the reactor). The chapter describes the information necessary to tackle the above-mentioned major problems in the successful design and operation of industrial-scale cavitational reactors. The discussion will cover the following important scaleup aspects: 1. theoretical approach to understand the bubble behavior under the fluctuating sound field and predict the cavitational intensity from the numerical simulations of bubble-dynamics equations considering the geometrical aspects of the reactor; 2. optimization of operating and geometric parameters and some recommendations for effective design of the cavitational reactors;

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3. development of design equations for the prediction of collapse pressure and cavitational yields as a function of the operating parameters, which then can be linked to the experimental yields; 4. recommendations for identifying the cavitational activity distribution in the cavitational reactors and possible ways for eliminating the dead zones; 5. possible methods for intensification of the cavitational activity in the system. Another major problem and its possible solution, not discussed here in detail, is the erosion of the sonicator surfaces at the high power intensities required for industrial-scale operations. The problem is more dependent on the availability of knowledge from the field of material science. The aim should be to develop new transducers with materials of construction giving minimal erosion rates at higher power dissipation levels and at the same time with similar levels of energy transfer for cavitation events. Moreover, transducers with concave surfaces for irradiation can also be developed, which results in focusing of energy at other locations and not at the transducer surface, leading to minimal erosion rates at the irradiating surface. Such designs can be effectively used at higher scales of operation where, due to high levels of power dissipation, the erosion rates may be very high in the conventional flat-surface designs. Magnetostrictive transducers need a special mention here; these can be operated at much higher levels of power dissipation, which is a necessity when large-scale operation is considered. These can be easily used in gases as well as liquids. Moreover, these systems can be cooled by a jet of cold water thrown against the vibrating system or be completely water cooled and hence greater outputs can be obtained in this manner and also continuous operation of the ultrasonic equipments for longer time periods is possible. A further advantage is that these can be sterilized or can be operated under sterile conditions for chemical or medical work. 8.2.2 Theoretical Approach

Solution of the bubble-dynamics equations describing the variation in the size of the generated cavity as a function of time of irradiation should be a useful tool in deciding the type of cavitation that is likely to occur and also in sketching the design strategies in terms of the selection of the operating and geometric parameters. Differential equations based on the momentum and mass balance describing the various stages of the cavitation phenomena are available in the literature (Plesset, 1949, Neppiras, 1980; Tomita and Shima, 1986, Togel et al., 2002) differing in aspects arising out of the following important facts: 1. a cavity can either be considered as a single cavity in isolation or in the form of a cavity cluster in which the dynamics of the cavity will be affected by the presence of the surrounding cavities;

8.2 Design of Cavitational Reactors

2. bubble/cavity interior is composed of both vapor and gas in an unknown ratio; 3. many types of energy losses are involved in damped oscillations of a cavity; 4. a number of discontinuities such as heat conduction, viscosity, liquid-phase compressibility, surface tension, mass transfer, diffusivity and temperature occur at the gas/liquid interface and these have not been sufficiently quantified as yet. Neppiras (1980) has written an excellent review on the different aspects of the bubble dynamics in the case of acoustic cavitation and is considered one of the basic pioneering works in this field. The modeling of cavitation phenomena has been done over the years with a varied objective ranging from explaining the sonoluminescence of the cavitating bubbles to explaining the effects in the chemical processing applications. From the application viewpoint as well as for designing the sonochemical reactors, one has to be more interested in the chemical effects of cavitation. The extent of chemical transformations can be quantified in terms of the collapse pressure and/or temperatures generated (pyrolysis is the controlling mechanism for driving the chemical reaction) and in terms of the radical generation rates induced by cavitation for the case where free-radical attack is the controlling mechanism. Various models have been proposed for the quantification of the chemical transformations and to explain the link between the bubble– cavity motion and the chemical kinetics. It is important to note that the acceptability of any one model as well as the solution schemes strongly depend on the assumptions used in the derivation of the model and the approximations used in the solution schemes. All the earlier models were based on a common platform to explain the variation of the cavity radius with time, i.e. using the conventional Rayleigh Plesset equation of bubble dynamics, which has been given below for ready reference: 1 4 µ  dR  2σ  d2R  3 dR  − p∞  R  2  +  −   =  pi − R  dt  R ρl    dt  2  dt  2

(8.2.1)

Where pi is pressure inside the bubble at time t, p ∞ is fluctuating local pressure that drives different phases of cavitation, R is radius of the bubble, dR/dt is the bubble-wall velocity, d2R/dt2 is the bubble-wall acceleration, rl is the medium density, m is the medium viscosity and s is the medium surface tension. A most important disadvantage of this equation, though simple for the solution, was that the equation considers the incompressible nature of the medium, which results in significant deviations in the predictions of the collapse conditions from the realistic values. More recently, the gas dynamics inside collapsing bubbles has been studied considering the compressibility of liquid in Navier–Stokes equations (Moss et al., 1999, Storey and Szeri, 1999). The equation reported by Tomita and Shima (1986) also considers the compressibility of the liquid medium. The

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Tomita and Shima equation that has been also considered in the analysis of bubble dynamics is as follows: ɺ ɺ2 ɺ ɺ2 ɺɺ  1 − 2R + 23R  + 3 Rɺ 2 1 − 4R + 7R  RR   2  3C 5C 2  C 10C  2   R p − p ɺ ɺ  ∞(t ) 2(r =R ) + C ( p∞(t ) − p1(r =R ) ) +  1 1 −2RRɺ ( pɺ ∞(t ) − pɺ 1(r =R ) ) + ( p∞(t ) − +   1 2 ρl  2  C p1(r =R ) )(Rɺ 2 + 3 ( p∞(t ) − p 1(r =R ) )  ρl

   =0    

(8.2.2)

where C is the velocity of sound in liquid medium, p ∞ is the surrounding timedependent pressure, R is the radius of the bubble/cavity, r is the radial distance from the bubble wall, ρl is the density of liquid, Rɺ is the velocity of bubble wall, ɺɺ is the acceleration, P1 and P 2 are the initial and final pressure during the simuR lation step, respectively, and are given as follows: 3γ

R 2σ 4 µ ɺ − R p1(r =R ) = pv + pgo o  − R R R 

(8.2.3)

4µ ( pɺ ∞(t ) − pɺ 1(r =R ) ) 3ρC 2

(8.2.4)

p2(r =R ) = p1(r =R ) −

where p v is the vapor pressure, pgo is the initial gas pressure, Ro is the initial radius of cavity, s is the surface tension of medium and m is the viscosity of the medium. These detailed models have been used to show that the state of the gas on collapse is strongly influenced by processes such as heat transfer, mass transfer, chemical reactions and nonuniform pressure in the bubble interior. This entire set of physical phenomenon has not been included in the traditional Rayleigh– Plesset formulations. The important works in this area along with the highlights of the work have been depicted in Table 8.2.1. It can be seen from the table that though the earlier works on the numerical investigations of bubble dynamics, concentrated on explaining the links between the bubble motion, its composition and chemical kinetics, none of them was directed at systematically exploring the effect of various operating parameters on the cavity-radius profi les and hence the final cavitation intensity produced at the collapse of the cavities. Also, there are absolutely no design equations quantifying the cavitational intensity as a function of operating parameters. Exact quantification of the cavitational intensity is a must for efficient design of the cavitational reactors and for the optimum selection of the operating parameters. Extensive work has been done in this direction in recent years (Dahnke and Keil, 1998a,b, 1999, Keil and Swamy, 1999, Gogate and Pandit, 2000a; Vichare et al., 2000a;

8.2 Design of Cavitational Reactors Table 8.2.1 Overview of the important works dealing with

bubble dynamics in acoustic cavitation. Models

Important features

Kamat et al. (1993) Naidu et al. (1994)

Separated the chemical kinetics from Rayleigh–Plesset model to predict the production of OH• radicals. Simplified model based on Rayleigh–Plesset equation and chemical kinetics to predict OH• radicals generation rate and the effect of parameters such as reactant concentration, temperature, etc. Accounted for chemical reactions and nonequilibrium phase change in collapsing bubble under single bubble sonoluminescence (SBSL) conditions. Modeled free-radical production in mildly forced bubbles by accounting for nonequilibrium phase change and gas-vapor interdiffusion with Rayleigh– Plesset model. Modeled some trends observed in sonochemistry experiments by coupling chemical reactions with Rayleigh–Plesset equation. Combined chemical reaction with bubble motion as well as nonequilibrium phase change. Considered inclusion of water vapor as a cause of various phenomena observed in cavitation induced bubbles. Accounted for water vapor, chemical reactions, nonequilibrium phase changes, mass and heat transfer as well as radical recombination and used equations of continuity and motion for more realistic picture of bubble. Prediction of the spatiotemporal evolution of cavitation bubbles taking into account interactions between the sound field and bubble distribution. Various phases during the lifetime of the bubble including evolution, migration to the pressure nodes and breakage or evaporation after reaching the air/water interface have been discussed. Application to mono- (low- as well as high- frequency) and multiple-frequency systems have been discussed. Some understanding about distribution of the cavitating and noncavitating zones in the reactor has been developed. The employed model is based on a set of ordinary differential equations and accounts for the bubble dynamics, heat exchange, phase change of water vapor, chemical reactions of the various gaseous species in the air–water system, and diffusion/dissolution of the reaction products in the liquid. Weber number criterion has been incorporated for the stability of the growing bubble.

Yasui (1997a,b)

Sochard et al. (1997) Gong and Hart (1998) Colussi et al. (1998) Moss et al. (1999) Szeri and Storey (2000)

Servant et al. (2000, 2003)

Togel et al. (2002)

Kanthale et al., 2003; Gogate et al., 2003a; Gogate and Pandit, 2004a) and based on the results obtained in the work, we now present some guidelines useful for the efficient design of cavitational reactors. 8.2.2.1 Identification of Stable or Transient Cavitation The radius–time history obtained from the simulations can be very well used to identify whether stable (cavities just oscillate generating small magnitude

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pressure pulses) or transient cavitation (cavities grow to a maximum size and then collapse violently) exits under a given set of operating cavitation (Figs. 8.2.1 and 8.2.2). This can then be used for the selection of operating parameters for the particular application in question. For chemical processing or other highintensity applications, transient cavitation is a must (magnitude of the collapse pressure pulse is much higher) whereas for physical processing applications such as emulsification and cleaning applications, stable cavitation, where continuous small magnitude pressure pulses are generated with each oscillation of the cavity, may be favorable. Typically, higher frequency coupled with lower intensity of irradiation and high initial size of the nuclei results in stable cavitation, and exactly the reverse operating conditions to the above should result in the generation of

Figure 8.2.1 Radius and pressure profiles in the case of transient cavitation (frequency of irradiation = 20 kHz, Intensity of irradiation = 0.12 W/m2 and initial radius of the nuclei = 0.001 mm).

8.2 Design of Cavitational Reactors

Figure 8.2.2 Typical radius and pressure profiles in the case of stable cavitation (frequency of irradiation = 300 kHz, Intensity of irradiation = 5 W/m2 and initial radius of the nuclei = 5 mm).

transient cavitation, though the exact transition is difficult to predict as this is a stochastic phenomena, i.e. both types of cavitation occur simultaneously and only the percentage differs from equipment to equipment and the operating conditions. 8.2.2.2 Effect of Compressibility of Medium Another flaw of the earlier investigations on the bubble dynamics was that al though there has been ample discussion on whether the compressibility of the cavitating medium plays an important role on the cavity motion, there have been no illustrations, where the difference between the predictions of equations considering the compressibility of the medium (e.g. Gilmore model [Gilmore, 1954]

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or the equation of Tomita and Shima [Tomita and Shima, 1986]) and the conventional Rayleigh Plesset equation [Plesset, 1949] that does not consider the compressibility of the cavitating medium, have been quantified as a function of operating parameters. The variation of the cavity size and the pressure generated at the collapse indeed depends on whether the compressibility of medium is considered, but the exact difference in the values may also depend on the range of the operating parameters. The predictions of the Rayleigh–Plesset equation [Plesset, 1949] and the equation given by Tomita and Shima (1986), solved using rigorous numerical simulations (details of the simulations including the set of assumptions, the solution methodology and the collapse criteria have been given in the earlier work; Gogate and Pandit, 2000a), clearly indicated that there is a substantial difference in the value of the maximum bubble size (indicative of the quantum of free radicals as well as the collapse pressure generated) obtained in the two cases beyond a certain critical ultrasonic intensity, which is dependent on the operating conditions (Fig. 8.2.3). Below the critical ultrasonic intensity, the predictions of the equation of Tomita and Shima are marginally lower as compared to the Rayleigh–Plesset equation and hence up to this point, it can be said that simplification of not considering the compressibility of the cavitating medium does not much affect the design of sonochemical reactors. However at larger intensities of operation, the simplification results in a significant underprediction of the cavitating conditions and will lead to a subsequent overdesign of the sonochemical reactor. For large-scale operations, usually higher intensities of irradiation are required (there has to be a certain minimum power input per unit volume of liquid and hence for large volumes, large power dissipation is a must) and hence consideration of the compressibility of the medium is a critical factor in the effective and optimum design of the reactors.

Figure 8.2.3 Dependence of the maximum size reached by the cavity on compressibility of the liquid medium.

8.2 Design of Cavitational Reactors

235

8.2.2.3 Optimization of Operating Parameters The magnitudes of collapse pressures and temperatures as well as the number of free radicals generated at the end of cavitation events are strongly dependent on the operating parameters of the equipment, namely, intensity and frequency of irradiation along with the geometrical arrangement of the transducers and the liquid-phase physicochemical properties, which affect the initial size of the nuclei and the nucleation process. Based on the bubble-dynamics analysis, important considerations regarding the selection of operating parameters have been presented in Table 8.2.2 whereas the details about the effect of these parameters can be obtained from the earlier work (Gogate and Pandit, 2000a; 2004a, Dahnke and Keil, 1998a,b, 1999, Keil and Swamy, 1999). This study indicates the ways and means to manipulate cavitating conditions for maximum effect. 8.2.2.4 Development of Design Equations Collapse Pressure The intensity of irradiation, frequency of irradiation and the initial size of the nuclei (fi xed by the vapor pressure of the cavitating media and the dissolved gases) significantly affect the magnitude of the pressure pulse generated at the collapse of the cavity. To develop the design equation, for the Table 8.2.2 Optimum operating conditions for the sonochemical reactors.

No.

Property

Affects

Favorable Conditions

1.

Intensity of irradiation (Range: 1 to 300 W/cm2 )

Number of cavities, collapse pressure of single cavity

2.

Frequency of irradiation (Range: 20 to 200 kHz)

3.

Liquid vapor pressure (Range: 40 to 100 mm of Hg at 30 ºC) Viscosity (Range: 1 to 6 cP) Surface tension (Range: 0.03 to 0.072 N/m) Bulk liquid temperature (Range: 30 to 70 ºC)

Collapse time of the cavity as well as final pressure/ temperature pulse Cavitation threshold, Intensity of cavitation, rate of chemical reaction. Transient threshold Size of the nuclei (cavitation threshold) Intensity of collapse, rate of the reaction, threshold/ nucleation, almost all physical properties.

Use power dissipation till an optimum value and over a wider area of irradiation Use enhanced frequencies till an optimum value

4. 5. 6.

7.

Dissolved gas A. Solubility B. Polytropic constant and thermal conductivity

Gas content, nucleation, collapse phase Intensity of cavitation events.

Liquids with low vapor pressures Low viscosity Low surface tension Optimum value exits, generally lower temperatures are preferable

Low solubility Gases with higher polytropic constant and lower thermal conductivity (monoatomic gases)

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prediction of the cavitational intensity expressed in terms of the collapse pressure, consideration of these important parameters is essential. The developed equation for the collapse pressure (Gogate and Pandit, 2000a) can be given as follows: Pcollapse = 114(Ro)−1.88(I )−0.17( f )0.11

(8.2.5)

The above correlation uses the initial cavity size in mm, frequency in kHz and intensity of ultrasound in W/cm2, while the collapse pressure is given in atmospheres and is developed for the following range of parameters: Initial cavity size = 0.01 to 0.5 mm, frequency of ultrasound = 20–200 kHz and intensity of ultrasound = 1–300 W/cm2 and when the final collapse radius is 5% of the initial radius, the cavity collapse is considered to be complete. The conditions at the collapse have been used to estimate the extent of compression and the collapse pressure as well as temperatures. The developed correlation is simple to use and valid over the range of parameters that are commonly used in the sonochemical applications. Moreover the development of correlation is based on the numerical results obtained from the solution of the Rayleigh–Plesset and Tomita–Shima equations and considers the effect of compressibility of the medium. Such a correlation will be helpful in the design of sonochemical reactors and will predict the magnitude of the pressure that will be generated following the collapse of the cavity irrespective of the type of the reaction for which the reactor is being used. The cavitational yield for the reaction will, however, be different depending upon the type of the reaction considered. A caution that needs to be raised is that the above equation can be used to relatively evaluate different scenario rather than the absolute values as the 5% final volume of the cavity at the collapse is chosen somewhat arbitrarily. We now discuss different approaches that can be taken to link the primary effect (collapse pressures) with the secondary effects (yield). Cavitational Yield The cavitational yield for the reactor indicates the ability of the equipment to produce the desired change based on the electric energy used for generating cavitation. The Weissler reaction was used for the quantification of the chemical changes. The different operating conditions used in the experimental studies (Gogate et al., 2001), i.e. operating intensity and frequency of irradiation, temperature, physicochemical properties of the liquid medium under the given conditions and an assumed initial size of the nuclei were then used in the numerical simulations for the exact quantification of the collapse pressure pulse generated. Quantitative correlation was obtained for the variation in the cavitational yield with collapse pressure and can be given as:

Cavitational yield = 1.931 × 10 −14(Collapse pressure) − 6.152 × 10 −7

(8.2.6)

The above equation uses the collapse pressure in units of atmospheres, while the cavitational yield is predicted in terms of g/J. The proportionality constant for the equation depends on the concentration of KI and the time of the treatment. It can

8.2 Design of Cavitational Reactors

also be seen from the above equation that a certain minimum intensity of cavitation (minimum collapse pressure) is required for the onset of the effects of cavitation in terms of the release of a measurable quantity of iodine. It should also be noted that the magnitude of the minimum intensity of cavitation will depend on the type of application in question and should be established with laboratory-scale studies unless data is available in the literature with a similar set of operating conditions as well as the geometry of the reactor. It is worthwhile at this stage to compare the predictions of the cavitational yield from the obtained relationship with some of the earlier studies on decomposition of potassium iodide. Naidu et al. (1994) have studied the decomposition of potassium iodide in an ultrasonic bath having a cross-sectional area of 0.0404 m2 and a height of 15 cm. The operating parameters used in the experimentation were 25 kHz driving frequency and 0.6188 W/cm2 intensity. The results obtained indicate a linear variation in the iodine liberation with the initial concentration of potassium iodide and time of the reaction, which confirms the consideration of inclusion of the concentration of the reactant (i.e. KI in this case) and the reaction time in the proportionality constant. Naidu et al. (1994) obtained an iodine liberation of 4.15 × 10 −4 g/l for a reaction time of 10 min and initial KI concentration of 5%. The correlation developed in the present work predicts an iodine liberation of 1.79 × 10 −4 g/l, which is two to three times lower than that predicted by Naidu et al. (1994). The variation in the obtained values can be attributed to the following two reasons: 1. The system used by Naidu et al. (1994) is a KI solution saturated with air. The rates of reaction are much larger when the system is saturated with the gas due to the availability of gas-bubble nuclei for easy cavitation and it decreases with the degassing effect (Senthilkumar et al. 2000). Vichare et al., (2000b) have also shown that the degradation rates are much higher (2- to 3-fold) in the initial stages when the system is completely saturated with gas and as deaeration occurs, the system becomes stable giving comparatively much lower rates of reaction and linear variation with time of operation has been observed. The work to develop the correlation considers a completely degassed system and hence will give somewhat lower rates of degradation as compared to the saturated system. 2. The development of the correlation is based on the global rates of reaction, i.e. considering the entire available volume for the reaction in the equipment, whereas Naidu et al. (1994) placed the test tube containing the reacting medium at a particular location where there was a maximum intensity of cavitation as detected by the hydrophones. The intensities of cavitation are different at different points in the ultrasonic reactor, which will result into different rates of reaction. The mapping studies

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indicate 2- to 10-fold variation in the observed decomposition rates (Gogate et al., 2002) with the location of the test tube used for experimentation and thus the above difference between our studies and those of Naidu et al. (1994) can easily be explained. A detailed discussion about the distribution of the cavitational activity in the sonochemical reactors follows later. Nevertheless, the order of magnitude appears to be similar and gives some confidence in the authenticity of the developed design correlation. There is always an uncertainty associated with the exact quantification of the collapse pressure generated. Use of the Rayleigh–Plesset equation will dictate the termination condition as bubble-wall velocity exceeding the velocity of sound in the medium, whereas for the case of equations considering the compressibility of liquid, new termination criteria will have to be considered. Thus, for some other conditions with better computational facilities, a different collapse criteria (cavity size lower than 1 or even 0.1% of the initial size) may look feasible. Another collapse criteria based on Vander Wall’s equation of state has also been considered [Gastagar, 2004]. The criteria considers that the cavity is assumed to be collapsed when the volume occupied by the cavity is equal to the material volume given by the product of the constant “b” in the Vander Wall’s equation of state and the number of moles. The exact predictions of the collapse pressure pulse is always a matter of debate, nevertheless, a new proportionality constant to avoid this uncertainty can always be developed based on the relative rates of the reaction. The maximum size of the cavity reached during the expansion phase also depends on the operating parameters. Moreover, it can be taken as the indicator of the volume of the cavity at the start of the collapse phase, giving an indirect indication of the number of free radicals that are likely to be generated at the end of the collapse phase. Thus, the maximum size reached by the cavitating bubble is a useful correlating factor for the design of the cavitational yield. It also has a limitation; the time of the collapse is not considered in the design equation for collapse pressure/cavitational yield and will affect the net magnitude of the collapse pressure/temperature. Thus, for reactions where pyrolysis is the controlling mechanism or for the applications controlled by the physical effects of cavitation, the design equation in terms of collapse pressure or oscillating pressure pulse magnitude looks to be more comprehensive. The data set obtained with the local measurements of cavitational yield for the Weissler reaction (Gogate et al., 2002) in the case of an ultrasonic bath can be effectively used for the development of the design equation for cavitational yield in terms of the maximum size of the cavity (recommended when free-radical attack is the controlling mechanism). The mathematical equation relating the two can be given as follows:

8.2 Design of Cavitational Reactors

Cavitational yield = 8.562 × 10 −6(R max Ro )3 − 3.847 × 10 −4

(8.2.7)

The equation is valid for acoustic pressure amplitude in the range of 2 to 7 atm and an operating frequency of 22 kHz. More details about the equation in terms of measurement of cavitational yield and calculation of the R max/Ro ratio from numerical simulations have been given in the earlier work of Gogate et al. (2002). The constants obtained in the equation depend on the type of equipment and the operating parameters. This is because (R max/Ro)3 gives the indication of the extent of free radicals generated theoretically as well as the volume of the reactor getting affected by the cavitation. All the free radicals generated in the bubble may not be available for the chemical reaction; this depends on the type of mechanism of the destruction of the bubble. In the case of multibubble situations, if the bubbles are unstable on collapse and break apart, in some way dispersing their total OH• content into the liquid (Colussi et al., 1998), peak amount of radicals will be available for the sonochemical reactions. However, if the broken-up bubbles are stable and the OH• radicals enter the liquid by uptake across the interface, the amount of radicals available will be substantially lower. Storey and Szeri (2000) have indicated an uptake coefficient of 0.001, which means that only 0.1% of the total radicals formed due to the compression and collapse of the cavitating bubbles actually take part in the chemical reactions. The type of bubble collapse will necessarily depend on the equipment used for generation of the cavitation and also on the operating conditions. Takami et al. (1998) have also highlighted the concept of uptake coefficient for the free radicals. It should be noted that the developed correlations, unique of its kind are only trend setting and by no means generalized. There is ample scope for further work, which may include establishing the validity of the equations over a wider range of operating parameters (with experiments in different reactors with varying power dissipation and/or frequency of irradiation), exactly quantifying the number of free radicals generated during the collapse (more importantly estimating the actual number taking part in the reaction, by considering the number of free radicals generated, time of collapse of the cavities and the lifetime of the free radical; also the relative rates of reaction of free radicals with the reactants and recombination reaction of free radicals), estimating the size of the nuclei/cavity that is dependent on the type of equipment used for the generation of cavitation (using techniques such as phase Doppler [Tsochatzidis et al., 2001] as well as a theoretical approach based on the thermodynamic analysis considering the lattice structure of the cavitating medium and the intermolecular distance of the cavitating medium), measuring the total bubble activity/transient bubble activity in different grades of violence, i.e. cavitational activity, combining the effects of collapse pressure generated and the maximum bubble size reached, etc. There is indeed a need for the development of these engineering equations for different systems and the present work forms a start in that direction.

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8.2.3 Distribution of the Cavitational Activity in the Reactors

Distribution of the cavitational activity in the sonochemical reactors is a very important issue considering the net overall effects in terms of yields of chemical reactions. The cavitational activity is not uniformly distributed in all the conventional designs of the sonochemical reactors and is mostly concentrated nearer the transducers. It is very important to consider the distribution of cavitational activity in the design of industrial-scale cavitational reactors and the optimum design should be such that near-uniform activity at all the locations is obtained. In the past, attempts have been made to understand the uneven distribution of the cavitational activity existing in the reactor, both using experimental techniques as well as numerical simulations. Gogate et al., (2002) have summarized these findings in terms of the technique used, type of the reactor considered and the important results as obtained in the studies. In general, these attempts have tried to identify the primary (measurements in terms of pressure amplitudes or the local temperatures) or the secondary effects (either physical (measurement of mass transfer coefficients) or chemical (sonoluminiscence and Weissler reactions) of the propagation of the wave in the medium either based on experimental work or rigorous theoretical analysis. Following important considerations for developing an effective scaleup strategy can be highlighted based on the detailed analysis of the findings presented in the work of Gogate et al., (2002): 1. There exists a wide distribution of cavitational activity in the sonoreactor with the maximum intensity observed just above the center of the transducer (for the standard arrangements such as ultrasonic horn/bath). The intensity varies both axially as well as in the radial direction with decreasing trends as we go away from the transducer (in the axial direction) and away from the axis passing through the center of the transducer (in the radial direction). The intensity has been found to increase by a small amount at distances corresponding to l /2 in the axial direction, where l is the wavelength of the driving sound source (Pugin, 1987, Romdhane et al., 1995a, Romdhane et al., 1997, Dahlem et al., 1998, 1999). 2. Immersion-type transducers are the poorest, when scaleup possibilities are considered, though very high intensities (pressures of the order of a few thousand atmospheres) are observed very near to the horn. The intensity decreases exponentially as one moves away from the horn and vanishes at a distance of as low as 1 cm away as observed by Chivate and Pandit (1995). It should be also noted that the active zone in the case of an ultrasonic horn also depends on the maximum power input to the equipment

8.2 Design of Cavitational Reactors

and also on the operating frequency. Romdhane et al. (1995a) have shown that the ultrasonic activity falls from a maximum of 120 ºC temperature difference measured with a thermal sensor to less than 10 ºC within a distance of 1 cm for the ultrasonic horn with operating frequency 20 kHz and Pmax = 160 W, whereas for another horn with operating frequency 40 kHz and Pmax = 600 W, the distance required is around 2 cm for the same overall decrease in intensity. Thus, using higher power dissipation does not necessarily result in directly increasing the active cavitation volume (an approximately 4-times increase in power dissipation results in only a 2-times increase in the active volume). Furthermore, the ultrasonic activity in the radial plane passing through the tip of the horn diminishes at a distance of 3 cm away from the axis through the center of the vibrating area. Thus, the active zone is restricted to a small zone around the irradiating surface. Dahlem et al. (1998, 1999) have found the local velocities due to acoustic streaming existing near the surface of ultrasonic horn using experimental measurements using the PIV technique and also with the help of numerical models (Dahlem et al., 1999). They have reported that the active zone is restricted to near the surface only and very low velocities are obtained at the bottom of the reactor as well as near the wall of reactor in the radial direction. Recently, Ajaykumar et al. (2006a) measured the mean and fluctuating fluid velocities using LDA and reported that the circulation velocities (mean) are dominant in the zone nearer to the source of energy and are substantially lower at the walls and at the bottom of the reactor. 3. In the case of an ultrasonic bath, when the bottom of the reactor is irradiated with a single transducer, the active zone is restricted to a vertical plane just above the transducer with maximum intensity at the center line of transducer. A recent study of Balasubrahmanyam and Pandit (2006) has clearly demonstrated different zones of cavitation at different planes away from the transducers with the help of mapping the erosion patterns of aluminum foil under the influence of ultrasonic irradiations. The area of the irradiating surface should be increased so as to get better distribution/dissipation of energy in the reactor. This has a twin advantage in terms of the decreased ultrasonic intensity (defined as power dissipation per unit area of the irradiating surface), which will increase the magnitude of the pressure pulse

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generated at the end of the cavitation events (Gogate and Pandit, 2000a). In the earlier work (Gogate et al., 2001), it has also been shown using decomposition of KI as the model reaction, that for similar operating frequencies, an ultrasonic bath gives better cavitational yields (defined as iodine liberation per unit power density [watts/m3 ] of the equipment) as compared to an ultrasonic horn. Mujumdar et al. (1997) have also obtained similar results with emulsification reactions. Dahlem et al. (1998, 1999) have shown better local ultrasonic intensities and iodineliberation rates for the radially vibrating horn (1000 W dissipated through an area of 365 cm2 ) as compared to a conventional horn (longitudinal vibrations, 300 W dissipated through an area of 0.8 cm2 ). 4. To increase the active zones existing in the reactor, one can easily modify the position of the transducers (if multiple transducers have been used, which is likely to be the case at large-scale operation due to the fact that it is quite difficult to successfully operate a single transducer with very high power and frequency due to limitations in the material of construction of the transducers) so that the wave patterns generated by the individual transducers will overlap, also resulting in uniform and increased cavitational activity. Arrangements such as triangular pitch in the case of ultrasonic bath (Soudagar and Samant, 1995, Gogate et al., 2001), tubular reactors with two ends either irradiated with transducers or one end with a transducer and the other with a reflector (Gonze et al., 1998), parallelplate reactors with each plate irradiated with either the same or different frequencies (Thoma et al., 1997, Gogate et al., 2001, 2003b) and transducers on each side of a hexagon (Romdhane et al., 1995a) can be constructed. It is of the utmost importance to have a uniform distribution of the ultrasonic activity in order to get increased cavitational effects. Romdhane et al. (1995a) have compared different ultrasonic equipments and have shown that a hexagonal reactor with a single transducer at each face of the hexagon gives better homogeneity as compared to conventional ultrasonic horn and parallelepiped geometry. Thus, the need for the future is to develop newer designs giving uniform and enhanced ultrasonic activity over the entire region of the cavitational reactor. One such design will be discussed in detail later. 5. The work of Keil and coworkers (Keil and Dahnke, 1996, Keil and Dahnke, 1997a, 1997b, Dahnke and Keil, 1998a,

8.2 Design of Cavitational Reactors

1998b, 1999, Dahnke et al., 1999a,b) appears to be pioneering in terms of simulations of the pressure fields existing in the reactor. The main features of the work have been given in Table 8.2.3. Such a detailed analysis can be used to identify the regions with maximum pressure fields in a large-scale reactor and then may be small reactors can be placed strategically at these locations in order to get maximum benefits. It might happen that the threshold required for certain transformation application is obtained at these locations but if considered globally these effects will be marginalized, resulting in much lower yields based on the total volume of the cavitation reactors. Thus, the location of the transducers on the irradiating surface and the location of microreactors will also depend on the type of application, which decides the required cavitational intensity. It should be noted that one-to-one correspondence between the simulated pressure fields and experimental reactions must be established before reaching any firm conclusions. The work done by Keil and coworkers can be taken as a starting point for obtaining a clearer picture of the cavitation phenomena. The experimental verification of the obtained simulated pressure fields and its effects on the physical and chemical effects of ultrasound needs to be established in the future. Simple techniques that can be used for the said measurements can be thermochemical/electrochemical probes (Contamine et al., 1994, Trabelsi et al., 1996), thermoelectrical probes (Romdhane et al., 1995b, Romdhane et al., 1997 and Faid et al., 1998), thermistor probes (Martin and Law, 1980), thermocouples (Gonze et al., 1998), sensors based on the piezoelectric effect (Soudagar and Samant, 1995, Dahlem et al., 1998, 1999, Moholkar et al., 2000a), colorimetric measurements (Ratoarinoro et al., 1995), chemical reactions, such as fading of color intensity of phenolphthalein (Rong et al., 2001), sonoluminiscence of luminol (Gonze et al., 1998), decomposition of KI (Weissler, 1950, Contamine et al., 1994, Faid et al., 1998, Dahlem et al., 1998, 1999), PIV measurements for the local velocity due to acoustic streaming (Dahlem et al., 1998, 1999), Michael reaction (Contamine et al., 1994), etc. Faid et al. (1998) have compared different sensors that can be used for the measurements of local cavitational effects, namely electrochemical, thermo-electrical and chemical probes and have showed that the results of cavitational

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distribution are the same irrespective of the type of sensor used for the measurement. 6. Use of multiple frequencies results in a relatively better distribution of cavitational activity and provides a better scaleup option. As pointed out by Dahnke and Keil (1998a), for obtaining better distribution of energy one has to modify the sound source in terms of number and properties and in this context multiple frequencies offer even enhanced flexibility. Moholkar et al. (1999, 2000b) have also shown that by adjusting the phase difference and magnitude of two waves, transient cavitation can be obtained at a larger number of reactor sites, which results in better overall cavitational activity (based on the total reactor volume). Moreover, due to the use of multiple frequencies the violent collapse of the cavities is restricted near the transducer surface resulting in decreased erosion of the surface (Moholkar et al., 2000b), which was another major problem in the scaleup as outlined earlier. In the earlier works, it has also been shown using different chemical reactions (decomposition of KI [Gogate et al., 2001], destruction of p-Nitrophenol [Sivakumar et al., 2002b], destruction of Rhodamine B, a typical effluent in the effluent from the dye industry [Sivakumar and Pandit, 2001b], degradation of formic acid [Gogate et al., 2003b]) that the reactors based on multiple frequencies give better cavitational yields as compared to single-frequency reactors. Moreover, Tatake and Pandit (2002) have also indicated, with the help of numerical simulations of bubble dynamics, using the Rayleigh–Plesset equation, that the collapse of cavities is more violent for the case of multiplefrequency systems as compared to single operating frequency of the same or double the intensity (similar total power dissipation). 7. Dahnke and Keil (1998a) have also shown that merely increasing the acoustic power input to the reactor or increasing the transducer frequency cannot enhance/ improve the distribution of the cavitational activity. Romdhane et al. (1997) have also reported that the local ultrasonic intensity as measured with the help of thermoelectric probes remains unchanged with increasing the power input (in the range 12.5 to 39 W). Thus, increasing the power input to the reactor in order to achieve higher cavitational yields certainly cannot be a good scaleup strategy. Confirmation can also be obtained from the experimental evidences (Couppis and Klinzing,

8.2 Design of Cavitational Reactors

1974, Gutierrez and Henglein, 1990, Ratoarinoro et al., 1995 and Ondruschka et al., 2000) that indicate that an optimum power input is always required for a particular reaction beyond which the beneficial effects are not observed. The detailed discussion about the optimum intensity has already been presented in the earlier work of Gogate et al., 2001. The presence of optimum intensity has been attributed to the decoupling effect that restricts the effective propagation of the incident sound waves into the system thereby decreasing the total available energy for the cavitation events. 8. The local ultrasonic activity is also dependent on the agitation, presence of solids (size and volume fraction) or the flow regimes used in the circulating type of reactors (liquid is being pumped at a definite rate that passes through a zone of ultrasonic irradiation and back to the storage tank). In a recent study carried out in this department, a dual-frequency flow cell was operated in a continuous loop and in combination with the hydrodynamic-cavitation reactor (orifice plate setup; for details refer to the earlier study [Gogate et al., 2001]) with an aim to increase the active cavitational volume in the reactor and examine the synergism between the two different types of cavitation. The cavitational yield due to the combined operation of the dual-frequency flow cell and hydrodynamic-cavitation setup depends not only on the individual contributions but also on the synergism between the two. Preliminary results indicate that the rates of iodine liberation at higher recirculation flow rates were lower as compared to the operation with lower flow rates, indicating that the contribution of the ultrasonic irradiation to the overall cavitational effect at higher flow rates is less, possibly due to disturbance of the sound field created by ultrasound in the reactor by the flow. This fact is further confirmed with the observation of higher cavitational yields at increased flow rates for the case of the hydrodynamic-cavitation setup alone (Vichare et al., 2000). Romdhane et al. (1997) have also shown that the local ultrasonic activity decreases with agitation and also with an increase in the Reynolds number (defined as = d × v × r / m , where r is the density of liquid, m is the viscosity, v is the recirculation velocity and d the inner diameter of the equipment used). Moreover, the peaks observed at distances of l /2 in the case of stagnant liquid medium (standing-wave pattern) diminishes in the case of agitated

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medium, thereby confirming the disturbance caused in the propagation of the sound wave in a specified direction. Similar results were obtained in another setup where the liquid was recirculated through the irradiation zone (Romdhane et al., 1997). To give a quantitative idea, an increase in the Reynolds number from 219 to 10 656 decreases the measured temperature difference from 121.5 ºC to 7.6 ºC, indicating that there is an almost linear effect of the Reynolds number on the ultrasonic activity. A quick analysis of the data given in their work indicates a temperature difference of around 127 ºC if the fluid was stagnant (temperature difference = −0.01 × Reynolds number + 127 with R 2 the value for the equation fitting as 0.95). It must be noted that the mathematical relationship just gives an indication of the effect of recirculation flowrate on the cavitational activity and may not be linear in all the cases. The effect of flow regime can also be linked to the standing-wave pattern existing in the reactor due to reflection of sound waves, which also contribute to the enhancement of cavitational activity. At higher operational flow regimes, standing waves will necessarily be disturbed or destroyed. However, the exact effect of the flow regime on the propagation of the sound wave and sound field existing in the reactor is far from well understood and quantified, still it can be said that to obtain enhanced benefits from the continuous operation of sonochemical reactors, one should operate with flow rates as low as possible. It might be worthwhile to do a computational fluid-dynamics study to understand the changes in the pressure-field distribution due to the presence of flow. Nevertheless, this is a very important point that needs to be considered particularly for applications involving multiple phases such as chemical reactions and operations involving solid–liquid extraction, etc. On the one hand, where the increased agitation or Reynolds number results in less cavitational activity, it also results in better mixing of the two phases, which might enhance the rates. The same trend was observed with the size and the volume fraction of the solid particles present. On the one hand the larger particles were observed to attenuate the wave propagation to a lower extent, whereas smaller size particles give better interfacial areas for chemical reactions as well as extraction. Also, an increase in the volume fraction results in lower local ultrasonic activity but may

8.2 Design of Cavitational Reactors

have a positive effect on the cavitational activity (additional surface and hence a greater number of nuclei for cavitation phenomena) and also on chemical reactions where solid particles used are acting as catalyst for the reaction. Thus, optimization needs to be done considering the two opposing effects. 9. For better scaleup possibilities with the ultrasonic reactor it is also important to design new reactors, which can be operated in a continuous mode. The traditional immersion or ultrasonic bath-type reactors cannot be operated in a continuous mode with high cavitational yields. Romdhane et al. (1995a) and Faid et al. (1998) have described a sonitube reactor that can be operated in a continuous manner. Mapping of this reactor showed the presence of some cavitational activity even beyond the zones under direct influence of the transducers. Moreover, as compared to various reactors studied in their work (ultrasonic horn, ultrasonic cleaner (bath), hexagonal reactor), sonitube gives a better homogeneity in terms of the cavitational activity; negligible variation in the radial direction has been reported. The dual- and triple-frequency flow cell designed in this department can also be operated in a continuous manner, which forms a part of the future investigation. In the case of continuously operated sonochemical reactors, selection of operating flow rates is a crucial factor, particularly for the multiphase systems as discussed earlier. The design used by Dahlem et al. (1998, 1999) also needs a special mention here. A telsonic horn, which has radial vibrations as against conventional longitudinal vibrations for the immersion system gives dual advantages of higher irradiating surface (lower intensity of irradiation resulting in better yields) coupled with good distribution of the energy in the radial direction. Moreover, even if the horn is radially vibrating, local measurements just below the horn also give high cavitational activity, which will be again more beneficial in enhancing the global sonochemical yields. Similar results in terms of superiority of the radially vibrating horn have been reported by Bhirud et al. (2004) and Ajaykumar et al. (2006b). 8.2.4 Intensification of Cavitational Activity in the Sonochemical Reactors

At times, the net rates of chemical/physical processing achieved using ultrasonic irradiations are not sufficient so as to prompt towards industrial-scale operation of sonochemical reactors. This is even more important due to the possibility of even more uneven distribution of the cavitational activity in the large-scale

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Table 8.2.3 Details regarding the work of Keil and coworkers

[Keil and Dahnke, 1996, Keil and Dahnke, 1997a, 1997b, Dahnke and Keil, 1998a, 1998b, 1999, Dahnke et al., 1999a,b] in the area of prediction of the cavitational intensity in sonochemical reactors. Details about equipment and measurement

Important findings

Numerical simulations of the local pressure fields using different modeling approaches such as Helmholtz and Kirchoff integral equation with homogenous and inhomogeneous distribution of cavitation bubbles, consideration of effects of small amplitude sinusoidal oscillations of the gas bubbles present in the liquid.

Cavitational activity was found to be maximum at the center axis of the transducer and diminishing away from the centerline in the same plane and also as one moves away from the transducers in the direction perpendicular to the tank bottom. For the case of a homogenous distribution of the bubbles, the influence on the pressure fields was negligible for a bubble fraction < 10 −3, whereas above this value a remarkable reduction in the pressure amplitude was observed. For reactors of type 1 and 3, intensity at the opposite end nearly vanishes whereas considerable attenuation was observed for the reactor 2. For very high volume fractions of gas/vapor (0.2), the intensity vanishes even near the transducer surface.

Different sonochemical reactors: one with simple cylindrical geometry with circular ultrasonic transducer at the bottom (1), second a circular tube with open ends with irradiation with three ultrasonic horns placed in an equilateral triangle in same plane around the tube (2) and a third one where an immersion transducer is used in a cylindrical vessel (3). Two different irradiating frequencies, 25 and 50 kHz and assumption of pure liquid have been considered in simulations.

For an inhomogeneous distribution of the bubbles, which assumes a very high bubble concentration near the ultrasound source and lower concentration away from the source, the damping effect decreases considerably as compared to a homogenous distribution. For a volume fraction of 0.2, the acoustic field keeps its original structure in the vicinity of the beam source, decreases to a low value and then an undamped propagation of intensity is observed due to the low concentration away from source. Thus, a decrease in intensity is only observed very close to the surface of the transducer. An important characteristic of the findings that can be highlighted is that the active cavitation region is not a simple function of the transducer frequency or surface amplitude. Thus optimization needs to be done in terms of the number and the properties of sound sources and properties of the liquid medium. The trends in the pressure fields existing in the reactor obtained are quite similar to the earlier experimental results of Soudagar and Samant (1995) although quantitative agreement is not seen. The presence of a PPIMP transducer alters the sound field in its surroundings and also additional standing waves are formed between the probe surface and the bottom of the reactor. This influence has not been considered in the theoretical simulations, which is also based on a number of approximations leading to inaccuracies.

reactors as discussed earlier. It is thus important to look into supplementary strategies with an aim of intensification of the cavitational intensity. Two types of operating strategies can be recommended depending on the type of applications:

8.2 Design of Cavitational Reactors

1. Use of process-intensifying parameters such as the presence of dissolved gases and/or continuous sparging of gases such as air, ozone and argon (to a limited extent), the presence of salts such as NaCl, NaNO2 and NaNO3, the presence of solid particles such as TiO2, CuO and MnO2 that can also act as a catalyst in some cases. 2. Use of a combination of cavitation and advanced oxidation processes such as ozonation, chemical oxidation using hydrogen peroxide and photocatalytic oxidation. 8.2.4.1 Use of Process-intensifying Parameters The presence of gases and solid particles mainly provide additional nuclei for the cavitation phenomena and hence the number of cavitation events occurring in the reactor is enhanced resulting in a subsequent enhancement in the cavitational activity and hence the net chemical/physical effects. It must be noted that the presence of both these parameters also has a negative effect on the cavitational activity (aeration gives a cushioning effect and incomplete collapse resulting in a decrease in the collapse pressure, whereas solid particles result in scattering of the sound waves thereby decreasing the focused energy transferred into the system). The net effect of these two phenomena will be dependent on the system in question and hence optimization is a must before operating parameters are selected for actual operation. The earlier work of Gogate et al. (2006) is recommended in this case to obtain an idea about the methodology to be used for the optimization of operating parameters. The readers are also requested to refer to earlier literature illustrations (Entezari et al., 1997; Seymore and Gupta, 1997; Kang et al., 1999; Krugar et al., 1999 Wakeford et al., 1999; Nagata et al., 2000) for a greater insight into the effect of the presence of gases on the cavitational applications. Considering the specific application of chemical synthesis, the presence of solid catalyst (particles/salts in a typical concentration range of 1 to 10% by weight of the reactants; optimization is recommended in the majority of the cases using laboratory-scale studies) in the sonochemical reactors results in intensification due to the following mechanisms; 1. Formation of increased cavitation nuclei due to a higher number of discontinuities in the liquid continuum as a result of the presence of particles to give a larger number of collapse events resulting in an increase in the number of free radicals. 2. In a biphasic solid–liquid medium irradiated by ultrasound power, major mechanical effects are the reduction of particles size, leading to an increased surface area, and the formation of liquid jets at solid surfaces by the asymmetrical inrush of the fluid into the collapsing voids. These liquid jets not only provide surface cleaning but also induce pitting and surface activation effects and increase

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3. 4. 5.

6.

the rate of phase mixing, mass transfer and catalyst activation. Enhanced generation of free radicals due to some catalysts such as FeSO4 or elemental iron. Better distribution of the organic pollutants increasing the concentration at reaction sites. Alteration of physical properties (vapor pressure, surface tension) facilitating generation of cavities and also resulting in more violent collapse of the cavities. The presence of salts might also result in preferential accumulation of the reactants at the site of cavity collapse, thereby resulting in an intensification of the cavitational reactions (Seymore and Gupta, 1997).

8.2.4.2 Use of Combination of Cavitation and Advanced Oxidation Processes Intensification can be achieved using this approach of combination of cavitation and advanced oxidation process only for chemical-synthesis applications where free-radical attack is the governing mechanism. For reactions governed by a pyrolysis-type mechanism, use of process-intensifying parameters that result in an overall increase in the cavitational intensity such as solid particles, sparging of gases, etc., is recommended. Usually, the use of hydrogen peroxide in conjunction with ultrasound is beneficial only up to an optimum loading (Chen et al., 1990, Chemat et al., 2001, Teo et al., 2001). The optimum value will be dependent on the nature of the chemical reactions and the operating conditions in terms of power density/operating frequency (these fi x the rate of generation of the free radicals) and laboratory-scale studies are essential to establish this optimum for the specific application in question. Literature reports may not necessarily give correct solutions (for optimum concentration) even if matching is done with respect to the primary components as it is the auxiliary components of the streams (most notably radical scavengers such as bicarbonate, carbonate ions, t-butanol, naturally occurring material, humic acids) that are crucial factors in deciding the effectiveness of the free-radical attack. The synergistic effect of combining ozonation with ultrasonic irradiation is observed only when the free-radical attack is the controlling mechanism and the rate of generation of free radicals due to ultrasonic action alone is somewhat lower (at lower frequencies of operation and power dissipation levels). At higher frequencies of operation, an optimum exists for the ozone concentration and it should be established by laboratory-scale studies for the application under question. The effect of the presence of radical scavengers such as bicarbonate, chloride, sulfate ions, naturally occurring material will be again dependent on the specific reaction in question. For volatile materials, such as chloroform, MTBE, the reactions takes place in the cavitating bubble and hence the effect will not be felt. But for the majority of the other reactants, the reaction takes place usually in the liquid bulk or at the gas/liquid interface and hence the decrease in the rates of synthesis processes will be significant. Acid pretreatment (only with a weak

8.2 Design of Cavitational Reactors

acid for the selective removal of ionic radical scavengers; also other types of treatment may be required if some of the nonionic radical scavengers are present even under acidic conditions) along with sparging with neutral gas can be done for increasing the reactivity. It must also be noted that a large concentration of the weak acid will be required for achieving similar acidic conditions as compared to the stronger acid but if the selection of the weak acid is made in such a way that radical scavenging action of the dissociated acid is not present (strong acid counterions, i.e. Cl−, SO4 −2 indeed have a scavenging action of the hydroxyl radicals) a large number of free radicals will be present in the system available for the specific application. Thus, the use of a suitable weak acid is justified for the pretreatment. The extensive work done by the group of Hoffmann (Kang and Hoffmann, 1998; Kang et al., 1999; Weavers et al., 1998, 2000), although in the area of wastewater treatment, is recommended for understanding more details about using a combination of ozone and cavitation synergistically. The synergistic effects of combining photocatalytic oxidation with cavitation can be possibly attributed to: 1. Cavitational effects leading to an increase in the temperatures and pressure at the localized microvoid cavity implosion sites. 2. Cleaning and sweeping of the TiO2 surface due to acoustic microstreaming allows for an access to more active catalyst sites at any given time. 3. Mass transport of the reactants and products is increased at the catalyst surface and in the solution due to the facilitated transport as a result of shockwave propagation. 4. Surface area is increased by fragmentation or pitting of the catalyst. 5. Cavitation-induced radical intermediates participate in the destruction of organic compounds. 6. The organic substrate reacts directly with the photogenerated surface holes and electrons. 7. Cavitation-induced turbulence also enhances the rates of the desorption of intermediate products from the active catalyst sites and helps in continuous cleaning of the catalyst surface. Toma et al. (2001) have given a brief overview of different studies pertaining to the effect of ultrasound on photochemical reactions concentrating on the chemistry aspects, i.e. mechanisms and pathways of different chemical reactions. It should be noted that in the situations where the adsorption of reactants at the specific sites is the rate-controlling step, ultrasound will play a profound role due to a substantial increase in the number of active sites and also due to the increased surface area available due to fragmentation of the catalyst agglomerates under the action of turbulence generated by acoustic streaming along with an increase in the diffusional rates of the reactants. It is of the utmost importance to operate simultaneously rather than having sequential irradiation of ultrasound followed

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by photocatalytic oxidation. This is because the catalyst surface is kept clean continuously due to the cleaning action of ultrasound in the operation as a result of which maximum sites are available for the photocatalytic reaction. Also, the number of free radicals generated in simultaneous irradiation (more dissociation of water molecules due to more energy dissipation and also the presence of solid particles enhances the ultrasound effects due to surface cavitation and adsorption of contaminants) will be at a maximum as compared to the sequential operation. The details regarding the operating and design strategies for maximizing the synergistic effects have been described by Gogate and Pandit (2004b,c). 8.2.5 Design of a Pilot-scale Reactor and its Experimental Evaluation 8.2.5.1 Rational for the Design of a Reactor Tatake and Pandit (2000) and Prabhu et al. (2004) have investigated the effect of multiple frequencies on the bubble dynamics and the maximum size of the cavity reached in the cavitation phenomena and indicated that the use of multiple frequencies results in higher growth and more violent collapse of the cavities as compared to the single-frequency operation. Sivakumar et al. (2002b) and Thoma et al. (1997), using experimental investigations, have also confirmed that multiple-frequency irradiation is superior in terms of generating higher cavitational intensity and hence enhanced cavitational effects. Gogate and Pandit (2000a) have performed bubble-dynamics simulations considering the compressibility of the liquid and reported that a decrease in the intensity by way of dissipating same power input through wider areas also results in an increase in the intensity of cavitation (higher growth and more violent collapse). Use of wider irradiating surfaces also results in the added advantage of better mixing characteristics due to higher acoustic streaming in the reactor as predicted by Vichare et al. (2001). Experimental studies, as discussed earlier, have also confirmed that the ultrasonic horn, which operates on the principle of irradiation through a single transducer gives the maximum variation in the cavitational activity in the reactor and there is a very small zone where the maximum cavitational intensity is present. Furthermore, it was also observed that in the case of an ultrasonic bath (three transducers situated at the bottom of the reactor), the cavitational intensity is comparatively more uniform as compared to the ultrasonic horn and the active cavitational volume of the ultrasonic reactor is also substantially increased. Thus, a larger number of transducers with large area of dissipation is recommended for energy-efficient operation. Dahnke and Keil (1998, 1999) have investigated the pressure fields existing in the reactor with the help of numerical simulations using homogeneous and nonhomogenous distributions of the cavitating bubbles and have indicated that there exists a large spatial variation in the cavitational activity. Balasubrahmanyam and Pandit (2006) clearly demonstrated different zones of cavitation at different planes away from the transducers with the help of mapping the erosion patterns of aluminum foil under the influence of ultrasonic irradiations. The intensity of cavita-

8.2 Design of Cavitational Reactors

tion is maximum very near to the transducer surface and diminishes away from the transducer in both axial and radial directions. Use of multiple sound sources and optimization of the power input to the systems helps in achieving uniform cavitating conditions. Mapping studies, as discussed earlier, have indicated that placement of transducers on parallel plates in a hexagonal configuration results in near uniform distribution of the cavitational activity. Taking a lead from this theoretical analysis, it was decided to use multiple transducers arranged in a hexagonal configuration in the novel reactor with a wider dissipating area and variable power dissipation. 8.2.5.2 Actual Design of the Novel Reactor Considering all the points mentioned above, a triple-frequency hexagonal flow cell was designed with each side of the hexagon hosting multiple transducers (3 on each side). The hexagonal triple-frequency flow cell has a total capacity of 7.5 liters and can be operated in batch as well as continuous mode. A schematic representation of hexagonal flow cell is given in Fig. 8.2.4. Transducers (3 in number in each set per side) having equal power rating of 150 W per side have been mounted (thus the total power dissipation is 900 W when all the transducers with a combination of 20 + 30 + 50 kHz frequencies are functional). The two opposite faces of the flow cell have the same irradiating frequency. The operating frequency of the transducers is 20, 30 and 50 kHz and they can be operated in

Figure 8.2.4 Schematic representation of triple-frequency hexagonal flow cell.

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different combinations (7 in total) either individually or in combined mode. It should also be noted that there is provision for simultaneous irradiation with UV light as photocatalytic oxidation has been reported to act in synergism with cavitation (Gogate and Pandit, 2004b) and the results for combined irradiation are much better as compared to the individual operation. 8.2.5.3 Comparison of the Efficacy of the Hexagonal Flow-cell Reactor with Conventional Designs The efficacy of the hexagonal flow cell as against the conventional designs has been evaluated using two unified criteria of energy efficiency and cavitational yield. Independent mapping studies were also carried out in this reactor to compare the distribution of the cavitational activity in this hexagonal configuration with the conventional designs. A calorimetric method has been used to determine the energy efficiency of the equipments under study. In this method, the rise in temperature of a fi xed quantity of water in an insulated container for a given time was measured. Using this information, the actual energy (power) dissipated into a liquid was calculated from the following equation,

Power (Watts) = m Cp (dT dt )

(8.2.8)

where Cp is the heat capacity of the solvent (J kg−1 K−1), m is the mass of solvent (kg), dT is the temperature difference between the initial temperature and the final temperature after a specific reaction time (K), and dt is time (s). The energy efficiency can then be calculated as follows: Energy efficiency = Power dissipated in the liquid/Electric power supplied to the system The energy efficiency gives an indication of the quantity of energy effectively dissipated in the system, a fraction of which is utilized for the generation of cavities and should be as high as possible for the particular equipment. The second parameter that has been used for the characterization of the cavitational reactors is the cavitational yield that is given by the following correlation: Cavitational yield =

Yield of the desired product in g/L Power density in W m3

(8.2.9)

where the power density is defined as the energy supplied (actual electrical energy) to the medium per unit volume of the medium. The cavitational yields for cavitational equipments indicate the ability of the equipment to produce the desired physical and/or chemical change based on the fraction of the electric energy supplied for generating the cavitation.

8.2 Design of Cavitational Reactors

The aim of the designers should be to maximize both the energy efficiency as well as the cavitational yield of the reactor. This can be done on the basis of manipulation of the operating conditions (frequency, intensity and temperature) and geometric parameters (shape, number and location of the transducers) of the reactor resulting in the required magnitude of pressure pulses from the cavitational events to give the desired effect in terms of the observed chemical change. An attempt has also been made to evaluate the efficacy in terms of mapping of the cavitational activity at different radial and axial locations in the reactor, as uniform distribution of the cavitational activity is a must for an efficient large-scale operation. The local pressure amplitudes have been measured using a hydrophone (Bruel & Kjaer Ltd., Type 8103, Denmark). The details of the measurement techniques can be obtained in the earlier study of Kanthale et al. (2003). Energy-efficiency Results The results of the study for the energy efficiency as obtained in various acoustic cavitation equipments and under different operating conditions are given in Table 8.2.4. From the table, it can be observed that the triple-frequency flow cell is the most energy efficient (energy efficiency of 75%) due to the uniform energy dissipation over a wider area and through multiple transducers (3 each on six sides of the hexagonal cross section) rather than the concentrated energy dissipation in the horn (energy efficiency 100 µm are used the model is not applicable because then the diffusion inside the particles cannot be neglected [46]. The consideration of this effect leads to the general rate model, which is described in the following section. 9.4.1.5 General Rate Model The general rate model represents the most complex case of the mathematical modeling. From this model, two sets of partial balance equations result: one for the region of the free fluid at the particle surface and one for the region inside the pores. The balance equation of the mobile phase will remain unchanged. For the mass-transfer parameter now the concentration of the pore fluid at the edge of the particle cp,i,r=rp is used.

∂ c f,i ∂ 2c f,i 3ki (1 − ε 0 ) ∂ q ∂ (uint c f,i ) +F i + − Dax + ⋅ ⋅(c f,i − c p,i ,r =rp ) = 0 rp ∂t ∂t ∂z ∂ z2 ε0 The particle diffusion can be described with the model of Gu [53]:

(9.22)

9.4 Modeling of Chromatographic Processes

∂ (ε pc p,i + (1 − ε p )qi )  1 ∂  ∂ c p,i   = ε pDp,i  2  rx2  ∂t  rx ∂r  ∂rx  

(9.23)

The first two terms stand for the accumulation in the pores and on the solid surface. The term on the right side describes the diffusion inside the pores from Fick’s 2nd law. Thereby, a parabolic concentration profi le is assumed. The concentration of the substances adsorbed on the solid surface is balanced with the concentration in the particle pore cp and this equilibrium balance is given by the adsorption isotherm. The general rate model was used as a basis for the development of a computer program for the simulation of chromatographic processes by Gu [53]. The solution of the partial differential equations in the nonlinear range of the adsorption isotherms can be obtained by application of numerical methods. One drawback for the modeling of real chromatographic separations with this model is the multitude of physical parameters, which cannot be determined experimentally and have to be estimated by approximations. In practice, these parameters are often only inaccurately fittable, so that a reasonable calculation is impossible. This model is rather applicable for theoretical studies [54]. 9.4.1.6 Initial and Boundary Conditions for Single-column Chromatography At the beginning of the elution it is assumed that only pure mobile phase is flowing through the column and the concentration is zero everywhere. Thereby, the initial condition for an unloaded column is given by:

c f,i (t = 0, z) = 0

(9.24)

The boundary conditions depend on the injection mode. For any injection profile the boundary conditions can be written as: c f,i (t, z = 0) = c i ,in (t )

(9.25)

∂ c f,i ∂z

(9.26)

and =0 t ,z =L

If the sample is injected as a rectangle impulse, as is often the case in elution chromatography, the boundary conditions can be defined as: c i ,in (t ) = c i ,inj 0 ≤ t ≤ tinj , t > tinj c i ,in (t ) = 0

(9.27)

If the rectangle function is approximated by a modified Gaussian function, the boundary conditions can be written as:

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c i ,in (t ) = c i ,inje −[K a (t −t0 )]100 , t > 0

(9.28)

The constant K a in Eq. (9.28) is fitted to the injected mass and the injection concentration. Analogous to the procedure recommended by Seidel–Morgenstern [46] all backmixing effects outside of the column are neglected. 9.4.2 Modeling of SMB Chromatography

Due to the complexity of the process, the modeling of SMB chromatography is the only acceptable possibility for optimizing the separation towards productivity and purity of fractions. In the following, the fundamental models for an SMB separation will be presented. The starting point for the SMB modeling is the modeling of a chromatographic separation on a single column. The “triangle theory”, which can be used to determine a suitable set of parameters for the operation of an SMB process with some simplifying assumptions, will be described in detail. 9.4.2.1 Model Classification In the literature various proposals for the mathematical description of SMB processes have been made. A summary of the models is given by Ruthven and Ching [7]. Criteria for a classification are: 1. Nature of the process Many model approaches assume a continuous countercurrent process neglecting the mixing of streams at the switching point. In contrast to these are the models that take into account the sectioning in discrete columns and thereby the switching procedure. 2. Description of the separation column The chromatographic bed can be described by discrete equilibrium stages or by a plug-flow reactor model. 3. Mass-transfer resistances Between the solid and the fluid phase either equilibrium can be assumed or a mass-transport model that takes into account mass-transfer resistances can be included. The complexity of these models can be variable. 4. Temperature distribution along the column The process can be considered as isotherm or nonisotherm. 5. Pressure gradient along the column

The models suitable for modeling of the SMB process (Fig. 9.8) can be classified accordingly if either a true countercurrent flow (TMB) for simplification is as-

9.4 Modeling of Chromatographic Processes

Figure 9.8 Classification of models for modeling of SMB processes [55].

Figure 9.9 Continuous model of a real SMB apparatus with different elements: mixer, plug-flow reactor, separation column and stirrer tank.

sumed or the discrete switching procedures are taken into account in an SMB model [55]. Both of these types can be calculated either continuously as plug-flow reactor or discontinuously as stage model. In a real SMB apparatus besides the actual separation columns additional elements as piping, pumps and valves are present, which have to be taken into account for the modeling. As a consequence, additional models have to be used that take into account the residence-time behavior of the total apparatus and the mixing of streams, besides the actual model for the separation column. In the model of the mixer, the mixing of the streams at the column inlet is taken into account. The residence-time behavior of the piping can be described as a plug-flow reactor. Backmixing effects outside the column can be described by an ideal stirrer tank (Fig. 9.9). In order to get a satisfactorily description of the processes with minimal effort, the ideal model is used as a starting point for the modeling only taking into account convection and adsorption equilibrium. On that basis the model is extended as far as is needed, until a model results that is suitable to describe the SMB process with computational costs as low as possible. 9.4.2.2 Modeling of the TMB Process In the mass balances of the fluidized-bed countercurrent process the true movement of the solid has to be taken into account. Furthermore, the process is divided into diverse separation zones with different flow rates of the fluid phase due to different in- or outgoing streams.

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Under SFC conditions these flow rates are no longer constant because of the pressure drop and the corresponding change in density inside the zones. Simplifying, the boundary conditions are written for a constant fluid velocity in the zones. Analogous to the models for a single column different model approaches exist. Here, only the approach for ideal chromatography will be explained. The more complex kinetic models result from this model in an analogous way as shown in Section 9.4.1. For the ideal TMB process the mass balance of the fluid phase for a separation zone j can be written as:

∂ c f,i ,j ∂ c f,i ,j ∂u j (1 − ε total ) ∂ qi ,j (1 − ε total ) ∂ qi ,j +uj + c f,i ,j + us + =0 ∂t ∂z ∂z ε total ∂ z ε total ∂t

(9.29)

The balance equation of the solid phase is omitted due to the assumption of the solid and the fluid phase being permanently in equilibrium state. The boundary conditions for the ideal TMB process for j = 1, . . . 4 are given by: c f,i ,j (t, z = 0) = c i ,j ,in (t )

(9.30)

∂ c f,i ,j ∂z

(9.31)

and

t ,z =L j

The inlet concentration ci,j,in(t) in each zone j are calculated from the mass balances at the inlets and outlets: c i ,1,in (t ) =

u4 c f,i , 4,z =L j u1

c i ,2,in (t ) = c f,i ,1,z =L j c i ,3,in (t ) =

u u2 c f,i ,2,z =L j + feed c i ,feed u3 u3

c i , 4,in (t ) = c f,i ,3,z =L j

(9.32) (9.33) (9.34) (9.35)

Because of the analogy between the TMB and the SMB process, this model can be applied for the determination of the operating parameters of an SMB process in the case of certain adsorption isotherms. In SMB-HPLC this so-called “triangle theory” is established for the generation of operating parameters because it enables the estimation of operating points by means of simple equations with analytical solutions. This method, and its expansion to the conditions of SFC, will be shown in more detail in Section 9.4.2.4.

9.4 Modeling of Chromatographic Processes

9.4.2.3 Modeling of the SMB Process In contrast to the TMB process in the process with simulated countercurrent flow no transport of the solid takes place. The balance equations obtained for a single column are valid further on. For a separation cascade NC single columns with length L are connected with each other and linked by boundary conditions. In particular, for the simulation of SMB-SFC the assumption of constant volume flow rates has to be dropped. For j = 1, . . . 4 zones and k = 1, . . . , NC( j) columns per zone j the following boundary conditions result:

c f,i ,j ,k (t, z = 0) = c i ,j ,k ,in (t )

(9.36)

∂ c f,i ,j ,k ∂z

(9.37)

and

t ,z =L

The concentrations at the inlet ports are given by: c i ,1,1,in (t ) =

Vɺ4,NC (z = L) c f,i , 4,NC,z =L Vɺ1,1(z = 0)

(9.38)

c i ,2,1,in (t ) =

Vɺ1,NC(z = L) − Vɺextract c f,i ,1,NC ,z =L Vɺ2,1(z = 0)

(9.39)

c i ,3,1,in (t ) =

Vɺ2,NC (z = L) Vɺfeed c f,i ,2,NC,z =L c ifeed Vɺ3,1(z = 0) Vɺ3,1(z = 0)

(9.40)

c i , 4,1,in (t ) =

Vɺ3,NC (z = L) − Vɺraffinate c f,i ,3,NC,z =L Vɺ4,1(z = 0)

(9.41)

In SFC, the volume flow between two directly coupled columns can change due to pressure drops between the columns caused by valves or tubings also without any ingoing or outgoing streams. Thus, for all other columns: c i ,j ,k ,in (t ) =

Vɺ j ,k −1(z = L) c f,i ,j ,k −1,z =L Vɺ j ,k (z = 0)

(9.42)

9.4.2.4 Solution of the TMB Model with the Triangle Theory If an SMB process is discretized by an increasing number of columns in the functional zones, the concentration profile converges to that of the TMB model. Thus, the TMB model represents a boundary case of the simulated moving-bed process. If, additionally, only the solution in the steady state of the system is considered, the balance equations in the formulation of the stage model can be simplified in a way that only one nonlinear system of equations has to be solved. Such

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a model was shown by Charton and Nicoud [52] among others. If, furthermore, an infinite number of theoretical stages in the columns is assumed, an analytical solution of the resulting balance equations can be found for particular adsorption isotherms. Moreover, this solution can be illustrated in a so-called triangle diagram (or (m2, m3 )-diagram). This method, known as “triangle theory”, was first shown by Ruthven and Ching for the case of linear isotherms [7]. The mathematically complex extension for nonlinear isotherms is based on the work of Storti et al. [6] and Mazzotti [56, 57]. An extension for different linear distribution coefficients in the functional zones of the TMB, as is the case under SFC conditions, was also published by Mazzotti [26]. The triangle theory is already described in the literature in detail. Therefore, here only the most important results and equations will be presented. All equations assume an incompressible mobile phase. The effects of a variable density within one zone cannot be taken into account by the triangle theory. Due to the analogy between TMB and SMB the operating parameters of the processes can be converted into each other. Thus, from the solution of the TMB model operating parameter of an SMB process can be obtained. The rules for calculation are given by Mazzotti [57]: QS =

Vc (1 − ε 0 ) tshift

= Q TMB + Q SMB j j

QS (1 − ε 0 )

V j = VCNC( j )

(9.43)

(9.44) (9.45)

With these equations the most important parameter for the description of the TMB process, the net flow ratio mj, can be written in terms of the SMB process: mj =

− Q sε P Q SMB Q TMB tshift − Vcε total j j = with j = 1, . . . , 4 zones Q s (1 − ε P ) Vc (1 − ε total )

(9.46)

Q j TMB : volume flow rate in zone j for the TMB model Q jSMB : volume flow rate in zone j for the SMB model Q S: volume flow rate of the solid Neglecting backmixing effects, the extra-column volume between the columns Vext has an effect by increasing the run time of the components only. This can be taken into account by an additional term in Eq. (9.46): m j ,corr =

Q SMB tshift − Vcε total Vext j − Vc (1 − ε total ) Vc (1 − ε total )

(9.47)

9.4 Modeling of Chromatographic Processes

The solution of the TMB model is given by a set of inequalities, where the complete separation of both components is required as a further condition. For several usual isotherm equations analytical terms for these inequalities exist. In the simplest case of linear isotherms these equations are: K A < m1 < ∞ K B < m2 < K A K B < m3 < K A −ε p < m4 < K B 1 − εp

(9.48)

As a positive feed flow rate is applied, it is m3 > m2. From this it follows: K B < m2 < m3 < K A

(9.49)

This set of inequalities can be illustrated graphically in a (m2, m3 )-diagram assuming that m1 and m 4 meet the conditions given above (Fig. 9.10). The triangular area along the diagonal describes the region of complete separation of the components. As the productivity is for (m3 − m2 ) → max!, the point “w” is the operating point with maximal productivity. Besides the region of complete separation other characteristic regions are found in the (m2, m3 )-diagram.

Figure 9.10 (m2, m3)-diagram for a linear adsorption isotherm (here: K A = 0.88; KB = 0.5). Region 1: complete separation of A and B Region 2: extract pure A, raffinate impure Region 3: raffinate pure B, extract impure Region 4: raffinate and extract impure Region 5: extract pure solvent, A and B in the raffinate Region 6: raffinate pure solvent, A and B in the extract

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In order to ensure a robust operation, the actual operating point is chosen inside the triangle, as in reality the optimal point of an SMB plant is not exactly at point “w”. The advantage of the triangle theory is not only the easy determination of optimal operating points. But also real operating states of SMB processes, which underlie the influences of mass-transfer resistances and a finite number of columns, can be interpreted with it. For the selection of new operating points a view on the total (m2, m3 )-diagram leads to a reasonable direction. In order to achieve the effects described before for the separation of the components inside the SMB plant, the following conditions have to be fulfilled: Q 1 ≥ Q SK A Q 2 ≥ Q SK B Q 3 ≤ Q SK A Q 4 ≤ Q SK B

(9.50)

With the introduction of a safety factor b ≥ 1 the inequalities can be rewritten as: Q 1 = Q SK A β Q 2 = Q SK Bβ Q 3 = Q SK A / β Q 4 = Q SK B / β

(9.51)

If the safety factor is equal to one, the SMB system is operated at its optimum, but is sensitive to small variations of flow rates. For the flow rates, the following relations are obtained: Q recycling = Q 4 Q raffinate = Q 3 − Q 4 Q feed = Q 3 − Q 2 Q extract = Q 1 − Q 2

(9.52)

Taking into account the safety factor, the flow rates for raffinate, extract and the stationary phase can be calculated by transformation of the equations. Q feed K A / β − K Bβ Q 1 = Q SK A β Q raffinate = Q S(K A − K B )/β Q extract = Q S(K A − K B )β Qs =

(9.53)

KA . KB Due to the equivalence of TMB and SMB process the switching time tshift can be calculated from:

From this is follows that the safety factor b is in the range 1 ≤ β ≤

9.4 Modeling of Chromatographic Processes

tshift = [(1 − ε 0 )Vc,single column ]/Q S

(9.54)

Together with the fact that only about 500 theoretical stages are needed for achieving the highest purities, this method makes possible the calculation of the appropriate flow rates from analytical injection into the present SMB apparatus. Only the feed volume flow rate has to be given. The shape of the region of complete separation in the (m2, m3 )-diagram illustrated for the case of linear isotherms (Fig. 9.11a) takes for the case of nonlinear isotherms (Fig. 9.11b) the shape of a distorted triangle. For the illustration in a (m2, m3 )-diagram the nondimensional parameter of the net flow rate ratio is used (Eq. (9.46)), [26]. If a compressible mobile phase is used, the mj in the zones are no longer constant since they depend on density. In order to keep a simple illustration of the separation regime in a (m2, m3 )-diagram, a new ratio m*j is defined, which uses the mass flow rate inside each zone (Eq. (9.55)), [58]. The reference density ρ0 is set to 1000 kg/m3. For the special case of constant densities the parameter can be converted into each other according to Eq. (9.56). m *j =

ρjQ SMB tshift − ρ0Vcε total j ρ0Vc (1 − ε total )

(9.55)

m *j =

m j ρj (ε total − 1) + ε total ( ρ0 − ρj ) ρ0(ε total − 1)

(9.56)

9.4.2.5 Consideration of Different Adsorption Coefficients in the Zones As the adsorption equilibrium in SFC is dependent on density, the application of a pressure gradient along the zones of the process can be used for a directed

Figure 9.11 (m2, m3)-diagram for linear (a) and nonlinear (b) isotherms.

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influence on the elution strength of the mobile phase. This corresponds to the application of a solvent-concentration gradient in HPLC. Taking into account the set of inequalities (Eq. (9.48)) it is obvious that the application of a pressure gradient is reasonable only if the highest elution strength is realized in zone 1 and the lowest in zone 4. For the change of the linear adsorption coefficients Ki on density Mazzotti et al. [26] suggested: ρ  K i ,j = K i ,0  0   ρj 

bi (9.57)

Thereby, Ki,0 is the linear adsorption coefficient at a reference density r0. With it, the linear adsorption coefficient Ki can be calculated for any density r from the information of a reference state. The exponent bi is an empirical parameter, which can be obtained from experimental data. Typical values for bi are in the range between 1 and 10. As a consequence, the adsorption coefficients for a component i in the four zones 1 to 4 meet the inequality: K i ,1 < K i ,2 < K i ,3 < K i ,4

(9.58)

If the method of the triangle theory is applied in this case, the region of complete separation can be determined in the same way as described before. However, the shape of the region of complete separation changes. Starting from a triangle along the diagonal a more or less formed rectangle is shaped up in the (m2, m3 )-diagram. The phenomenology differentiates between two cases: either K A,2 > K B,3 (Fig. 9.12a) or K A,2 < K B,3 (Fig. 9.12b). The coordinates of the point of maximal productivity w are analogous to the isobaric case (K B,2, K A,3 ). The point w has, in comparison to the isobaric operation at an average density, a larger distance from the diagonal, which is equivalent to a higher obtainable productivity. If the separation regimes for the gradient operation are compared to the one for the isobaric operation with respect to the purities of the products, the separation regions described for the isobaric case can be found. Notable are the new regions 7 to 9, in which one or both components accumulate in the apparatus. The assumptions of the triangle theory of linear isotherms and infinite solubility of the components lead to a nonsensical prediction of a permanent accumulation in the apparatus. In reality, the accumulating component will break through after a certain time. For modeling of real SMB-SFC processes the results of the triangle theory are applicable only in a limited way. But the model provides a boundary case for the validation of the dynamic SMB-SFC model. Furthermore, the application of the triangle theory enables the derivation of parameters for the evaluation of the quality of products and the separation process.

9.4 Modeling of Chromatographic Processes

Figure 9.12 (a) (m2, m3)-diagram for pressure gradient mode if K A,2 > KB,3 (b) (m2, m3)diagram for pressure gradient mode if K A,2 < K. Region 1: complete separation of A and B Region 2: extract pure A, raffinate impure Region 3: raffinate pure B, extract impure Region 4: raffinate and extract impure Region 5: extract pure solvent, A and B in the raffinate

Region 6: raffinate pure solvent, A and B in the extract Region 7: extract pure solvent, A accumulates, pure B in the raffinate Region 8: raffinate pure solvent, B accumulates, pure A in the extract Region 9: raffinate and extract pure solvent, A and B accumulate

Complete separation of the components is obtained if both extract and raffinate stream show purities of 100%. The purity of the extract is calculated by Pextract =

c A,extract 100 E c A,extract + c B,extract

(9.59)

and the purity of the raffinate by Praffinate =

c B,raffinate 100 R c A,raffinate + c B,raffinate

(9.60)

The enrichment E of a component is defined as the ratio of the concentrations of the components in the outlet stream and in the feed stream. For a binary, complete separation the enrichments can be calculated from the differences between the net flow rates [26]: EA =

c A,extract m3 − m2 = c A,feed m1 − m 2

(9.61)

EB =

c B,extract m3 − m2 = c B,feed m3 − m 4

(9.62)

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9.4.2.6 Modeling of Pressure Gradient in SFC In elution chromatography the maximal acceptable pressure drop and the configuration of the apparatus often limit the flow rate of the mobile phase. Indeed, the number of theoretical stages changes with the velocity of the mobile phase, but the purity of the products is not as strongly dependent on the flow velocity as it is for an SMB separation. A relatively small change of the volume flow in one of the functional zones can lead to a significant decrease in product purity [59]. For the design and optimization of a separation by SMB-SFC it is therefore more important to take into account the pressure drop and by that the change in volume flow. For simplifying of the simulation of elution chromatography with a single column, a linear pressure gradient is assumed and interpolated between the densities at the column inlet pressure and the column outlet pressure. But, for the modeling of SMB-SFC the pressure drop and the resulting density gradient are calculated from packing geometry, volume flow rates and the thermodynamic data of the mobile phase. For calculation of the pressure drop the relation after Darcy can be used, in which the constant hK is fitted to experimentally determined pressure drops.

∆p = ∆zhK

36(1 − ε 0 )2 w(z)η(z) ε 03d p2

(9.63)

The viscosity h and the superficial velocity w are dependent on pressure and density, respectively. Therefore, a new density has to be calculated for each pressure by means of an equation of state, e.g. for CO2 by the equation of Span and Wagner [60]:

ρ = f ( p(z), T )

(9.64)

For calculation of the viscosity an approach of Chung can be used, the influence of pressure is taken into account by the method of Reichenberg [61]. The viscosity of a gas at low pressure and a temperature T > Tc can be calculated after Chung by:

η0 = 40785

FC (MT )0.5 Vc2 / 3Ω v

(9.65)

where: Fc = 1 − 0.2756ω

(9.66)

Ω v = [ A(T * )−B ] + C [exp(−DT * )] + E [exp(−FT * )]

(9.67)

T * = 1.2593

T Tc

Vc: critical volume [cm3/mol] w : acentric factor [–] Tc: critical temperature [K]

(9.68)

9.4 Modeling of Chromatographic Processes

Starting from the viscosity at low pressure, the influence of pressure can be calculated after Reichenberg by: A( p / pc )2 / 3 η = 1+Q η0 B( p / pc ) + (1 + C( p / pc )D )−1

(9.69)

The variables A, B, C, D are functions of the reduced temperature Tr = T/Tc, and Q = 1 for nonpolar substances. Via the continuity equation, velocity and density are linked to each other: w=

ɺ m AC ρ(z)

(9.70)

The set of equations for the modeling of the SMB-SFC can be solved by simple discretization; a partition into 100 length intervals is used per column. pn +1 = pn − ∆zhK

36(1 − ε 0 )2 w(z)η(z) ε 03d p2

(9.71)

As the mass flow rates in the functional zones of the SMB apparatus are constant within one run, the set of equations can be solved for an operating point and the calculated velocities of the mobile phase can be recorded. This leads to shorter computation times in comparison to a new solving of the complete set of equations for each discrete length. Besides the pressure drop inside the columns there can be an additional pressure loss due to the piping and the valves between the columns. In experiments, a large pressure drop was found in an SMB-SFC apparatus due to these flow resistances [55]. Taking into account the pressure drop as a function of mass flow rates, a pressure-loss coefficient z for the total flow resistance (analogous to pipeline construction) between two columns is determined by fitting to experimental pressure drops. Then the pressure drop can be calculated from: ∆p = ζ

w2 ρ 2

(9.72)

If eight columns are used eight pressure-loss coefficients are needed in total. 9.4.3 Numerical Algorithms

From the literature, several possible methods for the solution of the balance equations presented are known. A classification firstly resulted from the fact that for some equations analytical solutions are available depending on the adsorption isotherm and boundary conditions chosen, while other models can be solved only numerically. A detailed overview of the numerical algorithms for the solution of

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balance equations is given by LeVeque [62]. The numerical algorithms developed can be divided into two groups [46]. On the one hand the equations can be solved numerically by the application of collocation schemes, on the other hand, difference methods can be applied. 9.5 Simulations of SMB-SFC Chromatography

Chromatographic separations of phytol isomers, ibuprofen enantiomers, and tocopherol homologs by SMB-SFC have been performed experimentally and were further investigated by simulations [29, 55, 58, 65–67]. The apparatus for experimental separations by SMB-SFC as well as the methods and models for determination of the adsorption equilibrium as the basis for chromatography simulation have been described elsewhere [27, 68]. For the dynamic simulation of the SMB-SFC process a plug-flow model with axial dispersion and linear mass-transfer resistance was used. The solution of the resulting mass-balance equations was performed with a finite difference method first developed by Rouchon et al. [69] and adapted to the conditions of the SMB process by Kniep et al. [70]. The pressure drop in the columns is calculated with the Darcy equation. The equation of state from Span and Wagner [60] is used to calculate the mobile phase density. The density of the mobile phase is considered variable. 9.5.1 Separation of Cis/trans-Isomers of Phytol

Phytol is a fatty alcohol with 20 carbon atoms (3, 7, 11, 15-tetramethyl-2hexadecene-1-ol, C20H40O) from the group of terpenes. Phytol is used in the synthesis of vitamins E and K1. In addition, it is a lipophilic component of chlorophyll. Because of a double bond cis- and trans-isomers exist. The trans-isomer is used as a fi xener in the perfume industry. The separation of cis- and trans-phytol by SMB-SFC was performed on a silica gel stationary phase (LiChrospher Si 60, 15 µm) and CO2 including 4.5 mass% modifier 2-propanol as the mobile phase at an inlet pressure of 23.0 MPa pressure and a temperature of 313 K. From the first experimental SMB-SFC runs a specific productivity of 1.3 kgfeed/lSPd with purities of 97% trans- and 98% cis-phytol was found [55]. By simulation, the influence of the number of theoretical plates, the number of columns and the feed concentration was determined. For most of the simulations the following parameters were kept constant: column configuration 2/2/2/2, column length 10 cm, column diameter 3 cm and number of theoretical stages 100 per column. In order to reach a steady state for each operating point 20 cycles (160 switching intervals) were simulated. Before analyzing the effect of operating variables on the process performance, the model for the dynamic simulation of the SMB-SFC process has to be verified. For validation of the SMB-SFC model the triangle theory illustrated in Section

9.5 Simulations of SMB-SFC Chromatography

9.4.2.4 is applied for different adsorption coefficients in the functional SMB zones. Assuming infinite numbers of theoretical stages and columns as well as the system being in a steady state, the triangle theory provides a checkable case for a system with a linear adsorption isotherm. For comparison of the separation regions predicted by the triangle theory with the results of the dynamic simulation, the mass flow rates in zones 2 and 3 are varied systematically. A low feed concentration of 0.05 ml/min was chosen in order to operate in the linear range. The purities resulted by the simulation will be recorded. A product is assumed to be pure if it has a purity of more than 99.9%. With the purities of each product the type of separation region is determined according to the classification given in Section 9.4.2.4 and registered in the (m *2, m3*)-diagram. In Fig. 9.13 a comparison of the separation regions predicted by the triangle theory with those resulting from dynamic simulations is shown. The agreement between simulation and triangle theory is excellent despite the restriction to 500 theoretical stages per column. A typical characteristic of SMB separations in comparison to separations in elution mode is the lower sensitivity against a decreasing number of theoretical stages, which can occur due to irreversible adsorption processes or changes in bed homogeneity during operation [52]. The simulations for the SMB process result even for only 100 theoretical stages per column to a good agreement between the SMB simulation and the TMB prediction by the triangle theory for a column configuration of 2/2/2/2. A further increase of the number of columns

Figure 9.13 Comparison between the separation regions calculated by the triangle theory (lines) and the dynamic SMB simulation (squares) for phytol in a (m 2*, m *3)-diagram. Classification of separation regions according to Fig. 9.12. Simulation parameter: column configuration 2/2/2/2, column length 10 cm, feed concentration 0.05 mg/ml, the number of stages is 500 per column.

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does not have any benefits for the system investigated. However, if the masstransport resistances are high the achievable purities and productivities can be enhanced by a finer discretization, which is always related to an increasing time and effort [17]. Conversely, the use of only 4 columns is the roughest discretization of the SMB process. In Fig. 9.14 the separation regions calculated with 4 columns of double length are shown. The region of complete separation of both fractions is significantly smaller and the maximal productivity is only 50% compared to the 8column SMB process. Important for the incomplete separation are larger backmixing effects by the switching of the inlet and outlet ports for the 4-column configuration. On closer look on the product purities of an operating point outside the region of complete separation (point B) it is found that the products still have purities of 98.2% and 98.7%, respectively. Compared to the complete separation by the 8-column SMB process this is indeed an obvious decrease. Nevertheless, due to lower technically complexity the application of a 4-column SMB process can be reasonable if lower purities meet the requirements. Besides the switching time, the feed concentration is the most important parameter for enhancing productivity of an SMB process. The shape and the location of the region of complete separation in a (m2, m3 )-diagram are mainly influenced by the feed concentration. Therefore, the feed concentration for the simulation was gradually increased. Quadratic Hill-adsorption isotherms dependent on pressure were determined for the phytol-isomers. The mixture interactions were taken into account by the ideal adsorbed solution theory (IAST) [58].

Figure 9.14 Separation regions calculated by the dynamic SMB simulation (squares) in a (m 2*, m *3)-diagram compared with the linear case (lines) for phytol. Classification of separation regions according to Fig. 9.12. Simulation parameter: column configuration 1/1/1/1, column length 20 cm, feed concentration 0.05 mg/ml, the number of stages is 200 per column.

9.5 Simulations of SMB-SFC Chromatography

Figure 9.15 Separation regions calculated by the dynamic SMB simulation (squares) in a (m 2*, m *3)-diagram compared with the linear case (lines) for phytol. Classification of separation regions according to Fig. 9.12. Simulation parameter: column configuration 2/2/2/2, column length 10 cm, feed concentration, 10 mg/ml, the number of stages is 100 per column.

From Fig. 9.15 the influence of the quadratic isotherm can easily be seen for a feed concentration of 10 mg/ml. In the figure, the boundaries of the region of complete separation for the linear isotherm are plotted as lines in order to show the difference between the linear and nonlinear separation regions. Point w, which has the maximal feed flow rate, moves to smaller m *2 and m3*. The size of the region of complete separation decreases. From Fig. 9.16 it can be seen that a further increase of feed concentration to 20 mg/ml leads to a region of complete separations of completely different size and shape. After the extensive optimization a specific productivity of 6.0 kgphytol/lSPd was obtained for an SMB column configuration of 6 × 5 cm. The same specific productivity was also obtained after optimization of the elution process. However, the specific solvent consumption was significantly lower for the SMB process (elution mode: 280 gCO2/gphytol; SMB mode: 69 gCO2/gphytol) [58]. 9.5.2 Separation of Enantiomers of (R,S)-Ibuprofen

Ibuprofen is a nonsteroidal anti-inflammatory agent with antipyretic and analgesic properties belonging to the pharmacologically important group of profens. Ibuprofen (2-(4-isobutylphenyl)propionic acid, C13H18O2 ) has a chiral center and therefore exists as S(+)- and R(−)-enantiomers. Up to now, ibuprofen is applied mostly as the racemate, although the pharmacological activity of the S(+)enantiomer is higher by a factor of about 100 than that of the R(−)-enantiomer

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Figure 9.16 Separation regions calculated by the dynamic SMB simulation (squares) in a (m 2*, m *3)-diagram compared with the linear case (lines) for phytol. Classification of separation regions according to Fig. 9.12. Simulation parameter: column configuration 2/2/2/2, column length 10 cm, feed concentration 20 mg/ml, the number of stages is 100 per column.

[71, 72]. The medication with the active S(+)-ibuprofen instead of the racemic (R,S)-ibuprofen is advantageous due to lower dose rates needed, lower loadings in metabolism and no inversion from the R(−)- to the S(+)-form. The chromatographic separation of the enantiomers of (R,S)-ibuprofen was performed by SFC on a chiral stationary phase (Kromasil CHI-TBB, 10 µm) and 4.5 mass% 2-propanol in CO2 as mobile phase at a medium pressure of 15 MPa and a temperature of 313 K. The first experimental SMB-SFC runs with low feed concentration lead to a specific productivity of 0.05 kgracemate/lSPd and purities of >98% R(−)-ibuprofen and >98% S(+)-ibuprofen. By simulation, the influence of the injected amount of racemate and the column configuration was determined. For most of the simulations the following parameters were kept constant: column length 9.6 cm, column diameter 3 cm, and the number of theoretical stages is 100 per column. In order to reach a steady state for each operating point 20 cycles (160 switching intervals) were simulated. Cubic Hill-adsorption isotherms dependent on pressure were determined for the ibuprofen enantiomers. The mixture interactions were taken into account by the IAST [29]. To show the influence of the injected amount of racemate per minute on the separation of the two ibuprofen enantiomers, the region of complete separation (purity > 99%) was calculated for a column configuration of two columns per zone and for different amounts of racemate. As shown in Fig. 9.17 increasing the

9.5 Simulations of SMB-SFC Chromatography

Figure 9.17 Region of complete separation in a (m 2*, m *3)diagram for feed injection of 20, 40, and 60 mg/min ibuprofen, column configuration: 2/2/2/2, column length 9.6 cm.

Figure 9.18 Region of complete separation in a (m 2*, m *3)diagram for feed injection of 60, 100 and 140 mg/min ibuprofen, column configuration 2/2/3/1, column length 9.6 cm.

amount of injected racemate to 60 mg/min leads to a much smaller region of complete separation compared to the calculations with an injected amount of racemate of 20 mg/min. In the case of a smaller region of complete separation, under real conditions, the process becomes unstable. Only small changes of flow rates can lead to a decrease of product purity. Further studies show that the productivity can still be enhanced if an unequal number of columns in the zones are used. A change of the 2/2/2/2-configuration (two columns per zone) to 2/2/3/1-configuration permits an increase in the injected amount of racemate to 140 mg/min for complete separation. In Fig. 9.18

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Figure 9.19 Internal concentration profile of the SMB separation between ibuprofen enantiomers for 140 mgracemate/min, column configuration 2/2/3/1, column length 9.6 cm (simulation and experiment).

the calculated region of complete separation for injection up to 140 mg/min is shown. If the amount of injected racemate is increased to 180 mg/min, a complete separation will be impossible. In Fig. 9.19 the internal concentration profile of an experimental SMB run (140 mgracemate/min) is shown. A comparison between the experimental and the simulated concentration profi le shows a good agreement. After optimization of the SMB-SFC process by simulation the specific productivity could be increased from 0.05 to 0.36 kgracemate/lSPd also experimentally while the consumption of CO2 and 2-propanol referring to the injected racemate could be cut in half [29]. 9.5.3 Separation of Tocopherols

The separation of substances of highest purity from natural mixtures by chromatography is of increasing interest. An example is the production of natural tocopherols (vitamin E) from vegetable oils. The purity that can be obtained by conventional processes like distillation or extraction is limited due to the multiplicity of components with similar extraction behavior. On the other hand by chromatographic separations it is even possible to obtain each single tocochromanol with high purity. The chromatographic separation of α - and δ -tocopherol as a model mixture was performed by SFC on a silica gel stationary phase (Kromasil Si 60, 10 µm) and 5 mass% 2-propanol in CO2 as mobile phase at a medium pressure of 20 MPa and a temperature of 313 K. For the optimization of the preparative chromatographic separation, the knowledge of adsorption isotherms is necessary. Therefore, the adsorption isotherms were determined with the perturbation method. The single-component adsorption behavior shows anti-Langmuir behavior and can be described using the cubic Hill isotherm [67]. In the following, different parameter studies aiming at a productivity increase will be presented. For the simulations, the following parameters were kept constant: column diameter 3 cm and

9.5 Simulations of SMB-SFC Chromatography

number of theoretical stages at 1400 per meter. The influence of feed concentration, column length and column configuration on the SMB separations was investigated. At first, the simulation program was used for the calculation of the (m *2, m3*)diagram for a 2/2/2/2-column configuration and a constant column length of 14.4 cm. The feed concentration was increased stepwise from 2 to 20 mg/ml. From Fig. 9.20 it can be seen that with increasing feed concentration the optimal operating point is shifting and the orthogonal distance to the diagonal is decreasing. The difference m3* − m *2 is decreasing and therewith the maximal feed flow rate. In Fig. 9.21 the specific productivity and the solvent consumption are shown as

Figure 9.20 Regions of complete separation between α - and δ -tocopherol in a (m*2, m 3*)-diagram for different feed concentrations (cfeed = 2.5, 5, 10, 15, and 20 g/l), column configuration 2/2/2/2, column length 14.4 cm.

Figure 9.21 Productivity and solvent consumption as a function of feed concentration. Separation system: α - and δ -tocopherol on Kromasil 60–10, column length 14.4 cm, column configuration 2/2/2/2.

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functions of the feed concentration. The maximum of productivity and the minimum in solvent consumption were found for the same feed concentration, so that no conflict of aims appeared. For a 2/2/2/2-column configuration and a feed concentration of 6 mg/ml the column length was stepwise decreased from 14.4 to 4 cm. In Fig. 9.22 the regions of complete separation for different column lengths are shown in a (m *2, m3*)diagram. Down to a column length of 6 cm no influence was found on the region of complete separation. Thus, a decrease of column length leads directly to an increase in productivity. For shorter columns a conflict of aims was found (Fig. 9.21): A decrease of column length from 6 cm to 4 cm enhances the productivity but increases the solvent consumption. In another parameter study the influence of column configuration on the productivity and solvent consumption of the α - and δ -tocopherol separation by SFC was investigated. By simulation the region of complete separation in a (m*2 , m*3 )diagram was calculated for a 2/2/2/2, 1/3/2/2, 1/3/3/1, and 1/2/2/1-column configuration. The size and shape of the region of complete separation was equal for the 1/2/2/1 and 2/2/2/2-configuration. This means that a 6-column configuration is adequate for the separation of α - and δ -tocopherol. For the cleaning of adsorbent in zone 1 and desorbent in zone 4, respectively only one column is needed. For the 1/3/2/2- ad 1/3/3/1-configuration the region of complete separation is slightly larger than for the 1/2/2/1- and 2/2/2/2-configuration, respectively (Fig. 9.23). Nevertheless, for all configurations the same optimal operating point was obtained. Finally simulations were done with all the optimized parameter (feed concentration: 15 mg/ml; column length: 6 cm, column configuration: 1/2/2/1). This enhanced the specific productivity to more than 1.3 kg/lSPd. A comparison of the optimized SMB process with the also optimized elution process showed, on the

Figure 9.22 Regions of complete separation between α - and δ -tocopherol in a (m 2*, m *3)-diagram for different column lengths, column configuration 2/2/2/2, feed concentration 6 mg/ml.

9.6 Conclusion

Figure 9.23 Regions of complete separation between α - and δ -tocopherol in a (m 2*, m *3)-diagram for different column configurations (1/2/2/1 and 1/3/3/1), column length 14.4 cm, feed concentration 6 mg/ml.

one hand, that for separation of the binary model mixture the specific productivity of the elution process is higher (elution mode: 2.9 kgToco/lSPd; SMB mode: 1.3 kgToco/lSPd). On the other hand the solvent consumption in the SMB mode is lower (elution mode: 620 gCO2 /gToco; SMB mode: 343 gCO2 /gToco). The solvent consumption and the needed solvent recovery, respectively, is one of the major cost factors of liquid chromatographic processes. In SFC the solvent consumption is less important. Due to high investment costs for several chromatographic columns in SMB the elution chromatographic process with supercritical fluids is very competitive [65].

9.6 Conclusion

In the context of intensive research work on SMB chromatography during the last decade a broad understanding of chromatographic theory was achieved. Various simulation tools have been developed, which can be used for optimization not only of SMB chromatography but also of elution chromatography. For optimization of SMB chromatography the influence of column length, column configuration, flow rates, feed concentration, and switching times is studied. During optimization it is of major importance to take into account the costs of the process, because in industry often the technical optima is not the economic optima. Therefore, for each large-scale application a comparison of discontinuous elution chromatography and the continuous SMB process should be made. After optimization of both processes they often show similar specific productivities, but there is lower solvent consumption in SMB chromatography.

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Symbols Latin Symbols

A act Ac c cf cinj cp d dp D Dap Dax Dm Dp E F H keff K L m minj mj m*j M N p P q Qj QS rp rx t tinj tM tR tshift T u uc uint

total active surface area of the adsorbent column cross-sectional area concentration in the mobile phase concentration in the free fluid phase concentration injected concentration in the particle pores of the stationary phase inner column diameter particle diameter diffusion coefficient apparent dispersion coefficient axial dispersion coefficient molecular diffusion coefficient particle diffusion coefficient enrichment phase ratio height of a theoretical stage effective mass-transfer coefficient linear adsorption coefficient column length mass injected mass net volume flow rate in zone j net mass flow rate in zone j molecular mass number of theoretical stages pressure purity loading of the stationary phase volume flow rate of the fluid phase volume flow rate of the solid phase particle radius radial coordinate for particle time injection time holdup time retention time switching time for an SMB process temperature linear velocity migration velocity of a concentration interparticle velocity of the fluid

cm2 cm2 mg/ml mg/ml mg/ml mg/ml cm µm cm2/min cm2/min cm2/min cm2/min cm2/min – – mm – – mm mg mg – – g/mol – MPa mg/ml ml/min ml/min µm – min min min min min K cm/min cm/min cm/min

References

uS V VC Vext Vf V0 VP w z

migration velocity of the shock front volume column volume extracolumn volume volume of the mobile phase outside the particles interparticle void volume pore volume superficial velocity longitudinal coordinate

cm/min ml ml ml ml ml ml cm/min cm

Greek Symbols b e0 e total ep z h r rP r0

ω

safety factor interstitial porosity total porosity particle porosity pressure-loss coefficient dynamic viscosity density packing density reference density (set to 1000 kg/m3 ) acentric factor

– – – – – g/(cm s) kg/m3 g/ml kg/m3 –

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Dissertation Technische Universität Hamburg-Harburg, 2000. Mazzotti, M.; Storti, G.; Morbidelli, M.: Robust design of countercurrent adsorption separation: 2. Multicomponent systems; AIChE Journal, 1994, 40, 11, 1825–1842. Mazzotti, M.; Storti, G.; Morbidelli, M.: Optimal operation of simulated moving bed units for nonlinear chromatographic separations. J. Chromatogr. A, 1997, 769, 3–24. Giese, T.: Simulation der Chromatographie mit überkritischem Kohlendioxid am Beispiel der Trennung eines Diterpens. Dissertation Technische Universität HamburgHarburg, 2002. Strube, J.: Simulation and Optimierung kontinuierlicher Simulated-Moving-Bed (SMB)-Chromatographie-Prozesse. Dissertation Universität Dortmund, 1996. Span R.; Wagner, W.: A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa. J. Phys. Chem. Ref. Data, 1996, 25, 1509–1596. Reid, R.C.; Prausnitz, J.M.; Poling, B.E.: The Properties of Gases and Liquids. McGraw Hill, New York, 1987. LeVeque, R.J.: Numerical methods for conservative laws. Birkhäuser, Basel, 1990. Strube, J.; Haumreißer, S.; SchmidtTraub, H.; Schulte, M.; Ditz, R.: Comparison of optimized Batch and SMB Chromatography. Org. Process Res. Dev., 1998, 1, 3, 264–271.

64 Jupke, A.; Epping, A.; Schmidt-Traub, H.: Optimal design of batch and simulated moving bed chromatographic separation processes. J. Chromatogr., 2002, 944, 1, 93–118. 65 Peper, S.; Johannsen, M.; Brunner, G.: Preparative Chromatography with Supercritical Fluids – Comparison of SMB and Batch Process. Submitted for publication in J. Chromatogr. A. 2007. 66 Johannsen, M.: Präparative Chromatographie mit überkritischen Gasen. Habilitation Technische Universität Hamburg-Harburg. Mensch & Buch Verlag, Berlin, 2004. 67 Peper, S.: Präparative Chromatographie mit überkritischen Fluiden. Dissertation Technische Universität HamburgHarburg, 2006. 68 Lübbert, M.: Adsorption aus überkritischen Lösungen. Dissertation Technische Universität HamburgHarburg, 2004. 69 Rouchon, P.; Schnauer, M.; Valentin, P.; Guiochon, G.: Numerical simulation of band propagation in nonlinear chromatography. Sep. Sci. Technol., 1987, 22, 1793–1833. 70 Kniep, H.; Falk, T.; Seidel-Morgenstern, A.: PREP, Basle, 1996. 71 Goto, M.; Noda, S.; Kamiya, N.; Nakashio, F.: Enzymatic resolution of racemic ibuprofen by surfactant-coated lipases in organic media. Biotechnol. Lett., 1996, 18, 839–844. 72 Trani, M.; Ducret, A.; Pepin, P.; Lortie, R.: Scale-up of enantioselective reaction for the enzymatic resolution of (R,S)ibuprofen. Biotechnol. Lett., 1995, 17, 1095–1098.

323

10 Modeling of Reactive Distillation Eugeny Y. Kenig and Andrzej Górak

10.1 Introduction

In recent decades, a combination of separation and reaction inside a single unit has become more and more popular. This combination has been recognized by the chemical process industries as having favorable economics of carrying out reaction simultaneously with separation for certain classes of reacting systems, and many new processes (called reactive separations) have been invented based on this technology [1–3]. One of the most important examples of reactive separation processes is reactive distillation (RD) in which reaction and distillative separation take place within the same zone of a distillation column. Reactants are converted to products with simultaneous separation of the products and recycling of unused reactants. The RD process can be efficient both in size and cost of capital equipment as well as in energy used to achieve complete conversion of reactants. Since reactor costs are often less than 10% of the capital investment, the combination of a relatively cheap reactor with a distillation column offers great potential for overall savings. Among suitable RD processes are etherifications, nitrations, esterifications, transesterifications, condensations and alcylations [1]. Optimal functioning of RD depends largely on relevant process design, properly selected column internals, feed locations, placement of catalyst as well as on sufficient understanding of the process behavior. All this unavoidably necessitates application of well-working, reliable and adequate process models [4]. The objective of this chapter is to give a comprehensive overview of both basics and peculiarities of RD modeling. A detailed description covers balance equations, mass and heat transfer, reaction kinetics including reaction-mass-transfer coupling, as well as steady-state and dynamic modeling issues. The achievements in the theoretical description are illustrated with several case studies supported by laboratory- and pilot-scale experimental investigations. Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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10 Modeling of Reactive Distillation

10.2 Characteristics of Reactive Distillation

Reactive distillation is especially advantageous for equilibrium-limited reactions. Several reviews have been published in the last decade that give an excellent introduction to and overview of RD processes [1, 5–7]. Among the most attractive features of RD processes are: • increased yield due to overcoming of chemical and thermodynamic equilibrium limitations; • increased selectivity through suppression of undesired consecutive reactions; • reduced energy consumption through direct heat integration in case of exothermic reactions; • avoidance of hot spots by simultaneous liquid evaporation; • separation of close boiling components. These advantages result in reduced capital investment and operating costs. However, the application of RD is somewhat limited by constraints, like, e.g.: • necessity for a common operation window (temperature and pressure) for distillation and reaction; • proper boiling-point sequence; • difficulties in providing proper residence time characteristics. Chemical reactions in RD take place in the liquid phase and, depending on the catalyzing mechanism, RD processes can be divided into homogeneous ones, either autocatalyzed or homogeneously catalyzed, and heterogeneous processes, in which the reaction is catalyzed by a solid catalyst. The latter, often referred to as catalytic distillation, permits an optimum configuration of the reaction and separation zones in a RD column, whereas expensive recovery of liquid catalysts may be avoided. However, it should be mentioned that catalytic distillation shows general drawbacks of heterogeneous catalysis like, e.g., limited catalyst lifetime due to deactivation usually caused by undesired reaction products and increased operating temperature. Table 10.1 gives a short overview of RD applications published to date. 10.2.1 Column Internals for Reactive Distillation

The design of internals for RD is more severe than for conventional nonreactive countercurrent vapor-liquid processes [5–7]. Feasibility and efficiency of a particular RD process strongly depend on the appropriate choice of internal characteristics like liquid residence time, separation efficiency, liquid holdup and pressure drop. Even if column internals for homogeneous RD are similar to noncatalytic column internals, modifications are often necessary to meet optimum reactor design demands. If the use of solid catalysts is necessary, specific bifunctional

10.2 Characteristics of Reactive Distillation

325

Table 10.1 Applications of RD processes (hom: homogeneously catalyzed, het.: heterogeneously catalyzed).

Reaction type

Synthesis

Catalyst

Reference

Esterification

Methyl acetate from methanol and acetic acid Methyl acetate from methanol and acetic acid Ethyl acetate from ethanol and acetic acid Butyl acetate from butanol and acetic acid Ethyl acetate from ethanol and butyl acetate Diethyl carbonate from ethanol and dimethyl carbonate Acetic acid and methanol from methyl acetate and water MTBE from isobutene and methanol ETBE from isobutene and ethanol TAME from isoamylene and methanol Cumene from propylene and benzene Diacetone alcohol from acetone Bisphenol-A from phenol and acetone Monosilane from trichlorsilane Monoethylene glycol from ethylene oxide and water 4-Nitrochlorobenzene from chlorobenzene and nitric acid

hom. het. no data hom. hom. het. het. het. het. het. het. het. no data het. hom. hom.

[8] [9, 10] [11] [12] [13] [14] [15] [16, 17] [18] [19] [20] [21] [22] [23] [24] [25]

Transesterification Hydrolysis Etherification

Alcylation Condensation Dismutation Hydration Nitration

internal structures capable of combining the separation and catalytic functions, the so-called catalytic internals, are needed. The design of catalytic internals for RD is presently based on two general concepts, namely immobilization of commercial catalyst pellets and catalytic activation of conventional internals for vapor-liquid contactors. When using small solid catalyst particles under countercurrent flow conditions, the following additional requirements should be met: • uniform liquid flow in the catalyst bed without stagnant zones and liquid bypassing; • wide vapor and liquid loading ranges without flooding; • limited catalyst abrasion; • possibility of variable catalyst amount; • simple catalyst exchange. One of possible ways to immobilize the catalyst pellets is to use wire gauze mesh, from which objects of different shape can be manufactured. These objects, together with the catalyst itself, form certain structures that should meet the demands of optimum hydrodynamic behavior under vapor-liquid countercurrent flow conditions. Examples of such structures are different random packings made of cylindrical baskets [26], wire-gauze boxes [27], wire-mesh bales [28], etc. In addition to this type of random packings, a significant effort has been made in the last decade towards the development of structured sandwich-type packings. Here, catalyst pellets are immobilized between two sheets of corrugated wire gauze forming a sandwich. These arrangements show hydrodynamic characteristics similar to those of traditional structured packings, e.g. reduced pressure

326

10 Modeling of Reactive Distillation

drop and optimum flow conditions within a wide operating range (KATAMAX® [29], KATAPAK-S® [30]). Furthermore, it is possible to combine catalyst sandwiches with conventional corrugated wire gauze layers. This hybrid sandwich structure meets the demands of flexible catalyst amount and separation efficiency (MULTIPAK® [31], KATAPAK-SP® [32]). Wire-gauze envelopes fi lled with catalyst pellets can also be placed on trays, either across the tray [33] or in the downcomer section [34]. Catalyst beads can be immobilized, as in a fi xed-bed reactor, between two nonreactive distillation trays [35, 36] as well as in an external sidestream reactor [37, 38]. The second concept for catalytic column internals is the use of catalytically active structures instead of those fi lled with catalyst. Such structures are either carrier-supported catalysts or solid catalytic structures. Carrier supports can be coated with any kind of catalyst (e.g. GPP rings and some specific structured packing [39], KATAPAK-M® [40]). Moreover, it is possible to develop solid catalytic structures without any carrier. The so-called BP-rings, for example, are produced by polymerization in an annular gap [39], whereas the monolithic structures are made by extrusion of catalytic material [41].

10.3 Modeling Principles of Reactive Distillation 10.3.1 General Aspects

Reactive distillation occurs in multiphase fluid systems, with an important role of the interfacial transport phenomena. It is an inherently multicompo nent process with much more complexity than similar binary processes. Multicomponent thermodynamic and diffusional coupling in the phases and at the interface is accompanied by complex hydrodynamics and chemical reactions [4, 42, 43]. As a consequence, an adequate process description has to be based on specially developed mathematical models. However, sophisticated RD models are hardly applicable for plant design, model-based control and online process optimization. For such cases, a reasonable model reduction should be applied [44]. An overview of possible modeling approaches for RD is shown in Fig. 10.1. A process model consists of submodels for mass transfer, reaction and hydrodynamics whose complexity and rigor vary within a broad range. For example, mass transfer between the vapor and the liquid phase can be described on the grounds of the most rigorous rate-based approach, with the Maxwell–Stefan diffusion equations, or it can be accounted for by the simple equilibrium-stage model assuming thermodynamic equilibrium between the two phases. Homogeneously catalyzed RD, with a liquid catalyst acting as a mixture component, and autocatalyzed RD present essentially a combination of transport

10.3 Modeling Principles of Reactive Distillation

Figure 10.1 Modeling approaches for reactive distillation.

phenomena and reactions taking place in a two-phase system with an interface. In these systems, reactions may occur both in the bulk and in the fi lm region. For slow reactions, a reaction account in the bulk phase only is usually sufficient. For heterogeneous systems, an additional consideration of the phenomena in the solid catalyst phase is usually necessary. In this case, very detailed models using intrinsic kinetics and covering mass transport inside the porous catalyst arise (see, e.g. [45–47]). However, it is often assumed that all internal (inside the porous medium) and external mass-transfer resistances can be lumped together [10, 48]. The catalyst surface is then totally exposed to the liquid bulk conditions and can be completely described by the bulk variables. This results in the socalled pseudohomogeneous models. By this name, a similarity to a simpler homogeneous bulk-phase reaction is reflected (see Fig. 10.1). If the reaction (either homogeneous or heterogeneous) is fast, it can be described using the data on chemical equilibrium only. Modeling of hydrodynamics in multiphase vapor-liquid contactors includes an appropriate description of axial dispersion, liquid holdup and pressure drop. The correlations giving such a description have been published in numerous papers and are collected in several reviews and textbooks (e.g. [49, 50]). Nevertheless, there is still a need for a better description of hydrodynamics in catalytic column internals, this is being reflected by research activities in progress [51–53].

327

328

10 Modeling of Reactive Distillation

In order to model large industrial reactive separation units, a proper subdivision of a column apparatus into smaller elements is usually necessary. These elements (the so-called stages) are identified with real trays or segments of a packed column. They can be described using different theoretical concepts, with a wide range of physicochemical assumptions and accuracy.

10.3.2 Equilibrium-stage Model

In recent decades, modeling and design of RD has usually been based on the equilibrium-stage model. Since 1893, as the first equilibrium-stage model was published by Sorel [54], numerous publications discussing various aspects of model development, application and solution have appeared in the literature [55]. The equilibrium-stage model assumes that the vapor stream leaving a tray or a packing segment is in thermodynamic equilibrium with the correspondent liquid stream leaving the same tray or segment. In the case of RD, the chemical reaction has to be additionally taken into account, either via reaction equilibrium equations, or via rate expressions integrated into the mass and energy balances. Different model assumptions reflect the relation between the mass transfer and reaction rates. The definition of the Hatta number, representing the maximum reaction rate with reference to that of the mass transfer, helps to discriminate between very fast, fast, average and slow chemical reactions [56, 57]. If the reaction system considered is fast, the process can be satisfactorily described assuming reaction equilibrium. Here, a proper modeling approach is based on the nonreactive equilibrium-stage model, extended by the chemical equilibrium relationship. An alternative approach proposed by Davies and Jeffreys [13] includes two separate steps. First, the concentrations and flow rates of the leaving streams are calculated with the simple nonreactive equilibrium-stage model. Afterwards, the leaving concentrations are adapted by using an additional equilibrium reactor concept. However, the latter approach does not consider direct interactions between the chemical and thermodynamic equilibrium. Such descriptions can be appropriate for instantaneous and very fast reactions. In contrast, if the chemical reaction is slow, the reaction rate dominates the whole process, and therefore, the reaction kinetics expression has to be integrated into the mass and energy balances. This concept has been used in a number of RD studies (e.g., [58, 59]). In practice, thermodynamic equilibrium can seldom be reached within a single stage. Therefore, some correlation parameters, like tray efficiencies or HETS values, have been introduced to adjust the equilibrium-based theoretical description to the reality. For multicomponent mixtures, however, the application of this concept is often difficult due to diffusional interactions of several components [60, 61]. These effects cause an unpredictable behavior of the efficiency factors, which are different for each component, vary along the column height and show a strong dependency on the component concentration [42, 61, 62].

10.3 Modeling Principles of Reactive Distillation

The acceleration of mass transfer due to chemical reactions in the interfacial region is often accounted for via the so-called enhancement factors [57, 63, 64]. They are either obtained by fitting experimental results or derived theoretically on the grounds of simplified model assumptions. It is not possible to derive the enhancement factors properly from binary experiments, and significant problems arise if reversible, parallel or consecutive reactions take place. 10.3.3 Rate-based Approach

A more physically consistent way to describe a column stage is known as the ratebased approach [42, 65, 66]. This approach implies that actual rates of multicomponent mass and heat transfer and chemical reactions are taken into account directly. Mass transfer at the vapor-liquid interface can be described using different theoretical concepts [4, 42]. Most often, the film model [67] or the penetration/surface renewal model [63, 68] are used, whereas the model parameters are estimated via experimental correlations. In this respect the fi lm model is advantageous, since there is a broad spectrum of correlations available in the literature, for all types of internals and systems. For the penetration/surface renewal model, such a choice is limited, and therefore, in this chapter we focus on the film model. In the fi lm model (Fig. 10.2), it is assumed that all of the resistance to mass transfer is concentrated in thin fi lms adjacent to the phase interface and that transfer occurs within these fi lms by steady-state molecular diffusion alone. Outside the fi lms, in the bulk fluid phases, the level of mixing is so high that there is no composition gradient at all. This means that in the fi lm region we have one-dimensional diffusional transport normal to the interface. Multicomponent diffusion in the fi lms can be rigorously described by the Maxwell–Stefan equations derived from the kinetic theory of gases [69]. The

Figure 10.2 Film model for a differential packing segment.

329

330

10 Modeling of Reactive Distillation

Maxwell–Stefan equations connect diffusion fluxes of the components with the gradients of their chemical potential. With some modification these equations take a generalized form in which they can be used for the description of real gases and liquids [42]: n

x iN j − x j N i

j =1

c t Dij

di = ∑

; i = 1, . . . , n

(10.1)

where di is the generalized driving force: di =

x i dµi ; i = 1, . . . , n ℜT dz

(10.2)

Similar equations can be also written for the vapor phase. Along these lines, the vapor-liquid mass transfer is modeled as a combination of the two-fi lm model presentation and the Maxwell–Stefan diffusion description. In this stage model, the equilibrium exists only at the interface. A reasonable simplification for RD is represented by the effective diffusivity approach, provided that the effective diffusion coefficients are estimated properly. These coefficients can be obtained, for instance, via a relevant averaging of the Maxwell–Stefan diffusivities [42]. The fi lm thickness represents a model parameter that can be estimated from the mass-transfer coefficient correlations. Such correlations govern the masstransport dependence on physical properties and process hydrodynamics and are available in the literature (see, e.g., [42, 50, 57, 70]). 10.3.3.1 Balance Equations Let us first consider the steady-state RD operation mode. The mass-balance equations of the traditional multicomponent rate-based model (see, e.g. [42, 43]) are written separately for each phase, and, as chemical reactions take place in the liquid phase, the steady-state liquid-phase balance equation should be extended to include the reaction source term:

0=− 0=

d (LxiB ) + (N LiB aI + RLiB φL ) Ac ; i = 1, . . . , n dl

d (GyiB ) − NGiB aI Ac ; i = 1, . . . , n dl

(10.3) (10.4)

In Eqs. (10.3) and (10.4), it is assumed that transfers from the vapor to the liquid phase are positive. Equations (10.3) and (10.4) represent the mass balances for continuous systems (packed columns). For discrete systems (tray columns), the differential terms transform to finite differences and the balances are reduced to algebraic equations [42, 43].

10.3 Modeling Principles of Reactive Distillation

The bulk-phase balances are completed by the summation equations for the liquid and vapor bulk mole fractions: n

∑x

B i

=1

(10.5)

i =1 n

∑y

B i

=1

(10.6)

i =1

The volumetric liquid holdup fL depends on the vapor and liquid flows and is calculated from empirical correlations (e.g., [50]). For the determination of axial temperature profi les, differential energy balances are formulated including the product of the liquid molar holdup and the specific enthalpy as energy capacity. The energy balances written for continuous systems are as follows: 0=− 0=

d (LhBL ) + (Q BL aI − RBLφL∆HRL0 ) Ac dl

d (GhGB ) − (Q GBaI ) Ac dl

(10.7) (10.8)

If the dynamic process behavior has to be considered, Eqs. (10.3), (10.4), (10.7) and (10.8) become partial differential equations including derivatives of the holdup with respect to time (see more details in Section 10.3.3.4). The molar fluxes Ni are expressed via the diffusional fluxes by n

N i = J i + x iN t ; i = 1, . . . , n; N t = ∑ N i

(10.9)

i =1

To relate the multicomponent mass-transfer rates to binary mass-transfer coefficients, the method of Krishna and Standart [71] can be used. By this method, the diffusional fluxes for the liquid phase are calculated from J = −c tav,L ⋅kLav  ⋅Γ Lav  ( x B − x I )

(10.10)

for which the matrix of mass-transfer coefficients is defined as: kLav  = RLav 

−1

(10.11)

with av = RLii

n x iav x av + ∑ k ; i = 1, . . . , n − 1 κ in k = i κ ik k≠ j

 1 1  av = − x iav  − RLij  ; i, j = 1, . . . , n − 1 κ κ  ij in 

(10.12)

331

332

10 Modeling of Reactive Distillation

Equations (10.9)–(10.12) can also be written for the vapor phase, for which [Γ Gav ] is usually taken as unit matrix. The binary mass-transfer coefficients k ij can be extracted from suitable binary mass-transfer correlations, using the appropriate Maxwell–Stefan diffusion coefficients Dij . A number of mass-transfer correlations for both catalytic and noncatalytic column internals are available in the literature (see, e.g., [42, 50, 57, 70, 72–74]). According to the linearized theory of Stewart and Prober [60] and Toor [61], the matrices [R av] and [Γav] are evaluated using an average molar fraction defined as x iav =

yiI + yiB ; i = 1, . . . , n 2

(10.13)

All required physical properties, for example diffusivities, are calculated on the same basis. At the interface, phase equilibrium is assumed: yiI = K ieq x iI ; i = 1, . . . , n

(10.14)

where vapor-liquid equilibrium constants K ieq are determined using selected thermodynamic models, such as UNIQUAC or NRTL, and a suitable equation for the saturated vapor pressure. The heat fluxes comprise a convective and a conductive part: n

n

i =1

i =1

Q GB = κ GT (TGB − T I ) + ∑ N i hGi = Q LB = κ LT (T I − TLB ) + ∑ N i hLi

(10.15)

whereas κTL and κTG are heat-transfer coefficients that can be, e.g., determined using the Chilton–Colburn analogy [75, 76]. 10.3.3.2 Mass Transfer and Reaction Coupling in the Fluid Film The component fluxes NBi entering into Eqs. (10.3) and (10.4) are determined based on the mass transport in the fi lm region. As the key assumptions of the fi lm model result in the one-dimensional mass transport normal to the interface, the differential component balance equations including simultaneous mass transfer and reaction in the fi lm are as follows:

dN Li − RLi = 0; i = 1, . . . , n dz

(10.16)

Equations (10.16) that are generally valid for any liquid fi lm phase represent simply a differential mass balance for the fi lm region with the account of the source term due to the reaction. To link this balance to the process variables, like component concentrations, some additional relationships, often called constitutive relations (see [42]) are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (10.1) and (10.2)) whereas for the source terms, they follow from the reaction kinetics description.

10.3 Modeling Principles of Reactive Distillation

The latter strongly depends on the specific reaction mechanism, stoichiometry, and presence or absence of parallel reactions schemes [64]. The rate expressions for R i usually represent nonlinear dependences on the mixture composition and temperature. Equations (10.16) should be completed by the boundary conditions relevant to the fi lm model. These conditions specify the values of the mixture composition at both liquid fi lm boundaries: x i ( z = 0) = x iI , x i ( z = δ L ) = x iB ; i = 1, . . . , n

(10.17)

Combining Eqs. (10.16) with the boundary conditions (10.17) written in a vector form and using the constitutive relations, like Eqs. (10.1) and (10.2), we obtain a vector-type boundary-value problem that permits the component concentration profi les to be obtained as functions of the fi lm coordinate. These concentration profi les, in turn, allow the component fluxes to be determined. Thus the boundary-value problem describing the film phenomena has to be solved in conjunction with all other model equations. The bulk values in both phases are found from the balance relations, Eqs. (10.3) and (10.4). The interfacial liquid-phase concentrations x Ii are related to the relevant vapor-phase concentrations y Ii by the thermodynamic equilibrium relationships and by the continuity condition for the molar and heat fluxes at the interface [42, 43]. The system of equations is completed by the necessary linking conditions between the bulk and fi lm phases. For problems with reaction and mass-transfer coupling, e.g., like in Eqs. (10.16), it is always essential whether the reaction rate is comparable with the rate of diffusion [56, 57] or not. In most RD processes developed up to now, reaction rates are rather slow, and thus they determine the whole RD process rates, at least in the reaction zone. For such cases, the influence of the film reactions is usually negligible. This may, however, change for faster reactions. 10.3.3.3 Nonideal Flow Behavior in Catalytic Column Internals The mass balances (Eqs. (10.3) and (10.4)) assume plug-flow behavior for both the vapor and the liquid phase. However, real flow behavior is much more complex and constitutes a fundamental issue in multiphase reactor design. It has a strong influence on the column performance, for example via backmixing of both phases, which is responsible for significant effects on the reaction rates and product selectivity. Possible development of stagnant zones results in secondary undesired reactions. To ensure an optimum model development for catalytic distillation processes, we performed experimental studies on the nonideal flow behavior in the catalytic packing MULTIPAK® [77]. The experimental results confirm that the fluid flow in the packing deviates from the plug-flow behavior (Fig. 10.3). Calculated axial dispersion coefficients are about 10 −4 –10 −2 m2/s, which is several orders of magnitude larger than that of molecular diffusion (Fig. 10.4). Therefore, in the investigated operating range, nonideal mixing effects are caused by hydrodynamic rather than by molecular diffusion effects. Calculated Bodenstein numbers for the packing are an order of magnitude smaller than for fi xed-bed reactors, which may be caused by two

333

334

10 Modeling of Reactive Distillation 1,6 B = 5,09 m3m-2h-1 ADM-Model B = 12,73 m3m-2h-1 ADM-Model

1,4

E θ [-] 1,2 1 0,8 0,6 0,4 0,2 0 0

0,5

1

1,5

2

2,5

3

θ [-]

Figure 10.3 Comparison between the experimental RTD curve for the catalytic packing MULTIPAK® (dC = 0.1 m) and the axial dispersion model (ADM) [82].

Figure 10.4 Axial dispersion coefficients of the catalytic packing MULTIPAK® (dC = 0.1 m) calculated with axial dispersion model [82].

effects: the occurrence of stagnant zones in the catalyst layer and liquid bypassing due to the hybrid structure of the catalytic packing [77]. Regarding the reactive tray hydrodynamics, the concentration distribution on a crossflow tray is also not even. The vapor concentration changes gradually when it rises through the liquid on the tray. The liquid concentration also changes

10.3 Modeling Principles of Reactive Distillation

gradually from the inlet to the outlet of the tray. Traditionally, this phenomenon has been lumped together with many other factors affecting performance of the tray to yield the stage efficiency. The situation becomes even more complicated if a reaction takes place on the plates, and there have been only very few attempts to tackle this issue. Among those, Alejski [78] presented a mixed pool model for RD. However, in his model, the mass-transfer modeling applied to the individual pools was based on the traditional equilibrium-stage model and efficiency concept. Recently, Higler et al. [79] suggested a nonequilibrium cell model for RD tray columns. In this work, a single distillation tray is represented by a series of cells along the vapor and liquid flow paths, whereas each cell is described by the fi lm model. Using different number of cells in both flow paths allows various flow patterns to be described. However, a consistent experimental determination of necessary model parameters (e.g. cell fi lm thickness) appears difficult, whereas the complex iterative character of the calculation procedure in the dynamic case limits the applicability of the nonequilibrium cell model. In the horizontal direction on the vapor side, it may be assumed that either vapor is totally mixed before it enters the tray, or that, after being separated from the liquid on the tray below, the vapor does not mix at all. The real situation is obviously between these two limiting cases. In small diameter columns, it is very close to complete mixing, while in large-diameter columns the vapor is less mixed. The horizontal liquid flow pattern is very complicated due to the mixing by vapor, dispersion, and the round cross section of the column. On single-pass trays, the latter results in the flow path, which first expands and then contracts. A rigorous modeling of this flow pattern is very difficult, and usually the situation is simplified by assuming that the liquid flow is unidirectional and the major deviation from the plug flow is the turbulent mixing or eddy diffusion. In [80], two different models, the eddy-diffusion model and the mixed pool model were developed and tested in the context of the rate-based approach for RD trays. The details of these models can be found in [81]. For packed columns, a promising approach is represented by differential models like the axial dispersion model [82] and the piston flow model with axial dispersion and mass exchange [83]. Experimental studies show that the axial dispersion model gives an appropriate description of the nonideal flow behavior of the liquid phase in catalytic packings (see Fig. 10.4) [77]. When applying the axial dispersion model to cover this nonideality, the liquid-phase mass balances (Eqs. (10.3)) transform to the following equations: 0=

∂ Dax ∂ 2 (LxiB ) − ∂ l (LxiB ) + (N LiB aI + RLiB φL ) Ac ; i = 1, . . . , n uL ∂ l 2

(10.18)

A thorough investigation of the influence of flow nonideality in catalytic packings on the process behavior of specific RD processes has been published in [84].

335

336

10 Modeling of Reactive Distillation

10.3.3.4 Dynamic Modeling The steady-state modeling is not sufficient for batch and semibatch RD processes or if one tries to optimize the start-up and shut-down phases of the process. In this case, a knowledge of dynamic process behavior is necessary. Further areas where the dynamic information is crucial are the process control as well as safety issues and training. In the dynamic rate-based stage model, molar holdup terms have to be considered in the mass balance equations, whereas the change of both the specific molar component holdup and the total molar holdup are taken into account. For the liquid phase, these equations are as follows:

∂ ∂ U Li = − ( Lx iB ) + (N LiB a I + RLiB φL ) Ac ; i = 1, . . . , n ∂t ∂l

(10.19)

U Li = x iBU Lt ; i = 1, . . . , n

(10.20)

The vapor holdup can often be neglected due to the low vapor-phase density, and the component balance equation reduces to Eq. (10.4) (see also [85]). The dynamic formulation of the model equations requires a careful analysis of the whole system in order to prevent high-index problems during the numerical solution [86]. As a consequence, a consistent set of initial conditions for the dynamic simulations and a suitable description of the hydrodynamics have to be introduced. For instance, pressure drop and liquid holdup must be correlated with the vapor and liquid flows.

10.4 Case Studies

In this section, four examples illustrating the application of the rate-based approach discussed above to the RD modeling are presented. The systems selected are methyl acetate synthesis, MTBE synthesis, ethyl acetate synthesis and transesterification of dimethyl carbonate. In the first example, dynamic process modeling is highlighted, whereas in three other examples, different aspects of steady-state modeling are discussed. 10.4.1 Methyl Acetate Synthesis 10.4.1.1 Process Description The synthesis of methyl acetate from methanol and acetic acid is a slightly exothermic equilibrium-limited liquid-phase reaction:

CH3OH + (CH3 )COOH ↔ (CH3 )COO(CH3 ) + H2O

(R10.1)

The low equilibrium constant and the strongly nonideal system thermodynamics that gives rise to the binary azeotropes methyl acetate/methanol and methyl ace-

10.4 Case Studies

tate/water make this reaction system interesting as a possible RD application [8]. Therefore, methyl acetate synthesis has been chosen as a test system and investigated in a semibatch RD column. Since the process is carried out under atmospheric pressure, no side reactions in the liquid phase occur [87]. A semibatch distillation column with a column diameter of 100 mm and a reactive packing height of 2 m (MULTIPAK I®) in the bottom section and an additional meter of conventional packing (ROMBOPAK 6M®) in the top section was used. The flow sheet of the column is shown in Fig. 10.5. At first, the distillation still is charged with methanol – the low boiling reactant – and heated under total reflux until steady-state conditions are achieved. At this moment, acetic acid – the high boiling reactant – is fed above the reaction zone to the second distributor. After 30 min the reflux ratio is turned from infinity to two and the product withdraw at the top of the column begins. During the column operations, the liquid-phase concentration profi les along the column and the temperature profi les are measured. For the determination of the liquid-phase composition, two methods are applied simultaneously. On the one hand, samples

Figure 10.5 Flow sheet of the catalytic RD column for methyl acetate synthesis (taken from [84]).

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10 Modeling of Reactive Distillation

are taken and analyzed by gas chromatography. On the other hand, an online NIR spectrometer is used to determine the concentration without taking any samples [88]. 10.4.1.2 Process Modeling The model is based on the film theory and comprises the material and energy balances of a differential element of the two-phase volume in the packing. The classical fi lm model shown in Fig. 10.2 is extended here to consider the catalyst phase (Fig. 10.6). A pseudohomogeneous approach is chosen for the catalyzed reaction (see also [89, 90]), and the correspondent overall reaction kinetics is determined by fi xed-bed experiments [9]. This macroscopic kinetics includes the influence of the liquid distribution and mass-transfer resistances at the liquidsolid interface as well as diffusional transport phenomena inside the porous catalyst. The reaction kinetics is integrated into the mass balances, and the liquid holdup, as an accumulation term, is accounted for simultaneously, as in Eqs. (10.19) and (10.20). As mentioned earlier, in the dynamic simulations, the vapor holdup can be neglected due to the low vapor-phase density. Ranzi et al. [91] found that the full energy balances including the accumulation term have to be considered in the liquid phase in order to predict the correct dynamic process behavior. Therefore, the differential dynamic liquid-phase energy balance is applied as follows:

∂ EL ∂ 0 = − ( LhLB ) + (Q LBa I − RLBφL∆HRL ) Ac ∂t ∂l

(10.21)

with QBL defined by Eq. (10.15) Similar to the mass-balance equation, the vaporphase energy balance simplifies to Eq. (10.8).

interface gas bulk phase

liquid bulk phase

NGi = NLi

yiB

catalyst

xiI xiB yiI

gas

δG

δL

Figure 10.6 Extended film model for a differential packing segment with heterogeneous catalyst.

liquid

10.4 Case Studies

Experimental studies were carried out to derive correlations for mass-transfer coefficients, reaction kinetics, liquid holdup and pressure drop for the new catalytic packing MULTIPAK (see [9, 10]). Suitable correlations for ROMBOPAK 6M® were taken from [70] and [92]. The vapor-liquid equilibrium is calculated using the modification of the Wilson method [9]. For the vapor phase, the dimerization of acetic acid is taken into account using the chemical theory to correct vaporphase fugacity coefficients [93]. Binary diffusion coefficients for the vapor phase and for the liquid phase are estimated via the method purposed by Fuller et al. and Tyn and Calus, respectively (see [94]). Physical properties like densities, viscosities and thermal conductivities are calculated from the methods given in [94]. Heat losses through the column wall are measured at pilot scale. 10.4.1.3 Results and Discussion Figures 10.7 and 10.8 show the liquid-phase compositions for the reboiler and condenser as functions of time. After column start-up, the concentration of methanol decreases continuously whereas the distillate mole fraction of methyl acetate reaches about 90%. A comparison of the rate-based simulation using the Maxwell–Stefan diffusion equations (Eq. (10.1)) and experimental results for the liquid-phase composition at the column top and in the reboiler demonstrates their satisfactory agreement. Figure 10.9 shows the simulation results obtained after an operation time of 10 000 s with different modeling approaches: the model including the Maxwell–Stefan diffusion description, the model with effective diffusion coefficients, and the equilibrium-stage model. Both the Maxwell–Stefan

Figure 10.7 Liquid bulk mole fractions in the column reboiler: lines – simulations; dots – experiments.

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10 Modeling of Reactive Distillation

Figure 10.8 Liquid bulk mole fractions in the column condenser: lines – simulations; dots – experiments.

Figure 10.9 Axial liquid bulk concentration profiles for the semibatch column (t = 10 000 s).

10.4 Case Studies

and effective diffusion model show similar results, whereas the equilibrium-stage model is able to describe the process behavior qualitatively. This can be explained by the low reaction rate dominating the entire process kinetics. 10.4.2 Methyl Tertiary Butyl Ether 10.4.2.1 Process Description The synthesis of methyl tertiary butyl ether (MTBE) has been one of the most important applications of RD. MTBE is produced via an acid-catalyzed reaction between methanol and isobutylene as shown by (R10.2):

CH3OH + C(CH3 )2CH2 ↔ C(CH3 )3OCH3

(R10.2)

This reaction has been extensively investigated by several authors, e.g. [95–97]. MTBE synthesis has been investigated both theoretically and experimentally [80, 98]. In this paper, we present some results for a pilot-scale RD column, with a catalytic section in the middle part. This section consists either of a packed bed of catalytically active rings (see [99]) or of catalytic packing MULTIPAK® [31]. The rectifying and stripping sections are fi lled with Intalox Metal Tower Packing. The methanol is fed just above and the hydrocarbon feed just below the catalyst section. 10.4.2.2 Process Modeling The mathematical description considered in Section 10.3.3 was used as a modeling basis for the specially developed completely rate-based simulator [80]. This tool consists of several blocks including model libraries for physical properties, mass and heat transfer, reaction kinetics and equilibrium as well as specific hybrid solver and thermodynamic package. It also contains different hydrodynamic models (e.g., completely mixed liquid – completely mixed vapor, completely mixed liquid – vapor plug flow, mixed pool model, eddy diffusion model [80]) and a model library of hydrodynamic correlations for the mass-transfer coefficients, interfacial area, pressure drop, holdup, weeping and entrainment that cover a number of different column internals and flow conditions. The simulator is able to treat different heterogeneous reaction kinetics, depending on the reaction rate and its character. For example, a detailed model for the heterogeneous catalyst mass-transfer efficiency can be used, which is based on the approach of [99]. When applying this type of kinetic model, the intrinsic kinetics data are needed (see Section 10.3.1). Another way is the pseudohomogeneous approach with effective kinetics expressions, by which the kinetics description is introduced as source terms into the balance equations (see Eqs. (10.3) and (10.4)). For the system considered here, the reaction is slow as compared to the masstransfer rate. For this reason the pseudohomogeneous approach is used, the reaction being considered in the liquid bulk only.

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The thermodynamic equilibrium is based on the UNIQUAC model [100]. The liquid-phase binary diffusivities are determined using the method of Tyn and Calus (see [94]) for the diluted mixtures corrected by the Vignes equation [42] to account for finite concentrations. The vapor-phase diffusion coefficients are assumed constant. The reaction kinetics parameters are taken from [101]. The vapor and liquid binary mass-transfer correlations were calculated for the inert packing and the catalytic rings with the correlation of Onda et al. [102].

10.4.2.3 Results and Discussion Figure 10.10 demonstrates the simulated and measured concentration profi les for the pilot column with the reactive section filled with catalytically active rings. In the simulations, four components, namely, methanol, isobutene, MTBE and 1butene, were chosen to represent the chemical system under consideration. Here, segment 1 corresponds to the reboiler. A satisfactory agreement between calculated and measured values can be clearly observed. In Fig. 10.11, the simulation results for the column packed with MULTIPAK® are shown. Here, 16 components are considered, and, again, the liquid bulk composition profi les agree well with the experimental data. Figure 10.12 demonstrates a comparison of experimental and simulated results for all 11 test runs. Generally, good agreement between the calculated and experimental conversion of isobutene can be established, with an average deviation of less than 5%. The model is also able to predict well the distillate compositions (Fig. 10.13).

Figure 10.10 Calculated and experimental liquid compositions for experiments with catalytically active rings.

10.4 Case Studies

Figure 10.11 Calculated and experimental liquid compositions for experiments with catalytic-structured packing.

Figure 10.12 Experimental and simulated conversion of isobutene (all 11 test runs) for the column with the reactive section filled with catalytic-structured packing Montz Multipak.

10.4.3 Ethyl Acetate Synthesis 10.4.3.1 Process Description Ethyl acetate is a commodity chemical substance produced and used in facilities worldwide [103]. Ethyl acetate is primarily used as a solvent. Other applications of ethyl acetate as well as conventional production processes are highlighted in [103, 104].

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Figure 10.13 Experimental and simulated liquid distillate compositions (all 11 test runs) for the column with the reactive section filled with catalytic structured packing Montz Multipak.

Table 10.2 Calculated azeotropic data for the ethyl acetate system.

Azeotrope

ETAC [wt%]

ETOH [wt%]

H2O [wt%]

T [ºC]

ETOH-H2O ETAC-ETOH ETAC-H2O ETAC-ETOH-H2O

– 70.2 91.3 82.0

95.5 29.8 – 9.8

4.5 – 8.7 8.2

78.1 71.8 70.9 70.4

The formation of ethyl acetate is equilibrium limited and hence conversion can be increased via RD. The main chemical reaction is given as: (CH3 )COOH + CH3CH2OH ↔ (CH3 )COO(CH2CH3 ) + H2O

(R10.3)

The boiling points of the pure components at atmospheric pressure are as follows: ethyl acetate (ETAC) 77.2 ºC; ethanol (ETOH) 78.3 ºC; water (H2O) 100.0 ºC; acetic acid (HAC) 118.0 ºC. There are three binary azeotropes and one ternary azeotrope summarized in Table 10.2, with respective boiling points at atmospheric pressure. The normal boiling points for the pure components as well as the compositions of the azeotropes are obtained from ASPEN Properties Plus® using UNIQUAC and show satisfactory agreement with the data available elsewhere [105]. A feasibility study based on RD lines was presented in [105]. Due to the presence of minimum-boiling azeotropes it is not possible to obtain pure ethyl acetate

10.4 Case Studies

Figure 10.14 Sketch of the basic column configuration for ethyl acetate synthesis.

at the top of the column. A liquid–liquid phase separation of the distillate stream allows a further enrichment of ethyl acetate. Phase separation is only possible at low ethanol concentrations, since a higher ethanol content prevent phase splitting. Thus, a sufficient ethanol conversion is important for the process. This can be achieved by an excess of acetic acid in the column feed [106]. A homogeneously catalyzed RD for the ethyl acetate synthesis was investigated by Kenig et al. [105], whereas a successful model validation was performed using the experimental data obtained in a glass bubble cup tray column. In recently developed processes for the ethyl acetate synthesis, heterogeneously catalyzed RD has also been applied. Kolena et al. [107] and Wu and Lin [108] suggested the combination of a pre-reactor and a RD column to carry out the reaction. The only difference between these processes is the location of the feed to the RD column. The process of Kolena et al. [107] is nowadays commercialized by Sulzer Chemtech Ltd. [109]. Here we give another example of heterogeneously catalyzed ethyl acetate synthesis via RD, with two different column scales studied, namely a 50-mm and a 162-mm diameter columns (see [52]). The principal column setup is shown in Fig. 10.14. The columns consist of three packed sections, whereas the middle part is equipped with structured catalytic internals. The reactants are fed above and below the reactive section. The RD columns studied are coupled with a liquid–liquid separator at the top, to process the distillate streams. This permits a further enrichment of ethyl acetate. Two different types of packings are selected for investigation, namely KATAPAK®-S and MULTIPAK®. Both types combine the benefits of modern structured packings, such as low pressure drop and high throughput, and offer the advantages of heterogeneous catalysis. Details on the applied internals are given in Table 10.3.

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Table 10.3 Characteristics of the applied structured catalytic

internals for the ethyl acetate system.

Diameter [mm] Specific surface area [m2/m3 ] Material Catalyst fraction [vol%] HETP [m]

KATAPAK®-S Laboratory

MULTIPAK®-1

MULTIPAK®-2

KATAPAK®-S 250.Y

50 270 wire gauze 0.158 0.3

50 317 wire gauze 0.160 0.25

50 262 wire gauze 0.277 0.3

162 125 wire gauze 0.250 0.9

Table 10.4 Pilot-plant characteristics for the ethyl acetate synthesis.

Column diameter Rectifying section (A) Reactive section (B)

Stripping section (C) Operating pressure Decanter operated at Feed 1 (above (B)) Feed 2 (below (B)) Bottom product Organic distillate

Laboratory-scale column

Pilot-scale column

50 mm 1.0 m (Sulzer DX) 1.0 m (KATAPAK®-S Lab) or 1.0 m (MULTIPAK®-1) or 1.0 m (MULTIPAK®-2) 1.0 m (Sulzer BX) Atmospheric 20 ºC

162 mm 4.3 m (MELLAPAK® 500.Y) 5.3 m (KATAPAK®-S 250.Y)

3.1 m (MELLAPAK® 500.Y) Atmospheric 40 ºC Acetic acid Ethanol Acetic acid (water) Ethyl acetate (water, ethanol, acetic acid)

The separation efficiency for the different laboratory-scale packings is considerably high (HETS between 0.33 and 0.25m), while for the industrial-type KATAPAK®-S 250.Y it is lower because of the smaller specific surface area (see Table 10.3). The laboratory-scale internals mainly differ in catalyst content. Further information on KATAPAK®-S and MULTIPAK® is given elsewhere (see, e.g. [105]). The relevant information on feeds and products as well as the details on the column scales is given in Table 10.4. For both column scales, there is a separation section below and above the catalytic section. The bottom section is used to separate acetic acid from the other components and the top section is mainly used to keep acetic acid in the reaction zone. The columns are operated at atmospheric pressure. The first column is a laboratory-scale apparatus, with 50 mm diameter and a total packing height of 3 m. The column is made from glass and has a silver-lined vacuum jacket. At this scale, different catalytic column internals were tested.

10.4 Case Studies

The second column is a pilot-scale unit, with 162 mm diameter and a packing height of about 12 m. It has a setup similar to that of the laboratory column (see Table 10.4). The distillate flows through a decanter. In the case of liquid–liquid phase separation, the aqueous phase is withdrawn completely. A part of the organic phase is fed back to the column as reflux. The acetic acid feed (in some experiments enriched with reaction products) is located between the rectifying and catalytic sections. The ethanol feed is located between the catalytic and stripping sections. For the laboratory-scale column, the feed is at room temperature, while for the pilot-scale setup, the feed is preheated. To avoid an additional purification step, it is desirable to withdraw nonconverted excess acid at the bottom that can then be recycled to the acetic acid feed. Therefore, in most experiments, an excess of acetic acid was fed to the column. 10.4.3.2 Process Modeling The steady-state rate-based model used in this case study was similar to the model from Section 10.4.2, with the pseudohomogeneous mode for the reaction kinetics and the Maxwell–Stefan diffusion description. The model was implemented into ASPEN Custom ModelerTM together with another, simpler rate-based model, with effective diffusivities. The latter allows starting values for the Maxwell–Stefanbased model to be generated, thus enhancing convergence. Effective diffusivities were used for the calculation of the mass-transfer coefficients. In contrast to the binary Maxwell–Stefan diffusivities, the effective diffusivities were calculated via available procedures in ASPEN Custom ModelerTM, whereas the Wilke–Chang model was used for the liquid phase and Chapman– Enskog–Wilke–Lee model for the vapor phase [94]. In the full model, computationally intensive matrix operations for the Maxwell–Stefan equations are necessary. The model has been further extended to consider the presence of liquid–liquid separation [110, 111]. As mentioned earlier, the UNIQUAC method is used for the calculation of the activity coefficients, since the method shows a satisfactory agreement with the data available in [105]. For the description of the vapor phase, the Hayden– O’Connell equation of state is used to account for the nonidealities due to the dimerization of acetic acid. Liquid–liquid equilibrium data from the DECHEMA data series [112] is applied for the ternary subsystem ethyl acetate/ethanol/water in combination with the liquid–liquid data available in Properties PlusTM. Relevant hydrodynamic and mass-transfer correlations were determined, for example, by Górak and Hoffmann [10] for MULTIPAK® at the laboratory scale and by Kolodziej et al. [113] at the pilot scale, whereas in [74] KATAPAK®-S is investigated at the pilot scale. 10.4.3.3 Results and Discussion A series of simulations was performed for the laboratory-scale column, and a very good agreement between simulated and experimental data was obtained (see

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Figure 10.15 Simulated (lines with empty symbols) and experimental (solid symbols) column profiles (ethyl acetate system, lab-scale column, no liquid–liquid phase separation).

[110]). In Fig. 10.15, concentration and temperature profi les along the column are presented for a selected experiment. The average deviation for conversion and product purity is about 2%. For the experiments with KATAPAK®-S, no liquid– liquid phase separation was observed at the laboratory scale, mainly because there was still too much ethanol in the distillate fraction preventing the phase splitting. The reason is that the catalytic section is only 1 m high and the catalyst fraction of KATAPAK®-S Lab is lower as compared to MULTIPAK®-2 or the industrial KATAPAK®-S 250.Y (see Table 10.3). It was possible to achieve the phase separation in the liquid–liquid separator with the laboratory-scale setup. On the other hand, the size of the catalytic section at the pilot scale enabled even higher conversions and thus liquid–liquid separation, due to the small ethanol fraction in the distillate. An important reason for the simulation study of the ethyl acetate process is that experimental investigations at the laboratory scale reveal an extremely low conversion under certain operating conditions [110]. In Fig. 10.16, product purity and conversion are depicted vs. heat duty varied over a wide range. The influence of different catalytic column internals is clearly seen. Here, the molar feed ratio of acetic acid and ethanol is 1.02, the reflux ratio is 4.41, and the total feed rate is 1.4 kg/h. In this example, no liquid–liquid phase separation is considered. The process behavior is very complex, regardless of the applied internals, and the investigated heat duty region can be subdivided into three parts. At low heat duties, ethanol is mainly accumulated in the catalytic section of the column, and, as expected, conversion increases with increasing reboiler heat duty, while the

10.4 Case Studies

Figure 10.16 Product purity (w) and conversion (X) vs. heat duty for different catalytic internals (ethyl acetate system, lab-scale column).

product purity slightly decreases. In the middle part, at moderate heat duties, water, the intermediate boiling component, accumulates in the catalytic section of the column, this results in a steep decrease in conversion and product purity. A further increase in the heat duty results in an increasing accumulation of acetic acid in the catalytic section, and conversion increases more rapidly due to the exponent 1.5 dependence of the acetic acid concentration in the kinetic expression. The last regime is undoubtedly the most suitable operating frame for the column at high distillate-to-feed ratios. With respect to the performance of the two internals, differences in maximum conversion and maximum product purity are attributed to different catalyst contents (significantly higher in MULTIPAK-2, see Table 10.3). In some regions, however, these differences decrease or fade out. Generally, at lower feed rates, the influence of the catalyst is less significant. Figure 10.17 represents a comparison of the concentration profi les in two different ethyl acetate synthesis modes, with and without a decanter. The investigation is performed for the pilot-scale column, with a molar feed ratio acetic acid/ethanol equal to 1.2, reflux ratio equal to 3, and a total feed rate equal to 30 kg/h. The distillate-to-feed ratio is set to 0.9. The simulations reveal that the conversion with the liquid–liquid separator is about 5% higher that without a decanter, since there is less water and more acetic acid in the catalytic section. Improved conversion and product enrichment due to liquid–liquid separation result in a significant (29%) improvement of the product purity. Finally, because there is less condensed water in the reflux to be evaporated, the heat duty is reduced by up to 26%.

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Figure 10.17 Concentration profiles with (solid lines) and without (dashed lines) liquid–liquid separation (ethyl acetate system, pilot scale column).

10.4.4 Dymethyl Carbonate Transesterification 10.4.4.1 Process Description The transesterification of dimethyl carbonate (DMC) to diethyl carbonate (DEC) represents a complex reaction system including five components and three binary azeotropes shown in Table 10.5. The characteristics of some binary mixtures of the substances involved in the reaction system have been investigated by Franchesconi and coworkers [114–116], Luo et al. [117–119] and Rodriguez et al. [120, 121], resulting in a good description of the required thermodynamic properties. The chemical reaction system consists of the two reaction steps shown by Eqs. (R10.4) and (R10.5):

(CH3O)2CO + CH3CH2OH ↔ (CH3O)CO(OCH2CH3 ) + CH3OH

(R10.4)

(CH3O)CO(OCH2CH3 ) + CH3CH2OH ↔ (CH3CH2O)2CO + CH3OH (R10.5) In a first step, the intermediate ethyl methyl carbonate (EMC) is produced from the reactant DMC and ethanol (EtOH), releasing methanol. The equilibrium constant of this reaction suffers a step change with temperature within the range of the boiling-point temperatures of all chemical components present (see Table 10.5) between 1.9 and 2.1 [117].

10.4 Case Studies Table 10.5 Boiling point temperatures of

pure components and binary azeotropes (UNIFAC, 1 bar). Temp (ºC)

Components

63.8 64.5 74.9 77.8 78.5 90.2 109.2 126

DMC-MeOH MeOH EtOH-EMC DMC-EtOH EtOH DMC EMC DEC

In a second reaction step, the intermediate EMC again reacts with ethanol, producing DEC and, again, releasing methanol. The equilibrium constant for this reaction step varies between 0.44 and 0.46 as the temperature changes within the range of the boiling-point temperatures of the reactants [117]. The second reaction step can be regarded as a parallel reaction to the first reaction with respect to EtOH or, alternatively, as a consecutive reaction with respect to the intermediate EMC. Due to the chemical equilibrium limitations, the reaction inside a batch stirredtank reactor would lead to a conversion rate of DMC X DMC = 0.45 and a selectivity of DMC towards DEC SDMC,DEC = 0.27. In particular, the low selectivity is caused by the unfavorable equilibrium constant of the second reaction step. Reactive distillation offers a possibility to increase this value through the removal of methanol from both reaction steps. This chemical system renders a possibility to influence the process selectivity via RD. Both products can be used as solvents in lithium-ion accumulators or as intermediates for the manufarturing of other products like polycarbonates and pesticides [122–124]. Starting with the given thermophysical and kinetic data, a heuristic-numeric process synthesis has been carried out. This procedure led to the specific column configuration displayed in Fig. 10.18 for the production of DEC. 10.4.4.2 Process Modeling For the study of the DEC synthesis via RD, a rate-based model was developed and implemented into the simulation environment gPROMS [125]. Along these lines, dynamic modeling methods could be used to describe the time-depended behavior of the process. As a first step, a model database was established, containing submodels for catalytic and conventional stages, redistributors, condenser and reflux drum. The rigor of the implemented process models ranges from simple equilibrium-stage consideration up to detailed rate-based treatment [126]. Furthermore, models of different accuracy and complexity for the calculation of

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Figure 10.18 Column configuration for the synthesis of DEC.

physical properties (activity coefficients, diffusion coefficients, etc.) are applied. Using these models, extensive sensitivity studies on different column configurations were performed; e.g. the impact of the reflux ratio and the heat duty on conversion and selectivity of this reaction system were investigated. A very important issue for this test system is a proper choice of the catalyst. Therefore, in [127], screening tests for a number of heterogeneous catalysts were carried out and correspondent kinetic equations were estimated based on the Arrhenius relation. The determined kinetic parameters are used in the simulations of catalytic RD. Modified potassium carbonate, Lewatit K1221 and Nafion SAC-13 were found to be the most active heterogeneous catalysts for the transesterification of DMC. Regarding the catalyst implementation into the catalytic packing MULTIPAK in a RD column, the most suitable catalyst seems to be Lewatit K1221 that was also found to be easily regenerated without any adverse effect on its activity [127]. For the estimation of equilibrium parameters, an adapted NTRL model was used. 10.4.4.3 Results and Discussion Extended simulation studies have been performed, and it has been found that the conversion of DMC increases with increasing heat duty of the reboiler, reaches a nearly constant value, and then decreases. The selectivity for the main product DEC also increases, but in a different way, namely it reveals a minimum (see Figs. 10.19 and 10.20). This process behavior becomes clear if one takes a closer look at the column composition profi les at the corresponding operating points. When the column is being operated under a given constant reflux ratio, increasing the heat duty leads to a shift in concentrations, which at first favors both reaction steps, because DMC

10.4 Case Studies

Figure 10.19 Conversion and selectivity for the synthesis of DEC (n = 3).

Figure 10.20 Conversion and selectivity for the synthesis of DEC (n = 9).

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10 Modeling of Reactive Distillation

and EtOH are enriched inside the catalytic section. The high conversion rate and relatively low selectivity are depicted in Fig. 10.19. A slight increase in the heat duty leads to a higher amount of EMC inside the catalytic section, and therefore, also the selectivity increases. This process continues until EtOH disappears from the catalytic section. As shown in Figs. 10.19 and 10.20, beginning from this important point, the conversion decreases continuously with the heat duty. The same is true for the selectivity towards DEC, except a very small range (this range is, however, larger in Fig. 10.20 for the reflux ratio equal to 9). Such behavior results from the fact that, at this particular heat duty, all of the EMC produced inside the catalytic section is immediately converted to DEC. With a further increase in heat duty, all reactants pass through the catalytic section so rapidly that the reaction can hardly take place, and conversion rates as well as the selectivity fall to zero. The influence of a loss in activity on the conversion and the selectivity of the process has been studied. It can be derived from the simulation results shown in Fig. 10.21 that the selectivity towards DEC decreases faster than the conversion rate. The decrease in catalytic activity first affects the second reaction step, and thus, it influences the overall selectivity towards DEC. As shown in Fig. 10.21, a slight decrease in activity influences the selectivity of DMC towards DEC significantly, whereas the conversion rate remains nearly constant. This is true for a loss in activity up to nearly 20%, at which the conversion decreases by 2% only and the selectivity by 8%.

Figure 10.21 Influence of loss of catalyst activity on conversion and selectivity.

10.5 Conclusions and Outlook

The results demonstrate that the two-step transesterification represents a suitable reaction system for a heterogeneously catalyzed RD process. The catalyst screening shows that there are several heterogeneous catalysts available to produce EMC and DEC, with sufficient yields for this application. Optimal operational conditions can be identified based on appropriate selectivity studies. 10.5 Conclusions and Outlook

This chapter reviews one of the most important reactive separation processes – reactive distillation (RD). This operation, combining the separation and reaction steps inside the same zone of a single column, is often advantageous compared to traditional sequential unit operations. The chapter gives a detailed discussion of the process basics and peculiarities, considers the up-to-date applications and thoroughly discusses the RD modeling and design issues. The theoretical description is illustrated by several case studies and supported by the results of laboratory-, pilot- and industrial-scale experimental investigations. Both steadystate and dynamic issues are treated, and in addition, the design of column internals is addressed. Reactive distillation occurs in multicomponent multiphase fluid systems, and there exist different possible ways to build a modeling framework for RD. This chapter advocates the rate-based approach as the most rigorous and appropriate way. This approach provides a direct consideration of the diffusional and reaction kinetics. In terms of the rate-based approach, the peculiarities of the specific process applications and the different solution strategies are treated while discussing the case studies. A detailed general description covers mass and heat transfer, reaction kinetics including the reaction-mass-transfer coupling, as well as steady-state and dynamic modeling issues. The modeling of RD processes is illustrated by the heterogeneously catalyzed syntheses of methyl acetate, methyl tertiary butyl ether (MTBE), ethyl acetate and transesterification of dymethyl carbonate using different catalytic internals. These processes are described based on the pseudohomogeneous approach for the reaction kinetics. For the methyl acetate synthesis, dynamic modeling effects are investigated, whereas for other systems, the focus is on different steady-state issues, for example the influence of liquid–liquid separation, operational conditions and different column internals (ethyl acetate) or selectivity effect (dimethyl carbonate transesterification). The comparison between the simulation and experimental data made for all RD case studies proves that the rate-based approach is capable of predicting correct process behavior, both steady state and dynamic. The key RD topics to be addressed in the near future are a proper hydrodynamic modeling for catalytic internals including residence-time distribution account and scale-up methodology. Further studies on the hydrodynamics of

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catalytic internals are essential for a better understanding of RD behavior and the availability of optimally designed catalytic column internals for them. In this regard, the methods of computational fluid dynamics appear very helpful (see [51, 53]). The development of new methodologies enabling creation of intelligent, tailormade column internals and consequent RD optimization constitutes one of the burning present-day challenges. Such a development has already been started (see [51, 52, 128]). Despite a fast recent development of computer technology and numerical methods, the rate-based approach in its current realization still requires a significant computational effort, with related numerical difficulties. This is one of the reasons why the application of rate-based models to industrial tasks is rather limited. Therefore, further work is required in order to bridge this gap and provide chemical engineers with reliable, consistent, robust and comfortable simulation tools for reactive distillation processes.

Acknowledgments

We would like to thank our colleagues at the Chair of Fluid Separation Processes and all project partners who have been involved in the research activities. We are also grateful to the German Research Foundation (DFG, Grant No. Schm 808/51), the European Commission (CEC Project No. BE95-1335 and GROWTH Project No. GRD1 CT199910596), and to BASF AG, Bayer AG and Degussa AG for financial support.

Nomenclature

Ac aI B c d dC D Dax E Eq Fc G ∆HR0 HETS h

column cross-sectional area specific vapor-liquid interfacial area liquid load molar concentration generalized driving force column diameter Maxwell–Stefan diffusion coefficient axial dispersion coefficient specific energy holdup dimensionless residence-time distribution gas capacity factor gas molar flow rate reaction enthalpy height equivalent to a theoretical stage partial molar enthalpy

m2 m2/m3 m3/(m2s) mol/m3 1/m m m2/s m2/s J/(mol s) – Pa0.5 mol/s J/mol m J/mol

Nomenclature

J J K ieq [k] l L n N Q R ℜ S t T uL U w x X y z

diffusion flux column vector consisting of Ji vapor-liquid equilibrium constant matrix of mass-transfer coefficients axial coordinate directed from column top to bottom liquid molar flow rate number of mixture components molar flux heat flux total component reaction rate gas constant selectivity time temperature liquid-phase velocity length-specific molar holdup mass fraction liquid mole fraction conversion gas mole fraction fi lm coordinate

mol/(m2 s) mol/(m2 s) – m/s m mol/s – mol/(m2 s) W/m2 mol/(m3 s) 8.3144 J/(mol K) % s K m/s mol/m kg/kg mol/mol – or % mol/mol m

Greek Letters

[Γ] d f κik κT m

matrix of thermodynamic correction factors fi lm thickness volumetric holdup binary mass-transfer coefficient heat-transfer coefficient chemical potential

Subscripts

G i,j L t

gas phase component/reaction indices liquid phase total

Superscripts

av B I

average bulk phase phase interface

– m m3/m3 m/s W/(m2 K) J/mol

357

358

10 Modeling of Reactive Distillation

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63 P. V. Danckwerts, Gas-Liquid Reactions, McGraw-Hill, New York, 1970. 64 K. R. Westerterp, W. P. M. van Swaaij, A. A. C. M. Beenackers, Chemical Reactor Design and Operation, Wiley, Chichester, 1984. 65 J. D. Seader, The Rate-Based Approach for Modeling Staged Separations, Chem. Eng. Prog., 1989, 85(10), 41–49. 66 S. S. Katti, Gas-Liquid-Solid Systems: An Industrial Perspective, Trans. IChemE, 1995, 73, Part A, 595–607. 67 W. K. Lewis, W. G. Whitman, Principles of Gas Absorption, Ind. Eng. Chem., 1924, 16, 1215–1220. 68 R. Higbie, The Rate of Absorption of a Pure Gas into a Still Liquid During Short Periods of Exposure, Trans. Am. Inst. Chem. Engrs., 1935, 31, 365–383. 69 J. O. Hirschfelder, C. F. Curtiss, R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964. 70 S. Pelkonen, Multicomponent Mass Transfer in Packed Distillation Columns, Ph.D. Thesis, University of Dortmund, Germany, 1997. 71 R. Krishna, G. L. Standart, Mass and Energy Transfer in Multicomponent Systems, Chem. Eng. Commun., 1979, 3, 201–275. 72 J. L. Bravo, J. R. Fair, Generalized Correlation for Mass Transfer in Packed Distillation Columns, Ind. Eng. Chem. Process Des. Dev., 1982, 21, 162–170. 73 J. A. Rocha, J. L. Bravo, J. R. Fair, Distillation Columns Containing Structured Packings – 2. Mass Transfer Model, Ind. Eng. Chem. Res., 1996, 35, 1660–1667. 74 A. Kolodziej, M. Jaroszynski, I. Bylica, Mass transfer and hydraulics for KATAPAK-S, Chem. Eng. Process., 2003, 43, 457–464. 75 L. U. Kreul, A. Górak, C. Dittrich, P. I. Barton, Dynamic Catalytic Distillation: Advanced Simulation and Experimental Validation, Comp. Chem. Eng., 1998, 22, S371–S378. 76 A. Hoffmann, C. Noeres, A. Górak, Scale-Up of Reactive Distillation Columns with Catalytic Packings, Chem. Eng. Process., 2004, 43, 383–395. 77 C. Noeres, C. Benvenuti, A. Hoffmann, A. Górak, Reactive Distillation: Nonideal

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88 R. Schneider, C. Noeres, L.-U. Kreul, A. Górak, Dynamic Modeling and Simulation of Reactive Batch Distillation, Comp. Chem. Eng., 2001, 25, 169–176. 89 F. G. Helfferich, Y.-L. Hwang, Ion Exchange as Catalysts, in Ion Exchange for Industry (Ed. M. Streat), Ellis Horwood, Chichester, 1988, pp. 585–596. 90 Z. P. Xu, K. T. Chuang, Kinetics of Acetic Acid Esterification over Ion Exchange Catalysts, Can. J. Chem. Eng., 1996, 74, 493–500. 91 E. Ranzi, M. Rovaglio, T. Faravelli, G. Biardi, Role of Energy Balances in Dynamic Simulation of Multicomponent Distillation Columns, Comp. Chem. Eng., 1988, 12, 783–786. 92 G. Ronge, Überprüfung unterschiedlicher Modele für den Stoffaustausch bei der Rektifikation in Packungskolonnen, VDIVerlag, Düsseldorf, 1995. 93 J. Gmehling, B. Kolbe, Thermodynamik, Wiley-VCH, Weinheim, 1992. 94 R. Reid, J. M. Prausnitz, B. E. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 1987. 95 A. Gicquel, B. Torck, Synthesis of Methyl Tertiary Butyl Ether Catalyzed by IonExchange Resin. Influence of Methanol Concentration and Temperature, J. Catal., 1983, 83, 9–18. 96 A. Rehfinger, U. Hoffmann, Kinetics of Methyl Tertiary Butyl Ether Liquid Phase Synthesis Catalyzed by Ion Exchange Resin. I. Intrinsic Rate Expression in Liquid Phase Activities, Chem. Eng. Sci., 1990, 45, 1605–1617. 97 D. Parra, J. Tejero, F. Cunill, M. Iborra, J. F. Izquierdo, Kinetic Study of MTBE Liquid-Phase Synthesis Using C4 Olefin Cut, Chem. Eng. Sci., 1994, 49, 4563–4578. 98 E. Kenig, K. Jakobsson, P. Banik, J. Aittamaa, A. Górak, M. Koskinen, P. Wettmann, An Integrated Tool for Synthesis and Design of Reactive Distillation, Chem. Eng. Sci., 1999, 54, 1347–1352. 99 K. Sundmacher, U. Hoffmann, Development of a New Catalytic Distillation Process for Fuel Ethers via a Detailed Nonequilibrium Model, Chem. Eng. Sci., 1996, 51, 2359–2368.

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111 M. Klöker, E. Y. Kenig, A. Hoffmann, P. Kreis, A. Górak, Rate-Based Modeling and Simulation of Reactive Separations in Gas/Vapour–Liquid Systems, Chem. Eng. Process., 2005, 44, 617–629. 112 J. Gmehling, U. Onken, DECHEMA Chemistry Data Series, 2nd edition, Schön & Wetzel, Frankfurt/Main, 1991. 113 A. Kolodziej, M. Jaroszynski, W. Salacki, W. Orlikowski, K. Fraczek, M. Klöker, E. Y. Kenig, A. Górak, Catalytic Distillation for the TAME Synthesis with Structured Catalytic Packings, Chem. Eng. Res. Des., 2004, 82, 175–184. 114 F. Comelli, R. Francesconi, S. Ottani, Isothermal Vapor-Liquid Equilibria of Dimethyl Carbonate Plus Diethyl Carbonate in the Range (313.15 to 353.15) K, J. Chem. Eng. Data, 1996, 41, 534–536. 115 R. Francesconi, F. Comelli, Excess Molar Enthalpies, Densities, and Excess Molar Volumes of Diethyl Carbonate in Binary Mixtures with Seven N-Alkanols at 298.15 K, J. Chem. Eng. Data, 1997, 42, 45–48. 116 F. Comelli, R. Francesconi, C. Castellari, Excess Molar Enthalpies and Excess Molar Volumes of Binary Mixtures Containing Dialkyl Carbonates Plus Pine Resins at (298.15 and 313.15) K, J. Chem. Eng. Data, 2001, 46, 63–68. 117 H. P. Luo, W. D. Xiao, A Reactive Distillation Process for a Cascade and Azeotropic Reaction System: Carbonylation of Ethanol with Dimethyl Carbonate, Chem. Eng. Sci., 2001, 56, 403–410. 118 H. P. Luo, W. D. Xiao, K. H. Zhu, Isobaric Vapor-Liquid Equilibria of Alkyl Carbonates with Alcohols, Fluid Phase Equilibria, 2000, 175, 91–105. 119 H. P. Luo, J. H. Zhou, W. D. Xiao, K. H. Zhu, Isobaric Vapor-Liquid Equilibria of Binary Mixtures Containing Dimethyl Carbonate under Atmospheric Pressure, J. Chem. Eng. Data, 2001, 46, 842– 845. 120 A. Rodriguez, J. Canosa, A. Dominguez, J. Tojo, Isobaric Vapor-Liquid Equilibria of Dimethyl Carbonate with Alkanes and Cyclohexane at 101.3 kPa, Fluid Phase Equilibria, 2002, 198, 95–109.

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields in Chemical Multiphase Processes Faïçal Larachi

11.1 Background

The main goal that is sought by process intensification is an implementation, in the broadest assertion, of highly efficient chemical processing setups able to fulfi ll sizable reduction in reactor dimensions, concomitantly with maximization, of one if not all, of the catalytic, mass- and heat-transfer efficiencies of the system. Process intensification can be perceived as a paradigm in the chemicalengineering arena, which is potentially shaping the notion of chemical processing, leading to small, safe, energy-saving and ecofriendly processes [1]. As a matter of fact, reducing development time from bench-scale to commercial-scale production by using new methods capable of achieving better conversion or selectivity is one of the chief priorities of process-intensification studies. This approach would be especially beneficial for the sectors of the fine and the pharmaceutical industries that often deal with higher-value added and low production rate processes, that is, less then a few metric tons per year [2]. From a process-intensification standpoint, the ability to influence the process behavior by application of external gradient magnetic fields is of potential practical interest for the operation of chemical reactors and for the course of catalytic reactions as well. Broadly speaking, any material can be labeled as being either nonmagnetic or magnetic. In chemical flow processes, fluids, obviously, are of chief importance, while very often they exhibit nonmagnetic behavior as assessed from their considerably low volumetric magnetic susceptibility values, usually of the order of 10 −6 –10 −7. Flows of such fluids might eventually be stimulated provided strong gradient magnetic fields, such as those produced by means of superconducting magnets, are used. Conversely, magnetic fluids (or ferrofluids), per se, are not readily available. These non-naturally occurring ferrofluids are known to possess magnetic susceptibilities larger by several orders-of-magnitude (up to 10 6 ) in comparison to those of nonmagnetic fluids. This enables ferrofluids Modeling of Process Intensifi cation. Edited by F. J. Keil Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31143-9

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

magnetic stimulation by means of weak gradient magnetic fields such as those generated in any conventional solenoid. 11.1.1 Nonmagnetic Fluids

Let us first turn our attention to the nonmagnetic fluids commonly encountered in chemical-engineering applications. These encompass organic, often nonelectrically conducting, or aqueous, often exhibiting low to moderate ionic strengths. The magnetization (or Kelvin) force generated by inhomogeneous magnetic fields is a body force and in that sense it is analogous to the gravitational force. The main characteristic of the magnetization body force compared with the gravity force is the directional control it allows for achieving artificially nonterrestrial gravitational conditions, such as hypogravity (down to micro) or macrogravity conditions. The Kelvin force acts upon nonelectrically or weakly electrically conducting nonmagnetic fluids and must be distinguished from the classical Lorentz braking force that is the cornerstone of the physical and engineering areas of magnetohydrodynamics (MHD) and concerns fluids that are electrically conducting and nonmagnetic, for example, liquid metals, strong electrolytes and plasmas; see, for instance, the excellent introductory book to MHD by Davidson [3]. Over the past few years, there have been some reports indicating that the magnetization force can compensate or, on the contrary, amplify the apparent effect of gravity. Beaugnon and Tournier [4] and Ikezoe et al. [5] through applying a strong inhomogeneous magnetic field were able to levitate water, acetone, ethanol and organic materials. Similarly, using an inhomogeneous magnetic field, Wakayama et al. [6] showed that vertical acceleration can be continuously tuned from normal gravity to near zero gravity, mimicking space weightlessness conditions in completely earthbound experimentations. The flow and diffusion characteristics for oxygen gas under inhomogeneous magnetic fields were reported by Tagawa et al. [7]. Recently, Wang and Wakayama [8] reported detailed numerical simulations for controlling natural convection in non- and low-conducting diamagnetic fluids contained in cubical enclosures under inhomogeneous magnetic fields oriented in different directions. The magnetic-field effects on biochemical systems were also studied. Lin et al. [9] found that the quality of protein crystals can improve or deteriorate depending on the magnetization force applied to protein-containing solutions. It seems that application of this type of force to multiphase catalytic reactors is still poorly explored. In the proposed work, we will illustrate the application of strong inhomogeneous magnetic fields to a small-scale trickle-bed reactor that consists of two-phase gas-liquid cocurrent flow through a fi xed bed of catalyst layer. Most commercial trickle-bed reactors operate adiabatically at high temperatures and high pressures and generally involve hydrogenation, oxidation, desulfurization, hydrocracking, etc. The most important hydrodynamic properties for trickle-bed reactors are i) the liquid holdup that controls the liquid-to-gas reactant

11.1 Background

flux ratios, ii) the wetting efficiency that controls the accessibility to active sites from gaseous (dry spot) or liquid (wetted spot) outer catalyst surface, and iii) the two-phase pressure drop that often is used as the energy-consumption indicator for assessing the extent of heat- and mass-transfer coefficients. All three previous factors are known to affect catalytic reaction performances (conversion and selectivity). Liquid maldistribution, hot-spot formation, and selectivity loss are serious problems that threaten reactor operation. We will see later through experiments and numerical simulations concerning the behavior of trickle beds under inhomogeneous magnetic fields [10] that liquid holdup can either be improved or reduced thus directly afflicting the wetting efficiency of the catalytic particles. This latter is a crucial operational parameter when liquid-limited or gas-limited catalytic reactions are being processed inside the reactor. For liquid-limited reactions, the highest possible wetting efficiency and particle–liquid mass-transfer rates result in the fastest transport of the liquid-phase reactant into the catalyst. Conversely, for gas-limited reactions, it is advantageous, while avoiding running the risk of gross liquid maldistribution and hot-spot formation, to reduce the extra mass-transfer resistance contributed by the liquid phase. 11.1.2 Magnetic Fluids

Ferrofluids are magnetic-liquid suspensions obtained by seeding surfactantstabilized single-domain superparamagnetic nanoparticles dispersed in appropriate organic or aqueous carrier matrices [11–13]. Such stabilized ferrocolloidal suspensions exhibit a number of intriguing behaviors when subjected to external magnetic fields. Owing to a giant single-domain magnetic moment, ca. 10 4 Bohr magnetons, each nanoparticle stands as a small permanent magnetic dipole. Collective response of these nanoparticles to magnetic-field stimulation gives rise to strong and coherent magnetization that, for motionless ferrofluids, aligns along the direction of the magnetic field, and that obeys the equilibrium Langevin magnetization law. However, most fluids, being in essence noninviscid, give rise to a vorticity that tends to cause misalignment between the fluid magnetization and the magnetic field vectors [11]. The noncollinearity between magnetization and magnetic field results in a magnetostatic torque counteracting the mechanical torque. We will see later that two magnetoviscous behaviors, as a consequence, could arise, namely, the magnetoviscothickening and the magnetoviscothinning. Pioneering studies by the Soviet school provided a large body of knowledge confirming the special behavior of magnetic fluids experiencing inhomogeneous or oscillating magnetic fields in Hagen–Poiseuille capillary flows or in the Darcy regime with very restrictive conditions on the filtration law in porous media [14–17]. Other recent studies of porous media ferrofluid flows highlight interesting geoapplications of migration of magnetic fluids in remote locations that preclude direct physical control such as in environmental subsurface situations [18, 19].

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

We propose to explore in this chapter the potential applications of magnetoviscosity in chemical engineering. The demonstration example will highlight the control of wall channeling in porous media exhibiting low column-to-particle diameter ratio. It is well known that such low-ratio columns may occasion severe maldistribution due to large radial permeability contrast with preferential flow short-circuiting alongside the high-permeability wall area. The challenge will be to identify which configurations are propitious to creating sufficient magnetoviscosity near the vessel wall to reduce the bypass and to reroute more fluid flow towards the core of the bed.

11.2 Nonmagnetic Fluids 11.2.1 Principle

Figure 11.1 illustrates a typical experimental setup producing an inhomogeneous magnetic field up to 9 T obtained with a superconducting NbTi solenoid magnet system. To reach the superconducting state, the solenoid temperature in routine commercial designs is usually decreased to ca. 4 K using liquid nitrogen and liq-

Figure 11.1 Magnetic field and magnetic-field gradient distributions in a 9-T superconducting magnet (trickle bed inside magnet bore).

11.2 Nonmagnetic Fluids

uid helium. Then, a vertical magnetic field is generated and controlled by a computerized control system. The magnetic field strength is controlled by changing the current density through the magnet. The system shown in Fig. 11.1 produces a maximum product gradient (BzdBz /dz) of 650 T 2/m around point A or A’. A miniature trickle-bed reactor equipped with nonmagnetic inlet and outlet valves is positioned in the maximum product gradient area of the atmospheric open magnet bore and allows for a two-phase flow, e.g., air-water, to take place. The magnetic-field intensity inside and outside of the bore can be measured using a Hall-effect gaussmeter, whereas the volumetric magnetic susceptibilities of the fluids can be determined using, for instance, an alternating gradient magnetometer. The reactor can slide up and down to locate it in the region of the desired magnetic field strength and magnetic-field gradient. During the flow of gas and liquid throughout the bed, the Kelvin force affects the fluids differently depending on their respective magnetic susceptibility values. By orienting this force vertically, a directional control can be exerted on the flow achieving thus artificially micro- or macrogravity conditions: FM α =

χα dBz Bz µ0 dz

(11.1)

where ca is the volumetric magnetic susceptibility, m0 is the absolute magnetic permeability of vacuum and B is the magnetic induction or magnetic flux density. To quantify the relative importance of gravity versus magnetization for each fluid phase, a gravitational-amplification factor can be defined [10] for each phase in order to commute the magnetization force into an apparent artificial gravity acceleration. Assuming the direction of nonhomogeneity of the magnetic field parallel to the gravitational field (radial magnetic field nonhomogeneities neglected), an artificial-gravity factor can be defined as follows:

γα =

χα ρα g + FMα dB =1+ Bz z ρα g ρα gµ0 dz

(11.2)

Depending on the signs of magnetic susceptibility (>0 for paramagnetic and 1 corresponds to macrogravity or hypergravity, ii) 1 > g a > 0 corresponds to subgravity (hypogravity or microgravity), iii) g a = 0 coincides with the case of levitation. There is also another case corresponding to g a < 0 when the weight force is largely counterbalanced by the magnetization force. This case will not be considered here. Figure 11.2 demarcates for dihydrogen at atmospheric pressure and ambient temperature (volumetric magnetic susceptibility = −2.05 × 10 −9 ), the regions of hypergravity, hypogravity and levitation as a function of the product BzdBz /dz inside the magnet. For example, in the bore region where BzdBz /dz neighbors a

369

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

Figure 11.2 Artificial-gravity factor g versus Bz dBz/dz for diamagnetic dihydrogen H2 at ambient conditions.

value of 350 T 2/m, dihydrogen experiences a gravitational effect equivalent to that felt on the planet Mars for which Eq. (11.2) gives g ≈ 0.38 or g Mars = 3.72 m/s2. 11.2.2 Theory

To estimate the liquid-holdup and pressure-drop variations in the inhomogeneous magnetic field, the artificial-gravity concept is used. The magnetic body force is considered in terms of an artificial gravity force. Performing a one-dimensional force balance for both phases gives a couple of relationships between pressure gradient, phase holdups, body-force densities and interfacial liquid-solid and gasliquid drag force densities. Assuming complementary liquid and gas holdups, equality between gas-side and liquid-side pressure gradients, and neglecting in the formalism the partial-wetting-subtended terms (by assuming full wetting), the force-balance equations become: −ε g

ε g χ g dBz dP Bz + ε g ρg g + − Fgl = 0 dz dz µ0

(11.3)

−ε l

dP dB εχ + ε l ρ l g + l l Bz z + Fgl − Fls = 0 dz dz µ0

(11.4)

After substitution of Eq. (11.2) into Eqs. (11.3) and (11.4), one obtains for each phase:

11.2 Nonmagnetic Fluids

−ε g

dP + ε g ρgγ g g = Fgl dz

(11.5)

−ε l

dP + ε l ρ lγ l g = −Fgl + Fls dz

(11.6)

In the case of trickle flow, it has been shown that under certain conditions the slit-flow approximation yields a very satisfactory set of constitutive equations for the gas-liquid and the liquid-solid drag forces [20, 21]. As a matter of fact, the slit flow becomes well representative of the trickle-flow regime when the liquid texture is contributed by solid-supported liquid fi lms and rivulets. This generally occurs at low liquid flow rates that allow the transport of fi lm-like liquids [20]. We will assume, without proof though, that such hypotheses also hold in the case of artificial-gravity operation. The validity of these assumptions and of the several others outlined above will be evaluated later in terms of model versus experiment comparisons. Choosing the drag force closures of the simplified Holub slit model [20], the equations system becomes: 3

Ψg = −

Re2g   ε   Reg dP 1 + 1 =    E1 + E2 Gag  dz ρgγ g g  ε g   Gag

Ψl = −

dP 1 Re2   ε   Re + 1 =    E1 l + E 2 l   ε l   Ga l dz ργl l g Ga l 

(11.7)

3

(11.8)

Note that in Eqs. (11.7) and (11.8), the Galileo numbers use the corresponding artificial-gravity values for liquid and for gas phase. Gaα =

ρα2γ α gd p3ε 3 ηα2 (1 − ε )3

(11.9)

Equations (11.7)–(11.9) are an adapted form of the slit model of Holub et al. [20] to a trickle bed experiencing artificial-gravity conditions. 11.2.3 Experimental Results and Discussion

The reactor is inserted in region A of the solenoid bore (Fig. 11.1). Experiments are first made to measure the liquid holdup, the pressure drop and the wetting efficiency in the absence of magnetic fields. A sufficient time is allowed for the system to reach steady state before measurements are acquired. Experimental data are compared with the predictions of the Holub et al. [20] model. Figures 11.3a and b show the experimental data versus Holub’s model for the trickle-flow regime with the magnetic field off. The slit model restores the hydrodynamic behavior pretty well in terms of pressure-drop and liquid-holdup variations. Turning the magnetic filed on yields, in region A, a product gradient of 650 T2/ m at 9 T (Fig. 11.1). The magnetic body force is calculated for (diamagnetic) water

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

Figure 11.3 Comparison between trickle-bed slit model and experimental pressure-drop (a) and liquid-holdup (b) data for an air-water system for various gas and liquid superficial velocities. Magnetic field OFF, 1-mm glass beads.

( c = −9.0 × 10 −6 ), and for (paramagnetic) air ( c = 0.38 × 10 −6 ). For a BzdBz /dz value of +650 T 2/m, the magnetization force acts upwardly for water ( g = 0.52, hypogravity) whereas it acts downwardly for air ( g = 17.7, hypergravity). The values of the magnetization body forces are FM艎 = −4679 N/m3 for water and FMg = 196 N/m3

11.2 Nonmagnetic Fluids

for air. As it results from the calculated values, the liquid-phase magnetization force is more important then the gas magnetization force and it could be assumed that the liquid magnetization force is controlling. Figures 11.4a and b shows the pressure-drop and liquid-holdup experimental data for 1-mm glass beads for different liquid and gas flow rates. In the presence

Figure 11.4 Comparison between pressure-drop (a) and liquid-holdup (b) measurements for an air-water system for various gas and liquid superficial velocities with the magnetic field ON and OFF, 1-mm glass beads, lines show trends.

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

of a magnetic field, the liquid holdup increases because the resistance to liquid flow increases. In this case, the liquid magnetization force is much more important than air magnetization force and it becomes controlling. For the liquid phase, hypogravity conditions prevail. For two-phase flow, the pressure drop in the presence of a magnetic field increases comparatively with the pressure drop measured in the absence of magnetic field. For positive magnetic-field gradients, liquid magnetization force being controlling it reduces the effect of gravity but it increases the resistance of the liquid flow and enhances the pressure drop. Figures 11.5a and b present a comparison between the pressure-drop and liquid-holdup experimental data obtained in artificial-gravity conditions along with prediction with the apparent-gravity modified Holub et al. [20] slit model developed in Section 11.2.2. For air-flow-free single-phase water flow, model predictions are very close to the experimental values and, in this case, the magnetic-field effect on the hydrodynamics properties is described well by the artificial-gravity concept. For two-phase flow, the tendencies are similar but there are some discrepancies between the experimental data and model predictions. The experimental values of liquid holdup and pressure drop are generally underpredicted by the model especially at the higher gas superficial velocities. Reasons for such mismatch between experiments and prediction are the assumptions inherent to the model that might no longer be valid when the magnetic field is on. For instance, the slit model neglects the velocity and shear slip factors that characterize the degree of phase interactions between gas and liquid. Another factor could be the so-called Moses effect that is responsible for the deformation of the liquid surface profi le by strong magnetic fields. The Moses effect was studied before by Sugawara et al. [22] who found that the interfacial shape between two immiscible nonmagnetic liquids deforms under an applied inhomogeneous magnetic field. In the liquid fi lm, a parabolic liquid velocity profi le is used. At the interfaces between the catalyst and liquid phase and between liquid and gas phases, the Moses effect is present and the surface shapes could be deformed, modifying the liquid velocity profi le. Another reason for the mismatch between experimental and predicted data could be the assumption of homogeneity of the magnetic field in the radial direction. Also, the peak value for the axial magnetic field product gradient (+650 T 2/m) is used in the simulations for the entire bed length and is considered constant, while in reality the axial magnetic field product gradient value is not constant and depends on the point position relative to solenoid center (see magnetic field longitudinal profi le in Fig. 11.1). The wetting efficiency experiments are performed in the same system, using a colorimetric method. The glass beads have been colored before the experiment with a crystal violet aqueous solution of known concentration. The reactor with the colored beads inside is inserted into the solenoid bore. Depending on the liquid and gas flow rates, the glass beads color is washed out by the liquid and the outgoing solution transmittance is measured spectrophotometrically. The final concentration (no color on glass beads) is considered as 100% when the glass beads are fully washed and the wetting is total. The wetting efficiency is calculated as the quotient between the intermediate and final concentrations.

11.2 Nonmagnetic Fluids

Figure 11.5 Comparison between slit model and experimental pressure drop (a) and liquid holdup (b) for an air-water system for various gas and liquid superficial velocities with the magnetic field ON, 1-mm glass beads.

Figure 11.6 presents the wetting efficiency experimental data for the 1-mm glass beads the system air-water+34.5%w. manganese chloride system ( c = 4.2 × 10 −4 ). Note now that both the gas and the liquid are paramagnetic. By displacing the trickle-bed reactor to a region where BzdBz /dz = −27 T 2/m allows conditions

375

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

with g ≈ 0.38 for both fluids to be mimicked. Unlike the previous experiments with air and water, the present combination allows to realize true Martian gravitational conditions for both gas and liquid. The concentration of MnCl2 was indeed adjusted in the water solution so as to attain a magnetic susceptibility yielding a gravity amplification factor identical to that of air at the same magnet location. It is known that wetting efficiency increases with increasing liquid mass velocity, and at high liquid velocities, the catalyst surface area becomes totally wetted. To unveil the magnetic-field effects and to prevent the system very quickly reaching full wetting, very low liquid superficial velocities are used in this work down to ca. 10 −4 m/s. The wetting efficiency increases in the presence of inhomogeneous magnetic field and this is ascribed to the magnetization force that corresponds to hypogravity conditions for the liquid and gas phases, thereby allowing longer liquid residence time inside the trickle-bed reactor. The average wetting efficiency increase is quite significant for Mars conditions indicating that the gravity scale-down of trickle-bed reactors for space flights or operation on planets other than Earth require precautions to achieve a similar wetting efficiency of the bed. Improvement of wetting efficiency due to the magnetic field is directly related to simultaneously increasing liquid holdup as shown earlier for the air-water sys-

Figure 11.6 Effect of Mars gravity on the wetting efficiency at various liquid and gas superficial velocities, 1-mm glass beads, air/water+34.5%w./w. MnCl2 salt. Lines show trends.

11.2 Nonmagnetic Fluids

tem. Wetting efficiency of the catalytic particles plays a role in the control of mass and heat transfer between gas, liquid and catalyst. Larger wetting efficiency values help the bed to exhibit a better mesoscale liquid distribution that is beneficial against the risk of hot spots or when liquid-limited reactions are being carried out in the reactor. To assess the phenomenology of existing wetting-efficiency correlations to handle the change of gravity conditions, we chose to simulate an amended version of the Al-Dahhan and Dudukovic [23] wetting-efficiency correlation:

3

η l = 1.1 Re l

3

−

dP + ργl l g dz ργl lGa l g

(11.10)

The artificial-gravity concept is used in the correlation and normal gravity is converted into an artificial-gravity value. Fig. 11.7 shows the wetting-efficiency evolution for different values of the artificial-gravity factor between unity and 0.01. The correlation appears to capture the evolution towards increasing wetting efficiencies the lower the artificial-gravity factor. Therefore, this correlation can be used as a basis, after appropriate recalibration using experimental data in non-Earth gravity conditions, for extending its use to any value of the artificialgravity factor.

Figure 11.7 Anticipation of the evolution of wetting efficiency as a function of artificial-gravity factor by means of AlDahhan and Dudukovic [23] correlation, 1-mm glass beads and gas velocity of 0.08 m/s.

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

11.3 Magnetic Fluids 11.3.1 Principle

As introduced earlier in the chapter, when ferrofluids are brought to motion in shear flows, misalignment between the magnetization M and magnetic-field H vectors arises due to an asynchrony between the nanoparticle spin w and the 1 ferrofluid vorticity, –2 ∇ × v [11]. For illustration purposes, the magnetic-field vector and the fluid vorticity vector are assumed orthogonal as depicted in Fig. 11.8, and in first approximation, both angular acceleration and diffusive couple dyadic contributions are ignored. Hence, this spin-vorticity asynchrony gives rise to a mechanical torque that will tend to pull the dynamic magnetization vector out from the direction of equilibrium (or equivalently, the direction of the local magnetic field in the state of motionlessness). Noncollinearity between dynamic magnetization and magnetic field in return results in a counteracting magnetostatic torque, which will try to pull in the magnetization vector back to its equilibrium position. The mechanical torque is proportional to a vortex viscosity, which proportionates the magnetostatic torque response. The two magnetoviscous behaviors that arise are as follows. Magnetoviscothickening, a magnetically driven apparent inflation in local ferrofluid suspension viscosity, occurs when the fluid vorticity and the magnetized nanoparticle spin are in contrarotation or are in corotation with fluid vorticity faster than the nanoparticle spin velocity (Fig. 11.8a). The retarding effect by the nanoparticle spin velocity results locally in an increased dissipation in the flow and, eventually, a slow down in fluid linear velocity. Similarly, Magnetoviscothinning, similar to an apparent decrease in local ferrofluid suspension viscosity, occurs when fluid vorticity and magnetized nanoparticle spin are in corotation with the nanoparticle spin velocity is higher than the fluid vorticity (Fig. 11.8b). Part of the nanoparticle internal angular momentum is transmitted to the fluid, reducing fluid dissipation. An intermediate case, corresponding to the classical fluid-mechanics situation, also exists when no spin lag between fluid element and magnetized nanoparticle yields null magnetostatic and hydrodynamic torques. This cancels the internal angular momentum equation and eliminates spin coupling with a linear momentum balance. In porous media, such as the one illustrated in Fig. 11.9, high fluid vorticity prevails near the column wall where often an unobstructed gross wall flow can develop. On the contrary, the fluid vorticity in the bed core is the lowest due to porosity damping, which implies weaker local velocity variations. It is anticipated that application of magnetic fields to ferrofluids would yield magnetoviscothickening mainly located in the wall region. It is hoped that nanoparticle spin retardation would therefore increase the resistance to flow in this region to yield a lower wall-bypass fraction. One simple arrangement to induce magnetoviscosity corresponds to axisymmetric magnetic field and geometry to have mag-

11.3 Magnetic Fluids

Figure 11.8 Illustration of the magnetoviscous effect (a) magnetoviscothickening (in contrarotation and in corotation when fluid vorticity is higher than nanoparticle spin) and (b) magnetoviscothinning (in corotation when nanoparticle spin is higher than fluid vorticity).

netostatic torque, vorticity and spin velocity orthoradial as shown in Figs. 11.8 and 11.9. In what follows the magnetoviscosity phenomenon is analyzed by formulating the local ferrohydrodynamic model, the upscaled volume-average model in porous media with the closure problem, and solution and discussion of a simplified zero-order steady-state isothermal incompressible axisymmetric model for nonDarcy–Forchheimer flow of a Newtonian ferrofluid in a porous medium of the

379

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

Figure 11.9 Ferrofluid flow through packed bed in lineargradient d.c. magnetic field.

Müller type [24, 25]. No turbulence will be accounted for in the present treatment. The Shliomis [26] magnetization relaxation equation has been written in the fluid compressible form according to Felderhof [27]. 11.3.2 Theory 11.3.2.1 Local Description Isothermal and incompressible single-phase Newtonian ferrofluid ( γ -phase) upflow in a nonmagnetic ( κ-phase) rigid Müller porous medium (Fig. 11.10) is assumed. The boundary-value problem describing the flow in the region V is stated in Table 11.1. The open set V, bounded by the ∂Vγ e and ∂Vκe phase-associated entrance/exit contours on V faces, is composed of the ferrofluid part, Vγ, the porous medium part, Vκ, and the fluid/solid interface, Γγκ, inside V (Fig. 11.10b): V = Vγ + Vκ + Γγκ . For the sake of brevity, the boundary conditions on ∂V = ∂Vγ e + ∂Vκe of the velocity (vγ ), the spin ( w γ ), the magnetization (Mγ ), the induced magnetic (hγ, hκ ) field vectors and the initial conditions are omitted, even though these are usually not known a priori. We will assume that there exists for the model porous medium, a spatially periodic representative elementary volume of the order of volume V, such that ignorance of these boundary conditions does not harm the closure problem of the volume-averaged equations [28]. Table 11.1 summarizes the continuity, linear momentum, internal angular momentum, magnetization relaxation, Maxwell-flux law and Ampère–Maxwell-law local equations in V, along with Γγκ boundary conditions and constitutive tensor and vector equations. Table 11.2 gathers the auxiliary equations for the relaxation time, t , ferrofluid dynamic

11.3 Magnetic Fluids

Figure 11.10 Porous medium (a), averaging volume, (c) radial porosity profile.

and vortex viscosities, h and z , suspension volume fraction, f h, scalar moment of inertia density, I, fluid initial magnetic susceptibility, c0, linear magnetic material (LMM) approximation for demagnetizing and induced fields, X, radial profi le porosity model, e , external magnetic field H0, volume-average˙ Langevin magnetization, mᐉ, under the resultant external and equilibrium demagnetizing fields, and the wall-bypass fraction, BPw. It is well known that low-ratio columns may occasion severe maldistribution due to large radial permeability contrast with preferential flow short-circuiting alongside the high-permeability wall area. The challenge will be to identify configurations that are propitious to creating sufficient magnetoviscosity near the vessel wall to reduce the bypass and to reroute a portion of fluid flow towards the bed center. Hence, to assess the comparative performances among the simulated configurations, BPw, is defined as a measure of the amount of ferrofluid flux escaping alongside the wall region over a distance corresponding to half the grain diameter when the fluid velocity exhibits an inhomogeneous radial profi le with respect to the ideal situation when the radial profi le is flat. 11.3.2.2 Upscaling Upscaling is an unavoidable step allowing the passage from complex pore-level solutions of the above physical phenomena. Upscaling consists in formulating an equivalent set of volume-average equations describing the ferrofluid macroscopic behavior at the porous-medium level from a description using the local

381

Continuity

{

}

∇⋅vγ = 0 in Vγ ∂ Linear momentum ρ v γ + ∇ ⋅ ρ v γ ⊗ v γ = ∇ ⋅ T γ + ∇ ⋅ Mγ + ρ g in Vγ ∂t ∂ Internal angular momentum ρI ωγ + ∇ ⋅ ρI ωγ ⊗ v γ = ∇ ⋅ C γ − E : T γ + G γ in Vγ ∂t ∂ Magnetization relaxation Mγ + ∇ ⋅ Mγ ⊗ v γ = ωγ × Mγ − τ −1(Mγ − Mℓ ) in Vγ ∂t Maxwell-flux law ∇⋅µ0 (H0 + hγ + Mγ ) = 0 in Vγ ∇⋅µ0 (1 + χκ ) (H0 + hκ ) = 0 in Vκ ∇⋅µ0 H0 = 0 in V Ampère–Maxwell law ∇ × (H0 + hγ ) = 0 in Vγ ∇ × (H0 + hκ ) = 0 in Vκ ∇ × H0 = 0 in V Boundary conditions on fluid/solid interface Γγκ inside macroscopic region V vγ = 0 on Γγκ (hκ − hγ ) × nγκ = 0 on Γγκ (11.22) (hκ − hγ ) ⋅ nγκ = (Mγ ⋅ nγκ ) on Γγκ µ0 (11.24) ωγ × Mγ = τ−1 (Mγ − M艎 ) on Γγκ ωγ = Mγ × (H0 + h γ ) on Γ γκ 4ζ Constitutive relations t Pressure-viscous stress tensor T I + zE I + h (∇ = γ = −p γ = =vγ + ∇ =vγ ) + λ∇ ⋅ vγ = ≡ ⋅ (∇ × vγ − 2ωγ ) µ0 2 Magnetic stress tensor Mγ = − H0 + h γ I + µ 0 (H0 + h γ + Mγ ) ⊗ (H0 + h γ ) 2 Couple stress dyadic C γ = η′(∇ ωγ + t∇ ωγ ) + l′∇ ⋅ ωγ I = = = = External-internal exchange angular momentum −E ≡ :T = γ = 2z(∇ × vγ − 2ωγ ) Body-couple density G γ = µ0Mγ × (H0 + hγ ) e −1  H0 + h γ 6 kT π µm  Langevin equilibrium (e) magnetization Mℓ = φm m coth dm3 0 H0 + hγe − 3 H0 + hγe  e 6 kT π dm µ 0 m   H0 + h γ

Table 11.1 Local ferrohydrodynamic model.

(11.31)

(11.28) (11.29) (11.30)

(11.27)

(11.26)

(11.25)

(11.21) (11.23)

(11.15) (11.16) (11.17) (11.18) (11.19) (11.20)

(11.14)

(11.13)

(11.12)

(11.11)

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

3

;

Wall-bypass fraction

Volume-average Langevin magnetization

BPw =

0

{

)

0

D /2

 D /2  r r v r dr ε ( ) ( ) z  ∫ rU 0dr  ∫  D / 2−d / 2  D / 2−d / 2

(

−1

}

External magnetic field

)

7.45 − 3.15 d  d D ε b = 0.365 + 0.22 ; a =  D 7.45 − 11.25 d  D t r 0 αz + β H0 = −α 2 π µm 6 kT  m ℓ = φm m coth dm3 0 H0 + he − 3 H0 + h e π dm µ0m 6 kT  t  H + he  H 0 z + hze m ℓ =  0r er m ℓ 0 mℓ   H +h  H + he

(

−1

 

(−1)i r 2i J 0 (r ) = ∑ 2i 2 i = 0 2 (i !) D ∈[2.02 − 13.0] d d ; b = 0.315 − 0.725 D D ≥ 13.0 d

∞

Bed porosity radial distribution (Müller model)

b(D −2r ) 2d

K π dm − 1 1 1 2kT = + = 3 + f 0e kT 6 τ τ B τ N π d hη0 −1 3 6 η  d −d  d − d   5   =  1 − φm   1 + h m  − 1.552φm  1 + h m    η0  dm  dm   2  f hd3m = f mdh3 5 5 5 1 ρp dm + ρs (d h − dm ) I= 3 3 3 10 ρp dm + ρs (d h − dm ) z = 1.5hf h π φm µ0m 2dm3 χ0 = kT 18 X = χ0 (3 + 2χ0 ) −1

a(D − 2r ) − ε(r ) = ε b + (1 − ε b ) J 0 e 2d

LMM approximation correction

Initial magnetic susceptibility

Vortex viscosity

Scalar moment of inertia density

Hydrodynamic-to-magnetic volume fraction

Ferrofluid suspension viscosity

Relaxation time

Table 11.2 Auxiliary equations used in ferrohydrodynamic model.

(11.42)

(11.41.1, 2)

(11.40)

(11.39.1–5)

(11.38)

(11.37)

(11.36)

(11.35)

(11.34)

(11.33)

(11.32)

11.3 Magnetic Fluids 383

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11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

ferrohydrodynamic equations. We have used the spatial averaging theorem and its corollaries, the theorem of derivative-integral interchange, a spatial decomposition around the intrinsic averages and truncation at the zeroth order of the Taylor expansion around the macroscopic volume centroid (Fig. 11.10b) of the intrinsic averages to build the volume-average ferrohydrodynamic model. Without going into the details of the procedure, the resulting volume-average equations contain several integrals of the spatially fluctuating microscale terms that need to be approximated regarding velocity, spin density, magnetization, external magnetic field, ferrofluid demagnetizing field and induced magnetic field in granular phase. The macroscopic ferrohydrodynamic model with the unknown closure terms is summarized in Table 11.3. Note that this model applies to the points belonging to the packed-bed interior domain excluding the vessel peripheral boundaries, which need appropriate boundary conditions. Formulation of the 15 model closure terms is a task of formidable complexity, and to the authors’ best knowledge, such closures have not been yet formulated for porous-media flows. 11.3.2.3 Closure Problem Assuming, as in Schumacher et al. [29], that spin viscosity shear and bulk coefficients are zero relaxes 4 closure terms in Eq. (11.46). In addition, the integrand of the last closure term in the linear momentum balance, Eq. (11.45), contains the sum of the ferrofluid dynamic and vortex viscosities multiplied by the gradient of the fluctuating velocity on Γγκ . This suggests that unlike classical drag formulation, the drag function here must account for the occurrence in the laminar Darcy or Ergun equation term of the total viscosity rather than only the dynamic viscosity. The closures, Eqs. (11.49) and (11.52), hide in the granular-induced magnetic field vector a monumental complexity associated with the hydrodynamic perturbation of the ferrofluid dynamic magnetization vector as well as the nonlinear Langevin equation between equilibrium magnetization, external and equilibrium demagnetizing fields (Eqs. (11.41.1-2)). In the general case, this virtually opens onto a deadlock when attempting to infer a mean-field effective magnetic permeability of the composite ferrofluid–granular system. When fluid and solid magnetic permeabilities do not differ much and a linear magnetic materials (LMM) behavior is valid (the case of the low magnetic field linear-Langevin limit), the effect of contacts between nonmagnetic grains is not very important and the Maxwell–Garnett theory can be used [30]. This theory assumes random distribution of grains in a carrier medium subject to a local Lorentz field. For spheres in densely packed beds with LMMs, the previous closures can be replaced by Eq. (11.54) that relate the ferrofluid demagnetizing field and the induced magnetic field in the granular phase. Due to lack of better alternatives, we extend in this work this approximation even to the nonlinear region. This artifice allows the closures associated with Eqs. (11.49) and (11.52) to be ignored. In addition, the solid magnetic susceptibility, cκ, is neglected with respect to the initial ferrofluid magnetic susceptibility, c0, yielding the LMM approximation correction Eq. (11.38). In addition, since ferrofluid magnetization is the chief determinant of

LMM approximation

Ampère–Maxwell law

Maxwell-flux law

Magnetization relaxation

Internal angular momentum

Linear momentum

Volume conservation Continuity

+ ρI ∇ w γ γ

vγ γ

) ⋅ n γκ dσ

• εγ

γ

 + ρI ∇ • wɶ γ ⊗ vɶ γ = +2ζε γ  ∇ × v γ  γ

+

∇ε γ × vγ εγ

γ

γ

 − 2 wγ γ  

+ η′

1 V

γ

∫ ∇wɶ

Γ γκ

+ ε γ µ0 Mγ

γ

⋅ n γκ dσ

γ

∇ ⋅ vγ

γ

ɶ γ ⊗ vɶ γ = ε γ w +∇⋅ M γ

γ

× Mγ

γ

ɶγ + wɶ γ × M

ɶ γ × (H ɶ 0 + hɶ γ ) + η ′ε γ ∆ w + (η ′ + λ ′)ε γ ∇∇ ⋅ w × (H0 + h γ γ ) + µ0 M γ γ γ γ    1  1 1 + (η ′ + λ ′)∇  ∫ wɶ γ ⋅ n γκ dσ  + (η′ + λ ′) V ∫ ∇ ⋅ wɶ γ n γκ dσ + η′∇ ⋅  V ∫ wɶ γ ⊗ n γκ dσ   V Γ γκ Γγκ Γ γκ

γ

γ

∫ (− pɶ I + (η + ζ )∇ vɶ

Γ γκ

∂ εγ w γ ∂t

1 V

∂ ε γ Mγ γ + ∇ε γ Mγ γ ⋅ v γ γ + ε γ Mγ ∂t εγ − ( M γ γ − Mℓ γ ) τ ∇ ⋅ (e γ〈hγ〉γ + e γ〈Mγ〉γ + e κ〈hκ〉κ ) = 0 1 ε κ ∇ ⋅ hκ κ − hɶ κ ⋅ n γκ dσ = 0 V Γ∫γκ ∇ ⋅ H0 = 0 ∇ × (e γ〈hγ〉γ + e κ〈hκ〉κ ) = 0 1 ε κ ∇ × hκ κ − n γκ × hɶ κ dσ = 0 V Γ∫γκ ∇ × H0 = 0 〈hκ〉κ = (1 + X )〈hγ〉γ

ρI

+

εγ + εκ = 1 ∇ ⋅ εγ〈vγ〉γ = 0 ∂ ρ ε γ v γ γ + ρε γ ∇ v γ γ ⋅ v γ γ + ρ∇ ⋅ vɶ γ ⊗ vɶ γ = −ε γ ∇ p γ γ + ε γ ρg + (η + ζ )ε γ ∂t ∆ε γ ∇ε γ  1   ∆ v γ γ + ε v γ γ + ∇ v γ γ ⋅ ε  + 2ζε γ ∇ × w γ γ + 2ζ V ∫ n γκ × wɶ γ dσ γ γ Γ γκ   ɶ 0 + hɶ γ ) ⋅ M ɶ γ + µ0  ε γ ∇(H0 + h γ ) + 1 (H ɶ 0 + hɶ γ ) ⊗ n γκ dσ  • Mγ + µ0 ∇(H ∫ γ V   Γ γκ

Table 11.3 Complete volume-average ferrohydrodynamic model.

(11.53) (11.54)

(11.52)

(11.50) (11.51)

(11.49)

(11.48)

(11.47)

(11.46)

(11.45)

(11.43) (11.44)

11.3 Magnetic Fluids 385

386

11 Experimental and Theoretical Explorations of Weak- and Strong-gradient Magnetic Fields

both solid-induced magnetic and ferrofluid demagnetizing fields, these should remain small with respect to the external field to attempt further simplifications. Closure of the 8 remaining terms entails mathematical complications and physical efforts beyond the scope of this study. Rather, we will restrict the discussion to solving the 0-order formulation and leave the complete formulation as an open problem for future research. 11.3.2.4 Zero-order Axisymmetric Volume-average Model Assuming that all the variables in the physical problem are independent of the azimuthal coordinate (axisymmetry), the projections of Table 11.3 model equations along the three coordinate axes are given in Table 11.4 together with the boundary conditions chosen at the four peripheral boundaries of the porous medium: z = 0, z = L, r = 0, r = R. We will assume without proof, uniqueness of solution for the system of equations describing this ferrohydrodynamic model. The pressure field is assumed to depend only on the axial coordinate. The external magnetic field is purposely chosen to exhibit only radial and axial dependences while fulfilling the divergenceless condition, thus H0 q = 0 (Eqs. (11.50) and (11.67.1)). This entrains that the azimuthal component of the equilibrium magnetization vector is zero. In addition, the boundary conditions (Eqs. (11.79)– (11.90), Eqs. (11.103)–(11.106)) as well as Eqs. (11.58)–(11.60) and (11.64)) are verified by the trivial set w r = w z = 0, m q = 0 and v q = 0. These are taken as solutions for our model. As for the equilibrium magnetization in the e q direction, the last equality m q = 0 is also coherent with H0 q = 0. Therefore, according to Eqs. (11.41.1-2) it is not unrealistic to assume also that h q = 0, Eq. (11.67.2). Preliminary simulations reveal that typically ||h|| < 10%||H0||, suggesting a further simplification of the model towards the decoupling of the magnetostatics equations (Eqs. (11.65)–(11.68.1–3)) from the ferrohydrodynamic equations (Eqs. (11.55)–(11.64)). This requires that the norm condition (t∇ h : ∇ h)½ 0) or bed entrance (